1 Recent Space Group Publications

Transcription

1 1/224 1 Recent Space Group Publications This pdf files contains our publications from the last 5 years at USF including two that are currently accepted for publication, and our Chemical Reviews article that is currently under review. It has 224 pages that can most easily be accessed by the Bookmarks on the lefthand side of the pdf viewer. The (composite) document is numbered sequentially on the lower lefthand part of the page in green numbers from 1/224 to 224/224. The top bookmarks has a sub-bookmark that can be accessed by clicking on the plus sign next to it to see the hidden files the name of the files can be seen in full by holding the cursor over the icon.

2 2/224 Theoretical Modeling of Interface Specific Vibrational Spectroscopy: Methods and Applications to Aqueous Interfaces Angela Perry, Christine Neipert and Brian Space Department of Chemistry, University of South Florida, Tampa, FL Preston B. Moore Department of Chemistry and Biochemistry University of the Sciences in Philadelphia, Philadelphia, PA Corresponding author. Tel. (813) Fax:(813) URL: In partial fulfillment of the requirements for the Ph.D. both C.N. and A.P. were the primary graduate student authors and contributed equally to this work. (Dated: June 16, 2005) 1

3 3/224 Contents I. Introduction 4 II. Theory of the Nonlinear Polarization 6 III. Calculating the Polarization in Limiting Cases Monochromatic and Impulsive Light 11 IV. The Measured Intensity Including Dielectric Effects from the Interfacial Boundaries 14 V. Wave Vector and Phase Matching Considerations 17 VI. SFG Detection Techniques Including Those Beyond the Monochromatic Limit 18 A. Alternative Polarization Conditions: Polarization Mapping 18 B. Beyond the Monochromatic Limit Comparison of Time Domain and Frequency Domain Results Mixed Time Frequency Results 22 VII. A Microscopic Expression for χ (2) 24 A. χ(ω 1, ω 2 ) in the Dipole Approximation 24 B. Quadrupole Contributions to χ (2) from the Bulk 29 C. Third Order Contributions to the Sum Frequency Response: Charged Surfaces in Centrosymmetric Media 33 VIII. Applications of Theoretical SFG Spectroscopy to Aqueous Interfaces 34 A. Theoretical Frequency Domain Approaches to the SFG Spectrum Applications to the Water/Vapor Interface Applications to other Aqueous Interfaces 38 2

4 4/224 B. Theoretical Time Domain Approaches to The SFG Spectrum Applications to the Water/Vapor Interfaces Applications to Salt Water/Vapor Interfaces 48 IX. Conclusion 51 X. Acknowledgments 52 XI. Works Cited 53 References 53 XII. Appendix A: Possible Second Order Nonlinear Processes 61 XIII. Tables 62 XIV. List of Captions 64 XV. Figures 68 3

5 5/224 I. INTRODUCTION Liquid water interfaces are ubiquitous and important in chemistry and the environment. Thus, with the advent of interface specific nonlinear optical spectroscopies, such interfaces have been intensely studied both theoretically 1 15 and experimentally Sum frequency generation (SFG) spectroscopy is a powerful experimental method for probing the structure and dynamics of interfaces. SFG is a second order polarization experiment, and the more common electronically nonresonant experiment is the main focus of the review (although the theory of other second order processes is discussed). SFG spectroscopy is dipole forbidden in centrosymmetric media such as liquids. Interfaces serve to break the isotropic symmetry, and produce a dipolar second order signal that is sensitive only to the interface in most cases. Contributions from bulk allowed quadrupolar effects have been demonstrated to be negligible in some cases, 44,45 but can be included if necessary, 12 and, in that case, contributions from the bulk and interface are obtained in the sum frequency signal. The SFG experiment typically employs both a visible and infrared (IR) laser field overlapping in time and space at the interface, and can be performed in the time or frequency domain. 16,22,46 53 In the absence of any vibrational resonance at the instantaneous IR laser frequency, a structureless signal due to the static hyperpolarizability of the interface is obtained. 4,19,25 When the IR laser frequency is in tune with a vibration at the interface, a resonant lineshape is obtained with a characteristic shape that reflects both the structural and dynamical environment at the interface. 2,13,54 Recent years have seen a great increase in the number of experimental groups performing SFG investigations. In contrast, molecularly detailed theoretical simulations of SFG spectra are comparatively few, and have only recently begun making a significant impact. Like all vibrational spectroscopies, the goal of SFG spectroscopy is to infer structural and dynamical properties from the observed spectroscopic signatures. In contrast to more traditional vibrational spectroscopies, SFG lineshapes tend to be more complex (reflecting the unique environment that is present at an interfacial boundary), and are not nearly as well 4

6 6/224 understood. Thus, the advent of effective theoretical simulation techniques promises to help realize the potential of SFG spectroscopy to permit detailed characterization of interfaces on par with that done in the bulk. Further, in analogy with condensed phase experiments, SFG experiments have recently begun being performed using a variety of time and frequency domain techniques taking advantage of the flexibility inherent in measuring a second order polarization signal. 24,55,56 An impediment to progress in interpreting SFG spectra has been the difficulty the theoretical community can experience in understanding crucial experimental issues inherent in these measurements. Conversely, the theoretical machinery needed to describe a SFG measurement is itself complex, and sometimes seemingly far removed from the essence of the experiment. Therefore, the purpose of this review is to give a unified description of the experimental and theoretical considerations that are necessary to describe SFG lineshapes theoretically in the context of extant measurements and simulations. Section II presents a general theory of nonlinear polarization starting with the N th order formulas, and specializing to second order processes. This formalism is needed to theoretically describe certain SFG experiments (especially time domain measurements) that do not utilize effectively monochromatic fields. Appendix A (Section XII) details all the possible second order signals that result from a three-wave mixing experiment. Section III describes how the general formulas simplify in idealized limits in which most extant experiments have been performed or interpreted. The expression derived there are those frequently presented in the SFG literature. Next, we present relevant considerations concerning optical experiments at interfaces including the origin and importance of Fresnel factors, and the phenomenological expression for the measured second order (SFG) intensity in terms of the signal field (Section IV). The relationship between the common experimental polarization conditions of the experimental fields (SSP, SPS, PPP, and PSS) and microscopic Cartesian susceptibility tensor elements is also presented. The wave vector and phase matching conditions that need to be satisfied for 5

7 7/224 coherent nonlinear optical experiments are discussed in Section V. Experimental designs, including those other than the common monochromatic frequency domain SFG experiment, are discussed in Section VI. Section VI A details the innovative method of polarization mapping, and how this method helps to resolve spectral features. For experiments performed outside of the monochromatic limit, the formulas presented in Section II are needed to formally theoretically describe the SFG response for time domain techniques that use spectrally broad femtosecond laser excitation pulses. A brief discussion of the experimental results from such measurements are presented in Section VI B. Next, Section VII presents formal expressions for dipolar (Section VII A), quadrupolar (Section VII B), and static field induced third order contributions (Section VII C) to the sum frequency signal. The microscopic formulas for the dipolar and quadrupolar SFG susceptibility tensors are also presented along with a discussion of the rotating wave approximation in this context. These expressions provide the tie between the earlier phenomenological expression, and the formulas needed to relate a systems dynamics to an SFG signal. Section VIII discusses theoretical simulations and their results with a focus on aqueous interfaces; comparison with experiment is stressed. Section VIII A presents a frequency domain approach to calculating SFG signals that represented the first attempt to directly model an SFG signal from a liquid interface. It also discusses other applications of this frequency domain approach. Section VIII B discusses time domain approaches to calculating SFG spectra including applications to the water/vapor and salt water/vapor interfaces. Results, including the identification of novel species at the water/vapor interface, are presented. Section IX presents conclusions, and a brief discussion of future directions for theoretical studies of SFG spectroscopy. II. THEORY OF THE NONLINEAR POLARIZATION SFG experiments are also referred to by terms like Sum Frequency Vibrational Spectroscopy 16,22 (SFVS) to distinguish interfacial electronically nonresonant IR-visible ex- 6

8 8/224 periments from other SFG experiments. 24,57 For example, recent experiments that are doubly, both electronically and vibrationally, resonant have been performed. 58 SFG experiments measure a second order polarization generated coherently in a direction given by the experimental wave vector and phase matching conditions. 24,55,57 It is one of several second order processes that are possible when two electric fields interact with a medium. While the formalism presented here is more general, we will focus on SFG experiments in our presentation, and will explicitly state when another second order process is being discussed. Such measurements are interface specific because even order polarization generating terms are forbidden in centrosymmetric media. This can be understood by considering reversing the direction of all the electric fields in an experiment for an isotropic system. Doing so must change the sign of the polarization because all directions are equivalent on average. 57 However, even numbers of fields will make the polarization equal to its negative a condition that insists the polarization is zero, i.e. P = P = At an interface, or in certain noncentrosymmetric solids, 24 the isotropy of the system is broken. This leads to a second order signal within the dipole approximation, and in this case, the signal is proportional to the product of the susceptibility and the electric fields as described below. It should also be noted, even in centrosymmetric media, bulk quadrupolar contributions to SFG signals are possible, but have been shown to be negligible in most cases involving liquid interfaces in the common SFG reflected geometry. 44,45,53,59 (When light impinges on an interface, a SFG signal is generated. This signal is both reflected from the boundary, and transmitted through the interface.) 60 They can, however, be important for experiments performed in the transmission geometry. 45 In the case of bulk quadrupoles, their contribution to the second order signal is proportional to derivatives of the field which invalidates the above symmetry argument. Molecular dynamics (MD) simulations of the SFG signal, including both bulk and surface terms, represent an excellent mechanism to test the importance of such contributions, 12,44 but have not been conducted as of yet. Like all nonlinear optical experiments, both time and frequency domain approaches to 7

9 9/224 SFG are possible. 46,55 To date, most SFG experiments have been performed in the frequency domain, and, effectively, in the limit of monochromatic fields. 16,22,50 53 However, there is growing interest in using both time domain, mixed time and frequency domain approaches, and also looking at other second order processes such as difference frequency generation (DFG) spectroscopy The theoretical methods, that will be discussed below, are capable of describing any of these second order processes. Thus, before specializing the theoretical expressions to the typical monochromatic frequency domain experiment, it is helpful to examine the formal theoretical structure of second order nonlinear processes. The resulting expressions will be required in calculating signals from experiments outside of the frequency monochromatic or time impulsive limit, e.g. typical time domain experiments. Such experiments are becoming increasingly more common because they can provide, in principle, information distinct from ideal frequency domain experiments. 47 Further, in the final analysis, it is often possible to model SFG experiments without reference to the detailed nature of the experimental measurement. But there is, however, often confusion in the theoretical community as to what experimental considerations are relevant, and, conversely, the theoretical methods may seem opaque and out of context to the experimental community. Thus, we seek to present the methods in a context that adds clarity for both communities. First, considering an N th order process (an N + 1 wave mixing experiment), an electric field is applied at time t at position r, and can be written as: N ( E(r, t) = En (t)e ikn r + E n(t)e ikn r) (1) n In Equation 1, k n is the wave vector specifying the field propagation direction. Equation 1 is partitioned into components that are slowly varying in space, and those that are spatially highly oscillatory. 24,55,65 The slowly varying spacial component, E n (t), can generally be further decomposed into temporally (ɛ n (t)) and spatially (E n ) dependent parts. 65 This subsequent separation allows the field to be rewritten in the form: E(r, t) = N ( En ɛ n (t)e ikn r + E nɛ n(t)e ikn r) (2) n 8

10 10/224 In Equations 1 and 2, the sum on n is included because, in the most general case, exact time ordering of the applied fields cannot be assumed. 55 In practice, experiments in the time domain typically use relatively short pulses that are separated and ordered in time while the frequency domain techniques employ nearly monochromatic laser fields that overlap in time and space such considerations simplify the required analysis considerably. Given the field, the observable nonlinear polarization, P (N), (within the dipole approximation where the response function and susceptibility tensors are independent of r and k) takes the form of a multiple time integration over the material response function, R (N), which contains all the system variables and information to be probed. (The description of the material system in the time domain is described by the response function, and is typically referred to as the susceptibility in the frequency domain.): P (N) (r, t) = 0 dτ 1 0 dτ N R (N) (τ 1,, τ N ) E(r, t τ 1 ) E(r, t τ N ) (3) In Equation 3, the vertical line represents N tensor contractions. In an N order experiment, there are N relevant times corresponding to the number of E(r,t) s in the expression for P (N) (r,t) that are each represented by the sum in Equations 1 or 2. Consequently, when exact time ordering of the applied fields cannot be assumed, and Equation 1 is used to describe the applied fields, a sum of (2N) N terms determine the N th order polarization. The polarization can then be written as: P (N) (r, t) = 2 N N N s P (N) (k s, t) = 2 N N N s P (N) (t)e iks r (4) In Equation 4, k s is the sum of the wave vectors associated with the applied fields, and represents the direction the generated signal will propagate. As is shown in Figure 1, considering a second order experiment probing an interface, after the nonlinear signal is generated, it will interact with the interface producing a reflected and transmitted signal with modified wave vectors (this issue will be discussed in Section V). Note, P (N) (k s, t) is a complex quantity, and it is one of the (2N) N processes that determines the total N th order polarization, P (N) (r, t). Further, P (N) (r, t) is a real quantity, and is the sum of all the P (N) (k s, t) terms 9

11 11/224 (Equation 4). However, once a particular k s is chosen (e.g., by the experimental geometry) the signal is a complex quantity, and the real and imaginary parts can be measured separately e.g., in a heterodyne detected experiment. 66 In principle, the sum of all (2N) N terms must be evaluated to calculate the total N th order polarization. Considering second order experiments, this leads to 16 distinct contributions that are described in Appendix A (Section XII). In practice, the polarization generated for a given experiment is associated with a particular wave vector and phase matching condition. This implies when the two incident wave vectors add, a second harmonic generation (SHG)/SFG signal is generated, and when the two incident wave vectors interact such that the resulting wave vector is equal to their difference, a DFG signal is generated. For surface probing spectroscopies, the direction the signal (DFG, SFG, and/or SHG) propagates in will be guided by Snell s (linear and nonlinear) refraction and reflection laws in conjunction with the original propagation directions of incident wave vectors, k 1 and k 2. Thus, with the proper experimental setup in which detectors are placed at the appropriate phase matched angle, it is typical to only detect one of the possible second order nonlinear processes, e.g. SFG or DFG. Phase matching criteria for surface probing spectroscopies will be discussed more throughly in Section V. Note, as shown in Appendix A (Section XII), there are two P (2) (k s, t) in which the incident wave vectors add, i.e. k s = k 1 +k 2. (The additional two terms that are classified under SFG in Appendix A (Section XII) are complex conjugates of the experimental wave vectors, and contribute to P (2) SF G (r, t).) Depending on the choice of excitation fields, it may be possible to explicitly detect the two individual P (2) (k s, t) contributions; in time domain experiments, given well separated time ordered pulses, it is possible to detect only one of the two k s SFG contributions. Conversely, in frequency domain experiments, time ordering of the applied fields is not possible, and, therefore, the individual P (2) (k s, t) contributions will always be simultaneously detected. In a similar manner, in formulating theoretical descriptions of the response function, R (2), 10

12 12/224 a given experiment may only be sensitive to a part of the response, and it is convenient to discard portions that do not contribute significantly. 61 This is accomplished by identifying terms in the response function that oscillate in time so as to phase cancel with those from the applied fields, and subsequently discarding the remaining terms. This is called the rotating wave approximation (RWA). Computationally, this approximation allows for inclusion of only fully resonant Louiville space pathways, and depends implicitly on the reference or model system being considered. 55,61,65 The RWA is a computational convenience; it is possible to include the entire response function, and perform the integration in Equation 3 (or presented specifically for second order processes such as SFG in Equation 5 below) explicitly. 67 Note, a particular experiment measures either the modulus of the complex P (N) (k s, t) (homodyne detection) or its real or imaginary parts (heterodyne detection). 66,68 Methods using interference effects (via homodyne detection) between bulk and interfacial contributions to an SFG signal have also been used to separately measure the real and imaginary contributions at aqueous quartz interfaces. 16,69 The real and imaginary parts of the response contain information that is not obtainable via measuring the modulus of the signal. 3,16 This will be discussed further in Section VIII. We proceed by demonstrating how the limits of ideal frequency and impulsive fields simplifies calculation of the polarization. III. CALCULATING THE POLARIZATION IN LIMITING CASES MONOCHROMATIC AND IMPULSIVE LIGHT The second order time dependent polarization is defined by: 24,55,57,70,71 P (2) (k s, t) = e iks r dτ 1 dτ 2 R (2) (τ 1, τ 2 ):E 1 (t τ 1 )E 2 (t τ 2 ) (5) 0 0 In Equation 5, R (2) (τ 1,τ 2 ) is the system s second order response function (a real quantity), τ 1 and τ 2 represent time delays between the first and second fields, and the second field and time of signal detection respectively. Note, the symbol : denotes contraction of the two dimensional tensor (response or susceptibility) with the fields. A direct relationship between 11

13 13/224 this quantity and the second order frequency dependent susceptibility can be established by representing the time dependent components of the applied fields as their Fourier transform: P (2) eiks r (k s, t) = 4π 2 dω 1 dω 2 dτ 1 dτ 2 R (2) (τ 1, τ 2 ):E 1 (ω 1 )E 2 (ω 2 )e iω 1(t τ 1 ) iω 2 (t τ 2 ) 0 0 (6) The double Fourier-Laplace transform of the system s response function is now identified as the second order susceptibility, χ: 55,70 χ (2) (ω 1, ω 2 ) = dτ 1 dτ 2 R (2) (τ 1, τ 2 ) e iω 1τ 1 e iω 2τ 2 (7) 0 0 Because the susceptibility results from a Fourier-Laplace transform of the (real) response function it is a complex quantity (the Fourier transform of the response function is, however, a real function). Substitution of Equation 7 into Equation 6 gives the time dependent polarization in terms of the back Fourier transform of the product of the frequency domain fields and susceptibility: P (2) eiks r (k s, t) = (2π) 2 dω 1 dω 2 χ (2) (ω 1, ω 2 ):E 1 (ω 1 )E 2 (ω 2 ) e it(ω 1+ω 2 ) (8) Fourier transforming the polarization with respect to time gives the frequency dependent polarization where the signal frequency, Ω s, is the transform variable conjugate to t, and ω s = ω 1 + ω 2 : P (2) (k s, Ω s ) = eiks r (2π) 2 dω 1 dω 2 χ (2) (ω 1, ω 2 ):E 1 (ω 1 )E 2 (ω 2 ) dt e it(ωs Ωs) (9) P (2) eiks r (k s, Ω s ) = 2π dω 1 dω 2 δ(ω s Ω s )χ (2) (ω 1, ω 2 ):E 1 (ω 1 )E 2 (ω 2 ) (10) Unlike Equation 7, causality does not require a Fourier-Laplace transform because no system function is directly involved in the transform. Integration of the complex exponential over t, the time of signal detection, results in the delta function in Equation 10. In the limit the applied fields are monochromatic (the limit in which most frequency domain SFG experiments 12

14 14/224 are performed), they may be represented as complex exponentials in the time domain, and will be delta functions in the frequency domain: E(ω i ) = 2πE i δ(ω i Ω i ). Thus, an ideal frequency domain experiment directly probes the susceptibility: P (2) (k s, Ω s ) = 2πe iks r χ (2) (Ω 1, Ω 2 ):E 1 E 2 δ(ω s Ω s ) (11) Note, as a result of the form of χ, Ω s = Ω 1 + Ω 2 is the signal frequency, and is at the sum of the input frequencies; χ is described in Section VII. In an SFG experiment, both frequencies would be positive, and a sum frequency detected. In a DFG experiment, one of the input frequencies would have a negative sign associated with it, and a difference frequency would be generated. The limit of ideal frequency (monochromatic applied fields) is just one common means of simplifying the integrals necessary to calculate the second order polarization. The opposite limit, the impulsive limit, occurs when the fastest material timescale is much longer than the durations of the applied fields. This limit is often assumed in theoretical developments for simplification purposes, but generally is not truly justifiable temporal pulse durations are not currently faster than even the shortest vibrational time scale. 55 In this limit, the applied field s temporal envelopes behave as delta functions - making the evaluation of Equation 5 trivial. The resulting expression for an impulsive field at some time τ is: E i (τ ) = E i δ(τ τ)e iω i τ (12) P (2) (k s, t) = e iks r R (2) (t τ 1, t τ 2 ):E 1 E 2 e i(ω 1 τ 1 +ω 2 τ 2 ) (13) Note, the applied fields in Equation 12 can also have an exponential phase factor associated with them. 72 While Equation 12 implies the experiment probes the entire response function, finite, yet short, pulses (that do not have the infinite frequency spectrum that a delta function pulse would contain) are only resonant with certain parts of the response function. To correct for this deficiency, it is useful to write the response function in terms of Louiville space pathways. 55 In this language, one should only include fully resonant Louiville space pathways 13

15 15/224 of the susceptibility (response) tensor in the polarization calculation this is equivalent to invoking the RWA. 55,65,73 In practice, this limit is commonly assumed for simplicity in calculating the polarization; it makes evaluation of the integrals in Equation 5 trivial. In this case, it is necessary to only include pathways that are expected to contribute for a given set of fields and a relevant model system. 55,72 IV. THE MEASURED INTENSITY INCLUDING DIELECTRIC EFFECTS FROM THE INTERFACIAL BOUNDARIES Because interfaces necessarily include dielectric boundaries, the equations derived thus far need to be modified accordingly the measured signal will include factors due to interactions with the boundaries The fields in the above derivations are local to the medium, and SFG excitation fields must travel through the vacuum before overlapping at the interface (for liquid/vapor or gas/solid interfaces) or through some other medium when considering a buried interface. When the fields combine at the interface, a second order nonlinear signal is produced that interacts with the dielectric boundaries. Hence, the observed fields must be related to the laboratory generated fields through Fresnel coefficients. 24,60 In a boundless medium, the Fresnel coefficients reduce to unity, and the laboratory and local fields are the same. 74 Experimentally, it is the intensity generated at the sum frequency of the two input beams that is measured (in typical homodyne detection experiments). Equation 14 24,45,70,74,77 describes the relationship between the field, E(r, ω s ), and the measured intensity, I(ω s ), generated at the sum frequency of the two input fields: I(ω s ) = c ɛ 1 (ω s ) E(r, ω s ) 2 2π (14) E(r, ω s ) is found through the use of (nonlinear) Maxwell s wave equation knowing the nonlinear polarization one can solve for the field. 24,78,79 The exact form of the relationship between the nonlinear polarization and the measured intensity generally depends on the 14

16 16/224 boundary conditions of the medium, the direction E(r, ω s ) propagates in, how well phase matching can be achieved, whether or not the slowly varying envelope approximation is made, and the form of the applied fields - i.e, monochromatic, Gaussian, etc. In general, the local fields are approximated as monochromatic, and the slowly varying envelope approximation is assumed to be valid. Explicit expressions using various approximations are available in the literature. 24,45,60,74 Generally, the intensity is found to be proportional to the square of the sum frequency multiplied by the amplitude of the nonlinear polarization: I(ω s ) ωs P(ω 2 s )e iks r 2 (15) The proportionality coefficients include the Fresnel factors that are typically calculated using an appropriate model of the experimental setup. 13 In order to directly compare theoretical and experimental spectra, it is necessary to include the Fresnel factors especially when comparing relative intensities from different polarization conditions. 2,13 Note, experimental spectra are sometimes presented as the second order susceptibility itself correcting for factors such as the leading ω 2 s dependence or more often as simply the observed SFG intensity. As demonstrated in Section III, in the limit of monochromatic fields, the polarization directly probes the susceptibility (Equation 11). Following from Equation 15, in this limit, the measured intensity will also directly probe the (the squared modulus of the) susceptibility tensor of the system. In total, the surface susceptibility tensor contains 27 elements. Consideration of symmetry conditions for a typical azimuthally isotropic interface requires all but 7 elements of the susceptibility tensor to vanish because the elements need to be invariant with respect to symmetry operations that preserve the (azimuthal) symmetry. 63 Additionally, of the 7 nonvanishing components, only four are, in general, unique (χ xzx = χ yzy, χ xxz = χ yyz, χ zxx = χ zyy, χ zzz ). Here, the subscripts on χ are the Cartesian directions in the laboratory frame. 77,80 By utilizing different polarization conditions, it is possible to directly probe three of the four nonvanishing susceptibility tensor components independently. 24,77,80 15

17 17/224 Each of the three light fields (with corresponding frequencies ω SF G, ω vis, ω IR ) in SFG experiments can be either S or P polarized. S polarized light has a polarization vector parallel to the interface, and the P polarization is at an angle tilted to the surface and lies in a plane that is perpendicular to the interface. If the xy plane is taken to be the interface, it is usually defined that S polarized light has a single Cartesian polarization vector component along the y axis bŷ while P the vector lies in the xz plane with components cˆx + dẑ. 77,81 Different combinations of S and P polarized fields allow for direct measurement of the following tensor elements: 77,80,82 χ (2) eff,ssp = sin(β IR)L yy (ω SF G )L yy (ω vis )L zz (ω IR )χ yyz (16) χ (2) eff,sp S = sin(β vis)l yy (ω SF G )L zz (ω vis )L yy (ω IR )χ yzy (17) χ (2) eff,p SS = sin(β SF G)L zz (ω SF G )L yy (ω vis )L yy (ω IR )χ zyy (18) χ (2) eff,p P P = cos(β SF G)cos(β vis )sin(β IR )L xx (ω SF G )L xx (ω vis )L zz (ω IR )χ xxz (19) cos(β SF G )sin(β vis )cos(β IR )L xx (ω SF G )L zz (ω vis )L xx (ω IR )χ xzx +sin(β SF G )cos(β vis )cos(β IR )L zz (ω SF G )L xx (ω vis )L xx (ω IR )χ zxx +sin(β SF G )sin(β vis )sin(β IR )L zz (ω SF G )L zz (ω vis )L zz (ω IR )χ zzz Here, L represents the Fresnel factors for the given fields (linear Fresnel factors for the visible and IR fields and a nonlinear factor for the sum frequency field) 22, and β(ω i ) is the angle that the field at frequency ω i makes with respect to the surface normal. We use χ (2) eff to denote the effective susceptibility - unlike χ (2) ijk, χ(2) eff explicitly accounts for the Fresnel factors. The S and P indicies on χ (2) eff,αβγ denote how the fields ω SF G, ω vis, and ω IR respectively are polarized - i.e., S or P. In the above expressions, because of the chosen experimental geometry, three of the polarization conditions (SSP, SPS, PSS) directly probe single susceptibility tensor components while the PPP condition has components of all unique allowed Cartesian tensor elements. 16

18 18/224 V. WAVE VECTOR AND PHASE MATCHING CONSIDERATIONS In coherent nonlinear optical experiments, the signal is generated at a well defined angle in the laboratory frame that is determined by the wave vector of the incident radiation. However, only certain experimental geometries will generate a desired P (N) (k s ) signal. The required geometries must satisfy phase matching conditions that are a consequence of the input wave vector choice. To understand the phase matching conditions that need to be met, consider a monochromatic plane wave, exp (ik j r iω j t), and its associate wave vector, k j. Its frequency, ω j, and wave vector are related by the complex refractive index, n(ω j )=Re{n(ω j )}+i Im{n(ω j )}: 57,62 k j = n(ω j)ω j u j c (20) Here, c is the speed of light, and u j is a unit vector which gives the direction of the wave vector. Each applied field then has an angle of incidence, θ j, and a distinct time dependent phase, φ j, associated with it(see Figure 1): φ j (t) = Re{n(ω i)}ω j u j r ω j t (21) c In the context of second order experiments, when the two incident wave vectors of the applied fields at an interface add, k 1 + k 2, an SFG/SHG signal is generated. Alternatively, when k s = ±k 1 k 2 a DFG signal is generated. Because the incident field s wave vectors are overlapped at the medium s interface, the field associated with k s will be transmitted through the medium, and also reflected from the surface of the medium (except for the case of total internal reflection). In the electric dipole approximation (in isotropic media), both the reflected and transmitted signals are interface specific. If bulk quadrupolar contributions are important, the transmitted signal may contain a significant contribution from the bulk that is not always separable from the interfacial signature. 44,45,83 Snell s Laws, in conjunction with medium specific properties and the incident field s wave vectors, must be considered when determining the wave vector of the reflected, k (r) s 17, and transmitted, k (T), signals. s

19 19/224 It should be noted, although the two incident field s wave vectors initially may combine to give k 1 + k 2 and/or k s = ±k 1 k 2, it is not guaranteed a measurable signal will be reflected. This is due to phase matching conditions, a consequence of energy and momentum conservation, that must satisfied. This consideration leads to SFG and DFG signals only being detectable at angles that satisfy the following equations (in a typical SFG experimental geometry detailed in Figure 1): 62 sin(θ DF G ) = ω 1sin(θ 1 ) + ω 2 sin(θ 2 ) ω 1 + ω 2 (22) sin(θ SF G ) = ω 1sin(θ 1 ) ω 2 sin(θ 2 ) ω 1 ω 2 (23) Note, although Equation 22 always has a solution, Equation 23 does not. Specifically, for the given conditions, this means SFG will always emit a signal while DFG will only emit a signal when Equation 24 is satisfied: ( ) 2 ω1 sin(θ 1 ) ω 2 sin(θ 2 ) 1 (24) ω 1 ω 2 VI. SFG DETECTION TECHNIQUES INCLUDING THOSE BEYOND THE MONOCHROMATIC LIMIT A. Alternative Polarization Conditions: Polarization Mapping Experimentally obtained SFG spectra heavily rely upon fitting techniques to deduce important spectral features even for relatively simple interfacial systems (see, e.g. Section VIII B 1); spectra generally have convoluted peaks, and spectral fitting techniques serve to further separate and resolve these peaks. Fitting of spectra is generally performed via a mathematical/theoretical approach, but it was recently shown 84,85 that polarization mapping experiments can benefit the spectral fitting process. This method is highly effective when there is a high density of vibrational modes present at the interface in a particular spectral range which makes resolution of individual vibrational peaks difficult. 18

20 20/224 Polarization mapping, 86 also used in SHG applications, 85,87 is accomplished by measuring spectra using a wide continuous range of different polarization conditions on the input and/or output field(s). Intermediate polarization conditions refer to light with a polarization vector rotated somewhere between the S and P polarization. This method is effective because different polarization conditions probe different parts of the susceptibility tensor providing sensitivity to different molecular orientations and vibrations in distinct chemical environments. Therefore, measuring and comparing spectra under different polarization conditions (that can be thought of as linear combinations of the independent polarization conditions) of the input/output field(s), can reveal individual peaks that might otherwise appear convoluted. 84,85 Chen et. al. recently 84 performed both theoretical and experimental investigations using the polarization mapping techniques on centrosymmetric bulk systems that were azimuthally isotropic at the interface. As mentioned above, it is possible to perform polarization mapping by changing several different experimental parameters. In this study, the intermediate polarization conditions all had the visible beam at 45 degrees from the S polarization, and the infrared field was P polarized. To map the polarization, the SFG signals were detected at different polarizations, ranging from S to P, via measuring the spectra at different polarization angles with a polarizer placed in front of the detector. Once the spectra were collected, the SSP and PPP spectra were fitted using standard techniques. Using the fitting parameters from this process allows the SFG spectra at other various polarization angles to be calculated. The calculated spectra for other various polarization angles should adequately reproduce their experimental spectra if the fitting parameters deduced from the SSP and PPP polarization conditions are correct. As Chen et. al. note, this is generally not the case for the initial SSP and PPP parameters deduced. They proceed by adjusting the fitting parameters, and refitting the SSP and PPP spectra. This process is repeated until the calculated spectra for the intermediate polarization conditions closely match those obtained experimentally thus validating the spectral fit. 19

21 21/224 Using this method, Chen et. al. were able to clearly show different vibrational modes in the same spectral region reached their maximum intensity at distinct polarization angles, and were, thus, able to perform improved spectral deconvolutions. To further test the method, an experimental model spectrum was constructed with overlapping vibrations. It was found that peak separation was even possible for two vibrational modes that had center frequencies only two wavenumbers apart (a.07 % difference), and had little difference between their τ zzz/yyz = χ (2) zzz/χ (2) yyz values. (In this case, 84 τ zzz/yyz is the ratio between the PPP and SSP susceptibility because the input and output fields were set to critical angles in a total internal reflection geometry which effectively makes the Fresnel factors of the other possible contributing tensor elements zero. 88 ) The polarization mapping method was also successfully applied to deuterated-polystyrene/air and histidine-tagged ubiquitin solution/deuterated-polystyrene interfaces proving its value for interpreting spectra at complex interfaces. B. Beyond the Monochromatic Limit To date, most SFG experiments have been conducted and interpreted in the limit of monochromatic laser sources thus, directly probing the susceptibility via Equation 11. However, taking advantage of the increasing availability of femtosecond pulsed lasers, it is possible to perform time domain (often referred to as FID free induction decay) SFG experiments In contrast to the frequency domain, using short laser pulses does not imply the impulsive limit that would directly probe the response function in time because material response times are not all long compared to the laser pulse durations. This implies, theoretically, it is necessary to evaluate the time integration in Equation 5 to calculate the nonlinear polarization. While this approach has not been adopted to date, using theoretical response functions calculated, e.g. via MD (see Section VIII), as inputs into Equation 5 can aid in both design and interpretation of useful time domain experiments. Typically this integration is avoided by still assuming the impulsive limit, but only including fully resonant 20

22 22/224 pathways of the response (susceptibility) tensor i.e., making the RWA. 55 Alternatively, if a specific functional form can be associated with the response (susceptibility) tensor, then it may be possible to simply analytically integrate Equation 5. However, in general, considering experiments outside of the monochromatic limit, it may be necessary to perform the numerical integrations explicitly. 1. Comparison of Time Domain and Frequency Domain Results An SFG experiment, under otherwise identical conditions, can be performed either in the time or frequency domain from a theoretical standpoint; both measurements represent equivalent spectroscopic methods, and can be simply related by Fourier transforming the polarization. For example, the time evolution of the polarization can be observed directly by using temporally short spectrally broad pulses. This can also be indirectly measured in frequency domain using temporally broad spectrally narrow pulses, and Fourier transforming the resulting spectra to the time domain. Both of these techniques should, in principle, allow the vibrational dephasing time to be calculated. Recently, it was shown, 46,48 while theoretically equivalent, time and frequency domain measurements for a given IR-visible SFG experiment can be sensitive to different physical aspects of the system. This difference arises from the convolution of the resonant and nonresonant components of the system s susceptibility, and how easily these experimentally measured components can be separated. Since the nonresonant susceptibility only contributes when the two fields overlap in time, as noted by Bonn et. al., 48 the resonant susceptibility, and hence the lineshape, is more readily separable in time domain experiments especially for time delays between the two incident fields which are > 500 fs. In a frequency domain experiment, time ordering of the pulses cannot be accomplished. Therefore, the resonant and nonresonant parts of the susceptibility will always be simultaneously present, but can be separated by a variety of methods; note the non-resonant contribution is typically constant (independent of frequency). Peak fitting is usually effective in separating these two 21

23 23/224 components 84 and isotopic dilution can also be performed. 19 By performing both theoretical and experimental investigations, 46 Bonn et. al. demonstrated systems with homogeneous distributions of adsorption sites showed nominal difference between time and frequency determinations of lineshape and vibrational dephasing times. Figure 2 details the same vibration (C-H stretch in acetonitrile) measured in both the frequency (top panel) and time (lower panel) regimes. The vibrational dephasing time was calculated from the time and frequency domain measurements, and were 0.61 and 0.66 ps respectively. Figure 3 shows the frequency (top) and time (bottom) domain measurements of the C-N stretch in acetonitrile. The calculated vibrational dephasing time using the frequency domain data was 0.68 ps. As the dashed line in the lower panel of Figure 3 illustrates, a vibrational dephasing time of 0.68 ps does not correctly reproduce the time domain experiment. The pronounced difference in the time and frequency domain data was shown to be attributed to an inhomogeneous distribution of adsorption sites. 46,48 In this case, the time domain SFG experiments provided more accurate vibrational dephasing times, and, thus, also more detailed spectral lineshapes than frequency domain techniques although it was only through the failure of the frequency domain data to accurately reproduce the time domain experiment that revealed the existence of an inhomogeneous distribution of adsorption sites. This investigation clearly demonstrates the utility and complimentary nature of using both time and frequency domain techniques. 2. Mixed Time Frequency Results Recently, Benderskii et. al. 47 developed a mixed time frequency domain technique, STiR- SFG (spectrally and time resolved SFG), to measure SFG spectra, and, specifically, analyze the time evolution of interfacial hydrogen bonds in liquid D 2 O systems by monitoring vibrational shifts in the OD stretch region; studies of this nature are critical because the time evolution of hydrogen bonds present at interfaces can profoundly affect the properties of water, and their characterization is, thus, necessary for a complete understanding of such systems. 22

24 24/224 In this technique, traditional time domain techniques are used (employing spectrally broad temporally short pulses), but the SFG signal is dispersed through a monochromator such that it is the SFG spectrum that is recorded as a function of the delay time between the two applied fields. (In typical time domain measurements, it is the SFG intensity that is measured as a function of delay time between the two incident fields.) Specifically, a temporally short resonant IR pulse ( 70 fs) is applied to the interface of the system of interest which creates a coherence, and, hence, a first order polarization. After a time delay, τ, a second temporally short off resonant visible pulse ( 40 fs) is applied. This pulse interacts with the coherence created by the IR pulse, and probes the second order polarization. In essence, this new technique uses a novel detection method to analyze information from experiments performed in the time domain. The described mixed time frequency technique (STiR-SFG) was used by Benderskii et. al. in a detailed study of a D 2 O/CaF 2 (SSP geometry). As shown in Figure 4, the STiR- SFG spectrum was measured using IR pulses tuned to the blue and red sides of the OD stretch. Dynamics observed on both the blue and red side of the OD stretch via the STiR- SFG method are shown in Figures 5 and 6 respectively. After careful deconvolution of the instrument response function, measurements on the blue side showed a distinct red shift ( 60 cm 1 ) to the OD stretch frequency on a time scale of fs. Conversely, measurements on the red side of the OD stretch showed a distinct blue shift to the OD stretch frequency ( 50 cm 1 ) in the first 130 fs, and a pronounced recursion at 125±10 fs. The differences in the spectra taken on the blue and red sides of the OD stretch highlight the heterogeneous nature of the hydrogen bond distribution in D 2 O. Specifically, in conjunction with theoretical simulations, Benderskii et. al. have been able to suggest there is not uniformity in the strength of hydrogen bonds present in interfacial water, and, rather, there exists distinct subensembles of relatively stronger (red side) and weaker (blue side) hydrogen bond structures. 23

25 25/224 VII. A MICROSCOPIC EXPRESSION FOR χ (2) A. χ(ω 1, ω 2 ) in the Dipole Approximation The systems susceptibility, χ, contains all the information about the material system, and, is thus, the focus of theoretical investigations into SFG interfacial vibrational spectroscopy. In order to calculate χ, or equivalently the system response function, R (2), it is necessary to develop a microscopic description of it. Further, it is desirable to represent the response function in a form amenable to calculation, and one that can exploit the power of MD interfacial simulations. MD is capable of accurately describing both the structure and dynamics of even complex interfaces. 2 5,9 11,15,89 91 Specifically, it will be shown the SFG response function is proportional to the imaginary part of the one time cross correlation function of the system dipole and polarizability. 1 4,6 In order to pursue this goal, starting from density matrix theory, and using perturbative techniques, a formal expression for the second order susceptibility in the dipole approximation can be derived. 4,24,57,92 Using this method, χ (2)SF G ijk (ω) is defined by a sum of six terms (shown below). 5,57 Four of the terms contribute to the resonant SFG signal (contained in R 1 and R 2 ), and the remaining contribute to the nonresonant portion of the signal (NR 1 and NR 2 ). The two terms in R 2 contain the expressions (ω IR + ω ng + iγ ng ), and may initially appear to be nonresonant; γ is an arbitrary convergence parameter in time 93 that is frequently interpreted physically as a dipole dephasing rate 5,57 that would be responsible for a single mode s homogeneous line width in the frequency domain. Inclusion of these terms in the resonant susceptibility is, however, necessary to develop a general theory. Neglecting these contributions results in an expression only valid when ω kt, where k is Boltzmann s constant and T is the system temperature (this is equivalent to making the RWA in this case). Note, although we use ω IR and ω vis, this is easily generalizable to two arbitrary applied fields. 24

26 26/224 In the frequency domain, χ (2)SF pqr G (ω) takes the form: χ (2)SF G pqr (ω SF G, ω vis, ω IR ) = g,n,m (ρ (0) g )(R 1 + R 2 + NR 1 + NR 2 ) (25) ( µ p gnµ q nm R 1 = (ω SF G ω ng + iγ ng ) µ q gnµ p ) ( nm µ r ) mg (ω vis + ω ng + iγ ng ) (ω IR ω mg + iγ mg ) ( µ q nmµ p mg R 2 = (ω SF G + ω mg + iγ mg ) µ p nmµ q ) ( mg µ r ) gn (ω vis ω mg + iγ mg ) (ω IR + ω ng + iγ ng ) NR 1 = µ q gnµ p mgµ r nm (ω SF G + ω mg + iγ mg )(ω vis + ω ng + iγ ng ) NR 2 = µ p gnµ q mgµ r nm (ω SF G ω ng + iγ ng )(ω vis ω mg + iγ mg ) In the above expressions that define the six components of the second order susceptibility, ω ng is the frequency corresponding to the energy difference between energy levels n and g. In Equation 25, ρ (0) g is the initial state thermal population, and the sum is over vibronic levels. µ η α,β is a dipole matrix element between states α and β for dipole vector component η. Approximating 1/ω sfg 1/ω vis, the resonant contributions can be simplified by rewriting them in terms of polarizabilities and dipoles. Given the definition of polarizability in Equation 26, the two resonant terms, R 1 and R 2, simplify to Equations 27 and 28 respectively. α pq (ω) = g,n [ µ p gnµ q ng µ q gnµ p ] ng + ρ (0) g (26) ω + ω ng iγ ng ω + ω ng + iγ ng α pq gmµ r mg R 1 = (ω IR ω mg + iγ mg ) (27) µ r gnα pq ng R 2 = (ω IR + ω ng + iγ ng ) (28) Let χ Res pqr denote only the sum of the resonant terms R 1 and R 2. Replacing the denominators in both of the resonant terms with the integral identities 0 dte it(ω ωo iγ) = i ω ω o iγ 25 and

27 27/224 dte it(ω+ωo+iγ) i = 0 ω+ω o+iγ, and then taking the implied limit that gamma goes to zero, gives Equation 29. Equation 30 follows as an exact rewrite of Equation 29, and expresses the susceptibility in terms of the cross correlation of the system dipole and polarizability. [ χ Res i pqr = e iωmgt e iωirt α pq gmµ r mgdt i ] e iωngt e iωirt α pq ngµ r gndt ρ (0) g (29) gm 0 ng 0 χ Res pqr = i 0 dt e itω IR < α pq (t)µ r (0) > i 0 dt e itω IR < µ r (0)α pq (t) > (30) In deriving Equation 29 from 30, α pq (t) is identified as the Heisenberg representation of the polarizability operator α at time t, and a sum over states is performed to remove a resolution of the identity. 94,95 Expressing the correlation function in Equation 30 explicitly as the sum of its real and imaginary components reduces Equation 30 to Equation 31, below. Note, < µ r (0)α pq (t) >= C R (t) + ic I (t) = (< α pq (t)µ r (0) >), and the subscripts R and I will be used throughout the article to represent the real and imaginary parts (both of which are themselves real) of complex quantities. 94 In the frequency domain the TCF is real, and takes the form C(w) = C R (ω) + C I (ω) where C R is even (C R (ω)=c R ( ω)) while C I (ω) is odd ( C I (ω)=c I ( ω)): 94,96,97 χ Res pqr (ω IR ) = 2 0 dte itω IR C I (t) (31) Note, χ Res pqr is presented as an explicit function of the IR frequency because the other optical frequencies are absorbed implicitly into the polarizability. Equation 31 is a nearly exact rewrite (exact other than substituting 1/ω sfg 1/ω vis ) of the perturbation expression, and is the central result of this section; it links the susceptibility to a TCF of the system s dipole and polarizability. The quantum mechanical TCF is amenable to calculation using classical MD supplemented by a suitable spectroscopic model via quantum correcting the classical TCF; 3,98,99 while Equation 31 includes the imaginary part of the TCF, classical TCF methods can only approximate the real part, but as will be detailed below, the real and imaginary 26

28 28/224 parts of the TCF are simply related in the frequency domain. 100,101 Equation 31 is also a starting point for possible calculation of the quantum TCF using reduced dimensional models 71,102 or quantum dynamical approaches. 103,104 The literature contains examples of using an expression similar to Equation 31, but written in the RWA. 4,12,15 It will now be demonstrated that only in the high frequency limit, and because of the exact frequency relationship between the real and imaginary components of the correlation function, can R 2 be excluded from the resonant component of the susceptibility. In this approximation, the resonant susceptibility is given by only the first term in Equation 29 i.e., the resonant susceptibility is then given as the Fourier transform of the full TCF: χ Res pqr = i gm 0 e iωmgt e iω IRt α pq gmµ r mg dt = i 0 dt e iω IR < α pq (t)µ r (0) > (32) To proceed, the correlation function is expressed as its real and imaginary time dependent components. Next, both the real and imaginary components of the correlation function are represented as their corresponding Fourier transforms, and the order of integration is switched: χ Res pqr = i 2π 0 dt e iω IRt χ Res pqr = i 0 dt e iω IRt (C R (t) ic I (t)) (33) dω e iωt C R (w) + 1 2π 0 dt e iωirt ( i) dω e iωt C I (w) (34) χ Res pqr = i dω C R (w) 2π 0 dt e it(ω IR+ω) i dω C I (w) 2π 0 dt e it(ω IR+ω) (35) The integration over dt in Equation 35 can easily be performed, and results in a delta function and principle part contribution: 93 χ Res pqr = i 2 C R(ω IR ) + i 2 C I(ω IR ) + P dω C I(ω) + C R (w) 2π ω IR + ω (36) Equation 36 contains both the Fourier transform of the real and imaginary parts of the TCF in contrast to the exact result; Equation 31 (after performing the time integration) 27

29 29/224 is proportional to only the sum of the Fourier transform of the imaginary part of the TCF plus a principle part contribution. However, because there exists an exact detailed balance relationship between the real components of the frequency domain correlation function, C I (w) = tanh(β ω/2)c R (w), Equation 36 can be rewritten as: χ Res pqr = i 2 [1 + coth(β ω IR)]C I (ω IR ) + P dω C I(ω) + coth(β ω/2)c I (w) 2π ω IR + ω (37) In the high frequency limit, where coth(β ω/2) 1, Equation 37 is the correct expression relating the resonant susceptibility to the imaginary component of the correlation function. Equation 32 has been used in the literature to calculate the resonant susceptibility, and, in those cases, the full quantum TCF is approximated as the classical TCF. 5,15 Equation 32 is accurate at sufficiently high frequency, but when adopting a classical approach, it is better to link the classical correlation function to the quantum TCF via quantum correction approaches 2,3,105 although quantum correction effects the magnitude of the signal more than the lineshape. Further, Equation 32 is not accurate at lower frequencies where many interesting interfacial phenomena occur. 106 However, to date, SFG experiments have focused on high frequency intramolecular vibrations due to technical limitations most tunable IR lasers are not yet capable of probing lower frequencies. Even though some nonlinear crystals (e.g. GaSe) can generate infrared beams with strong enough intensity in the lower wavenumbers, they are not widely applied, and current typical SFG experiments employ optical parametric amplifier (OPA) generated tunable IR radiation that have insufficient power to create measurable SFG signals below about 1000 cm 1. 17,52,107 There are, however, free electron laser sources that produce sufficiently intense light for SFG experiments into the far IR. 106 Computationally, these regions of lower frequency can be analyzed, 2,3 and have revealed novel low frequency species present at the water/vapor interface. 2,3,7 These results will be discussed in Section VIII. 28

30 30/224 B. Quadrupole Contributions to χ (2) from the Bulk In three-wave mixing experiments where the dipole approximation is valid, χ (2) vanishes for isotropic medium. 24,44,57,70,108,109 This property is what makes SFG a useful method for probing interfacial dynamics. When it is necessary to include higher order terms (i.e. quadrupole contributions) to the perturbed Hamiltonian, χ (2) does not vanish for isotropic media. 44,70,108,110 Under these conditions, SFG not only probes the interface, but also quadrupole contributions from the bulk. 12,44,83 (Bulk contribution in isotropic media via quadrupole interactions was first observed by Terhune and coworkers. 111 ) There is no universal set of criteria for determining when quadrupole interactions will significantly contribute to the polarization, and the magnitude of their contribution must be considered on a case by case basis. 44 Quadrupole contributions can be understood by considering the perturbed Hamiltonian, H, that includes both dipole and quadrupole contributions respectively: H (t) = i µ i E i (t) i,j q ij E i (t) r j (38) In Equation 38, r j represents a system coordinate, and q ij is the system s quadrupole moment. Starting from density matrix theory, and using perturbative techniques, the susceptibility including dipole and quadrupole contributions can be derived. 12,44,45 The microscopic quadrupolar susceptibility is a fourth ranked tensor, and is the sum of three components χ (2)Qs ijkl, χ (2)Q IR ijkl, and χ (2)Q vis ijkl. χ (2)Qs ijkl, defined in Equation 39, contributes to the quadrupole polarization. It is generated by a dipolar coupling to the first two applied fields, ω IR and ω vis, and a quadrupolar coupling to the sum frequency field, ω s = ω IR + ω vis. 44 χ (2)Q IR ijkl contributes to the dipolar polarization via dipolar coupling to E vis and E s, and a quadrupolar coupling to ω IR. χ (2)Q vis ijkl contributes to the dipolar polarization via dipolar coupling to ω IR and ω s, and a quadrupolar coupling to ω vis. Theoretically, it is possible to separate dipole and quadrupole contributions to the polarization, but experimentally it is not possible to fully separate all of the bulk s (quadrupole) contributions. 12,44,50,112 This is due to 29

31 31/224 the nonunique definition of what defines the surface and bulk portions of the material. We proceed by deriving the resonant portion of the quadrupolar susceptibility in terms of TCF s starting from perturbative density matrix expressions. 12 The molecules in the subsequent quadrupolar perturbative susceptibility terms is a sum over all molecules in the system. This summation arises because the perturbation expressions are based on the Hamiltonian of a single particle. As Morita notes, the summation on molecules must take into account local fields. 12 In analogy with presenting the interfacial dipolar contributions in TCF form, writing the quadrupolar contributions in terms of TCF s allows the use of classical MD techniques to calculate these contributions from molecularly detailed simulations. To date, no such calculation has been done. The explicit derivation is presented next for χ (2)Qs ijkl, and only the final result is shown for the other contributions. Perturbation theory gives χ (2)Qs ijkl as: χ Qs ijkl = molecules mno ρ (o) oo qonµ ij k nmµ l mo (ω no ω sfg iγ no )(ω mo ω IR iγ mo ) + q ij on µ l nmµ k mo (ω no ω sfg iγ no )(ω mo ω vis iγ mo ) qnmµ ij k onµ l mo (ω nm +ω sfg +iγ nm )(ω mo ω IR iγ mo ) + q ij nm µ l onµ k mo (ω nm +ω sfg +iγ nm )(ω mo ω vis iγ mo ) + + qnmµ ij l onµ k mo (ω mn ω sfg iγ mn )(ω no +ω IR +iγ no ) + q ij nm µ k onµ l mo (ω mn ω sfg iγ mn )(ω no +ω vis +iγ no ) qmoµ ij l onµ k nm (ω mo +ω sfg +iγ mo )(ω no +ω IR +iγ no ) + q ij mo µ k onµ l nm (ω mo +ω sfg +iγ mo )(ω no +ω vis +iγ no ) Equation 39 possesses intrinsic permutation symmetry with respect to the IR and visible fields. In an analogous method to the derivation above for the susceptibility in the dipole approximation, the third and sixth terms, and the fourth and fifth terms can be combined to reduce χ (2)Qs ijkl to the six terms in Equation 40: 30 (39)

32 32/224 Terms 3&6 = ( ) µ k on qij nmµ l mo 1 1 ω sfg +ω mn iγ mn ω vis +ω no+iγ no + ω IR ω mo+iγ mo = ( ) µ k on qij nmµ l mo ωsfg +ω nm+i(γ no+γ mo) (ω vis +ω no+iγ no)(ω IR ω mo+iγ mo) ω sfg +ω nm+iγ mo µ k onq ij nmµ l mo (ω vis +ω no+iγ no)(ω IR ω mo+iγ mo) Terms 4&5 = ( ) µ l onqnmµ ij k mo 1 1 ω sfg +ωmn iγ mn ω IR +ω no+iγ no + ω vis ω mo+iγ mo = ( ) µ l on qij nmµ k mo ωsfg +ω nm+i(γ no+γ mo) (ω vis ω mo+iγ mo)(ω IR +ω no+iγ no) ω sfg +ω nm+iγ nm µ l on qij nmµ k mo (ω vis ω mo+iγ mo)(ω IR +ω no+iγ no) χ Qs ijkl = 1 2 molecules mno ρ (o) oo qonµ ij k nmµ l mo (ω no ω sfg iγ no )(ω mo ω IR iγ mo ) + q ij on µ l nmµ k mo (ω no ω sfg iγ no )(ω mo ω vis iγ mo ) + µ l onqnmµ ij k mo (ω vis ω mo +iγ mo )(ω IR +ω no +iγ no ) + µ k onq ij nm µ l mo (ω vis +ω no +iγ no )(ω IR ω mo +iγ mo ) qmoµ ij l onµ k nm (ω mo +ω sfg +iγ mo )(ω no +ω IR +iγ no ) + q ij mo µ k onµ l nm (ω mo +ω sfg +iγ mo )(ω no +ω vis +iγ no ) (40) In Equation 40, only the terms which contain ω IR can become appreciably resonant since the visible beam is tuned far from resonance. Again, we approximate 1/ω vis 1/ω sfg. This approximation allows us to write the resonant portion of χ (2)Qs ijkl polarizability, β ijk : in terms of the quadrupole 31

33 33/224 β ijk = 1 on ( ρ (o) oo q ij onµ k no ω no ω iγ no + ) µ k onqno ij ω no + ω + iγ no (41) χ (2)Qs,Res ijkl = 1 molecules mo ( ρ (o) oo µ l moβ ijk om ω IR ω mo + iγ mo + ) βmoµ ijk l om ω IR + ω mo + iγ mo (42) In Equation 42, β ijk δ,ɛ is the matrix element of the quadrupolar polarizability. An exact rewrite of Equation 42 is possible using the same integral identities used in Section VII A. Note, this derivation closely follows that due to Morita, 12 but does not invoke the RWA in analogy with the derivations in Section VII A. χ (2)Qs,Res ijkl molecular simulation: now takes the form of a TCF suitable for χ (2)Qs,Res ijkl = i molecules ( dt e iωirt < β ijk (t)µ l (0) > 0 0 ) dt e iωirt < µ l (0)β ijk (t) > (43) In analogy, the resonant part of χ (2)Q IR ijkl and χ (2)Q vis ijkl may be written as follows: χ (2)Q IR,Res ijkl = i molecules ( dt e iωirt < α ij (t)q kl (0) > 0 0 ) dt e iωirt < q kl (0)α ij (t) > (44) χ (2)Q vis,res ijkl = i molecules ( dt e iωirt < β kli (t)µ j (0) > 0 0 ) dt e iωirt < µ j (0)β kli (t) > (45) The TCF s in Equations 43, 44, and 45 can be expanded into their real and imaginary components. In each of the three equations, direct expansion into their respected correlation function s real and imaginary parts simply equates the resonant susceptibility to the integral over the imaginary component (which is real) of the correlation function. This expansion, and subsequent simplification, is in exact analogy to the (previously shown) steps required to transform Equation 30 to Equation 31 for the susceptibility in the dipole approximation. 32

34 34/224 C. Third Order Contributions to the Sum Frequency Response: Charged Surfaces in Centrosymmetric Media When properly designed, three-wave mixing experiments can probe both the second order, χ (2), and third order, χ (3), susceptibilities. Techniques of this nature, have been utilized in interfacial studies of solids since the 1960 s, 113,114 but it has not been until more recently that liquid interfaces have been analyzed. 23,33,115,116 In the context of liquid interfaces, this method, sometimes referred to as Electric Field Induced SHG/SFG/DFG, relies on the presence of a charged species such as, surfactants 116 which can create a static field, E static, that lies in the region of the interface, and is local to the medium. 117 In this case, the observed polarization, in the limit of monochromatic fields, is given by: 23,117 P = P (2) + P (3) = χ (2) : E vis E IR + χ (3).E vis E IR E static (46) There are two main contributions to the third order susceptibility. (1) The presence of three fields (two incident + static) gives rise to a third order electronic nonlinear polarizability from the solvent. (2) The symmetry is broken by the presence of E static, and, thus, it further extends the anisotropic interfacial region into normally centrosymmetric regions of the bulk. 23,117 Hence, contributions of χ (3) to the observed polarization are inherently bulk in origin. 117 The value of theoretical and experimental electric field induced three-wave mixing investigations lies in their ability to deduce the electrostatic potential created by the charged species near the interfacial surface, and monitor how the electrostatic potential inherently changes the nature of the interface. Since many interfaces such as, air/water and oil/water are generally not exclusively present in a natural environment, such studies can provide highly relevant information for many environmentally important systems. 33

35 35/224 VIII. APPLICATIONS OF THEORETICAL SFG SPECTROSCOPY TO AQUE- OUS INTERFACES Theoretical modeling of SFG vibrational spectroscopy using MD based methods is a relatively young field with only a few existing studies all but two of which consider aqueous interfaces. 1 7,9,11 15,69,118 However, such studies clearly show great promise, and are growing in importance due to their ability to provide a molecularly detailed description of interfacial species and their vibrations. These early studies have also demonstrated the ability to link a particular spectroscopic SFG signature to an interfacial vibrational mode the principle goal of vibrational spectroscopy. Theoretical SFG spectra have also been used to identify multiple species in complex lineshapes, 3 and reveal lower frequency species that were previously undetected; 2,3 while experimental studies also have these abilities, in principle, they have been limited by technological barriers such as, difficulty in separately measuring the real and imaginary portions of the SFG signal, 16 and the lack of intense tunable IR radiation sources. 17,106 The use of MD based methods to describe condensed phase (intramolecular) vibrational spectroscopy is widespread and highly effective. 67,99, The utility of theoretical input, in the case of SFG spectroscopy, can be even greater given the complex lineshapes that are characteristic of interfaces. For example, Figure 7a shows the SSP SFG intensity for the O-H stretching region of the water/vapor interface obtained experimentally along with a deconvolution of the resonant part into possible component species (Figure 7b). 19,130 Clearly, the lineshape is far more complex than the corresponding bulk lineshape which is nearly symmetric. 3,131,132 Theoretical studies have also explicitly demonstrated the degree to which SFG spectroscopy is interface specific 5 something that is difficult to ascertain experimentally. This kind of insight is particularly important at highly ordered aqueous interfaces, e.g. those on charged solids, or when surfactants are present at interfaces. In such cases, third order polarization contributions can be important because the (relatively) static electric field at the 34

36 36/224 interface combines with the IR and visible fields to give a contribution at the sum frequency. 22 Simulations also have the ability to include other possible contributing phenomena like bulk quadrupolar contributions, 12,16,44,45 or bulk contributions from noncentrosymmetric solids. 24 This can help disentangle the relative contributions in a straight forward manner which is difficult to do so experimentally. 44 These issues will be considered separately below. The first attempt at approximately modeling SFG spectroscopy considered the IR spectrum of the water/vapor interface 9 that had been reported experimentally a short time earlier. 25 For a bulk system, IR spectroscopy is typically calculated using linear response theory via the dipole-dipole autocorrelation function. 94,95,99 However, this spectrum is not sensitive to the interface due to the much smaller number of oscillators resident there compared to the bulk. To obtain some surface selectivity, the study calculated the dipole correlation function for only molecules within a small distance of the Gibbs dividing surface; while this does not directly represent any physically measurable quantity, it is a reasonable approach to probe the interfacial vibrations. This study revealed the presence of free O- H oscillators at the interface, and at the correct frequency. The study also demonstrated that adding methanol quenches the free O-H peak in agreement with another early SFG experiment. 29 A similar approach was recently adopted by Mundy et. al. in an ambitious ab initio MD simulation of the water/vapor interface that also identified the free O-H moiety, and described the dipolar change as the water/vapor interface was approached. 7 This study also described the nature of the bonding at the water/vapor interface. The first attempts to directly calculate an SFG signal did not consider liquid interfaces, but rather the SFG spectrum of adsorbates at solid interfaces. 6,14 The authors first used a frequency domain approach. 14 Later, using a time dependent perturbation theory result, 55 they wrote down a TCF expression for the SFG signal, and evaluated it in terms of a TCF of the systems coordinates in the linear dipole and Placzek approximation; their expression appears to be correct only for the modulus of the signal. 6 35

37 37/224 A. Theoretical Frequency Domain Approaches to the SFG Spectrum 1. Applications to the Water/Vapor Interface The next theoretical study was due to Morita and Hynes, and examined the O-H stretching region of the ambient water/vapor interface. They adopted a frequency domain approach that was highly effective. 5,14 Their method was reminiscent of a very similar approach to calculating the IR spectrum of bulk water that was employed earlier To calculate the SFG spectrum, the authors evaluated the perturbation expression for the susceptibility 24 for harmonic normal mode, Q: [ ] χ (2) R (ω ω SF G ω IR SF G) ( µ i / Q) ( α jk / Q) (ω SF G ω IR ) 2 + γ [ 2 ] χ (2) γ I (ω SF G ) ( µ i / Q) ( α jk / Q) (ω SF G ω IR ) 2 + γ 2 (47) (48) Equations present the real and imaginary parts of the susceptibility where γ is a mathematical convergence parameter that physically can be interpreted as a homogeneous line width, and was estimated in their work. Note, the choice of γ has a large effect on the signal shape. Equations demonstrate the signal magnitude is proportional to the product of dipole and polarizability derivatives which provides the interface specificity. The equations can then evaluated for each molecule s normal coordinates in the molecular frame, and rotated into laboratory coordinates. The different molecular contributions are then summed for an ensemble of configurations. To calculate the nature of the (bulk and) interfacial normal coordinates, it was assumed a priori the O-H stretching modes were localized on single molecules this had been demonstrated theoretically for the bulk O-H stretching modes earlier. 98,125,126 Next, the mode shapes were calculated based on a water molecule s local environment. The O-H stretching mode was taken as a linear combination of the gas phase symmetric and antisymmetric stretching modes with weights based on a simple two state vibrational eigenvalue equation that included a calculation of the condensed phase O-H frequency shift (calculated by using the 36

38 38/224 force on the bond coordinate and a cubic anharmonic potential function parameterized to ab initio calculations) and the off diagonal gas phase mode couplings. The polarizability and dipole derivatives are then calculated by parameterizing the derivatives as a function of O-H bond length using ab initio calculations. Finally, the SFG intensity was calculated by multiplying the squared modulus of the susceptibility by the frequency squared, in accordance with Equation 15. The nonresonant contribution was assumed to be a constant. 5,19 The resulting SFG spectrum in the SSP geometry for the ambient water/vapor interface is presented in Figure 8 (the paper also presents the SPS spectrum). The resulting lineshape compares quite favorably to the experimental measurement in Figure 7a, and displays the essential features with a free O-H stretching mode at about cm 1, and a broad somewhat structured signature between about cm 1 and cm 1. This method is appealingly (relatively) simple, and was successfully adopted by experimentalists to model other aqueous interfaces. 11,118 Interestingly, this methodology can be interpreted as an approximation to INM methods to calculating spectra, 1 3,99,120,121, in which the relevant perturbation expression is also evaluated for harmonic oscillators. The difference in the INM approach is the normal modes are calculated as the exact normal coordinates of the instantaneous configuration of the system, and the homogeneous line width is neglected (it is effectively the bin size used to calculate the spectrum to approximate a delta function contribution). The results from the Morita and Hynes work are, however, different from the INM results 2,3 which are consistently broader (even though their work uses a relatively large line width of γ = 22. cm 1 ). 5 This implies that the method used to calculate the frequency shifts must not sample all the underlying frequency fluctuations, and effectively includes some motional narrowing thus giving a spectrum resembling that observed experimentally. Results from INM spectroscopy calculations of SFG spectra will be considered in more detail below, but INM methods do show the interfacial normal modes are largely localized on single molecules consistent with the ansatz used in the above work. 1 3 Their study also 37

39 39/224 plotted the real and imaginary parts of the susceptibility, shown in Figure 9, that is due to the upper and lower vibrational eigenstates from each molecule. 5 Note, it was also found that the eigenstates are approximately O-H local stretching modes although the lower (higher) frequency modes show symmetric (antisymmetric) mode character; this is consistent with results from INM studies at both the interface 1,3 and in the bulk. 99 Figure 9 demonstrates the higher frequency mode is largely responsible for the free O-H stretch, and the lower eigenstate on the same molecule makes up most of the rest of the O-H stretching spectrum. This evidence is consistent with an interpretation of the O-H stretching spectrum as a free O-H and donor O-H region. However, it will be shown below, by analyzing the real and imaginary parts of the susceptibility (on a better averaged signal) in more detail, the donor O-H region apparently contains a number of distinct oscillator species. 3,19,22 Figures 10a-b, are from the same work, and represent the first theoretical investigation of the interface specificity of the SFG signal. It is clear the free O-H stretches are nearly all localized at the interface, and the donor O-H region of the spectrum has a dominant contribution from the first layer of molecules yet exhibits nonnegligible intensity from the second layer. The ability to describe an SFG signal in molecular detail is a significant strength of SFG spectroscopy. 2. Applications to other Aqueous Interfaces Using the theoretical frequency domain techniques described by Morita and Hynes, Brown et. al. examined the hexane/water and CCl 4 /water interfaces. 11 The simulated SFG spectra for these interfaces are in excellent agreement with the experimental spectra. In both systems, a sharp free O-H peak is observed at cm 1 as well as a broad peak at cm 1. The relative intensities are comparable to experiment for both interfaces. Comparison of the two systems reveals the frequency of the free O-H vibration is slightly lower for the CCl 4 /water and hexane/water interface than for the water/vapor interface due to interactions at the interface between either CCl 4 or hexane and water. The intensity of 38

40 40/224 the free O-H peak is significantly decreased in the hexane/water spectra compared to the CCl 4 /water interface, and is speculated to be due to strong interactions between the free O-H oscillators at the interface and hexane. This interaction is also postulated to be the cause of some asymmetry in the free O-H peak for the hexane/water interface. B. Theoretical Time Domain Approaches to The SFG Spectrum The remaining theoretical SFG studies all adopted a time domain approach 1 4 to calculating the resonant susceptibility similar to that introduced originally by Girardet. 6 All of these approaches require, at their core, calculating the cross TCF of the system dipole and polarizability, < α pq (t)µ r (0) >. This, being the product of a first and second rank tensor, vanishes in isotropic media. 147 The following derivation follows that of Perry et. al.. The resonant part of the susceptibility is given by the imaginary part of this quantum mechanical TCF via Equation 31. The goal, in this context, is to rewrite Equation 31 in a form that is amenable to calculation using classical TCF theory in order to take advantage of the power of the molecularly detailed description offered by many body classical MD. To proceed, the imaginary part of the one time TCF is related in frequency space exactly to the real part: C I (ω) = tanh(β ω/2)c R (ω); where the subscripts denote the Fourier transform of the real and imaginary parts of the complex function C(t) which is a real function of frequency, i.e. C(ω) = ( 1 2π ) dte iωt (C R (t) + ic I (t)) = C R (ω) + C I (ω). Substituting into Equation 31 gives: C I (t) = χ Res (ω) = 2 0 dte iωt C I (t) (49) dω e iω t tanh(β ω /2)C R (ω ) Note, C I (t) is written in a form that can be calculated using the real part of the correlation function which is approximately obtainable, after quantum correction, using classical MD and TCF approaches. Due to causality, the Fourier-Laplace transform gives a real and imaginary part in Equation 49 as the cosine and sine transform of C I (t) respectively. Equation 39

41 41/ can be simplified by changing the order of integration performing the frequency domain integral first. Defining the real and imaginary parts of χ Res (ω) = χ Res R (ω) + iχres(ω): I χ I (ω) = 1 tanh(β ω/2)c R(ω) = 2 0 sin(ωt)c I (t)dt (50) χ R (ω) = 1 π P tanh(β ω/2)c R (ω ) dω = 2 ω + ω 0 cos(ωt)c I (t)dt (51) To obtain Equations 50-51, the identity 0 e iωt dt = ip ω principle part πδ(ω) was used. P designates the The current focus of SFG experiments is on intramolecular vibrations, and to calculate observables in this spectral region, classical mechanics is clearly invalid. Building on our previous work, the classical correlation function result, that is amenable to calculation using MD and TCF methods, is quantum corrected using a harmonic correction factor: C R (ω) = C Cl (ω)( β ω 2 coth(β ω/2)).105,148 This correction factor is exact in relating the real part of the classical harmonic coordinate correlation function to its quantum mechanical counterpart. It is worth noting it is not uncommon in modeling vibrational spectroscopy via TCF s to see the real part of the quantum TCF replaced by the classical TCF that has the same even time symmetry, and becomes equivalent classically (at low frequencies where ω kt ). This approach is reasonable in describing vibrational lineshapes, but does not give accurate intensities. It is generally better to use the harmonic correction factor. Similar caveats apply to replacing the full quantum TCF with its classical counterpart, but, in that case, one neglects the imaginary part of the TCF entirely (that may not matter very much when considering high frequency phenomena for which C I (ω) = C R (ω) because the hyperbolic tangent function is approaching unity). Using the classical harmonic coordinate quantum correction factor does not account for the fact the dipole and polarizability contain higher orders of the coordinates exact corrections for harmonic systems of this type are still possible, but unneeded (the linear dipole 40

42 42/224 and Placzek approximation are adequate). 105 Using this result, the TCF approximation to the resonant part of the SFG spectrum, χ Res, takes the form: χ I (ω) = βωπc Cl (ω) (52) χ R (ω) = βp C Cl (ω ) ω ω + ω dω (53) C Cl (t) =< µ i (0)α jk (t) > (54) In Equation 54, the angle brackets represent a classical TCF that can be computed using MD, and a suitable spectroscopic model. 4,149 Finally, Equations give the TCF signal in a form amenable to classical simulation. Note, while it easier to evaluate χ I (ω) using Equation 52, χ R (ω) is more easily computed by doing the cosine integral as in Equation Considering three possible independent polarization conditions, SSP, PPP, and SPS, for the TCF in Equation 54, the first index in the polarization designation corresponds to the last index in the TCF. For example, the SSP and PPP polarization conditions probe dipolar motions normal to the interface, and the SPS case is sensitive to dipolar changes parallel to the interface. Note, the PPP condition is sensitive to motions both parallel and perpendicular to the interface. 77 Further, the (SSP and PPP)/(SPS) probe diagonal/off diagonal polarizability matrix elements respectively. Morita and Hynes adopted a very similar approach to modeling the SFG spectrum, but quantum corrections were not included. Additionally, the RWA was invoked (see Section VII A). The RWA is equivalent to setting the hyperbolic tangent factor to unity in Equation 49. At lower frequencies, including this factor leads to expressions that are quite different, and the tanh factor produces a time derivative of the correlation function in the time domain. Perry et. al. also constructed an INM approximation to SFG spectra. To do so, it is sufficient to evaluate Equations for a harmonic system. To do so, it is convenient to invoke both the Placzek and linear dipole approximation to evaluate the resulting matrix elements (consistent with the frequency domain work in Section VIII A) 5 although higher 41

43 43/224 order contributions can be included, and simple analytic expressions result for these contributions. An equivalent approach is to evaluate C Cl (t) for classical harmonic oscillators, and quantum correct the resulting expression using the harmonic correction factor, given above, to relate C Cl (t) and C R (t). C Cl (ω) =< ( µ i / Q l ) ( α jk / Q l ) δ(ω ω l ) kt ω 2 > (55) In Equation 55, ω l is the frequency of mode Q l, and the angle brackets represent averaging over classical configurations of the system. C Cl (ω) is then back transformed into the time domain, and used in Equations in place of the classical TCF to obtain an INM approximation to the spectroscopy. 1. Applications to the Water/Vapor Interfaces The first application of TCF theory to water vapor interfaces was by Morita and Hynes 4 followed by a study by Perry et. al.. 1 Both investigations concentrated on the O-H stretching region of the SSP SFG spectrum. To calculate the requisite TCF in Equation 54, it is necessary to first create a series of time ordered MD configurations of the interface using a particular force field. Morita and Hynes employed both polarizable and nonpolarizable models while Perry et. al. used a (nonpolarizable) flexible simple point charge (SPC) 128 model. In calculating the TCF itself, a spectroscopic model is needed to calculate the time dependent dipoles and polarizabilities. Calculating accurate TCF s for SFG spectroscopy is difficult because not only is accurate calculation of the dipoles and polarizabilities required, but it is also essential to properly describe their derivatives which control intramolecular intensities. Morita and Hynes used a spectroscopic model based on extensive ab initio calculations to describe the change of the dipole and polarizability in the gas phase, and a bond polarizability description of the many body polarization contributions in the condensed phase. Perry et. al. used a many body polarization model (a point atomic polarizability model) for the polarizability and induced dipoles that naturally incorporates the polarizability derivatives 42

44 44/224 (fit to both ab initio and experimental Raman data). 154,155 The considerations in choosing a spectroscopic model are no different from those required for modeling condensed phase Raman or IR spectra, and need not be repeated here. Unfortunately, these early TCF studies produced very noisy spectra that were difficult to interpret because evaluating the TCF in Equation 54 presents a problem for interfacial systems; MD interfaces are typically constructed using a standard MD geometry with vacuum/vapor above and below the water. 5,9 This produces two interfaces with average net dipoles in opposite directions. Calculating the SFG spectrum of the entire system would lead to partial cancellation of the SFG signal, and meaningless results. Therefore, the system needs to be split into two pieces through the center of mass of the interfacial system along the direction normal to the interface. Each of the resulting subsystems is then handled separately, and each molecule assigned to one half or the other for the entire length of the calculation. Still, a problem arises in that molecules at one interface can diffuse to the other interface over time. This is a problem in calculating the TCF spectra where different molecular contributions are all added to form a single net dipole and polarizability at each step that are then correlated to form C Cl (t). 149 Using the bulk diffusion constant for water, it can be estimated it takes about ps for a molecule to diffuse from one interface to the other for the system sizes considered. It was, therefore, necessary to limit the length of a TCF correlation time to 15. ps (for a 108 particles) or 30. ps (for a 512 particles), and many correlations of this length were performed in calculating an averaged TCF. The need to do this can be understood from a single molecule perspective. If cross terms between the dipoles and polarizability elements could be neglected, C Cl (t) could be written in terms of single molecule contributions: C Cl (t) N < µ M i (0)αjk M (t) > where N is the number of molecules, and the superscript M is the molecule index. 94 Because bulk isotropic molecular motions give no SFG signal, it is the anisotropic dynamics of molecules at the interface that generate a signal. Both the polarizability tensor elements and dipole 43

45 45/224 moment are independent of translational origin thus, it is only necessary to distinguish between molecules exhibiting bulk and interfacial dynamics to understand the contribution of a molecule to the TCF. If a molecule were to reside at both interfaces during the duration of the TCF calculation, invalid results would be obtained; this is clearly undesirable. Further, if molecules were reassigned to a specific half of the system at each calculation time point, an asymmetry would be introduced at the dividing surface by including the dynamics of a molecule at only certain steps as it appears in, and disappears from, a given half. This would introduce an artificial inhomogeneity in truly bulk like isotropic dynamics that might generate an SFG signal. 1 The fact the correlation function calculations were limited to short times, and the SFG TCF, a cross correlation between the system dipole and polarizability elements (not invariants like in traditional Raman and IR experiments), is long lived, leads to poor averaging at longer times. This makes the signal difficult to accurately Fourier transform even though the focus is on extracting the short time high frequency behaviors. Nonetheless, the spectra that were obtained resembled the experimental data, and clearly showed a free O-H and donor O-H region. Figure 11 presents the TCF generated spectra for three distinct MD models. 4 It is clear, while the free O-H peak is relatively well averaged, the rest of the O-H spectrum is not sufficiently well resolved to reveal its structure. The author s do note the polarizable MD model produces relative intensities between the two regions of the spectrum reminiscent of the experimental results. Perry et. al. 1 present similar results, but erroneously conclude (largely due to the noisy data) dynamical effects (motional narrowing) are not represented in the SSP spectrum; this result is strongly contradicted by their later work discussed below. Their approach was to compare INM and TCF generated spectra. INM results represent an underlying spectral density that may or may not be motionally narrowed in the observed lineshape. 3,54,99 If the INM and TCF spectra are similar in breadth (and in that case in shape), then dynamical effects will not be represented in the lineshape. When dynamical effects are important, the INM lineshape will be broader, but, in both cases, will have the same integrated intensity. The author s also show an INM illustration 44

46 46/224 of a representative free and donor O-H on a single molecule demonstrating the power of the INM approach in revealing the molecular nature of vibrational modes. 1 In order to obtain better TCF results, long time (cross) correlations between the system dipole and polarizability need to be followed. To overcome the time limit problem discussed above, Perry et. al. 2,3 added a weak (laterally isotropic) restraining potential to the MD force field that effectively confined molecules over time to the half of the simulation box they start in (in the dimension normal to the interface). The external potential was chosen such that it did not significantly perturb the relevant dynamics; even though the molecular diffusion constant (normal to the interface) is changed, the molecule can only contribute to the spectrum while resident at the interface where it is free of any significant external potential. This modification permitted the calculation of TCF s out to arbitrarily long times resulting in sharp spectra that included intermolecular spectral lineshapes. As a check, it was noted the interfacial density profile was unchanged by the restraining potential demonstrating the restraining potential used did not perturb the average structure of the liquid that contributes to the interfacial spectroscopy. Figure 12 displays the theoretical TCF SFG spectra in the O-H stretching region for the three independent polarization conditions that are possible in the electronically nonresonant experiment (SSP, PPP, and SPS) when the TCF s were converged at long times. 2 The theoretical spectra have been adjusted in relative intensity to account for the Fresnel factors that modify the experimental intensities to directly compare with experiment; 3,13 the data also includes the nonresonant contribution, χ NRes (ω), which is a small negative constant. 4,130 The full signal is given by: χ (2) SF G (ω) 2 χ Res (ω) + χ NRes (ω) 2. The inset of Figure 12 displays experimental data for the O-H stretching region taken in the same polarization geometries. 13 The relative intensities agree nearly quantitatively between theory and experiment, and the lineshapes are far improved over the earlier attempts. Due to the use of the restraining potential and the ability to follow the TCF out to long times, the authors were able to obtain, for the first time either experimentally or theoretically, 45

47 47/224 low frequency SFG spectra. Figure 13 displays the theoretical SFG spectrum over the entire water vibrational spectrum. 2 The theoretical INM spectrum (for the SSP geometry) is also shown. The INM and TCF spectra were found to integrate to the same value (over the entire cm 1 range), and separately over the O-H stretching region ( cm 1 ). This behavior is strong evidence for the interpretation of the INM lineshape as an underlying spectral density that is motionally narrowed in the observed spectrum. This result also suggests SFG spectra are sensitive to both structure and dynamics. The INM spectrum clearly exhibits the same resonances, but is broader; implying the observed lineshapes are motionally narrowed, and dynamical contributions to SFG signals are important. 3,13 Most strikingly, Figure 13 reveals an intense intermolecular resonance at 875. cm 1. In contrast, the intermolecular spectrum of bulk water is relatively unstructured. 99 This symmetric lineshape indicates a spectroscopically distinct species, and represents like the free O-H stretch a population of water molecules unique to the interface. It is roughly as intense (considering the susceptibility and not the SFG intensity that includes an additional factor of the frequency squared) as the rest of intermolecular spectrum, and is about a sixth of the intensity of the free O-H peak within our model. (Note, the bending lineshape at higher frequency is much less intense.) Recent experiments 42,43 and theory 7 indirectly inferred the presence of a surface species a water molecule with two dangling hydrogens. The SSP and PPP spectra also show an intense intermolecular mode at 95. cm 1. Using instantaneous normal mode methods, the resonance was found to be due to hindered translational modes localized on single molecules oscillating perpendicular to the interface. The SPS spectra, which is sensitive to dipole derivatives parallel to the interface, shows an intermolecular mode at 220. cm 1. This mode is a result of translations parallel to the interface. These results highlight the importance of polarization sensitivity in SFG experiments. The authors also note these species could be experimentally measurable using SFG, but this has not been done to date due to a lack of intense IR laser sources in this spectral region. The fact these interfacial species have gone long undetected might be surprising given the 46

48 48/224 large numbers of MD simulations of the water/vapor interface that have been performed previously. This observation highlights the power of calculating spectroscopic observables in assessing interfacial structure and dynamics. Not only can the results be directly compared with experiment thus, validating a particular MD and spectroscopic model but the spectroscopic calculation serves as a filter of the dynamics extracting out the identity of collective coordinates with well defined frequencies that persist at the interface. The authors also show an MD generated snapshot of the interface which is detailed in Figure 14. This molecular snapshot highlights the different interfacial species that were identified in their studies. 3 The hindered rotational (wagging) and translational modes are clearly shown. These results demonstrate how the INM approach does not require a priori assumptions about the nature of interfacial modes, but does reveal their physical characteristics, and how different molecular motions contribute to the spectrum. It was also observed that examining the real and imaginary parts of the spectrum can offer insights unavailable from the modulus alone. 3 The real and imaginary parts could be measured experimentally via a heterodyne detection scheme, or by taking advantage of interference effects between bulk and interfacial contributions to the spectrum. 16 To see the advantages of separately examining the real and imaginary contributions, it is useful to reexamine Equations They imply a single type of mode will lead to an imaginary contribution that is a symmetric well defined peak (Lorentzian in character) while the real part will change sign, dipping below zero, at the maximum of the imaginary portion. If more than one species is contributing to the signal in a given region, a more complex lineshape would result from the overlapping signals. As pointed out by Morita and Hynes, 5 orientational information can also be deduced from the relative signs of the imaginary mode lineshapes given knowledge of the signs of the prefactors in Equations (the dipole and polarizability derivatives). Figure 15 presents the real and imaginary parts of the susceptibility in the O-H stretching region from approximately cm 1 to cm 1 calculated by Perry et. al.. 3 Careful 47

49 49/224 examination of the spectrum reveals three separate modes in this region centered at cm 1, cm 1, and cm 1. Remarkably, this agrees very well with previous experimental work that deconvoluted the spectrum in this region. That analysis revealed three modes present in the same region centered at cm 1, cm 1, and cm 1 nearly the same frequencies as shown in Figure 7. 19,130 This is strong evidence for distinct populations of water molecules in this donor O-H region of the spectrum. This results represents strong motivation for experimentalists to measure the real and imaginary parts of the susceptibility for the water/vapor interface. 2. Applications to Salt Water/Vapor Interfaces Interfacial electrolytes are import in biological, industrial, and atmospheric processes. Despite the importance of these interfaces, the atomic details of the surface of electrolyte solutions are unknown. Recent theoretical and experimental work, including SFG studies, by Jungwirth, Tobias, Allen, and coworkers have shed some light on the interfacial composition at simple inorganic salt solutions. 15,18, Historically, the view has been the interface is largely devoid of ions. This view comes from thermodynamic arguments, and experimental evidence such as, surface tension measurements. 159,160 The argument is that differences between the bulk and interfacial concentrations can be related by Gibbs equation, and also by considering the concept of Gibbs surface excess when the interfacial region s density is not constant. relating surface tension to surface excess is given by: Γ = β ( ) dγ d ln c The Gibbs equation (56) In Equation 56, Γ is the surface excess, γ is the surface tension, β is 1/k b T, and c is the concentration. 159,160 Using this treatment, a decrease in surface tension (relative to the pure substance) results from an increase in concentration at the interface (e.g. due to surfactants), and an increase in surface tension results from a decrease in concentration at the interface. 48

50 50/224 Thus, because simple inorganic salts such as aqueous NaCl increase the surface tension relative to pure water, a decrease in concentration at the interface is inferred. These thermodynamic arguments have been used for almost a century to conclude, considering simple salt solutions, their interfaces will be nearly devoid of ions. Recent theoretical 15,18,156,157,161,162 and sensitive experimental 18,130, work has called this traditional view into question. Specifically, the pioneering work of Jungwirth and Tobias 162 that predicted a surface enhancement of chloride ions at the solution/vapor interface for aqueous salts. Recent theoretical work has suggested the propensity for surface enhancement depends on the polarizability and size of ion. For example, F is repelled from the surface consistent with the traditional view. Conversely, Cl, Br, and I all show enhancement at the interface. The simulations that displayed this enhancement employed polarizable force fields in contrast to earlier simulation studies using traditional nonpolarizable forces. Specifically, if the aqueous anion is not polarizable during the simulation, enhancement of the anion at the surface is diminished or eliminated. Enhancement in interfacial concentration of the more realistically modeled polarizable anion is not in contradiction with the thermodynamic equations because they allow for a nonmonotonic ion concentration profile. 156 Indeed, the simulations suggest an overall net decrease of the ion concentration, in agreement with the thermodynamics, with a corresponding increase in surface tension. However, the profile is not monotonic; there is an increase in concentration relative to the bulk at the outer most layer, and a depletion relative to the bulk just below the surface to give an overall depletion of the ions at the interface. Because SFG is a sensitive probe of the interface, Brown et. al. 15 have calculated the SFG spectra arising from salt (NaI (aq) ) air interfaces to look for a signature characteristic of anions at the solution/vapor boundary. They calculate the SFG spectrum of the NaI (aq) interface using a time domain approach, but without the advantages of a restraining potential. 4 I was chosen as the anion because it showed the largest interfacial enhancement in previous simulations. 156 The authors were not able to obtain a reasonable frequency independent 49

51 51/224 nonresonant susceptibility, and, thus, scaled their neat water results to match what has been previously reported. They subsequently used these same scaling factors to deduce the interfacial spectra of the salt solution. Due to these scaling factors, the authors did not compare or draw absolute conclusions from the computed spectra, but did compare and contrast the neat water/vapor spectra with the salt water/vapor spectra. 15 The NaI (aq) /vapor calculated SFG spectrum differs in several ways from the neat water/vapor spectrum that was obtained. Relative to the neat interface signal, there is a slight decrease in the free O-H peak at about cm 1, a large increase in the peak near cm 1, and a slight decrease in the shoulder peak at cm 1. SFG experimental results on the NaI (aq) /vapor SFG, and other sodium halides, have been reported by Raymond and Richmond, 130 and independently by Liu et. al.; 164 both show similar spectra. These spectra are qualitatively similar to the SFG spectra calculated by Brown et. al., 15 but differ in the details. Both experiment and simulation show little difference in the free O-H peak (compared to the neat interface), a large increase in the peak at about cm 1, and a slight decrease in the shoulder peak at cm 1. However, the relative changes in intensity are different between the experimental and simulated spectra. Further, while the reported experimental spectra are very similar for the two groups, they interpret their data differently. Raymond and Richmond ascribe the difference between the neat water/vapor and salt water/vapor interface as evidence the anion is in the subsurface region. In contrast, Liu et. al. interpret the observed spectral changes as evidence the anion is at the surface similar to the MD simulations. Recently, Mucha et. al. have reported simulations and experimentally measured SFG signals of acid, base, and salt solutions 18. While the SFG signal for these systems has not been calculated, the experimental SFG are interpreted in terms of the extant ionic solution simulations. Using the methods outlined in this review, improvement in agreement between the computed and experimentally measured SFG spectra for the salt solution/vapor interface seems likely. Such calculations will allow for the confident interpretation of the spectra, and 50

52 52/224 atomistic resolution of the interfacial region of these important electrolyte systems. IX. CONCLUSION SFG experimental measurements are growing in number and importance; they are providing valuable information about interfacial structure and dynamics that would be difficult to measure, or are not obtainable otherwise. Theoretical studies are only now sufficiently sophisticated that they can begin to play the major role simulation has in modeling and interpreting condensed phase spectroscopy. In principle, SFG spectroscopy is capable of giving a complete picture of the interface including structure and dynamics (although fourth order polarization experiments, 166,167 would be required to analyze the detailed interface vibrational dynamics and unambiguously distinguish between homogeneous and inhomogeneous vibrational lineshapes). 55 Realizing this promise depends critically on the spectra being reliably interpreted, and the methods described in this review are capable of unambiguously characterizing the nature of SFG spectra including inferring subpopulations of molecules from complex lineshapes. Still, a vigorous interplay between theory and experiment is needed to further develop the interpretative and predictive power of theoretical studies. The investigation of more complex interfaces using the improved TCF methods, described here, will help to both interpret the large and growing body of experimental data, and to predict heretofore unexplored interfacial vibrational structure. Further, experimental advances are likely to extend the frequency range for SFG measurements into the far IR where theory predicts important interfacial species are present; lower frequency phenomena are important in contributing to processes such as, interfacial solvation reaction dynamics. Additionally, time domain SFG techniques are extending the abilities of SFG spectroscopy to probe interfacial vibrations in new novel ways. However, these methods require theoretical support in designing and interpreting the convoluted signals that result. Lastly, theoretical and experimental measurements of both the real and imaginary parts of the SFG signal (as opposed to measuring the squared modulus as in the typical homodyne detected experiment) 51

53 53/224 show great promise in helping unravel complex SFG lineshapes. It would be beneficial to the field if more experiments were conducted (either via heterodyne detection or use of interference effects) to separately measure these contributions. X. ACKNOWLEDGMENTS The research at USF was supported by an NSF Grant (No. CHE ), a grant from the Petroleum Research Foundation to Brian Space, and a Latino Graduate Fellowship to Christine Neipert. Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research. The authors would like to acknowledge the use of the services provided by the Research Oriented Computing center at USF. The authors also thank the Space Foundation for Basic and Applied Research for partial support. We also thank Dr. s Alex Benderskii, Chris Cheatum, Zhan Chen, Randy Larsen, and Tom Keyes for helpful discussions. Lastly, we thank Christina Ridley Kasprzyk and Tony Green for their contributions to the work. 52

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62 62/224 XII. APPENDIX A: POSSIBLE SECOND ORDER NONLINEAR PROCESSES For three-wave mixing experiments there are two relevant applied time dependent fields which result in a sum of (2N) N =16 terms when specific time ordering of the fields cannot be assumed. Each one of these 16 pathways corresponds to a specific P (2) (k s, t). Of the 16 possible contributions, only four distinct physical processes occur, SFG, DFG, SHG and optical rectification. Each of the four effects can be independently measured experimentally using the appropriate phase matching condition that corresponds to a particular experimental geometry and associated signal wave vector, k s. 55,57 Note, for SHG experiments, where two visible fields of the same frequency are typically used, 4 of the pathways (and their complex conjugates) become equivalent providing added intensity. SHG experiments are also extensively used to probe interfacial properties when a chromophore is present that can be excited by the visible light E(r, t τ 1 ) = E 1 (t τ 1 )e ik 1 r + E 1(t τ 1 )e ik 1 r + E 2 (t τ 1 )e ik 2 r + E 2(t τ 1 )e ik 2 r (57) E(r, t τ 2 ) = E 1 (t τ 2 )e ik 1 r + E 1(t τ 2 )e ik 1 r + E 2 (t τ 2 )e ik 2 r + E 2(t τ 2 )e ik 2 r (58) 61

63 63/224 XIII. TABLES TABLE I: The possible second order nonlinear optical processes given in terms of the incident wave vectors. Second Harmonic Generation (SHG) k s = ±2k i ω s = ±(ω 1 + ω 1 ) Sum Frequency Generation (SFG) k s = ±(k 1 + k 2 ) ω s = ±(ω 1 + ω 2 ) Sum Difference Generation (DFG) k s = ±(k 1 k 2 ) ω s = ±ω 1 ω 2 Optical Rectification k s = 0 ω s = 0 62

64 64/224 TABLE II: Columns 1 and 3 detail the sixteen terms resulting from the two incident fields. Columns 2 and 4 give their corresponding nonlinear optical process respectively. E 1 (t τ 1 )e k 1 r E 1 (t τ 2 )e ik 1 r k s = 2k 1 E 1 (t τ 1 )e k 1 r E 1 (t τ 2)e ik 1 r k s = 0 E 1 (t τ 1 )e k 1 r E 2 (t τ 2 )e ik 2 r k s = k 1 + k 2 E 1 (t τ 1 )e k 1 r E 2 (t τ 2)e ik 2 r k s = k 1 k 2 E 1 (t τ 1)e ik 1 r E 1 (t τ 2 )e ik 1 r k s = 0 E 1 (t τ 1)e ik 1 r E 1 (t τ 2)e ik 1 r k s = 2k 1 E 1 (t τ 1)e ik 1 r E 2 (t τ 2 )e ik 2 r k s = k 1 + k 2 E 1 (t τ 1)e ik 1 r E 2 (t τ 2)e ik 2 r k s = k 1 k 2 E 2 (t τ 1 )e ik 2 r E 1 (t τ 2 )e ik 1 r k s = k 1 + k 2 E 2 (t τ 1 )e ik 2 r E 1 (t τ 2)e ik 1 r k s = k 1 + k 2 E 2 (t τ 1 )e ik 2 r E 2 (t τ 2 )e ik 2 r k s = 2k 2 E 2 (t τ 1 )e ik 2 r E 2 (t τ 2)e ik 2 r k s = 0 E 2 (t τ 1)e ik 2 r E 1 (t τ 2 )e ik 1 r k s = k 1 k 2 E 2 (t τ 1)e ik 2 r E 1 (t τ 2)e ik 1 r k s = k 1 k 2 E 2 (t τ 1)e ik 2 r E 2 (t τ 2 )e ik 2 r k s = 0 E 2 (t τ 1)e ik 2 r E 2 (t τ 2)e ik 2 r k s = 2k 2 63

65 65/224 XIV. LIST OF CAPTIONS 64

66 66/224 FIG. 1: Coplanar geometry of the incident, reflected and transmitted beams. θ 1 (θ 2 ) is the angle of incidence with respect to the z-axis of the visible (IR) field. θ SF G (θ DF G ) is the angle the generated SFG (DFG) signal is radiated at. k 1 (k 2 ) is the wave vector of the visible (IR) field. k r s (k T s ) is the wave vector of the reflected (transmitted) field, and k s = k 1 + k 2. All incident fields are assumed to lie in the same xz plane which is normal to the surface. FIG. 2: Frequency domain and time domain SFG measurements in the C-H stretch region of acetonitrile. Top Panel: SFG spectra of a clean Au film (lower spectrum) and a Au film with acetonitrile. The weak feature around cm 1 corresponds to the asymmetric stretch vibration. Lower Panel: free induction decay of clean Au film (offset) and a Au film with acetonitrile. The dashed lines in both panels are fits to the data with very similar vibrational dephasing times. The inset shows the IR field, Ē IR and resonant susceptibility, χ (2) Res (t). Reprinted from reference.48 FIG. 3: Frequency domain and time domain SFG measurements in the C-N stretch region of acetonitrile. Top Panel: two (identical) SFG spectra of a Au film with acetonitrile. The upper (lower) spectrum is fitted assuming a homogeneous (inhomogeneous) distribution of adsorption sites. Lower Panel: free induction decay of a Au film with acetonitrile. The data shown consists of an average of nine different data sets, of which the error bars show the spread. The dashed (solid) line is a calculation assuming a homogeneous (inhomogeneous) distribution of adsorption sites. The inset shows the resonant susceptibility, χ (2) Res (t), used for theoretical comparisons. Reprinted from reference. 48 FIG. 4: Frequency domain SFG spectrum of the ν (OD) transition of D 2 O at CaF 2 interface (dotted line experimental data, solid line fit). IR pulse spectra used for the spectrally and time resolved SFG measurements (STiR-SFG) are shown in thin dashed and dotted lines. Reprinted from reference. 47 FIG. 5: Right Panel: STiR-SFG measurement of D 2 O on CaF 2 interface. IR excitation cm 1 (blue side). Left Panel: instrument response function (IR-visible SFG-FROG cross correlation). Vertical axis, IR-visible delay time (fs); Horizontal axis, SFG wavelength; Top, Converted IR Frequency Left Panel: instrument response function. Reprinted from reference

67 67/224 FIG. 6: Right Panel: STiR-SFG measurement of D 2 0 on CaF 2 interface. IR excitation cm 1 (red side). Left Panel: instrument response function (IR-visible SFG-FROG cross correlation). Vertical axis, IR-visible delay time (fs); Horizontal axis, SFG wavelength; Top, Converted IR Frequency Left Panel: instrument response function. Reprinted from reference. 47 FIG. 7: (a) SFG spectrum of the water/vapor interface taken under SSP polarization conditions. The smooth line is the best fit to the data. (b) Resonant SF response from the water/vapor interface. Also shown are the contributions from the different molecular species. The inset shows the 3200 cm 1 peak more clearly. Reprinted from reference. 130 FIG. 8: Simulated SFG spectrum of water of SSP polarization using a frequency domain method. Reprinted from reference. 5 FIG. 9: Susceptibility χ Res ssp per unit surface area decomposed into two vibrational eigenstates, lower energy state ( ) and higher energy state (...). Panels (a) and (b) show the real and imaginary parts, respectively. Reprinted from reference. 5 FIG. 10: Surface sensitivity of the susceptibility χ Res ssp per unit surface area of water. The solid lines denote the whole susceptibility, and the other lines correspond to virtual susceptibilities within restricted systems by various depths from the surface to examine the convergence. (a) and (b) show the real and imaginary parts respectively. Note the depths are given with respect to the calculated Gibbs dividing surface. Reprinted from reference. 5 FIG. 11: Calculated SSP SFG spectrum of the water surface. Three models are employed: (a) the Ferguson force field with no induced polarization, (b) the Kuchitsu and Morino force field with no induced polarization, and (c) the Ferguson force field with induced electronic polarization. Reprinted from reference. 4 FIG. 12: TCF SFG spectra in the O-H stretching region for three polarizations: SSP (thick, solid black line), PPP (dashed red line), and SPS (thin, solid green line). The inset is experimental data 13 for the same polarizations using the same color scheme. Reprinted from reference. 2 66

68 68/224 FIG. 13: TCF SFG spectra for the entire water vibrational spectrum for three polarizations: SSP (thick, solid black line), PPP (thick, dashed red line), and SPS (thin, solid green line). The SSP INM SFG spectra is also shown (thin, dashed blue line). The inset highlights the intermolecular resonance at 875. cm 1. Reprinted from reference. 2 FIG. 14: A snapshot of a water/vapor interface containing 216 water molecules featuring INM s from different regions of the spectra. The water molecule shown in blue is representative of a free O-H mode at cm 1. The water molecule shown in green is representative of a wagging motion at 858. cm 1. The water molecule shown in yellow highlights a translation perpendicular to the interface at 46. cm 1. The water molecule shown in black highlights a translation parallel to the interface at 197. cm 1. Reprinted from reference. 3 FIG. 15: Real (solid green line) and imaginary (dashed blue line) components of the SFG SSP TCF spectra for the water/vapor interface for the O-H stretching region. The arrows highlight three separate modes centered at cm 1, cm 1, and cm 1. Reprinted from reference. 3 67

69 69/224 XV. FIGURES 68

70 70/224 Figure 1 69

71 71/224 Figure 2 70

72 72/224 Figure 3 71

73 73/224 Figure 4 72

74 74/224 Figure 5 73

75 75/224 Figure 6 74

76 76/224 Figure 7 (a) VSF spectrum of the vapor/water interface taken under 75

77 77/224 Figure 8 76

78 78/224 Figure 9 77

79 79/224 Figure 10 78

80 80/224 Figure 11 ulated SFG spectrum of the water surface. Three models are employed: (a) the Ferguson force field with no ind 79

81 81/224 Figure (10 43 m 4 V 2 ) wavenumber (cm 1 ) (10 15 A 8 e 2 K 2 ) wavenumber (cm 1 ) 80

82 82/224 Figure (10 15 A 8 e 2 K 2 ) (10 15 A 8 e 2 K 2 ) wavenumber (cm 1 ) wavenumber (cm 1 ) 81

83 83/224 Figure 14 82

84 84/224 Figure (10-8 A 4 ek -1 ) wavenumber (cm -1 ) 83

85 85/224 A Theoretical Description of the Polarization Dependence of the Sum Frequency Generation Spectroscopy of the Water/Vapor Interface Angela Perry, Christine Neipert, Christina Ridley Kasprzyk, Tony Green and Brian Space Department of Chemistry, University of South Florida, Tampa, FL Preston B. Moore Department of Chemistry and Biochemistry University of the Sciences in Philadelphia, Philadelphia, PA (Dated: July 7, 2005) 1

86 86/224 Abstract An improved time correlation function (TCF) description of sum frequency generation (SFG) spectroscopy was developed and applied to theoretically describing the spectroscopy of the ambient water/vapor interface. A more general TCF expression than was published previously is presented it is valid over the entire vibrational spectrum for both the real and imaginary parts of the signal. Computationally, earlier time correlation function approaches were limited to short correlation times that made signal processing challenging. Here, this limitation is overcome, and well averaged spectra are presented for the three independent polarization conditions that are possible for electronically nonresonant SFG. The theoretical spectra compare quite favorably in shape and relative magnitude to extant experimental results in the O-H stretching region of water for all polarization geometries. The methodological improvements also allow the calculation of intermolecular SFG spectra. While the intermolecular spectrum of bulk water shows relatively little structure, the interfacial spectra (for polarizations that are sensitive to dipole derivatives normal to the interface SSP and PPP) show a well defined intermolecular mode at 875. cm 1 that is comparable in intensity to the rest of the intermolecular structure, and has an intensity that is approximately one-sixth of the magnitude of the intense free O-H stretching peak. Using instantaneous normal mode methods, the resonance is shown to be due to a wagging mode localized on a single water molecule, almost parallel to the interface, with two hydrogens displaced normal to the interface, and the oxygen anchored in the interface. We have also uncovered the origin of another intermolecular mode at 95. cm 1 for the SSP and PPP spectra, and at 220. cm 1 for the SPS spectra. These resonances are due to hindered translations perpendicular to the interface for the SSP and PPP spectra, and translations parallel to the interface for the SPS spectra. Further, by examining the real and imaginary parts of the SFG signal, several resonances are shown to be due to a single spectroscopic species while the donor O-H region is shown to consist of three distinct species consistent with an earlier experimental analysis. Author to whom correspondence should be addressed. space@cas.usf.edu 2

87 87/224 I. INTRODUCTION Liquid water interfaces are ubiquitous and important in chemistry and the environment. Thus, with the advent of interface specific nonlinear optical spectroscopies, such interfaces have been intensely studied both theoretically 1 9 and experimentally Sum frequency generation (SFG) spectroscopy is a powerful experimental method for probing the structure and dynamics of interfaces. SFG spectroscopy is one of several experimental methods that measure a second order polarization, and the more common electronically nonresonant experiment is considered here. SFG spectroscopy is dipole forbidden in isotropic media such as liquids. Contributions from bulk allowed quadrapolar effects have been demonstrated to be negligible in some cases, 36,37 but can be included if necessary. 38 Interfaces serve to break the isotropic symmetry, and produce a dipolar second order signal. The SFG experiment employs both a visible and infrared laser field overlapping in time and space at the interface, and can be performed in the time or frequency domain. 14,39 In the absence of any vibrational resonance at the instantaneous infrared laser frequency, a structureless signal due to the static hyperpolarizability of the interface is obtained. 3,11,17 When the infrared laser frequency corresponds to a vibration at the interface, a resonant lineshape is obtained with a characteristic shape that reflects both the structural and dynamical environment at the interface. 2,40,41 In this paper, classical molecular dynamics (MD) methods are used to model the dynamics of the water/vapor interface. Two complementary theoretical approaches quantum corrected time correlation function (TCF) and instantaneous normal mode (INM) methods use the configurations generated by MD as input to describe the SFG spectrum of the interface, and to ascertain the molecular origin of the SFG signal; both INM and TCF methods rely on a suitable spectroscopic (dipole and polarizability) model. This dual approach was demonstrated to be highly useful in understanding condensed phase spectroscopy of water, other liquids, and interfaces classical mechanics, especially in the context of quantum corrected TCFs, has proven to be surprisingly effective in modeling intramolecular vibrational spectroscopy. 1,2 In particular, TCF methods have provided a quantitative description of the O-H stretching lineshape in ambient liquid water, and INM methods have served to identify the molecular motions that result in the observed signal; these complementary techniques are equally effective for modeling water interfacial spectroscopy. 1 4,42 48 An INM approximation to SFG spectroscopy is quantum mechanical by construction, but offers a limited 3

88 88/224 dynamical description. As a result, in bulk water (and other liquid state intramolecular lineshapes), INM intramolecular resonances are broader than their TCF counterparts, but have the same central frequency and integrated intensity. This observation suggests the intramolecular INM spectra represent an underlying spectral density that is dynamically motionally narrowed in the actual lineshape. 40 This is also found to be the case here for SFG spectra in all polarization conditions. This result contrasts with a previous report by us, 1 and evidence from the literature. 3,4 Previous TCF and INM calculations of the (SSP polarization) SFG O-H stretching spectrum of the water/vapor interface were very noisy, and suggested the spectra had equal breadth thus, suggesting motional narrowing effects were not apparent in the spectra. The success of an approximate, non-dynamical, frequency domain technique, 4 and the similarity of the spectra to those obtained using TCF methods, 1,3 appeared to be further evidence of spectra that could be described in the inhomogeneously broadened limit. 49 That method, 4 however, contains an empirically adjustable line width that effectively accounts for some motional narrowing making it difficult to draw conclusions. Because of methodological advances, it is now possible to calculate well averaged TCF and INM spectra, and they unambiguously demonstrate SFG O-H stretching lineshapes (at least at the water/vapor interface) are significantly motionally narrowed to a degree reminiscent of the bulk. 50,51 This observation suggests dynamical motional narrowing effects are important at interfaces, and the dynamics are best described as intermediate between the fast and slow modulation limits of motional narrowing. In the slow modulation inhomogeneously broadened limit, all frequency fluctuations of the oscillator are represented in the lineshape. 40,49,51 A recent study in the Shen group also suggested motional averaging effects may well be significant in the SPS geometry, and, in that case, the free O-H stretching peak is greatly diminished. Although that study did not address motional narrowing, the presence of motional averaging suggests motional narrowing is important because it is due to fast reorientational motions within the vibrational relaxation time for the mode that would also be expected to result in motional narrowing. In order to obtain better TCF results, long time (cross) correlations between the system dipole and polarizability need to be followed. Because molecular simulations of interfaces in Cartesian space necessarily produce two interfaces, simulation times were limited to the molecular diffusion time between interfaces so molecules could not contribute to the signal at both interfaces during one MD run. 1 This leads to TCFs 4

89 89/224 without long time decays that are difficult to Fourier transform accurately. In this work, a weak restraining potential is added that confines the molecules over time to the half of the simulation box they start in (in the dimension normal to the interface) without significantly perturbing the (relevant short time) dynamics and average structure of the liquid that contributes to the interfacial spectroscopy; even though the molecular diffusion constant (normal to the interface) is changed, the molecule is only contributing to the spectrum while resident at the interface, and is free of any significant external potential. This modification permits the calculation of TCFs out to arbitrarily long times resulting in sharp spectra that include intermolecular spectral lineshapes. Surprisingly, a well defined intermolecular mode was found to be prominent in the spectrum. 2 It is centered at 875. cm 1, and is comparable in (integrated) intensity to the rest of the intermolecular lineshape the lineshape also has an intensity that is approximately one-sixth of the magnitude of the intense free O-H stretching peak for spectra taken in polarization geometries that are sensitive to dipole derivatives normal to the interface (SSP and PPP). Using instantaneous normal mode methods, the resonance is shown to be due to a wagging mode localized on individual water molecules. Water molecules contributing to this resonance are at a slight angle to the interface with their oxygen atoms anchored in the interface, and the hydrogen atoms wagging nearly normal to the interface. The presence of another population, aside from the free O-H stretch, of interfacial molecules was recently proposed via indirect evidence, 6,34,35 and that hypothesis is strongly supported by this work. Here, we have directly observed a spectroscopically distinct species, and clearly identified the vibrational mode responsible for the lineshape. Thus, experimental setups that permit taking spectra at relatively long wavelengths could probe this mode as a complement to the information contained in the free and donor O-H stretching modes At lower frequencies, well defined hindered translational modes are found both parallel and perpendicular to the interface. The perpendicular modes are prominent in the polarization conditions sensitive to dipolar changes normal to the interface (SPP and PPP) while the parallel modes are more pronounced in the SPS geometry which is sensitive to motions along the interface. Further, some of the time domain expressions given earlier for SFG spectroscopy were only correct for the modulus of the SFG signal (but not for the real and imaginary portions), 1,5 or were not entirely general (and only valid at high intramolecular frequencies). 3,38 Note, some of these works also equated the 5

90 90/224 (complex) quantum TCF with the (real) classical TCF, further complicating matters. 3,38 The correct general and exact expressions are given below and include both the resonant and non-resonant contributions. The previous expressions all give acceptable lineshapes (at high frequencies) for the modulus of the second order signal. They are not correct for calculating the amplitude of the signal that can be detected in a heterodyne experiment, 58 or by taking advantage of phase interference effects. 10 Here, it is shown by carefully examining the real and imaginary parts of the SFG signal, individual mode contributions to an observed lineshape can be identified. Using this approach, the free O-H and the newly discovered modes were identified as individual spectroscopic species (one type of oscillator at the interface), and the donor O-H region consists of three distinct species. This last conclusion agrees with results from a careful deconvolution of O-H stretching signal in an earlier experimental work that also found three species each with approximately the same central frequency. 11,59 Thus, as a prelude to more complex interfaces, this joint TCF/INM approach is applied to the water/vapor interface producing good agreement with the shape and relative amplitudes of SFG measurements for all independent polarization conditions. 41 The theoretical expression in terms of a TCF for the SFG signal is also presented including corrections from expressions published previously 1,3,5 in Section II and Appendix A. The MD, dipole, many body polarization methods, and associated parameters are also summarized in Section II. The theoretical results, and their comparison to experiment, are discussed in Section III. The paper is concluded in Section IV. II. MODELS AND METHODS The SFG signal consists of nonresonant (due to the static hyperpolarizability) and resonant contributions that are important when the infrared laser frequency distribution is resonant with a vibrational transition at the interface. The signal intensity, is proportional to the square of their sum: I χ (2) SF G (ω) 2 χ Res (ω) + χ NRes (ω) 2 and directly measures the modulus squared in the typical homodyne detected effectively monochromatic frequency domain experiment. The superscripts denote the resonant and nonresonant contributions respectively, and χ (2) SF G (ω) is the susceptibility tensor. The proportionality constants include a factor of ω 2 and Fresnel factors. 60 The vibrational information is contained in the resonant signal, and 6

91 91/224 the nonresonant was found to be a (negative) constant in the O-H stretching region; 3,11 this can still lead to the nonresonant contribution changing the frequency dependent intensities through cross terms in the squared modulus signal. Through isotopic substitution, the nonresonant contribution can be measured independently, and this permits the deconvolution of the full signal to extract χ Res (ω) ; this deconvolution was done for the SSP polarization geometry at the water/vapor interface. 11 The second order response is given theoretically by a combination of resonant and nonresonant terms. 3,16,61,62 The resonant terms can be grouped to give a simple expression in terms of the systems polarizability and dipole. A derivation including the resonant and nonresonant terms from perturbation theory is given in Appendix A. 16,62 Thus χ Res (ω) is given by: 5,61 χ Res (ω) = i 0 dte iωt T r{[ρ, µ i ]α jk (t)} (2.1) In Equation 2.1, ρ = e βh /Q for a system with Hamiltonian H and partition function Q at reciprocal temperature β = 1/kT, and k is Boltzmann s constant. µ is the system dipole and α its polarizability tensor where the subscripts represent the vector and tensor components of interest respectively. The operator evaluated at time t is the Heisenberg representation of the operator α jk (t) = e iht/ α jk e iht/ ; T r represents the trace of the operators. It is convenient to proceed by rewriting the Fourier-Laplace transform in Equation 2.1 as the Fourier transform of a correlation function that can then be interpreted in the classical limit, and quantum corrected. Evaluating the commutator in Equation 2.1 gives: T r{[ρ, µ i ]α jk (t)} = C(t) C (t) = 2iC I (t) (2.2) C(t) =< µ i α jk (t) >= C R (t) + i C I (t) In Equation 2.2, the superscript star is the complex conjugate, and the subscripts denote the real and imaginary parts of C(t) both of which are themselves real functions. The angle brackets are the trace of the operators divided by the partition function in the standard notation. 63 In the classical limit, C R (t) becomes the classical cross correlation function of the system dipole and polarizability tensor elements, i.e. lim β ω 0 C R (t) = C Cl (t) =< µ i α jk (t) >. Since only classical TCFs can be calculated using classical MD and TCF theory, the goal is to write the response function entirely in terms of the (quantum corrected) 7

92 92/224 classical TCF, C Cl (t). To proceed, the imaginary part of the one time correlation function is related in frequency space exactly to the real part: C I (ω) = tanh(β ω/2)c R (ω); where the subscripts denote the Fourier transform of the real and imaginary parts of the complex function C(t) which is a real function of frequency, i.e. C(ω) = ( 1 2π ) dte iωt (C R (t) + ic I (t)) = C R (ω) + C I (ω). Using the result obtained in Equation 2.2 for the trace in Equation 2.1 gives: C I (t) = χ Res (ω) = 2 0 dω e iω t tanh(β ω /2)C R (ω ) dte iωt C I (t) (2.3) Equation 2.3 demonstrates the SFG experiment probes the imaginary part of C(t). Note, C I (t) is written in a form that can be calculated using the real part of the correlation function which is obtainable from classical MD. Due to causality, the Fourier-Laplace transform gives a real and imaginary part in Equation 2.3 as the cosine and sine transform of C I (t) respectively. Equation 2.3 can be simplified by changing the order of integration performing the frequency domain integral first. Defining the real and imaginary parts of χ Res (ω) = χ Res (ω) + iχres(ω): R I χ I (ω) = 2π tanh(β ω/2)c R(ω) = 2 χ R (ω) = 2 P tanh(β ω/2)c R (ω ) ω + ω dω = sin(ωt)c I (t)dt (2.4) cos(ωt)c I (t)dt (2.5) To obtain Equations , the identity e iωt dt = ip 1 0 ω + πδ(ω) was used. P designates the principle part. 64 Due to technical constraints in producing intense tunable infrared laser light, the focus of SFG experiments is currently on high frequency spectra where ( ω kt ), and classical mechanics is clearly invalid. Building on our previous work, the classical correlation function result, that is amenable to calculation using MD and TCF methods, is quantum corrected using a harmonic correction factor: C R (ω) = C Cl (ω)( β ω 2 coth(β ω/2)).65,66 This correction factor is exact in relating the real part of the classical harmonic coordinate correlation function to its quantum mechanical counterpart. Here, we are using it to correct functions, the dipole and polarizability, that contain higher orders of the coordinates, and exact 8

93 93/224 corrections for harmonic systems of this type are still possible, but unneeded the linear dipole and Placzek approximation are adequate. 65 Using this result, the TCF approximation to the resonant part of the SFG spectrum, χ Res, takes the form: χ R (ω) = βp χ I (ω) = βωπc Cl (ω) (2.6) C Cl (ω ) ω ω + ω dω (2.7) C Cl (t) =< µ i (0)α jk (t) > (2.8) In Equation 2.8, the angle brackets represent a classical TCF that can be computed using MD, and a suitable spectroscopic model. 67 Finally, Equations give the TCF signal in a form amenable to classical simulation. Note, while it easier to evaluate χ I (ω) using Equation 2.6, χ R (ω) is more easily computed by doing the cosine integral as in Equation 2.5. Considering the three possible independent polarization conditions, SSP, PPP, and SPS, for the TCF in Equation 2.8, the first index in the polarization designation corresponds to the last index in the TCF. For example, the SSP and PPP polarization conditions probe dipolar motions normal to the interface, and the SPS case is sensitive to dipolar changes parallel to the interface. Note, the PPP condition is sensitive to motions both parallel and perpendicular to the interface. 68 Further, the (SSP and PPP)/(SPS) probe diagonal/off-diagonal polarizability matrix elements respectively. A similar TCF approach was adopted earlier by others 3,5 and us 1 for modeling the SFG spectrum of both solid 5 and liquid interfaces 1,3 note, quantum corrections were not included in the works by the other groups. The earlier papers, and our previous work, did not give the exact expression for the real and imaginary parts of the SFG signal. Two of the papers, 1,5 started with the time domain expression for the second order response function, 61 one improperly evaluated a contour integral which violated causality. This effectively eliminated the real frequency contribution, and doubled the imaginary frequency part of the susceptibility. The other work 3 used frequency domain perturbation theory, 16,62 and divided the terms into resonant and nonresonant contributions then recast the resonant contribution as a TCF. This led to a TCF expression that replaced C I (t) with the full TCF, C(t), in Equation 2.3 (to within a constant factor). Note, at high frequency, the tanh factor is almost unity, and their expression is a correct limiting expression by making the rotating wave approximation the expression is only correct at high frequencies. 61,69 In that work, if two of their non-resonant terms are included in the resonant contribution, then the exact Equation 2.3 is 9

94 94/224 obtained this result is demonstrated in Appendix A. At lower frequencies, the two expressions are quite different, and the tanh factor produces a time derivative of the correlation function in the time domain. To construct an INM approximation to Equation 2.1, it is sufficient to evaluate the trace in Equation 2.1 for a harmonic system. It is convenient to invoke both the Placzek and linear dipole approximation to evaluate the resulting matrix elements although higher order contributions can be included, and simple analytic expressions result for these contributions. An equivalent approach is to evaluate C Cl (t) for classical harmonic oscillators, and quantum correct the resulting expression using the harmonic correction factor, given above, to relate C Cl (t) and C R (t). C Cl (ω) =< ( µ i / Q l ) ( α jk / Q l ) δ(ω ω l ) kt ω 2 > (2.9) In Equation 2.9, ω l is the frequency of mode Q l, and the angle brackets represent averaging over classical configurations of the system generated. C Cl (ω) is then back transformed into the time domain, and used in Equations in place of the classical TCF to obtain an INM approximation to the spectroscopy. Below, it will be demonstrated the TCF approach, which does not invoke the Placzek and linear dipole approximation (except implicitly in quantum correcting the results), gives results in close agreement with the INM results and Equation 2.9 is therefore sufficient. MD simulations were performed using a code developed at the Center for Molecular Modeling at the University of Pennsylvania, and uses reversible integration and extended system techniques. 70 Microcanonical MD simulations were performed on ambient H 2 O with a density of 1.0 g /cm 3, and an average temperature of 298K. To create an interface, a cubic simulation box of equilibrated liquid water was extended (doubled) along the z axis, and the system was allowed to equilibrate creating two water/vapor interfaces. The interfaces were sufficiently far apart so as they did not interact strongly, and Ewald summation was included in three dimensions. 9 The density profile of the system was monitored to verify equilibration. 9 In all cases, the results were tested, and found to be system size independent. Most results were generated from 216 molecule simulations, and smaller system sizes down to 64 molecules were tried, and did not alter the results. 2 As in our previous work, MD simulations were conducted using a flexible simple point charge (SPC) model that included a harmonic bending potential, linear cross terms and Morse O-H stretching potentials, V (r) = D e (1 e ρr ) The Morse O-H stretching potential used here was slightly softer than previous 10

95 95/224 work; our ρ value is Å 1 instead of 2.566Å 1. 1,50 For a Morse potential, the force constant, k, can be approximated as k = 2D e ρ 2. Assuming a harmonic oscillator with frequency ω =, this implies the ratio ρ 1 :ρ 2 is proportional to the ratio ω 1 :ω 2. Therefore, a 2.5% change in the exponential Morse parameter implies a 2.5% shift in the spectral frequencies, and this behavior is demonstrated in Figure 1. This analysis assumes the relevant coordinates are simple one dimensional O-H stretching modes. If several distinctly different types of modes were present, a change in the shape of the broad O-H stretching signal would be expected. This is additional evidence that interfacial normal modes are well approximated as simple O-H stretches. 1,2,4 Figure 1 highlights the spectral changes resulting from using a softer Morse potential. The slightly softer potential does not alter the intermolecular region of the spectra as would be expected. The intermolecular portion of the spectrum has polarizability and dipole derivatives (changes) that are due primarily to reorientation. These changes then depend on the polarizability tensor and the dipoles themselves, and not their derivatives. On the other hand, the intramolecular region of the spectra is simply shifted to the red this point will be returned to when discussing the modal composition of the broad O-H stretching lineshape. This change resulted in the free O-H stretching frequency in better agreement with experimental values even though the Morse potential change is almost imperceptible to the naked eye. This implies the center of the lineshape is very sensitive to the local frequency along the Morse potential as the O-H stretching motion, perturbed by hydrogen bonding in the liquid, explores the highly anharmonic potential surface. In performing the MD, partial point charges were placed on the atoms that were chosen to reproduce the condensed phase dipole moment. At the water/vapor interface, the true water dipole falls from its condensed phase value, about 2.9 Debeye, to that in the gas phase, 1.8 Debeye, over a distance of only a few molecular layers. 71 It would seem polarizable dynamics would be essential to model the dynamics of aqueous interfaces, but the use of non-polarizable MD seems to adequately represent the structure of the water/vapor interface. A previous work using a polarizable model in this context is consistent with this observation. 3 Evaluating the TCF in Equations presents a problem for interfacial systems. The interface was constructed using the standard MD geometry with vacuum/vapor above and below the water. 4,8 Unfortunately, this produces two interfaces with average net dipoles in opposite directions. Calculating the SFG k m 11

96 96/224 spectrum of the entire system would lead to partial cancellation of the SFG signal, and meaningless results. Another problem arises in that molecules at one interface can diffuse to the other interface over time. In this case, simulation times are limited to the molecular diffusion time between interfaces so that molecules cannot contribute to signal at both interfaces during one MD run. This leads to TCFs without long time decays that are difficult to Fourier transform accurately. 1,3 In order to obtain better TCF results, long time (cross) correlations between the system dipole and polarizability need to be followed. A weak (laterally isotropic) restraining potential was added effectively confining the molecules over time to the half of the simulation box they start in (in the dimension normal to the interface) without significantly perturbing the relevant short time dynamics; even though the molecular diffusion constant (normal to the interface) is changed, the molecule is only contributing to the spectrum while resident at the interface, and is free of any significant external potential. This modification permits the calculation of TCFs out to arbitrarily long times resulting in sharp spectra that include intermolecular spectral lineshapes. The restraining potential is of the form V = ɛ (σ/r) 9 with ɛ = 2.3K, σ = 2.474Å, and r = 0 is at the center of the box. The restraining potential becomes negligible near the interface, and is only significant within 2.0 Å of the box center. The interfacial density profile was unchanged demonstrating the restraining potential used did not perturb the average structure of the liquid that contributes to the interfacial spectroscopy. The MD was performed without explicit polarization forces; when the SFG TCF or INM spectrum are calculated, polarizability is included in the calculations over 3 million 3 fs time steps were included in calculating the MD and TCFs. The model employed includes full many-body polarization effects included explicitly via a point atomic polarizability approximation (PAPA) polarizability model with point polarizabilities on the atoms (α O = Å 3, α H = Å 3 ). 75 The permanent dipoles were calculated based on ab initio data as described previously. 1,3 The SFG signal is sensitive to both dipole and polarizability derivatives. PAPA polarizability models naturally incorporate parameters that determine the polarizability derivatives. To implement this, it is sufficient to make the point polarizabilities on the atomic centers (O-H) bond length dependent The point polarizabilities then change as: α(r) = α 0 (r) + α r. r is displacement from the equilibrium bond length. The α parameters for hydrogen and oxygen (α O = 2.7Å2,α H = 1.06Å2 ) 12

97 97/224 are somewhat different than in our previous model, but still give reasonable values for the gas phase Raman and IR transition moments. 1 Figure 2c highlights the differences between the previous and current model for the SFG SSP TCF spectra (showing the same SFG SSP spectrum presented in Figure 1). It is necessary, but not sufficient, to simply match the gas phase spectroscopic data, and suggests that interfacial molecules explore geometries different from both the gas phase and the bulk (where the earlier polarizability model worked very well) 45,47. Fitting to ab initio data for these interfacial geometries is clearly desirable, and is being pursued. 3 Further, point atomic polarizability models, such as the one used here, 1,45,72,75,76 offer flexible and transferable parameters for both neat mixtures and liquids. They also offer a natural description of the induced dipole derivatives, and the ability to fit polarizability derivatives. 74 However, to produce an accurate description of interfacial polarizability derivatives, it may be necessary to make the point polarizabilities depend on bond angles, and not simply bond lengths as was done in this work. 1 It is interesting to note the new model captures the free O-H mode more accurately without significantly perturbing the intermolecular region of the spectra intramolecular spectra are sensitive to dipole and polarizability derivatives that do not significantly change the magnitude of the dynamically more important dipole and polarizability. (Note, the small differences in the intermolecular spectrum are likely due to the relatively poor averaging that was done in calculating the spectrum using the previous model. In this case only one-fifteenth of the number of configurations were included in the calculation, and the SFG TCFs were slow to converge. 1,38 ) Thus, even relatively small changes to these derivatives can greatly effect the spectroscopic observable without changing the essential physics of the problem e.g., the identity of the relevant modes and motions. Figure 2 also presents the (a) infrared and (b) isotropic Raman TCF spectra (relevant to the SSP polarization condition because it probes diagonal elements of the polarizability matrix) for liquid water using both models. Again, only the intramolecular region of the spectra changed. For the O-H stretching region, increased asymmetry in the lineshape is apparent for the new model with a shoulder on the blue side. This is consistent with previous work that identified this shoulder to be due to instances in which a hydrogen does not form a hydrogen bond in the bulk; 77,78 this would be analogous to the free O-H stretch found in interfacial spectra. The new polarizability model does a better job at highlighting this non-hydrogen bonded 13

98 98/224 frequency distribution for liquid water, and, consequently, allows for more accurate interfacial spectra. The figure also clearly demonstrates the power of calculating spectroscopic observables to analyze condensed phase and interfacial structure. Interestingly, the shoulder on the blue side of the bulk Raman and IR spectrum is at the same central frequency as the free O-H mode at the interface strongly suggesting the presence of free, non-hydrogen bonded, O-H modes in bulk water. III. DISCUSSION Figure 3 displays the theoretical SFG SSP spectra for the entire water vibrational spectrum derived from both TCF and INM methods. Both the TCF and INM results are in absolute units, and no parameters were adjusted in displaying the data. The INM and TCF spectra were found to integrate to the same value over the entire cm 1 range, and separately over the O-H stretching region ( cm 1 ) for all polarization conditions (the others are not shown). This behavior is strong evidence for the interpretation of the INM lineshape as an underlying spectral density that is motionally narrowed in the observed spectrum. 40 INM approximations to spectroscopy offer only a limited dynamical description, and correspond to an underlying spectral density that is typically broader than the observed lineshape when considering intramolecular modes. As an example, in bulk water (and other liquid state intramolecular lineshapes) INM intramolecular resonances were found to be broader than their TCF counterparts, but with the same central frequency and integrated intensity. Our TCF and INM spectra in Figure 3 unambiguously demonstrate SFG O-H stretching lineshapes at the water/vapor interface are significantly motionally narrowed to a degree reminiscent of the bulk. 50,51 This result also suggests SFG spectra are sensitive to both structure and dynamics. The INM spectrum clearly exhibits the same resonances, but is broader. This implies the observed lineshapes are motionally narrowed, and dynamical contributions to SFG signals are important. 41 Figure 4 presents TCF derived theoretical descriptions of the SFG spectra in the O-H stretching region for the water/vapor interface. The three possible independent polarization conditions, (SSP, PPP, and SPS), in the electronically nonresonant experiment are displayed. The first two indices can be interpreted as the element of the system polarizability tensor, and the second index as the element of the system dipole that is being probed. In the data, for all polarizations, we have included the SSP nonresonant contribution (this 14

99 99/224 is only strictly correct for the SSP polarization condition, and serves as an estimate in the other cases), χ NRes (ω), which is a small negative constant, 3,59 and the full signal is given by: χ (2) SF G (ω) 2 χ Res (ω) + χ NRes (ω) 2. In order to account for the Fresnel coefficients that modify the experimental intensities, we have adjusted the relative intensities of our theoretical spectra so they can be more easily compared with experimental results. 41 The spectrum in the SSP geometry that correlates the dipole moment component normal to the interface with diagonal polarizability matrix elements in the plane of the interface (e.g < µ z (0)α xx (t) >, with the z axis taken as the surface normal direction) leads to the most intense spectrum due to a relatively sizable, and changing, net normal dipole moment at the interface, and the relatively large diagonal polarizability elements; water has a nearly diagonal polarizability matrix with nearly equal elements in both the gas phase and bulk. (Note, the PPP polarization condition is sensitive to a combination of all allowed susceptibility tensor elements in contrast to SSP and SPS that only probe a single tensor element. 68 ) Previously, we showed the agreement between the TCF and experimental spectrum, including the relative intensities of the different polarization conditions, was excellent, and within the statistical error over most of the frequency range. 2 Thus, the essential features of the spectrum, and its polarization dependence, are captured very well by the TCF theory with the caveat that absolute intensities of the intramolecular modes are quite sensitive to the choice of polarizability parameters. The polarization dependence of the signal is demonstrated in Figure 4. For polarizations that are sensitive to dipole derivatives normal to the interface SSP and PPP the signal has an intense lineshape. In contrast, for the SPS geometry, which is sensitive to dipole derivatives parallel to the interface, only a hint of a signal is found. The SPS polarization condition also probes small off-diagonal polarizability matrix elements. These results also suggest by evaluation of the polarization dependence of the SFG spectra, given a knowledge of the expected nature of the polarizability and dipole derivatives, allows interfacial molecular geometries to be inferred via the spectra. 10,68,79 While the intermolecular spectrum of bulk water shows little structure, the interfacial spectra is complex as shown in Figure 5. The figure highlights the intermolecular SFG TCF spectra for the three independent polarization conditions, (SSP, PPP and SPS). The polarizations that are sensitive to dipole derivatives 15

100 100/224 normal to the interface, SSP and PPP, show a well defined intermolecular mode at 875. cm 1 that is comparable in intensity to the rest of the intermolecular structure and approximately one-sixth the intensity of the intense free O-H stretching peak. 2 Using instantaneous normal mode methods (looking at the nature of the INMs in the same spectral region), the resonance is shown to be due to a wagging mode localized on a single water molecule, at a slight angle to the interface, with two hydrogens vibrating/librating normal to the interface, and the oxygen anchored in the interface. 2 The hydrogens, pointing into the vapor phase, are hydrogen bonded to an oxygen atom at the interface. The SSP and PPP also show an intense intermolecular mode at 95. cm 1. Using instantaneous normal mode methods the resonance is found to be due to translations perpendicular to the interface. The SPS spectra, which is sensitive to dipole derivatives parallel to the interface, shows an intermolecular mode at 220. cm 1. This mode is a result of translations parallel to the interface. The importance of polarization sensitivity in SFG experiments is, thus, highlighted. Further, we have observed spectroscopically distinct species, and clearly identified the vibrational modes responsible for the lineshape. Hence, experimental setups that permit taking spectra at relatively long wavelengths could probe these modes as a compliment to the information contained in the free and donor O-H stretching modes. 52,54 These three distinct populations of water molecules at the interface were previously undescribed other works have inferred the existence of something like the wagging mode. 6,34,35 This might be considered surprising given the large numbers of MD simulations of the water/vapor interface that have been performed previously. This observation highlights the power of calculating spectroscopic observables in assessing interfacial structure and dynamics. Not only can the results be directly compared with experiment, thus validating the MD model, the spectroscopic calculation serves as a filter of the dynamics extracting out the identity of collective coordinates with well defined frequencies that persist at the interface. Figure 6 highlights the vibrational modes from the intermolecular and intramolecular region of the spectra. A typical free O-H mode, shown in blue, produces the high-frequency feature at cm 1. It is clear the oxygen atom is anchored in the interface, and the O-H is oscillating freely above the interface. The wagging mode giving rise to the spectral feature at 875. cm 1 is displayed in green at the opposite interface. Here, the oxygen atom is anchored in the interface, and the two hydrogens are vibrating into 16

101 101/224 the vapor phase. A representative perpendicular translational mode (with lineshape centered at 95. cm 1 ) is shown in yellow, and the roughly parallel translational mode (with lineshape centered at 220. cm 1 ) is shown in black. These results demonstrate how INM approach does not require a priori assumptions about the nature of interfacial modes but does reveal their physical characteristics, and how different molecular motions contribute to the spectrum. In future work, a quantification of the relative populations of these interfacial species is planned via this approach. Figure 7 displays the distribution of the direction cosine from the surface normal of O-H vectors pointing into the vapor. This result compares well with previous theoretical data. 4 We see an enhancement in probability at cos θ 1. We also find approximately 20% of surface water molecules have a free O-H bond pointing out of the liquid, and into the vapor which is consistent with previous theoretical work. 4,8 This analysis also points out it is necessary to talk of broad distributions of angles at the water/vapor interfaces, and that relatively less can be learned from single average values of orientations. Figure 8a displays the real and imaginary parts for the SSP spectrum calculated via equations 2.5 and 2.6. Examining the real and imaginary parts of the spectrum can offer insights unavailable from the modulus alone. The real and imaginary parts could be measured experimentally via a heterodyne detection scheme, or by taking advantage of interference effects between bulk and interfacial contributions to the spectrum. 10 To see the advantages of separately examining the real and imaginary contributions, it is useful to write the resonant SFG signal of a single harmonic mode, Q, (with linear dipole and polarizability) in frequency space as: 4 χ (2) R (ω) ( µ i/ Q) ( α jk / Q) χ (2) I (ω) ( µ i / Q) ( α jk / Q) [ ω ω IR ] (ω ω IR ) 2 + γ 2 [ ] γ (ω ω IR ) 2 + γ 2 (3.1) (3.2) In Equations , γ is a mathematical convergence parameter that physically can be interpreted as a homogeneous line width. The signal magnitude is seen to be proportional to the product of dipole and polarizability derivatives. Equations imply a single type of mode will lead to an imaginary contribution that is a symmetric well defined peak (Lorentzian in character) while the real part will change sign, dipping below zero, at the maximum of the imaginary portion. If more than one species is contributing to the signal 17

102 102/224 in a given region, a more complex lineshape will result from the overlapping signals. Examining the real and imaginary contributions in Figure 8a, it is clear several of the resonances are essentially single mode in character: the free O-H (3700. cm 1 ), the small bending contribution at the surface (1800. cm 1 ), the wagging mode (875. cm 1 ), and translational modes (95. cm 1 and 220. cm 1 ). There is some overlap in the translational modes, and it is instructive the higher frequency 220. cm 1 mode, that is pronounced only in the SPS modulus spectrum, also shows up in the SSP real and imaginary spectra. Figure 9 highlights the O-H stretching region from approximately cm 1 to cm 1. Careful examination of the spectrum reveals three separate modes in this region centered at cm 1, cm 1, and cm 1. Remarkably, this agrees very well with previous experimental work that deconvoluted the spectrum in this region. That analysis revealed three modes present in the same region centered at cm 1, cm 1 and cm 1 nearly the same frequencies. 11,59 This is strong evidence for distinct populations of water molecules in this donor O-H region of the spectrum. Further work is underway to identify the nature of these distinct O-H stretching species. It should be noted, that while the real and imaginary parts of the TCF derived SFG spectra do clearly indicate the presence of distinct subpopulations of oscillators, it is difficult to unambiguously identify the species responsible for the signals. This complication occurs because the modes are identified using INM methods, and the INM signal is broad in this region. Therefore, there is not an absolute correspondence between an INM frequency and the associated TCF spectrum (in this congested spectral region). This difficulty does not arise, however, in investigating spectra regions dominated by a single resonance like the free O-H or wagging mode. Further, the theoretical and experimental spectra have a somewhat different shape in this region, and this manifests itself in the relative intensities of the different contributions (considering the extant water/vapor SFG spectra that have similar features but not identical shapes in this region). 11,17,41,59,80 The differences are most likely due to the spectroscopic intensities of these species via our spectroscopic model rather than different populations of these species at the interface within the MD model. However, further investigation is required to definitively demonstrate this. It should also be noted, as pointed out in an earlier work, 4 orientational information can also be deduced from the relative signs of the imaginary mode lineshapes given knowledge of the the signs of the prefactors in Equations (the dipole and polarizability derivatives). 18

103 103/224 To further show the utility of the real and imaginary modal analysis, Figure 8b displays the real and imaginary parts of the bulk water O-H stretching region calculated as the Fourier-Laplace transform. While a linear IR experiment does not measure this observable, the transform can still be applied as an analysis tool. Figure 8b is strong evidence there are two distinct species in the bulk, and the higher frequency moiety arises from the bulk free O-H, non-hydrogen bonded molecules. 77,78,81 IV. CONCLUSION The combined use of improved TCF and INM approximations to SFG spectroscopy represent a powerful complementary approach. Achieving agreement with experimental measurements engenders confidence in the MD and spectroscopic models used to produce the theoretical spectrum. Many MD simulations of the water/vapor interface have been performed, but traditional analysis techniques do not easily uncover important interfacial subpopulations such as the wagging (hindered rotational ) and hindered translational motions. Thus, SFG spectroscopy may be capable of giving a complete picture of the interface including structure and dynamics. Realizing this promise depends critically on the spectra being reliably interpreted, and the methods employed in this study are designed to unambiguously characterize the nature of SFG spectra including inferring subpopulations of molecules from complex lineshapes. The plan is to investigate more complex and interesting interfaces using our improved combined INM/TCF approach. V. ACKNOWLEDGMENTS The research at USF was supported by an NSF Grant (No. CHE ) and a grant from the Petroleum Research Foundation to Brian Space, a Latino Graduate Fellowship to Christine Neipert, and a USF Presidential Fellowship to Christina Ridley Kasprzyk. Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research. The authors would like to acknowledge the use of the services provided by the Research Oriented Computing center at USF. The authors also thank the Space Foundation for Basic and Applied Research for partial support. 19

104 104/224 VI. APPENDIX A Starting from frequency domain perturbation theory for χ (2) SF G (ω), six terms are obtained, and are shown below. 3,16,62 Four of the terms contribute to the resonant signal (contained in R 1 and R 2 below), and two are nonresonant (NR 1 and NR 2 below). While two of the resonant terms may appear initially to be nonresonant, those with denominators containing the expression (ω IR + ω ng + iγ ng ), they contribute to the resonant susceptibility, and lead to the complex conjugate correlation function C (t). Ultimately, their inclusion is necessary to reproduce Equation 2.3. In the expressions below, ω IR and ω vis are the frequencies of the infrared and visible fields. ω SF G is the sum frequency of the infrared and visible fields, and ω ng is the frequency corresponding to the energy difference between energy levels n and g. In Equation 6.1, ρ (0) g is the initial state thermal population, and the sum is over vibronic levels. µ γ α,β is a dipole matrix element between states α and β for dipole vector component γ. χ pqr (ω SF G, ω vis, ω IR ) = g,n,m (ρ (0) g )(R 1 + R 2 + NR 1 + NR 2 ) (6.1) ( µ p gnµ q nm R 1 = (ω SF G ω ng + iγ ng ) µ q gnµ p ) ( nm µ r ) mg (ω vis + ω ng + iγ ng ) (ω IR ω mg + iγ mg ) ( µ q nmµ p mg R 2 = (ω SF G + ω mg + iγ mg ) µ p nmµ q ) ( mg µ r ) gn (ω vis ω mg + iγ mg ) (ω IR + ω ng + iγ ng ) NR 1 = µ q gnµ p mgµ r nm (ω SF G + ω mg + iγ mg )(ω vis + ω ng + iγ ng ) NR 2 = µ p gnµ q mgµ r nm (ω SF G ω ng + iγ ng )(ω vis ω mg + iγ mg ) The resonant contributions can be simplified by rewriting in terms of polarizabilities and dipoles, and approximation 1/ω sfg = 1/ω vis. Given the definition of polarizability in Equation 6.2, the two resonant terms, R 1 and R 2, simplify to 6.3 and 6.4 respectively. 20

105 105/224 α pq (ω) = [ µ p gnµ q ng µ q gnµ p ] ng + ρ (0) g (6.2) ω + ω g,n ng iγ ng ω + ω ng + iγ ng α pq gmµ r mg R 1 = (ω IR ω mg + iγ mg ) (6.3) R 2 = µ r gnα pq ng (ω IR + ω ng + iγ ng ) (6.4) Let χ Res pqr denote only the sum of the resonant terms R 1 and R 2. Replacing the denominators in both of the resonant terms with the integral identities 0 dte it(ω ωo iγ) = i ω ω o iγ and 0 dte it(ω+ωo+iγ) = i ω+ω o+iγ, and then taking the implicit limit that gamma goes to zero gives Equation 6.5. Equation 6.6 follows as an exact rewrite of Equation 6.5, and expresses the susceptibility in terms of the cross correlation of the system dipole and polarizability. χ Res pqr = [ i gm 0 e iωmgt e iω IRt α pq gmµ r mgdt i ng 0 e iωngt e iω IRt α pq ngµ r gndt ] ρ (0) g (6.5) χ Res pqr = i 0 dte iω IR < α pq (t)µ r (0) > i 0 dte iω IR < µ r (0)α pq (t) > (6.6) Expansion of the correlation functions in Equation 6.6 results in a significant simplification. Noting, < µ r (0)α pq (t) >= C R (t) + ic I (t) = (< α pq (t)µ r (0) >) gives Equation 6.7 below which is given in the text as Equation 2.3. χ Res pqr (ω IR ) = 2 0 dte (itω IR) C I (t) (6.7) 21

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108 108/224 VII. FIGURES 24

109 109/ (10-15 A 8 e 2 K -2 ) wavenumber (cm -1 ) FIG. 1: (Color online) SFG SSP TCF spectra for the water/vapor interface highlighting the spectral changes in the use of two different Morse potentials the original Morse potential (dashed blue line), and a softer Morse potential (solid green line). The softer potential results in a shift of approximately 100. cm 1 in the O-H stretching spectrum. a b c wavenumber (cm -1 ) FIG. 2: (Color online) The (a) IR TCF spectra for liquid water, the (b) isotropic Raman TCF spectra for liquid water, and the (c) SFG SSP TCF spectra for the water/vapor interface highlighting the spectral changes in the use of two polarizability models previous model (dashed blue line) and current model (solid green line). 25

110 110/ (10-15 A 8 e 2 K -2 ) wavenumber (cm -1 ) FIG. 3: (Color online) SFG SSP spectra for the water/vapor interface for the entire water vibrational spectrum using TCF (solid green line) method and INM (dashed blue line) method (10-15 A 8 e 2 K -2 ) wavenumber (cm -1 ) FIG. 4: (Color online) SFG TCF spectra for the water/vapor interface in the O-H stretching region for three polarizations: SSP (solid green line), PPP (dashed blue line), and SPS (dotted red line). 26

111 111/ (10-15 A 8 e 2 K -2 ) wavenumber (cm -1 ) FIG. 5: (Color online) SFG TCF spectra for the water/vapor interface in the intermolecular region for three polarizations: SSP (solid green line), PPP (dashed blue line), and SPS (dotted red line). FIG. 6: (Color online) A snapshot of a water/vapor interface containing 216 water molecules featuring INMs from different regions of the spectra. The water molecule shown in blue is representative of a free O-H mode at cm 1. The water molecule shown in green is representative of a wagging motion at 858. cm 1. The water molecule shown in yellow highlights a translation perpendicular to the interface at 46. cm 1. The water molecule shown in black highlights a translation parallel to the interface at 197. cm 1. 27

112 112/ Probability cos theta(oh) FIG. 7: The probability distribution of the direction cosine from the surface normal of O-H vectors pointing into the vapor. 3.0 a (10-8 A 4 ek -1 ) b wavenumber (cm -1 ) FIG. 8: (Color online) Real (solid green line) and imaginary (dashed blue line) components of the (a) SFG SSP TCF spectra for the water/vapor interface and for (b) bulk water calculated as the Fourier-Laplace transform. 28

113 113/ (10-8 A 4 ek -1 ) wavenumber (cm -1 ) FIG. 9: (Color online) Real (solid green line) and imaginary (dashed blue line) components of the SFG SSP TCF spectra for the water/vapor interface for the O-H stretching region. The arrows highlight three separate modes centered at cm 1, cm 1, and cm 1. 29

114 114/224 Applications of a Time Correlation Function Theory for the Fifth Order Raman Response Function I: Atomic Liquids Russell DeVane,Christina Ridley and Brian Space Department of Chemistry, University of South Florida, 4202 E. Fowler Ave., SCA400, Tampa, FL T. Keyes Department of Chemistry, Boston University, Boston MA, (Dated: July 27, 2005) Abstract Multidimensional spectroscopy has the ability to provide great insight into the complex dynamics and time resolved structure of liquids. Theoretically describing these experiments requires calculating the non-linear response function, which is a combination of quantum mechanical time correlation functions (TCF), making it extremely difficult to calculate. Recently, a new theory was presented in which the 2D Raman quantum response function, R (5) (t 1, t 2 ) was expressed with a 2- time, computationally tractable, classical TCF. Writing the response function in terms of classical TCF s brings the full power of atomistically detailed molecular dynamics (MD) to the problem. In this paper, the new TCF theory is employed to calculate the fifth order Raman response function for liquid xenon and investigate several of the polarization conditions for which experiments can be performed on an isotropic system. The theory is shown to reproduce line shape characteristics predicted by earlier theoretical work. Author to whom correspondence should beaddressed. space@cas.usf.edu 1

115 115/224 I. INTRODUCTION Spectroscopic techniques that can lend insight into the molecular structure and complex dynamics of liquids have been actively pursued for sometime now. The inability of techniques related to the third order nonlinear response function, such as optical kerr effect spectroscopy (OKE), to identify the physical origins of spectroscopic line broadening has made pursuit of higher order techniques, e.g., two-dimensional infrared (2DIR) and Fifth order Raman spectroscopy, of more importance. 1 Much effort has been put into the Fifth order Raman experiments both experimentally 2 9 and theoretically The already difficult experimental measurements are complicated by the challenge of separating third order cascade signals from the desired fifth order spectrum, although recent advances have shown promise. 3 9,13 To fully realize the power of multidimensional spectroscopy, a sound, computationally tractable theory is necessary; this presents a considerable theoretical challenge. An n- dimensional spectrum is given by an n-dimensional response function, a quantum average of nested commutators of n Heisenberg representation variables at different times. These quantum mechanical quantities are exceedingly difficult to calculate exactly and are nearly intractable in their classical limit replacing the commutators with Poisson brackets. In the classical limit, there is still a computational demand that is only practical for small simple systems. 21,29 A classical time correlation function (TCF) theory of the multidimensional spectroscopy permits the multidimensional techniques to be described using the same sort of TCF methods that are used successfully in describing linear (and related third order) spectra, e.g. obtaining linear infrared spectra from the classical dipole-dipole autocorrelation function, < µ(0)µ(t) >. A TCF theory also allows the description of the nonlinear response in terms of fully anharmonic molecular dynamics (MD) calculations that are supplemented by a suitable spectroscopic (dipole and polarizability) model something that is not possible in lower dimensional quantum calculations for model systems. However, it has been demonstrated that nonlinear spectra cannot be expressed exactly in terms of TCF s. 23 Nevertheless, we proposed approximate TCF theories of nonlinear spectroscopy that are 2

116 116/224 computationally tractable for complex molecular condensed phase systems and are capable of accurately describing both intermolecular and intramolecular vibrational spectra for both the fifth order Raman response 10,25 and the third order dipole response (responsible for 2DIR spectra). 30 We have presented explicit calculations of the 2D Raman spectrum of ambient liquid CS 2, determined by the fifth order response, R (5) (t 1, t 2 ). 25 The 2D Raman spectra were in excellent agreement with extant experiments and approximate theoretical results. 3,29,31 But the agreement does not represent a rigorous test because uncertainties still exist in the experimental measurements and the theoretical results use approximate methods and different molecular models. As a more rigorous test, recently we published results in which R (5) xxxxxx(t 1, t 2 ) was calculated for liquid xenon using the TCF theory and compared to exact results previously reported by Ma and Stratt for the fully polarized experimental geometry. 10,21 It was demonstrated that the TCF theory quantitatively reproduced the simulated spectrum. In addition, preliminary indications were obtained that the theory reproduced the polarization dependence predicted in an earlier analysis. 15 In this paper, results are presented for the fifth order response of liquid xenon for different polarization conditions to assess the physical information content of the nonlinear experiments. Companion papers will follow presenting results for ambient liquid water and CS 2 and the temperature dependence of the atomic spectra. The success of such rigorous tests of the TCF theory suggests that it can be applied to more complex systems; TCF calculations with more extensive molecular models are not computationally difficult. 25 Thus, it is demonstrated here that, while exact TCF expressions for nonlinear spectroscopy are not possible, effective TCF theories can be constructed. Such theories are powerful because they can be evaluated using atomistically detailed MD on complex liquids and solutions. TCF theories have distinct advantages over alternative approaches because they eliminate the need for costly classical calculations that are limited to small systems and intermolecular dynamics 21,29,31 and quantum calculations that rely on low dimensional model systems. 32 It then becomes possible to predict and interpret nonlinear spectra for a 3

117 117/224 wide variety of chemically important systems. In Sec.(II), the development of the TCF theory of R (5) (t 1, t 2 ) will be summarized; details are provided in our earlier work. 25 Section (III) will discuss technical aspects of the MD simulations and the polarizability model employed. Results from the application of the TCF theory to liquid xenon will be discussed in Sec. (IV). This will include discussion and analysis of several of the distinct polarization conditions that arise in isotropic systems for R (5) (t 1, t 2 ). Finally, conclusions will be given in Sec. (V). II. CONSTRUCTING THE TCF THEORY The quantum mechanical expression for the electronically non-resonant fifth order polarization response is given by 24,29 R (5) α,β,γ,δ,ɛ,φ (t 1, t 2 ) = (i/ ) 2 T r{π αβ (t 1 + t 2 )[Π γδ (t 1 ), [Π ɛφ (0), ρ]]}. (2.1) In Eq. (2.1), ρ = e βh /Q, for a system with Hamiltonian H and partition function Q at reciprocal temperature β = 1/kT, and k is Boltzmann s constant; T r represents a trace, square brackets denote commutators, Π is the system polarizability tensor, and the Greek superscripts denote the elements and thus polarization condition being considered. The classical limit of the trace is of order 2 and results from a combination of four two-time correlation functions that are themselves equivalent classically. Expanding the commutators in Eq. (2.1) gives R (5) (t 1, t 2 ) = ( 1/ 2) [g(t 1, t 2 ) f (t 1, t 2 ) f(t 1, t 2 ) + g (t 1, t 2 )]. (2.2) In Eq. (2.2) f(t 1, t 2 ) = Π (t 1 )Π(t 2 )Π, (2.3) g(t 1, t 2 ) = Π(t 2 ) Π Π (t 1 ) (2.4) f (t 1, t 2 ) = ΠΠ(t 2 )Π (t 1 ) (2.5) g (t 1, t 2 ) = Π (t 1 )Π Π(t 2 ) (2.6) 4

118 118/224 an asterisk denotes the complex conjugate. The superscript notation on Π is now suppressed and the results apply to all possible polarizations. The angle brackets represent an equilibrium average. The expression inside the square brackets is the difference between the real part of the functions g and f, g R (t 1, t 2 ) f R (t 1, t 2 ), that must be of order 2 in the classical limit, ω kt. It is easy to show that a multiplicative factor of leading order can be obtained exactly using frequency domain (detailed balance) relationships between g and f, e.g. e β ω 1 g(ω 1, ω 2 ) = f(ω 1, ω 2 ); the frequency domain functions are the Fourier transform of the time domain functions. Using these relationships and taking the classical limit gives Eq. (2.7). A sum of derivatives of the real and imaginary parts of a single two-time TCF then remains. A relationship between the real and imaginary part of g(t 1, t 2 ) is required to eliminate g I (t 1, t 2 ) for g R (t 1, t 2 ) in the second term, that can then be calculated in terms of a classical TCF; such a relationship would give an order contribution in the classical limit. R (5) (t 1, t 2 ) = β 2 2 g R (t 1, t 2 ) t iβ g I (t 1, t 2 ) t 1. (2.7) For a one time correlation function, C(t), a simple frequency domain relationship exists between the real and imaginary parts, 33 C I (ω) = tanh(β ω/2)c R (ω), here and later the subscripts denote the Fourier transform of the real or imaginary parts of the TCF s, both of which are real functions of frequency. If a similar relationship between the real and imaginary parts of the two-time correlation function existed, the fifth order response could be written as second derivatives in time of a classical TCF, but no exact analytic relationship is possible and, therefore, an exact TCF theory is also not possible. However, an approximate relationship between the real and imaginary parts of the quantum TCF can be found for a harmonic system with the polarizability expanded to second order in the harmonic coordinate, Q: Π = Π (0) + Π (1) Q + 1/2 Π (2) Q 2. (2.8) In Eq. (2.8) the superscripts on Π are the order of the derivatives with respect to the coordiante. This harmonic reference system is the simplest one that produces a fifth 5

119 119/224 order response and represents the leading contributions to the spectroscopy, analogous to IR intensities being given by the squared dipole derivative for a mode of a harmonic system with a linearly varying dipole. 14,34 The real and imaginary parts are found to be approximately related in this case as: g I (ω 1, ω 2 ) = tanh( β (ω 1 /4 + ω 2 /2))g R (ω 1, ω 2 ), (2.9) Using this result does not lead to a harmonic theory fully anharmonic dynamics are used to calculate the relevant TCF the approximation only serves to weight the different phonon processes implicit in the anharmonic MD as they would contribute in the harmonic system. 14,15,25 Employing this relationship, the quantum mechanical response function can be written in terms of a classical TCF. The response function, in the classical limit takes the form: R (5) (t 1, t 2 ) = β 2 /2 [ ] 2 g R (t 1, t 2 )/ t g R (t 1, t 2 )/ t 1 t 2, (2.10) with g R (t 1, t 2 ) = Π(0) Π(t 1 ) Π(t 1 + t 2 ), (2.11) the classical two-time TCF. Here, Π = Π Π and the TCF is written in terms of the correlated polarizability fluctuations. 25 A more general result may be obtained for high frequencies if one approximates the quantum mechanical TCF with the (quantum corrected) classical TCF. 30,35 Because fifth order Raman spectroscopy has primarily been applied to intermolecular dynamics, only the classical limit will be considered in the following text. R (5) (t 1, t 2 ), in this form, can be evaluated using MD and a model of the systems polarizability. Note Eq.(2.10) exhibits the correct limiting behaviors including giving zero signal at the origin and everywhere along t 2 = 0. III. MODELS AND METHODS The most rigorous test of this theory is to compare the resulting spectra to those from exact numerical results for the same model system. To evaluate R (5) (t 1, t 2 ) exactly, the 6

120 120/224 classical limit is taken directly by replacing the commutators with Poisson brackets. Brackets of variables at different times, which require the exceedingly difficult task of calculating the dependence of a many-body dynamical variable on its initial conditions; the approach is computationally feasible only for very small, simple systems. Ma and Stratt calculated R (5) (t 1, t 2 ) exactly for a system of 32 liquid xenon atoms. 21 To compare our theory to those results, microcanonical MD simulations were performed for a neat liquid xenon system consisting of 108 atoms. The atoms interacted via a Lennard-Jones pair potential with σ=4.099 Å and ɛ= 222 K. The reduced density and temperature were ρσ3 =.8 and kt/ɛ=1.0 respectively and the same model parameters were used in the exact calculation. 21 MD was used to generate 6,000,000 configurations with a 0.01 ps timestep. These configurations were then used to calculate the TCF. Derivatives were evaluated using numerical approximation methods. 36 For the evaluation of the derivatives at the boundaries (e.g. along t 2 = 0), it was necessary to use lower order algorithms making the results less precise. This is a result of calculating only the positive quadrant (+t 1, +t 2 ) of the TCF. Polarization forces were not explicitly included in the MD calculations. However, in calculating the system polarizability to be used in the TCF, full many body polarization effects were included by solving the dipole-induced dipole equations for a point atomic polarizability approximation (PAPA) and these results will be compared with an approximate first order evaluation described below. 37 The expression for the effective polarizability, α i, for site (atom) i is given by n α i = α i + α i T (r ij ) α j (3.1) j i where α i is the isotropic point polarizability for site i and T (r ij ) is the dipole field tensor between sites i and j in a system with n sites. 37 The total system polarizability is given by summing the effective polarizabilities for all sites: n Π = α i (3.2) i=1 Two forms of the many body polarizability model were used in our calculations; truncating Eq. (3.1) to terms first order in T (r ij ), the first order dipole induced-dipole model (FO- 7

121 121/224 DID), and the exact infinite order dipole induced-dipole (MBP). The MBP model requires iteratively solving Eq. (3.1) or a matrix inversion (matrix inversion was employed in the present calculations) while the FODID only requires a single iteration of Eq. (3.1). The exact classical calculation of R (5) (t 1, t 2 ) was performed only in the FODID approximation to make the calculation feasible. 21 Assessing the polarization dependence of the signal is greatly facilitated by the introduction of rotational invariants of the system polarizability. 15 The task of fully understanding the imlications of using an FODID approximation is made easier by considering the response function directly in terms of the rotational invariants. 15,21 When considering an isotropic system, products of three dimensional second-rank tensors, such as the many body polarizability that appears in R (5) (t 1, t 2 ) via Eq. (2.11), can be written as orientational averages. 15 For R (5) (t 1, t 2 ), the correlation function is a product of three such tensors and can be written in terms of three rotational invariants, the trace, the pair product and the triple product: T r(a) = i a ii (3.3) P P (a, b) = i,j a ij b ij (3.4) T P (a, b, c) = i,j,k a ij b ik c jk (3.5) In Eq. (3.3), a, b and c represent three dimensional second rank tensors and take the form of polarizabilities evaluated at different times in the present case. 15 Because correlated fluctuations of the total system polarizability are being considered (for an atomic liquid the single atom polarizability α is static) only the induced polarizability contributes. In the FODID approximation, in which the only terms kept are first order in T (r ij ) (a traceless tensor) this leads to T r( Π) = 0. To see how this manifests itself in the overall response function we have to consider how the invariants combine to give the orientational average of a product of three tensors 15 a xx b xx c xx = 1 8 T r(a)t r(b)t r(c) + T P (a, b, c) [T r(a)p P (b, c) + T r(b)p P (a, c) + T r(c)p P (a, b)] (3.6) 105 8

122 122/224 Equation (3.6) presents the fully polarized case and the line over the tensor product represents isotropic averaging. The important thing to note here is that P P terms always appear coupled to a T r term. The form of Eq. (3.6), linear combinations of invariant products, holds not just for the fully polarized case shown here but for all of the possible polarization conditions. The various contributions are weighted differently in the other cases and the coefficients are tabulated elsewhere. 15 Considering the FODID approximation the T r contributions are zero and this eliminates any contribution from the P P and leaves only the T P contribution (making all polarization conditions equivalent to within a constant). In the MBP case the T r contribution is small but the P P is sufficiently large that the product contributes significantly to the overall signal. Figure (1) presents a slice of the invariants for liquid xenon along t 1 = 0 for the MBP model. The relative magnitudes of the four distinct contributions along t 1 = 0 (where T r( Π(0))P P ( Π(t 1 ), Π(t 2 )) = T r( Π(t 1 ))P P ( Π(0), Π(t 2 ))) are clear. Specifically, the plot shows the T P (solid line), T P ( Π(0), Π(t 1 ), Π(t 2 )), (3.7) the (T r)(p P ) product combinations (line with circles and dashed respectively), T r( Π(0))P P ( Π(t 1 ), Π(t 2 )) (3.8) T r( Π(t 2 ))P P ( Π(0), Π(t 1 )) (3.9) and T r (dotted) T r( Π(0))T r( Π(t 1 ))T r( Π(t 2 )). (3.10) Also shown, for comaprison, in Fig. 1 is the T P using the FODID model (line with squares). Each of the (T r)(p P ) terms have amplitudes that are sizable and significant compared to the T P contributions. It is evident from Fig. 1 that neglecting the significant P P contributions in the FODID case severely alters the lineshape. In addition, the FODID approximation predicts that all of the polarization conditions will yield identical lineshapes (with varying 9

123 123/224 amplitudes) since only the T P appears (with different weightings). Indeed, it will be seen in Sec. (IV) that the two models yield vastly different spectra and that within the MBP model the various polarizations provide distinct information. It is worth noting that both the MBP and FODID models predict similar line shapes with a slight variation in magnitudes for the T P contributions. This result implies that the fifth order response function contains several terms, when calculated using a MBP model, that do not contribute to a FODID calculation (all the terms that couple to the T r, that is zero within a FODID approximation). This result is in stark contrast to the third order polarized spectrum, that is dominated, for noble gas liquids, by pair product contributions and therefore does not change significantly within a FODID calculation (although a small T r contribution is neglected within the FODID approximation). 21 Obtaining a third order isotropic spectrum does, however, depend on using a MBP model because it vanishes within a FODID treatment. It is therefore not suprising that the (fully polarized) fifth order signal changes significantly when a FODID model is used; such a treatment neglects terms that clearly are important. As a technical note, calculating the fluctuations in the system polarizability requires the calculation of the average system polarizability (tensor) in order to substract off the static contributions in calculating the polarizability fluctuations. The calculation of the fifth order Raman response function is extremely sensitive to the average system polarizability that appears in the Π terms of Eq 2.11 and getting sufficient averaging becomes a lengthy task. For example, to obtain the result that R (5) t1,t 2 =0 = 0, requires the condition that 2 / t 2 1g R = 2 2 / t 1 t 2 g R within our theory. While it is easy to prove that this is true, numerical error in calculating the derivatives is complicated by numerical error in < Π >. This can lead to spurious results characterized by non-zero contributions along t 2 = 0. Conversely, meeting the R (5) t1,t 2 =0 = 0 condition appears sufficient to obtain reliable spectra. This problem is amplified in calculating signals for polarization conditions other than the fully polarized. 10

124 124/224 IV. RESULTS FOR LIQUID XENON Results for the fully polarized response function, R (5) xxxxxx(t 1, t 2 ) 2, for both the FODID and MBP models were previously reported. 10 Due to space restraints in the previous publication, the FODID result was reported overlayed with the exact result from Ma and Stratt 21 and the MBP result was reported overlayed with the FODID TCF calculation. For clarity, the MBP and FODID fully polarized TCF theory results are shown here again. The fully polarized fifth order response function, R (5) xxxxxx(t 1, t 2 ) 2, within the FODID approximation, is shown in the upper panel of Fig Even though an exact TCF theory is not possible, 23 the present theory was shown to be remarkably effective by capturing the characteristic features present in the exact calculation including the same decay times, the lack of an echo signal along the diagonal and extreme asymmetry. 3,8,10,21,28,31 The lower panel of Fig. 2 shows the result for R (5) xxxxxx(t 1, t 2 ) 2 using the MBP model. 10 While the FODID model captures much of the response, the full many body treatment exhibits significant differences as expected by the invariant analysis from Sec. (III). (Note that it was previously demonstrated that the third order response for liquid xenon is also similar in shape for the FODID and MBP model. 21 ) The utility of multidimensional spectroscopy is the ability to select out specific system information due to the two-dimensional nature of the experiment. This can be accomplished by using the various polarization conditions to enhance or diminish specific Liouville pathways that contribute to R (5) (t 1, t 2 ). 15 The use of the FODID approximation removes this utility by predicting the same TCF for all polarization conditions as discussed in Sec. (III); however, the MBP model retains this capability. Table I shows a summary of the analysis performed by Fourkas et.al. to aid in evaluating the spectra presented here. 15 The left column indicates the experimental polarization conditions considered and the top row shows the contributions to the response function in terms of Liouville pathways; below each pathway is the effect of each polarization condition on the correlation function contributing to it. Ideally, enhancing the pathway leading to the echo-like signal, R (Π(1) Π (2) Π (1)) in 11

125 125/224 Table I, and/or diminishing the others would lead to interesting information. Note that the pathway notation used here is that of Fourkas et. al. and differs from other literature references inluding our earlier work. 18,25,28,38 A major reason for developing fifth order Raman spectroscopy was to probe intermolecular dynamics for modes with characteristics that would produce an echo signal. The utility of polarization conditions is shown in table I; each polarization condition, except the fully polarized, leads to a unique correlation function contributing for one pathway. In the fully polarized condition, the different Liousville space pathways are equally weighted suggesting that this is a poor choice for selecting out specific information; in addition, this makes it impossible to determine if a single pathway is dominant in the fully polarized signal. The other possible polarization conditions provide a means by which certain pathways can be enhanced while others are diminished and this analysis is based on a harmonic model of the dynamics where nonlinearity in the polarizability provides the signal. 15 To aid in identifying the dominant pathway for each polarization codition, Table I has been included. For semipolarized (xxxxzz,xxzzxx,zzxxxx) and depolarized (xzxxxz,xxxzxz,xzxzxx) polarization conditions, 14 the pathway with a correlation function that has the odd index (zz for semipolarized; xx for depolarized) on Π (2) leads to enhancement of that pathway. For example, for the xxxxzz polarization condition, the odd index, zz, is on Π (2) in the correlation function Π (1) xx Π (1) xx Π (2) zz. From Table I, this corresponds to the pathway R (Π(1) Π (1) Π (2)). The enhancement results from an emphasis of the dominant invariant combination T r(π (2) )P P (Π (1), Π (1) ) for those forms. The polarization conditions xxzzxx and xzxxxz lead to a correlation function for the echo-like pathway of the form Π (1) xx Π (2) zz Π (1) xx and Π (1) xz Π (2) xx Π (1) xz respectively, and thus enhance that pathway. The other pathways are enhanced by other polarization conditions in a similar way.it should be pointed out that the leading order terms in fifth order nonresonant spectroscopy involve Π (2), e.g. terms of the form Π (1) Π (2) Π (1). 15 In cases where there is dominant anharmonicity, terms of the form Π (1) Π (1) Π (1) can make contributions. 39 However, this contribution is not expected to be large in a Lennard-Jones fluid under these conditiosn. Figure (3) shows the previously reported result for R (5) xxzzxx(t 1, t 2 ) 2 using the MBP 12

126 126/224 model. 10 The most notable feature of this figure is the echo peak that exists along the diagonal. The echo signal is elliptical with the length along the diagonal being nearly twice as long as the width perpindicular to the diagonal. In fifth order Raman spectroscopy, the echo peak implies that an intermolecular mode, excited at time zero, is still oscillating at the time of the measurement one period later and suggests that the spectrum is dominated by inhomogeneous contributions. The appearance of an echo in such a simple liquid implies that intermolecular modes may generally live at least a period in more complicated liquids that have significant Lennard-Jones interactions. The signal along the diagonal of Fig. 3 was fit to a single exponential of the form e ( t/τ), where t is the time and τ is the lifetime, and was found to have a lifetime of approximately 190 fs. This serves as an estimate of mode lifetimes. Note that the echo signal is smaller in magnitude than the other contributions that hide it in other polarization geometries. To date, molecular liquids have not shown an echo feature in 2D Raman experiments. This implies that the other pathways are dominating the echo signal making it difficult to detect. The stronger signal in molecular liquids may be dominated by rotations but this remains to be examined. However, it maybe that the presence of a single molecule polarizability in that case will overwhelm the lesser magnitude echo contribution. Figure (4) represents the remaining semipolarized polarization conditions. The upper panel shows R (5) zzxxxx(t 1, t 2 ) 2 with its strong ridge at t 1 =0 along t 2. The signal is instantly nonzero along t 1 and begins to grow in along t 2 at around 200 fs and the ridge continues beyond the 1 ps shown here. In the t 1 direction the signal decays to zero by t fs. Figure (5) shows a slice of the zzxxxx polarization signal taken along t 2 with t 1 =0 where the decay time can be seen to extend to approximately 3 ps. The signal also appears to decay in phases with periods of slow or almost no decay followed by periods of rapid decay. The lower panel of Fig. 4 shows R (5) xxxxzz(t 1, t 2 ) 2 which nearly quantitatively reproduces the fully polarized result (in shape with a different amplitude due to the different weighting coefficients), R (5) xxxxxx(t 1, t 2 ) 2. This suggests that there is one dominant pathway that contributes to the fully polarized signal and it is enhanced in the R (5) xxxxzz(t 1, t 2 ) 2. Using 13

127 127/224 the analysis from above, it would be expected that the pathway enhanced in this polarization would be the pathway labeled R (Π(1) Π (2) Π (1)) in table I. Figure (6) represents the depolarized polarization conditions. The upper panel shows R (5) xzxxxz(t 1, t 2 ) 2 which resembles the fully polarized result. Using Table I, we can predict that this polarization would be expected to enhance the echo-like pathway by observing the fact that the correlation function contributing to that pathway takes the form Π (1) xz Π (2) xx Π (1) xz. The signal is slightly elongated along the diagonal suggesting that the pathway leading to the echo is enhanced although with less efficiency than the xxzzxx polarization. As discussed by Fourkas et.al., earlier theoretical analysis suggested that the signal should not be instantly nonzero along either axis, but should build in along both t 1 and t Although, the theory predicts a signal that is rigourously zero along t 2 = 0, the signal in Fig. 6 shows some noise along that axis due to numerical error in taking the required derivatives. The top panel of Fig. 7 shows slices along t 2 = 0 (dotted-dashed line) and t 1 = t 2 (solid line) for R (5) xzxxxz 2 plotted with error bars (±0.0081Å3 /ps 2 ). The error estimate was calculated along the t 2 = 0 axis representing the maximum estimated error due to the need to calculate a lower order numberical derivative on the zero axis and the lack of any real signal. The result for R (5) xxxzxz(t 1, t 2 ) 2 (middle panel) shows a combination of various polarization characteristics. From table I, we would expect a signal that enhances the pathway R (Π( 2)Π ( 1)Π ( 1)), the same enhanced in the zzxxxx polarization. Indeed, both xxxzxz and zzxxxx share the dominant ridge along t 2 with t 1 =0. The sharp, fast rising peak near the origin resembles that seen in the fully polarized result with similar decay times. The signal is instantly nonzero along t 1 and grows in rather quickly along t 2. The signal then continues in the t 2 direction well beyond the 1 ps time shown. In the t 1 direction the signal decays to zero by 200 fs. Note that the R (Π(2) Π (1) Π (1)) pathway is zero for a harmonic or a Brownian oscillator system. 18,28,38 Therefore the ridge contribution must dominated by anharmonic dynamical contributions as was suggested by an earlier MD and INM analysis. 21,22 The lower panel shows R (5) xzxzxx(t 1, t 2 ) 2. It would be expected that this polarization would enhance the same pathway enhanced in the xxxxzz polarization. Both polarization 14

128 128/224 conditions do bear strong resemblance to the fully polarized with shorter decay times along the diagonal in the xzxzxx signal. This again agrees with earlier theoretical work. 15. The small bump appearing around 250 fs is suggestive of an echo. The lower panel of Fig. 7 shows the t 1 = t 2 slice of R (5) xzxzxx 2 plotted with error bars (±0.05Å3 /ps 2 ). This suggests that the small bump is not an artifact of numerical error in the calculations. It is possible to enhance specific pathways more by combining signals from multiple polarizations. The upper panel of Fig. 8 shows R (5) xxzzxx(t 1, t 2 ) R (5) xxxxzz(t 1, t 2 ) 2 which enhances the echo signal relative to the other features. Figure 4 (upper) shows a ridge along t1=0 and (lower) shows a relatively sharp peak near the origin (a spike). The dominant pathways for these two figures are R (π(2) π (1) π (1)) (upper) and R (π(1) π (1) π (2)) (lower). 15 Therefore these two pathways will be referred to here as the ridge and spike pathways respectively. Taking the difference of xxzzxx and xxxxzz polarization signals acts to diminish the spike pathway contribution that is a dominant feature in both the xxxxzz (Fig.4) and xxzzxx (Fig.3) signals. As shown in table I the correlation functions that contribute to the ridge pathway are π xx (2) π xx (1) π zz (1) and π xx (2) π zz (1) π xx (1) for xxxxzz and xxzzxx respectively. Within each correlation function, the π s commute and so the correlation functions are identical for these two polarization conditions. This leads to perfect cancellation of the ridge contribution for this combination. Note, that for the zzxxxx polarization conditions, the correlation function that contributes to the ridge pathway is π (2) zz π (1) xx π (1) xx. This correlation function is not identical to those from either xxxxzz or xxzzxx, thus taking the difference of zzxxxx and xxxxzz or zzxxxx and xxxxzz would not lead to perfect cancelation of the ridge pathway. The ridge contribution can be seen in both the zzxxxx (Fig.4) and xxzzxx (Fig.3) signals. The lower panel of Fig. 8 shows R (5) xzxxxz(t 1, t 2 ) R (5) xzxzxx(t 1, t 2 ) 2 in which similar cancellation takes place. The spike pathway contribution is effectively removed along with exact cancellation of the ridge pathway contribution with this combination. Note the signal to noise ratio is significantly lower in the depolarized signals as seen in the lower panel of Fig. 8. Lastly, while it is possible to preferentially enhance a pathway relative to the other pathways, it is not possible to reduce the fifth order response to a 15

129 129/224 single pathway. The pathways are convoluted in such a way that no combination of various polarization condition signals will yield a result that depends on a single pathway. Although the analysis by Fourkas et al. was applied to existing experimental CS 2 results, it is general and applies equally well to liquid xenon. The differences in the semipolarized and depolarized spectra is not at all unexpected. A combination of factors could lead to these differences including diminished ability of the depolarized conditions to enhance/diminish the pathways relative to one another. If indeed there is one dominant pathway that leads to the fully polarized spectra, then it would be expected that the depolarized results would have fully polarized characteristics. In addition, the effect of the pathway R (Π(2) Π (2) Π (2)) has not been stressed in our analysis. The fact that this pathway contributes equally within each of these two polarization forms (semipolarized and depolarized) makes it somewhat more difficult to appreciate its contribution without further analysis. It is also remarkable the variety of lineshapes that can be obtained for the 2D Raman spectrum of even simple liquids. This suggests rich information content in the signal that requires better understanding to extract. V. CONCLUSIONS Given success of the present computationally tractable theory for R (5) (t 1, t 2 ), examining the temperature dependence of the signal for liquid xenon and for other liquids and solutions will help establish the nature of the fifth order Raman measurement and its variability deeply supercooled liquids might be especially interesting considering the onset of nearly harmonic dynamics in that regime. Application of these methods to R (3) (t 1, t 2 ) has also led to a practical TCF theory of the 2DIR experiment. 30 It would appear that although exact TCF theories are impossible, approximate yet effective TCF theories of all nonlinear spectroscopy maybe possible. These theories bring the power of MD simulations to bear the difficult task of theoretically modeling the spectroscopy of complex condensed phase chemical systems. When multidimensional 16

130 130/224 nonlinear optical experiments were first proposed it was expected they could have an impact similar to introducing higher dimensional techniques into NMR. A major impediment to developing and interpreting these spectroscopies has been the lack of tractable yet accurate molecularly detailed theory of the spectroscopy and TCF theories serve to fill this void. Acknowledgments The research at USF was supported by an NSF Grant (No. CHE ) and a grant from the Petroleum Research Foundation to Brian Space. The authors would like to acknowledge the use of the services provided by the Research Oriented Computing center at USF. The authors also thank the Space (Basic and Applied Research) Foundation for partial support. The research at BU was supported by NSF Grant (No. CHE ) to T. Keyes. Electronic address: space@cas.usf.edu 1 Y. Tanimura and S. Mukamel, J. Chem. Phys. 99, 9496 (1993). 2 D. J. Ulness, J. C. Kirkwood, and A. Albrecht, J. Chem. Phys. 108, 3897 (1998). 3 K. J. Kubarych, C. J. Milne, S. Lin, V. Astinov, and R. J. D. Miller, J. Chem. Phys. 116, 2016 (2002). 4 V. Astinov, K. Kubarych, C. Milne, and R. D. Miller, Chem. Phys. Let. 327, 334 (2000). 5 O. Golonzka, N. Demirdoven, M.Khalil, and A. Tokmakoff, J. Chem. Phys. 113, 9893 (2000). 6 L. J. Kaufman, J. Heo, L. D. Ziegler, and G. R. Fleming, Phys. Rev. Let. 88, (R) (2002). 7 D. A. Blank, L. J. Kaufman, and G. R. Fleming, J. Chem. Phys. 111, 3105 (1999). 8 D. A. Blank, L. J. Kaufman, and G. R. Fleming, J. Chem. Phys. 113, 771 (2000). 9 L. J. Kaufman, D. A. Blank, and G. R. Fleming, J. Chem. Phys. 114, 2312 (2001). 10 R. DeVane, C. Ridley, T. Keyes, and B. Space, Phys. Rev. E 70, (2004). 11 J. Kim and T. Keyes, Phys. Rev. E 65, (2002). 12 T. Steffen, J. T. Fourkas, and K. Duppen, J. Chem. Phys 105, 7364 (1996). 13 K. Okumura, A. Tokmakoff, and Y. Tanimura, J. Chem. Phys. 111, 492 (1999). 17

131 131/ R. L. Murry, J. T. Fourkas, and T. Keyes, J. Chem. Phys. 109, 7913 (1999). 15 R. L. Murry and J. T. Fourkas, J. Chem. Phys 107, 9726 (1997). 16 T. I. Jansen, J. Snijders, and K. Duppen, J. Chem. Phys 113, 307 (2000). 17 T. I. Jansen, A. Pugzlys, G. D. Cringus, J. Snijders, and K. Duppen, J. Chem. Phys 116, 9383 (2002). 18 S. Saito and I. Ohmine, J. Chem. Phys. 108, 240 (1998). 19 R. A. Denny and D. R. Reichman, Phys. Rev. E 63, 65101(R) (2001). 20 A. Ma and R. M. Stratt, Phys. Rev. Let. 85, 1004 (2000). 21 A. Ma and R. M. Stratt, J. Chem. Phys. 116, 4962 (2002). 22 A. Ma and R. M. Stratt, J. Chem. Phys. 116, 4972 (2002). 23 S. Mukamel, V. Khidekel, and V. Chernyak, Phys. Rev. E. 53, R1 (1996). 24 S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, Oxford, 1995). 25 R. DeVane, C. Ridley, T. Keyes, and B. Space, J. Chem. Phys. 119, 6073 (2003). 26 J. Cao, S. Yang, and J. Wu, J. Chem. Phys. 116, 3760 (2002). 27 S. Hahn, K. Park, and M. Cho, J. Chem. Phys. 111, 4121 (1999). 28 S. Saito and I. Ohmine, J. Chem. Phys. 119, 9073 (2003). 29 S. Saito and I. Ohmine, Phys. Rev. Lett. 88, (2002). 30 R. DeVane, C. Ridley,, A. Perry, C. Neipert, T. Keyes, and B. Space, J. Chem. Phys. 121 (2004). 31 T. I. Jansen, K. Duppen, and J. Snijders, Phys. Rev. B 67, (2003). 32 N.-H. Ge, M. T. Zanni, and R. M. Hochstrasser, J. Phys. Chem. A 106, 962 (2002). 33 J. Borysow, M. Moraldi, and L. Frommhold, Mol. Phys. 56, 913 (1985). 34 K. Okumura and Y. Tanimura, J. Chem. Phys. 107, 2267 (1997). 35 A. Perry, H. Ahlborn, P. Moore, and B. Space, J. Chem. Phys. 118, 8411 (2003). 36 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970). 37 J. Applequist, J. R. Carl, and K.-K. Fung, J. Am. Chem. Soc. 94, 2952 (1972). 38 T. Keyes and J. T. Fourkas, J. Chem. Phys. 112, 287 (2000). 39 K. Okumura and Y. Tanimura, J. Chem. Phys. 107, 2267 (1997). 18

132 132/224 VI. TABLES TABLE I: A summary of the analysis performed by Fourkas et.al. is shown here. 15 The top row represents the pathways that correspond to those in Fourkas s analysis and the effect of various polarization conditions (left column) is indicated in each column. The xxzzxx combination enhances the echo signal. R (Π(2) Π (1) Π (1)) R (Π(1) Π (2) Π (1)) R (Π(1) Π (1) Π (2)) R (Π(2) Π (2) Π (2) ) xxxxxx Π (2) xx Π (1) xx Π (1) xx Π (1) xx Π (2) xx Π (1) xx Π (1) xx Π (1) xx Π (2) xx Π (2) xx Π (2) xx Π (2) xx xxxxzz Π (2) xx Π (1) xx Π (1) zz Π (1) xx Π (2) xx Π (1) zz Π (1) xx Π (1) xx Π (2) zz Π (2) xx Π (2) xx Π (2) zz xxzzxx Π (2) xx Π (1) zz Π (1) xx Π (1) xx Π (2) zz Π (1) xx Π (1) xx Π (1) zz Π (2) xx Π (2) xx Π (2) zz Π (2) xx zzxxxx Π (2) zz Π (1) xx Π (1) xx Π (1) zz Π (2) xx Π (1) xx Π (1) zz Π (1) xx Π (2) xx Π (2) zz Π (2) xx Π (2) xx xzxxxz Π (2) xz Π (1) xx Π (1) xz Π (1) xz Π (2) xx Π (1) xz Π (1) xz Π (1) xx Π (2) xz Π (2) xz Π (2) xx Π (2) xz xzxzxx Π (2) xz Π (1) xz Π (1) xx Π (1) xz Π (2) xz Π (1) xx Π (1) xz Π (1) xz Π (2) xx Π (2) xz Π (2) xz Π (2) xx xxxzxz Π (2) xx Π (1) xz Π (1) xz Π (1) xx Π (2) xz Π (1) xz Π (1) xx Π (1) xz Π (2) xz Π (2) xx Π (2) xz Π (2) xz 19

133 133/ Invariants (A 3 ) Time (ps) FIG. 1: Time slices (for short times) of the rotational invariants for R (5) xxxxxx(t 1, t 2 ) from Eq. (3.6). Shown is the triple product, T P ( Π(0), Π(t 1 ), Π(t 2 )), for MBP (solid) and FODID (line with squares), the trace pair product combinations for MBP, T r( Π(0))P P ( Π(t 1 ), Π(t 2 )) (line with circles), T r( Π(t 2 ))P P ( Π(0), Π(t 1 )) (dashed), and T r( Π(0))T r( Π(t 1 ))T r( Π(t 2 )) (dotted). The slice was taken along t 1 = 0 where T r( Π(0))P P ( Π(t 1 ), Π(t 2 )) = T r( Π(t 1 ))P P ( Π(0), Π(t 2 )).

134 134/224 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps FIG. 2: TCF theory results for R (5) xxxxxx(t 1, t 2 ) 2, using the FODID approximation (upper) and the MBP model (lower) for liquid xenon. while the overall shapes of the first order and many body polarization signal look similar, the time scales involoved are significantly different. Contours are evenly spaced from (upper) and (lower).

135 135/224 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps FIG. 3: R (5) xxzzxx(t 1, t 2 ) 2 for liquid xenon using the MBP model. Six contour line are evenly spaced from

136 136/224 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps 0.2 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps 0.2 FIG. 4: Results for semipolarized polarization conditions of R (5) (t 1, t 2 ) 2, using the MBP model for liquid xenon. Shown are the zzxxxx (top panel) and the xxxxzz (lower panel) polarization conditions. Upper panel: Three contour lines are evenly spaced from ; lower panel: Four contour lines are evenly spaced from

137 137/ R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 2 (ps) FIG. 5: Time slice of R (5) (t 1, t 2 ) 2 for the zzxxxx polarization along t 2 with t 1 =0.

138 138/224 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps 0.2 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps 0.2 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps 0.2 FIG. 6: Results for depolarized polarization conditions of R (5) (t 1, t 2 ) 2, using the MBP model for liquid xenon. Shown are the xzxxxz (top panel), xxxzxz (middle panel) and the xzxzxx (lower panel) polarizations. Upper and middle panel: five contour lines are evenly spaced from ; lower panel: four contour lines are evenly spaced from

139 139/ R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 (ps) R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 = t 2 (ps) FIG. 7: Time slices of R (5) (t 1, t 2 ) 2 for the xzxxxz polarization (upper) along t 2 = 0 (dotteddashed line) and t 1 = t 2 (solid line) and for xzxzxx along t 1 = t 2 plotted with error bars. The error estimates for the slices were ± Å3 /ps 2 for the t 2 = 0 slice of xzxxxz and ± Å 3 /ps 2 for the t 1 = t 2 slice of xzxzxx.

140 140/224 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps 0.2 R (5) (t 1,t 2 ) 2 (A 3 /ps 2 ) t 1 /ps t 2 /ps 0.2 FIG. 8: R (5) xxzzxx(t 1, t 2 ) R (5) xxxxzz(t 1, t 2 ) 2 (upper panel) and R (5) xzxxxz(t 1, t 2 ) R (5) xzxzxxls (t 1, t 2 ) 2 (lower panel) for liquid xenon using the MBP model. Six contour lines are evenly spaced from (top panel) and (bottom panel)

141 141/224 RAPID COMMUNICATIONS PHYSICAL REVIEW E 71, R 2005 Identification of a wagging vibrational mode of water molecules at the water/vapor interface Angela Perry, Christine Neipert, Christina Ridley, and Brian Space* Department of Chemistry, University of South Florida, 4202 E. Fowler Ave., SCA400, Tampa, Florida , USA Preston B. Moore Department of Chemistry and Biochemistry, University of the Sciences in Philadelphia, Philadelphia, Pennsylvania 19104, USA Received 12 August 2004; revised manuscript received 7 January 2005; published 31 May 2005 An improved time correlation function description of sum frequency generation SFG spectroscopy was applied to theoretically describe the water/vapor interface. The resulting spectra compare favorably in shape and relative magnitude to extant experimental results in the OuH stretching region of water. Further, the SFG spectra show a well-defined intermolecular mode at 875 cm 1 that has significant intensity. The resonance is due to a wagging mode localized on a single water molecule. It represents a well-defined population of water molecules at the interface that, along with the free OuH modes, represent the dominant interfacial species. DOI: /PhysRevE PACS number s : g, Uv, Ky Liquid water interfaces are ubiquitous and important in chemistry and the environment. Thus, with the advent of interface specific nonlinear optical spectroscopies, such interfaces have been intensely studied, both theoretically 1 6 and experimentally 7 9. Sum frequency generation SFG spectroscopy is a powerful experimental method for probing the structure and dynamics of interfaces. SFG spectroscopy is dipole forbidden in isotropic media and interfaces serve to break the symmetry and produce a dipolar second-order polarization signal. The SFG experiment employs both a visible and infrared laser field overlapping in time and space at the interface, and in the absence of any vibrational resonance at the infrared laser frequency, a structureless signal due to the static hyperpolarizability of the interface is obtained 1,3,7,8. When the infrared laser frequency corresponds to a vibration at the interface a resonant line shape is obtained with a characteristic shape that reflects both the structural and dynamical environment at the interface 10. Here, classical molecular dynamics MD methods are used to model the dynamics of the water/vapor interface. Two complementary theoretical approaches quantumcorrected time correlation function TCF and instantaneous normal mode INM methods use the configurations generated by MD as input to describe the SFG spectrum of the interface and to ascertain the molecular origin of the SFG signal; both INM and TCF methods use a suitable spectroscopic dipole and polarizability model. This dual approach was demonstrated to be highly effective in understanding condensed phase spectroscopy of water, other liquids, and interfaces; classical mechanics, especially in the context of quantum-corrected TCF s, has proven to be surprisingly effective in modeling intramolecular vibrational spectroscopy 1,2, An INM approximation to SFG spectroscopy is quantum mechanical by construction but offers a limited dynamical description. As a result, e.g., in bulk water, INM intramolecular resonances are broader than their TCF counterparts, *Electronic address: space@cas.usf.edu but with the same central frequency and integrated intensity. This suggests that the intramolecular INM spectra represent an underlying spectral density that is dynamically motionally narrowed in the actual line shape 10. This is also found to be the case here for SFG spectra in all polarization conditions. This means that motional narrowing effects are important at interfaces and the dynamics is intermediate between the fast and slow modulation limits of motional narrowing. Most importantly this is a clear demonstration that SFG spectra contain both structural and dynamical signatures. Previous TCF investigations of the water/vapor interface were limited to only following short time correlations and obtaining high-frequency spectra and used incorrect theoretical expressions 1 3. In order to obtain better TCF results, especially at lower frequencies, long time correlations between the system dipole and polarizability need to be followed. Because molecular simulations of interfaces in Cartesian space necessarily produce two interfaces, simulation times were limited to the molecular diffusion time between interfaces so that molecules cannot contribute to signal at both interfaces during one MD run 1. In this paper, a weak restraining potential is added that confines the molecules over time to the half of the simulation box they start in without significantly perturbing the relevant dynamics and the average structure of the liquid that contributes to the interfacial spectroscopy. This modification permits the calculation of TCF s out to long times resulting in well-defined spectra that include intermolecular spectral line shapes. Surprisingly, a welldefined intermolecular mode was found to be prominent in the spectrum. It is centered at 875 cm 1 and is comparable in integrated intensity to the rest of the intermolecular line shape. The line shape also has an intensity that is about a sixth of the magnitude of the intense free OuH stretching peak. Thus, a large and distinct population of interfacial water molecules has been identified. These modes, along with the well known free OuH oscillators are the dominant constituents of water/vapor interfaces and determine the physical and chemical properties of the interface. Using INM methods the resonance is found to be due to a wagging mode localized on individual water molecules. The /2005/71 5 / /$ The American Physical Society

142 142/224 RAPID COMMUNICATIONS PERRY et al. water molecules responsible are nearly parallel to the interface with their oxygen atoms anchored in the interface and the hydrogen atoms wagging normal to the interface. The hydrogens, pointing into the vapor phase, are hydrogen bonded to oxygen atoms at the interface. The presence of another distinct population of interfacial molecules, other than the free OuH stretch, was recently proposed 4,15,16. That hypothesis is strongly supported by this paper. Here, a spectroscopically distinct species is directly observed and the vibrational mode responsible for the line shape is clearly identified. Thus, experimental SFG setups that permit taking spectra at relatively low wavelengths could probe this mode as a complement to the information contained in the free and donor OuH stretching modes. The second-order response is given theoretically by a combination of resonant and nonresonant terms 2,3,7,17,18. The resonant terms can be grouped to give a simple expression in terms of the systems polarizability and dipole. Res is given by 1,17,19 : Res = i dte 0 i Tr, i jk t. 1 In Eq. 1, =e H /Q for a system with Hamiltonian H and partition function Q at reciprocal temperature =1/kT, and k is Boltzmann s constant; is the system dipole, and its polarizability tensor where the subscripts represent the vector and tensor components of interest, respectively. The operator evaluated at time t is the Heisenberg representation of the operator jk t =e iht/ jk e iht/ ; Tr represents the trace of the operators. It is convenient to proceed by rewriting the Fourier-Laplace transform in Eq. 1 as the Fourier transform of a TCF that can then be interpreted in the classical limit and quantum corrected. Equation 1 can be simplified, defining the real and imaginary parts of Res = Res R +i Res I gives 2 : I = 2 tanh /2 C R = 2 0 sin t C I t dt, R = 2 P tanh /2 C R d + = 2 cos t C I t dt. 3 0 Note that previous expressions in the literature were mistaken although the expressions used did not significantly effect the modulus of the signal 1,3,19. InEq. 3, P designates the principle part and the TCF, C t = i 0 jk t =C R t +ic I t, is defined in terms of its real and imaginary parts and their real Fourier transforms C =C R +C I. The focus of SFG experiments is on high-frequency spectra where kt and classical mechanics are clearly invalid. The classical TCF result, C Cl t can be calculated using MD and TCF methods; C Cl is the same as C R in the 2 PHYSICAL REVIEW E 71, R 2005 classical limit. C R is obtained for high frequencies via quantum correction using a harmonic correction factor: C R = /2 coth /2 C Cl 1,2. This correction factor is exact in relating the real part of the classical harmonic coordinate correlation function to its quantum mechanical counterpart. C I is then obtained using the exact result C I =tanh /2 C R. Thus the procedure is to calculate a classical TCF, C Cl t, using MD and a suitable spectroscopic model, Fourier transform into frequency and quantum correct it to obtain C R and then C I that can be used in Eqs. 2 and 3. To construct an INM approximation to Eq. 1 it is sufficient to evaluate the trace in Eq. 1 for a harmonic system, giving in its classical form for consistency : C Cl = i / Q l jk / Q l l kt 2 4 In Eq. 4, l is the frequency of mode Q l and the angle brackets here represent averaging over classical configurations of the system generated. C Cl is then back transformed into the time domain and used in place of the classical TCF, in the procedure given above, to obtain an INM approximation to the spectroscopy. The MD methods and spectroscopic model have been described previously 1. Briefly, microcanonical MD was performed on an ambient water/vapor interface to generate a time ordered series of configurations ns data sets a flexible simple point charge SPC model was used. The simulation consisted of 64 water molecules, and the spectra were checked for convergence by comparing with a 216- water-molecule simulation and the spectra were statistically indistinguishable. The interface was constructed using the standard MD geometry with vacuum vapor above and below the water. Unfortunately this produces two interfaces with average net dipoles in opposite directions and if molecules were allowed to diffuse from one interface to the other the SFG signal would be perturbed 1 ; each interface is treated as a separate entity in calculating TCF s. This limited earlier investigations to relatively short time simulations and resulted in noisy spectra that were limited to the high frequency OuH stretching region 1,3. To overcome this limitation, a weak laterally isotropic restraining potential in the direction normal to the interface was employed to keep a molecule in the half of the box that it started in for the length of the simulation 2,20. The restraining potential becomes negligible near the interface and is only significant within 2.0 Å of the box center. The interfacial density profile was unchanged by the restraining potential and it only significantly affects the long time diffusion constant in the normal direction 2. The induced dipoles and polarizability tensor of each configuration is then calculated using a point atomic polarizability approximation PAPA polarizability model that includes many-body polarization effects explicitly and accounts for polarizability derivatives with bond-length-dependent point polarizabilities. The permanent dipoles are calculated based on ab initio data as described previously 1,3. The TCF, C Cl t, was calculated out to 50 ps where it has nearly de

143 143/224 RAPID COMMUNICATIONS IDENTIFICATION OF A WAGGING VIBRATIONAL PHYSICAL REVIEW E 71, R 2005 FIG. 1. Color online TCF SFG spectra in the OuH stretching region for three polarizations: SSP thick, solid black line, PPP dashed red line, and SPS thin, solid green line. The inset is experimental data 7 for the same polarizations using the same color scheme. cayed to zero and is used to calculate the SFG spectra as described above. Figure 1 displays the theoretical TCF SFG spectra in the O u H stretching region for the three independent polarization conditions that are possible in the electronically nonresonant experiment, i.e., SSP, PPP, and SPS. The theoretical spectra have been adjusted in relative intensity to account for the Fresnel factors that modify the experimental intensities 2,7. The first two last index can be interpreted as the element of the system polarizability tensor dipole that is being probed, respectively S denotes directions parallel to the surface and P perpendicular. In the data we have included the nonresonant contribution, N Res, that is a small negative constant 3,8 and the full signal is given by 2 SFG Res + N Res 2. In Fig. 1, the free OuH peak is prominent at 3700 cm 1 and the rest of the OuH stretching region has a more complicated shape. The inset of Fig. 1 displays experimental data for the OuH stretching region taken in the same polarization geometries 7. The relative intensities agree nearly quantitatively between theory and experiment. The free OuH stretching line shape is captured very accurately by the theory and the rest of the OuH region has a similar shape. The ratio of relative intensities between the free OuH and the rest of the OuH stretching band are about 2:1 for both experiment and theory. Clearly the theory captures the essential features of the spectrum and its polarization dependence. To quantitate the polarization dependence, the ratio of the SSP:PPP intensities for the free O u H stretch, where the signal to noise is best, is 13:1 10:1 for the theoretical experimental spectra, and the SPS is about a factor of three smaller than the PPP in both cases. The agreement is well within the relative error demonstrating the success of theoretical methods. Figure 2 displays the theoretical SFG spectrum over the entire water vibrational spectrum. The theoretical INM spectrum for the SSP geometry is also shown. The INM and TCF spectra were found to integrate to the same value over the entire cm 1 range and separately over the FIG. 2. Color online TCF SFG spectra for the entire water vibrational spectrum for three polarizations: SSP thick, solid black line, PPP thick, dashed red line, and SPS thin, solid green line. The SSP INM SFG spectra is also shown thin, dashed blue line. The inset highlights the intermolecular resonance at 875 cm 1. OuH stretching region cm 1. This behavior is strong evidence for the interpretation of the INM line shape as an underlying spectral density that is motionally narrowed in the observed spectrum. This result also suggests that SFG spectra are sensitive to both structure and dynamics the INM spectrum clearly exhibits the same resonances but is broader implying that the observed line shapes are motionally narrowed and dynamical contributions to SFG signals are important 2,7. Most strikingly, Fig. 2 reveals an intense intermolecular resonance at 875 cm 1. In contrast, the intermolecular spectrum of bulk water is relatively unstructured 10. This symmetric line shape indicates a spectroscopically distinct species and represents such as the free O u H stretch a population of water molecules unique to the interface. It is roughly as intense as the rest of intermolecular spectrum and about a sixth of the intensity of the free OuH peak within our model note that the bending line shape at higher frequency is much less intense. Recent experiments 15,16 and theory 4 indirectly inferred the presence of a surface species a water molecule with two dangling hydrogens. Here we have directly identified the species and its spectroscopic signature. The spectrum is expected to be experimentally measurable using SFG but there has been a lack of intense infrared laser sources in this spectral region. Figure 3 displays representative INM s from the OuH wagging region that give rise to the spectral signature. The mode is a wagging motion localized on a single water molecule, almost parallel to the interface, with two hydrogens displaced normal to the interface and the oxygen anchored in the interface. Further analysis revealed that, while the hydrogens appear to be freely oscillating into the vapor phase, they are both typically hydrogen bonded to an oxygen in the interface, based on standard hydrogen bonding criteria 4. Indeed, it is the hydrogen bonding that provides the vibrational restoring force that results in a relatively high frequency intermolecular resonance. The inset of Fig. 2 focuses in on the OuH wagging region and the polarization dependence of the signal is clear

144 144/224 RAPID COMMUNICATIONS PERRY et al. PHYSICAL REVIEW E 71, R 2005 FIG. 3. Color online Two snapshots of a water/vapor interface containing 216 molecules and 64 molecules featuring INM s from the OuH wagging region at 864 and 950 cm 1, respectively. The INM s are representative of a wagging motion localized on a single water molecule shown in blue, almost parallel to the interface, with two hydrogens displaced normal to the interface and the oxygen anchored in the interface. Both the SSP and PPP geometries show an intense line shape and are sensitive to motions dipole derivatives perpendicular to the interface. The SPS geometry shows only a hint of the signal consistent with its sensitivity to modes with dipoles changing parallel the surface. This analysis also highlights the ability of SFG spectra to infer intermolecular molecular geometries by examining the polarization dependence of the spectroscopy. Should appropriate infrared laser sources become available, this OuH wagging mode may represent another probe, in addition to OuH stretching, of interfacial structure and dynamics that couples to lower frequency dynamical processes. In summary, a spectroscopically distinct interfacial species has been identified at the water/vapor interface via improved theoretical SFG spectra. The mode was found to be an OuH wagging motion localized on a single molecule and a similar species was suggested from earlier investigations 4,15,16. The mode is currently difficult to detect using SFG spectroscopy. However, electronically and vibrationally doubly resonant second-order experiments may be able to detect the interfacial mode 21. Lastly, this paper points out the power of theoretical spectroscopy in providing insight into MD simulations many MD simulations of the water/vapor interface have been conducted but traditional analysis techniques do not easily reveal important interfacial subpopulations like the O u H wagging motions. The research was supported by NSF Grant No. CHE to B. S., a grant from ACS/PRF to B.S., and a grant from University of the Sciences in Philadelphia to P.B.M. Acknowledgment is made of financial support from the American Chemical Society Petroleum Research Fund. The authors would also like to acknowledge the use of the services provided by the Research Oriented Computing center at USF. 1 A. Perry, H. Ahlborn, P. Moore, and B. Space, J. Chem. Phys. 118, A. Perry, C. Neipert, C. Ridley, P. Moore, and B. Space unpublished. 3 A. Morita and J. T. Hynes, J. Phys. Chem. B 106, I. W. Kuo and C. J. Mundy, Science 303, P. Vassilev, C. Hartnig, M. T. Koper, F. Frechard, and R. A. v. Santen, J. Chem. Phys. 115, I. Benjamin, Phys. Rev. Lett. 73, X. Wei and Y. R. Shen, Phys. Rev. Lett. 86, E. A. Raymond and G. Richmond, J. Phys. Chem. B 108, M. Shultz, C. Schnitzer, D. Simonelli, and S. Baldelli, Int. Rev. Phys. Chem. 19, P. Moore, H. Ahlborn, and B. Space, in Liquid Dynamics Experiment, Simulation, and Theory, edited by M. D. Fayer and J. T. Fourkas, ACS Symposium Series Vol. 820 Oxford University Press, Oxford, R. DeVane, A. Perry, C. Neipert, C. Ridley, and B. Space, J. Chem. Phys. 121, H. Ahlborn, X. Ji, B. Space, and P. B. Moore, J. Chem. Phys. 112, X. Ji, H. Ahlborn, B. Space, P. Moore, Y. Zhou, S. Constantine, and L. D. Ziegler, J. Chem. Phys. 112, H. Ahlborn, X. Ji, B. Space, and P. B. Moore, J. Chem. Phys. 111, K. R. Wilson, M. Cavalleri, B. S. Rude, R. D. Schaller, A. Nilsson, L. M. Pettersson, N. Goldman, T. Catalano, J. Bozek, and R. J. Saykally, J. Phys.: Condens. Matter 14, L K. R. Wilson, R. Schaller, D. Co, R. J. Saykally, B. S. Rude, T. Catalano, and J. Bozek, J. Chem. Phys. 117, S. Mukamel, Principles of Nonlinear Optical Spectroscopy Oxford University Press, Oxford, R. W. Boyd, Nonlinear Optics Academic, London, V. Pouthier, P. Hoang, and C. Girardet, J. Chem. Phys. 110, The restraining potential is of the form V= /r 9, with =2.3 K, and =2.474 Å, and r=0 at the center of the box. 21 M. B. Raschke, M. Hayashic, S. H. Lind, and Y. R. Shen, Phys. Rev. Lett. 359,

145 145/224 RAPID COMMUNICATIONS PHYSICAL REVIEW E 70, (R) (2004) Tractable theory of nonlinear response and multidimensional nonlinear spectroscopy Russell DeVane, Christina Ridley, and Brian Space* Department of Chemistry, University of South Florida, 4202 E. Fowler Avenue, SCA400, Tampa, Florida , USA T. Keyes Department of Chemistry, Boston University, Boston, Massachusetts 02215, USA (Received 4 May 2004; published 16 November 2004) Nonlinear spectroscopy provides insights into dynamics, but the response functions required for its interpretation pose a challenge to theorists. We proposed an approach in which the fifth-order response function R 5 t 1,t 2 was expressed as a two-time classical time correlation function (TCF). Here, we present TCF theory results for R 5 t 1,t 2 in liquid xenon. Using a first-order dipole-induced dipole polarizability model, the result is compared to an exact numerical calculation showing remarkable agreement. In addition, R 5 t 1,t 2 is calculated using the exactly solved polarizability model, yielding different results and predicting an echo signal. DOI: /PhysRevE PACS number(s): Uv, p Multidimensional nonlinear spectroscopic techniques are emerging as powerful tools that can reveal the microscopic structure and dynamics of condensed phases and biomolecules on ultrafast (subpicosecond) time scales. For example, they offer the possibility of differentiating between homogeneous and inhomogeneous broadening processes, which is difficult with traditional spectroscopy [1]. The fifth-order, two-dimensional (2D) Raman spectrum, which has been the subject of many recent experimental [2 6] and theoretical [7 14] investigations, is well suited to the study of low frequency, intermolecular, motions in liquids. The circumvention of previous difficulties by using heterodyne detection has reinforced interest in the 2D Raman experiment [2 6]. The 2D Raman spectrum also was predicted to show an echo feature along the diagonal time slice if intermolecular modes were sufficiently long lived although no echo has been found to date. Multidimensional spectroscopy cannot achieve its full potential until an unambiguous theoretical framework is available; this presents a considerable theoretical challenge. An n-dimensional spectrum is given by an n-dimensional response function, a quantum average of nested commutators of n Heisenberg representation variables at different times. Even the classical limit yields quantities that are far more complex than familiar time correlation functions (TCF). In fact, it has been demonstrated that nonlinear spectra cannot be expressed exactly in terms of TCF [15]. Nevertheless, we recently proposed an approximate TCF theory of nonlinear spectroscopy that is computationally tractable for complex molecular condensed phase systems and is capable of accurately describing both intermolecular and intramolecular vibrational spectra. TCF theories are desirable because they allow nonlinear spectra to be calculated for realistic molecular dynamics (MD) models of condensed phase systems. It is essential that the theory be tested in a unambiguous manner. Previously we have presented explicit *Author to whom correspondence should be addressed. calculations of the 2D Raman spectrum of liquid CS 2, determined by the fifth-order response, R 5 t 1,t 2 [16]. The 2D Raman spectra were in excellent agreement with extant experiments and approximate theoretical results [2,9,17]. But the agreement does not represent a rigorous test because uncertainties still exist in the experimental measurements and the theoretical results use approximate methods and different molecular models. Using the TCF theory, here we present the 2D Raman spectrum for liquid xenon using both first-order dipole-induced dipole (FODID) and exact [many body polarizability (MBP)] solutions of the dipole-induced dipole polarizability model. Ma and Stratt [12] carried out a numerically exact simulation of the fully polarized, R xxxxxx 5 t 1,t 2, 2D Raman spectrum of liquid Xe using the FODID model. Exact calculations can be performed for simple, small systems [9,12,17] only because of the computational burden. In this paper it is demonstrated that the TCF theory of R 5 t 1,t 2 quantitatively reproduces the Ma-Stratt simulation, providing support for its further applications to more complex systems; TCF calculations with more extensive molecular models are not computationally difficult [16]. Further, the theory predicts a very different spectrum using the MBP 5 model, including a significant echo feature for the R xxzzxx t 1,t 2 polarization condition that was earlier suggested to be likely to emphasize an echo signature [18]. This echo occurs with a period that corresponds to frequencies in the heart of the xenon vibrational density of states (DOS), implying the existence of intermolecular vibrational modes that last for at least one full period [12]. The TCF theory describes the nonlinear response in terms of fully anharmonic MD calculations that are supplemented by a suitable spectroscopic (dipole and polarizability) model. Thus, it is demonstrated here, that while exact TCF expressions for nonlinear spectroscopy are not possible, effective TCF theories can be constructed. Such theories are powerful because they can be evaluated using atomistically detailed MD on complex liquids and solutions. TCF theories have distinct advantages over alternative approaches because they eliminate the need for costly classical calculations that are /2004/70(5)/050101(4)/$ The American Physical Society

146 146/224 RAPID COMMUNICATIONS DEVANE et al. PHYSICAL REVIEW E 70, (R) (2004) limited to small systems and intermolecular dynamics [9,12,17] and quantum calculations that rely on lowdimensional model systems [19]. It then becomes possible to predict and interpret nonlinear spectra for a wide variety of chemically important systems. In addition to its computational tractability, the TCF theory also makes a more manageable starting point for the development of theories based on the fifth-order response function. The development of the TCF theory of R 5 t 1,t 2 will be briefly revisited here; details are provided in our earlier work [16]. First consider the quantum mechanical expression for the electronically nonresonant fifth-order polarization response [13,17]: R,,,,, 5 t 1,t 2 = i/ 2 Tr t 1 +t 2 [ t 1, 0, ]. In this equation, =e H /Q for a system with Hamiltonian H and partition function Q at reciprocal temperature =1/kT, and k is Boltzmann s constant; Tr represents a trace, square brackets denote commutators, is the system polarizability tensor, and the greek superscripts denote the elements and thus the polarization condition being considered. The classical limit of the trace is of order 2 and results from a combination of four two-time correlation functions that are themselves equivalent classically. Expanding the commutators in the above equation gives R 5 t 1,t 2 = 1/ 2 g t 1,t 2 f * t 1,t 2 f t 1,t 2 +g* t 1,t 2. Here, f t 1,t 2 = * t 1 t 2, g t 1,t 2 = t 2 * t 1, f * t 1,t 2 = t 2 * t 1, and g* t 1,t 2 = * t 1 t 2 ; an asterisk denotes the complex conjugate. The superscript notation on is now suppressed and the results apply to all possible polarizations. The angle brackets represent an equilibrium average. The expression inside the square brackets is the difference between the real part of the functions g and f, g R t 1,t 2 f R t 1,t 2, that must be of order 2 in the classical limit. It is easy to show that a multiplicative factor of leading order can be obtained exactly using frequency domain (detailed balance) relationships between g and f, e.g., e 1g 1, 2 = f 1, 2 ; the frequency domain functions are the Fourier transform of the time domain functions. A sum of real and imaginary parts of a single two-time TCF then remains, and their O contribution is required for the classical limit to exist in the form of a TCF. For a one-time correlation function, C t, a simple frequency domain relationship exists between the real and imaginary parts, C I =tanh /2 C R [20]. If a similar relationship between the real and imaginary parts of the twotime correlation function existed the fifth-order response could be written as second derivatives in time of a classical TCF, but no exact analytic relationship is possible and, therefore, an exact TCF theory is also not possible. However, an approximate relationship between the real and imaginary parts of the quantum TCF can be found for a harmonic system with the polarizability expanded to second order in the harmonic coordinate, Q, = 0 + Q+1/2 Q 2. The real and imaginary parts are related as g I 1, 2 =tanh 1 /4+ 2 /2 g R 1, 2, where the subscripts denote the Fourier transform of the real or imaginary parts of the TCF s, both of which are real functions of frequency. This harmonic reference system is the simplest one that produces a fifth-order response and represents the leading contributions to the spectroscopy, analogous to IR intensities being given by the squared dipole derivative for a mode of a harmonic system with a linearly varying dipole [8,21]. The result is not a harmonic theory fully anharmonic dynamics are used to calculate the relevant TCF the approximation only serves to weight the different phonon processes implicit in the anharmonic MD as they would contribute in the harmonic system [8]. Using this relationship, the exact quantum mechanical response function can be written in terms of a classical TCF. The fifth-order response function, in the classical limit kt, takes the form R 5 t 1,t 2 = 2 /2 2 2 g R t 1,t 2 / t g R t 1,t 2 / t 1 t 2, and g R t 1,t 2 0 t 1 t 1 +t 2, the classical two-time TCF. Here, = and the TCF is written in terms of the correlated polarizability fluctuations [16]. This represents our TCF theory of R 5. The expression is the classical limit of a more general result which, in the case of intramolecular spectroscopy, can be evaluated for high frequencies if one approximates the quantum mechanical TCF with the (quantum corrected) classical TCF. Because fifth-order Raman spectroscopy has primarily been applied to intermolecular dynamics, only the classical limit will be considered here. R 5 t 1,t 2 can be evaluated using MD and a model of the systems polarizability. It exhibits the correct limiting behaviors, including giving zero signal at the origin and everywhere along t 2 =0. Also, the two-dimensional spectrum for ambient CS 2, calculated using our theory, was shown to be in excellent agreement with existing experimental and theoretical work [3,9,16]. The most rigorous test of this theory is to compare the resulting spectra to those from exact numerical results for the same model system. To evaluate this approach exactly, the classical limit is taken directly by replacing the commutators with Poisson brackets. Then, R 5 is seen to contain brackets of variables at different times, which requires the exceedingly difficult task of calculating the dependence of a manybody dynamical variable on its initial conditions. Calculating R 5 in this way is only practical for small simple systems and results for liquid xenon were reported previously [12]. To compare our theory to those results, microcanonical MD simulations were performed for the neat liquid xenon consisting of 108 atoms. The atoms interacted via a Lennard- Jones pair potential and the systems reduced density and temperature were 3 =0.8 and kt/ =1.0 and the same model parameters were used as in the exact calculation [12]. Results for the square of the fully polarized fifth-order 5 response function, R xxxxxx t 1,t 2 2, within the FODID approximation, are shown in Fig. 1. The result using our TCF theory is overlayed on the exact classical calculation performed by Ma and Stratt [12]; even though an exact TCF theory is not possible [15], the present theory is very effective. It captures the characteristic features present in the exact calculation, including the same decay times and the lack of an echo signal along the diagonal which implies that the intermolecular modes have lifetimes of less than a full period. The signal is sharply peaked around t 1 30 fs, t fs. Decay times vary along each axis with the signal

147 147/224 RAPID COMMUNICATIONS TRACTABLE THEORY OF NONLINEAR RESPONSE AND PHYSICAL REVIEW E 70, (R) (2004) 5 FIG. 1. Results for R xxxxxx t 1,t 2 2 using the FODID approximation for liquid xenon. The exact classical molecular dynamics calculation (red, dark), Stratt et al. was reported in arbitrary units and is normalized here to our TCF theory data (green, light) at the maximum point. dying out along t 1 by 400 fs; along t 2 there is a long time decay that continues beyond the 1 ps that is shown. In contrast to liquid xenon, the plot of the spectrum of CS 2 [2,16] is roughly symmetric, so the extreme asymmetry of the xenon simulation, with the peak practically on the t 2 axis, is striking [2,9,16]. The capability of the TCF theory to correctly yield symmetric or asymmetric spectra is further strong evidence of its general applicability. To highlight the effectiveness of the TCF theory, Fig. 2 5 shows a slice of R xxxxxx t 1,t 2 2 along t 2 with t 1 =0. The dashed line is from the exact classical calculation and the solid line marked with squares represents the same result shifted forward in time by 34.7 fs. The solid line without symbols is from our TCF theory. The TCF result and the time-shifted exact calculation show quantitative agreement. The difference between the exact result and the TCF theory, the shift in time, is very likely due to finite system size effects on the exact calculation. Ma and Stratt were limited to using only 32 atoms in their simulation even within the 5 FIG. 2. Slices of R xxxxxx t 1,t 2 2 along t 2 with t 1 =0. The dashed line represents the exact classical calculation, the line marked with squares represents the time-shifted exact classical calculation, and the solid line without symbols represents the result from the TCF theory. FIG. 3. The upper panel compares the TCF theory results for 5 R xxxxxx t 1,t 2 2 using the MBP model (red, dark, peaks earlier) vs the FODID approximation (green, light). The lower panel shows 5 R xxzzxx t 1,t 2 2 for liquid xenon using the MBP model. The solid lines parallel to the t 2 axis represent the width and location of (the beginning and end of) the echo peak along the diagonal (0.1 ps, 0.45 ps) using the diagonal to choose the lines leads to them overlapping off-diagonal features in the figure. The inset shows the INM DOS for liquid xenon with vertical lines placed to show the breath of the vibrational periods associated with the echo peak. FODID approximation that is far less computationally demanding than a MBP evaluation of the polarizability. To test the effect of the small system size they performed a simpler calculation of a one-dimensional slice of 2 g R t 1,t 2 / t 1 t 2 t2 =0 for 32 and 108 atoms because a non- Poisson bracket piece of the fifth-order signal with this form may be identified [14]. The resulting line shape had the same shape but with a time phase shift nearly identical to that apparent in Fig. 2 (see Fig. 4 of their paper), implying that the present theory may agree even better then the figure suggests [12]. Figure 3 shows R 5 t 1,t 2 2 for liquid xenon using the MBP model. The MBP result is a prediction of the experimental fifth-order response function xenon is highly polarizable and the FODID approximation is not strictly valid. The use of the FODID model effectively acts to remove contributions from the trace of the polarizability matrix [12] one of three matrix invariants (in isotropic media) that give rise, in different combinations, to the signals for different polarization conditions [18]. Although the trace contributions are small in 1D correlation functions [12], in 2D correlation functions the invariants appear as products and a zero trace invariant kills off other significant terms that include sizable invariants multiplied by the trace contribution [18]. Thus, the use of the FODID approximation in the 2D correlation func

148 148/224 RAPID COMMUNICATIONS DEVANE et al. PHYSICAL REVIEW E 70, (R) (2004) tions acts to remove significant contributions that lead to different signals including removing the echo contribution. The upper panel of Fig. 3 compares results for 5 R xxxxxx t 1,t 2 2 using the MBP model (red) and the FODID model (green). The most notable differences are that the MBP signal has faster decay times, along both the t 1 and t 2 axis, and the peak is shifted to earlier times along t 2. The signal is characterized by a strong peak around t 1 45 fs, t fs and dies out by 200 fs along both t 1 and t 2. As in the FODID approximation, no echo signal appears along the diagonal and the signal appears featureless beyond 300 fs in both t 1 and t 2 directions. While the FODID model captures much of the response, the distinct differences observed are a result of the exclusion of the polarizability tensor invariant contributions mentioned above. The lower panel of Fig. 3 shows R xxzzxx t 1,t 2 2 using the MBP model with the instantaneous normal mode (INM) vibrational DOS for xenon appearing in the inset [22]. The 5 presence of an echo signal for the R xxzzxx polarization condition was suggested by Fourkas et al. and only appears using the MBP model [18]. The use of the FODID approximation acts to remove distinguishing contributions to differing polarization conditions by only allowing the triple product invariant to contribute with varying magnitudes [12,18]. The echo peak implies that an intermolecular mode, excited at time zero, is still oscillating at the time of the echo. As shown in Fig. 3, associating a period with the oscillation leads to the first echo signature at 53 cm 1 (100 fs) and the end of the signal occurs at a frequency of 12 cm 1 (450 fs). The echo onset frequency is in the tail of the INM DOS and the last echo occurs near the maximum of the INM DOS and 5 the echo signature spans nearly the entire INM DOS. The appearance of an echo in such a simple liquid implies that intermolecular modes may generally live at least a period in more complicated liquids that have significant Lennard- Jones interactions. Given the present computationally tractable theory for R 5 t 1,t 2, examining the temperature dependence of the signal for liquid xenon and for other liquids and solutions will help establish the nature of the fifth-order Raman measurement and its variability deeply supercooled liquids might be especially interesting considering the onset of nearly harmonic dynamics in that regime. Using the current theory to examine other polarizations will provide further insights into the information content of fifth-order Raman responses of molecular liquids. TCF theories bring the power of MD simulations to bear on the difficult task of theoretically modeling the spectroscopy of complex condensed phase chemical systems. When multidimensional nonlinear optical experiments were first proposed it was expected they could have an impact similar to introducing higher-dimensional techniques into NMR. A major impediment to developing and interpreting these spectroscopies has been the lack of a tractable yet accurate molecularly detailed theory of the spectroscopy and TCF theories serve to fill this void. Lastly, further work is needed to explain the absence of echo features in the molecular liquids studied to date. The research was supported by NSF Grants No. CHE (B. S.) and No. CHE (T. K.), and a grant from ACS/PRF (B. S.). Finally, the authors are grateful for the use of data from earlier work by Richard Stratt and Ao Ma. [1] Y. Tanimura and S. Mukamel, J. Chem. Phys. 99, 9496 (1993). [2] K. Kubarych, C. Milne, S. Lin, V. Astinov, and R. Miller, J. Chem. Phys. 116, 2016 (2002). [3] V. Astinov, K. Kubarych, C. Milne, and R. D. Miller, Chem. Phys. Lett. 327, 334 (2000). [4] O. Golonzka, N. Demirdoven, M. Khalil, and A. Tokmakoff, J. Chem. Phys. 113, 9893 (2000). [5] K. Okumura, A. Tokmakoff, and Y. Tanimura, J. Chem. Phys. 111, 492 (1999). [6] L. J. Kaufman, J. Heo, L. D. Ziegler, and G. R. Fleming, Phys. Rev. Lett. 88, (R) (2002). [7] J. Kim and T. Keyes, Phys. Rev. E 65, (2002). [8] R. L. Murry, J. T. Fourkas, and T. Keyes, J. Chem. Phys. 109, 2814 (1999). [9] T. I. Jansen, K. Duppen, and J. Snijders, Phys. Rev. B 67, (2003). [10] R. A. Denny and D. R. Reichman, Phys. Rev. E 63, (R) (2001). [11] A. Ma and R. M. Stratt, Phys. Rev. Lett. 85, 1004 (2000). [12] A. Ma and R. M. Stratt, J. Chem. Phys. 116, 4962 (2002). [13] S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, Oxford, 1995). [14] J. Cao, S. Yang, and J. Wu, J. Chem. Phys. 116, 3760 (2002). [15] S. Mukamel, V. Khidekel, and V. Chernyak, Phys. Rev. E 53, R1 (1996). [16] R. DeVane, C. Ridley, T. Keyes, and B. Space, J. Chem. Phys. 119, 6073 (2003). [17] S. Saito and I. Ohmine, Phys. Rev. Lett. 88, (2002). [18] R. L. Murry and J. T. Fourkas, J. Chem. Phys. 107, 9726 (1997). [19] N.-H. Ge, M. T. Zanni, and R. M. Hochstrasser, J. Phys. Chem. A 106, 962 (2002). [20] J. Borysow, M. Moraldi, and L. Frommhold, Mol. Phys. 56, 913 (1985). [21] K. Okumura and Y. Tanimura, J. Chem. Phys. 107, 2267 (1997). [22] The INM DOS was calculated drawing 1000 statistically independent configurations of 108 Lennard-Jones particles from microcanonical MD. Using the configurations, the systems force constant matrix was calculated and diagonalized. The imaginary frequencies are shown on the negative abscissa

149 149/224 J. Phys. Chem. B 2004, 108, A Molecular Dynamics Study of Aggregation Phenomena in Aqueous n-propanol Alfred B. Roney and Brian Space* Department of Chemistry, UniVersity of South Florida, Tampa, Florida Edward W. Castner Department of Chemistry and Chemical Biology, Rutgers UniVersity, Piscataway, New Jersey Raeanne L. Napoleon and Preston B. Moore Department of Chemistry and Biochemistry, UniVersity of the Sciences in Philadelphia, Philadelphia, PennsylVania ReceiVed: December 18, 2003; In Final Form: March 1, 2004 Low-frequency Raman studies of various concentrations of aqueous n-propanol at room temperature indicate that both water and n-propanol form single-component aggregates in solution. Small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) studies also provide evidence of this tendency toward aggregation. Molecular dynamics simulations of 16% aqueous n-propanol, a concentration for which maximum segregation of n-propanol and water is observed, have been carried out in an attempt to elucidate the structure of these aggregates. Kirkwood-Buff integrals calculated from the radial distribution functions of the components show excellent agreement with experimentally derived values. Analysis of the atomic coordinates from the simulations reveal that approximately 50% of the n-propanol molecules are members of homogeneous hydrogen-bonded chains of up to 16 members in length, the majority of which are dimers. The g(r) data also indicate that a strong hydrophobic association exists between the hydrocarbon tails. This hydrophobic association is independent of hydrogen-bonding state, and results in the formation of an approximately 10-member micelle structure centered around the n-propanol chains. Water is excluded from the regions occupied by the n-propanol micelles. The water structure is largely unaffected except for a small amount of disruption at the interface between the bulk solvent and the n-propanol clusters, and the formation of small water clusters at the interface with the bulklike solvent that interact with hydroxyl groups at the ends of the propanol chains. 1. Introduction Aqueous solutions of aliphatic alcohols find many uses, from cleaning products and chromatography solvents to beverages and food additives. While much is known about the macroscopic chemical behavior of these solutions, the solvation structure is poorly understood for many aqueous alcohols. Multiple studies indicate that alcohol molecules aggregate in aqueous solution, but studies of the n-propanol aggregation indicated by Raman spectra, 1 small-angle X-ray scattering (SAXS), 2-3 small-angle neutron scattering (SANS), 4 and thermodynamic data 3,5,6 do not provide an atomistically detailed description of the microscopically heterogeneous solutions. The existence of hydrogen-bonded chains of solute molecules has been suggested by experiments on many aliphatic alcohols. 5,7 Significant hydrophobic interactions for alcohols possessing large hydrocarbon functionalities were also indicated. 1,2,7 While many studies confirm that some sort of clustering does occur in n-propanol, only the hydration structures of single molecules and dilute solutions of n-propanol have been reported. 8 The present molecular dynamics (MD) study investigates the structure of n-propanol aggregates in aqueous solutions. A description of the methods that are employed is provided in Section 2. Table 1 presents the nomenclature used throughout the paper to * Author to whom correspondence should be addressed. space@ cas.usf.edu. TABLE 1: Nomenclature. Atom Definitions and Cluster State Prefixes Used in This Paper symbol CA CM OH HO OW Atom Symbol Definitions definition R-carbon of n-propanol methyl carbon of n-propanol hydroxyl oxygen of n-propanol hydroxyl hydrogen of n-propanol oxygen of water Prefix Definitions prefix n-propanol H 2O c member of chain member of bulk (N cluster g 10) u nonchain nonbulk describe the molecular structure of aqueous n-propanol derived from MD simulation. 2. Background and Methodology 2.1. Raman Spectroscopy. Suggestive evidence of aggregation is present in a low-frequency Raman spectroscopy study of alcohol/water solutions. 1 The intensities of the absorption bands attributed to intermolecular O-H O stretching between hydrogen-bonded alcohols were observed to vary in a piecewiselinear fashion with alcohol mole fraction. Additionally, the intensity of the O-H O stretching band for water-water /jp037922j CCC: $ American Chemical Society Published on Web 04/22/2004

150 150/ J. Phys. Chem. B, Vol. 108, No. 22, 2004 Roney et al. hydrogen bonds varied linearly with the water mole fraction. This linear relationship permits the Raman spectra for aqueous solutions of methanol, ethanol, n-propanol, 2-propanol, and tertbutyl alcohol to be represented as simple linear combinations of the pure components spectra between 0 and 300 cm -1 according to the formula R(νj, χ) ) ar(νj,0)+ br(νj, 1) (1) where R(νj, χ) is the Raman signal at νj wavenumbers, χ is the mole fraction of alcohol, and a and b are coefficients. Since both bands of interest are intermolecular stretching modes, their linear variance with concentration indicates that the hydrogenbond structure of both components is largely unaffected in solution. This implies that despite the fact that n-propanol and water are infinitely miscible, the solution is far from homogeneous and suggests that both n-propanol and water form aggregates that mimic, at least spectroscopically, the structure of the respective pure components. For n-propanol, the intermolecular O-H O stretching band of interest is located at 69 cm -1. The coefficients a and b, while regionally linear, exhibit inflection points at 20 and 50% molefraction of n-propanol, in agreement with the respective minimum and maximum of the heat of mixing for n-propanol. These inflection points coincide with a suspected transition from n-propanol clusters in water solvent at χ p < 10%, to separate clusters of n-propanol and water at 20% e χ p e 60%, and finally to water clusters in the n-propanol solvent at χ p > 60%. 1 These distinctions, however, are only qualitative observations without a molecularly detailed explanation. Nonetheless, formation of different solvation structures with concentration is strongly suggested by these observations, and MD is an ideal method to identify the structures involved Aggregate Structures. Much of the existing work on aqueous solutions of low-molecular-weight aliphatic alcohols indicates that the hydroxyl groups from the alcohol molecules form hydrogen-bonded chains, with the alkyl tails extending into the solvent. In this structure, each hydroxyl oxygen donates and accepts one hydrogen from another hydroxyl group, for a total of two hydrogen bonds per molecule of alcohol. 5 X-ray diffraction and mass spectrometry studies 7 indicate that for methanol these chains take two forms, cis- and trans-, based upon the orientation of the alkyl groups. The number of molecules comprising these chains is directly proportional to the concentration of methanol, and the O-O spacing along the chains decreases 4% as the mole fraction varies from χ m ) 0% to χ m ) 100%. Ethanol, 2-propanol, and tert-butyl alcohol also exhibit similar behavior. Several differences in the aggregation behavior of n-propanol compared to other alcohols have been noted. One significant difference is the temperature dependence of the Debye correlation length, an indication of the correlation distance of density fluctuations in a solution. 9 While the correlation length of tertbutyl alcohol exhibits a temperature dependence, the correlation length of n-propanol is independent of temperature. The temperature dependence for tert-butyl alcohol is attributed to the melting of an ice-like cage structure surrounding the tertbutyl group with increasing temperature; therefore, the lack of temperature dependence for n-propanol suggests that no similar hydration structure exists around the n-propyl groups. 10,2 Additionally, no clathrate-hydrate crystal structure of n-propanol has been observed, despite the existence of these structures for 2-propanol and tert-butyl alcohol. This indicates a significant difference in the hydration structure of n-propanol when compared to these similar molecules, and suggests that the alkyl tails of n-propanol disrupt the water structure enough to prevent the formation of a hydrogen-bonded water network surrounding the hydrophobic functionalities. It also suggests that these functional groups may be excluded from the water structure by solute aggregation. Animations of the system were produced from the MD simulations and rendered in order to obtain a qualitative visual idea of the aggregation phenomena. Snapshots of individual configurations were taken highlighting different molecular solvation phenomena including, for example, sorting species by their hydrogen-bonding environment. As an example, Figure 1(a-c) presents a snapshot of a typical n-propanol-water MD simulation with all species 1(b) and the water 1(a) and n-propanol 1(c) removed. Details are presented below Hydrogen-Bonded Neighbor Search. Since hydrogen bonding is the primary method of aggregation indicated by spectroscopic data, a search for hydrogen bonds was performed on each atomic configuration using criteria from the literature. 11 The criteria were implemented as follows. A search for oxygen atom pairs possessing an interatomic radius of Å was performed. Among oxygen pairs identified as possessing the correct separation, the intermolecular hydrogen-oxygen distance (r OH ) was calculated for each hydrogen that was molecularly bonded to a member of the candidate O-O pair. If r OH fell between 1.5 and 2.2 Å, the prospective hydrogen-bond angle formed by the candidate O-H O triad (θ OHO ) was calculated. If θ OHO fell between 130 and 180, the respective molecules were considered to be hydrogen-bonded. From the identification of hydrogen-bonded pairs, coordination numbers were calculated and individual aggregates of molecules were identified. Detailed histograms of chain and cluster sizes were also calculated for various species and subspecies described below Nth-Nearest-Neighbor Histograms. A set of histograms were generated for various pairs of atoms i and j by calculating all radial distances r ij for each molecule of species i and saving them in an array. This array was then sorted into ascending order, and the seven smallest distances were used to increment the corresponding bin of seven separate histograms. The histograms were normalized by dividing the bin counts by the number of species i present, and averaged by dividing each bin by the number of configurations analyzed. This resulted in a separate histogram of the radial distance to each of the first seven neighbors of a given species; this analysis is useful in discerning the variation in solvation structure of a given species. For example, while many water molecules are tetrahedrally bonded, some are shown below to have between 0 and 5 hydrogen-bonded neighbors, and up to 7 neighbors at hydrogenbonding distance. The nearest-neighbor distributions for OW- OW, CA-CA, CM-CM, CA-OW, CM-OW, OH-OH, and OH-OW were calculated Radial Distribution Functions. To make connection with experiment and to further understand the solution structure, various radial distribution functions, g ij (r) were calculated. The radial distribution function is formally defined for isotropic systems by 12 g ij (r) ) F j (r ij ) F id (r ij ) ) F j (r ij ) F j where F j (r ij ) is the average number density of species j in a spherical shell of radius r ij centered at species i, and F id (r ij )is the density of an ideal gas at the same number density F j as molecule j. The radial distribution function is easily calculated from the coordinate output of a molecular dynamics simulation (2)

151 151/224 Aggregation Phenomena in Aqueous n-propanol J. Phys. Chem. B, Vol. 108, No. 22, Figure 1. Snapshots of two representative configurations from the molecular dynamics simulation. Water and n-propanol do not intermingle appreciably, in accordance with experimental evidence. Additionally, hydrogen-bonded chains of n-propanol molecules aggregate with other chains and with free n-propanol through a hydrophobic association of their alkyl groups. by creating a histogram of the radial distances r ij, calculating the average density of species j within the spherical shell defined by the bin location and dimensions, and dividing the resulting density by the ensemble average density of j to give g ij (r) at the location of the center of the bin. The species represented by i and j can be either the center of mass of a molecule, or any atom of interest on a molecule. In this paper they are also chosen to represent the center of mass or a specific atom from the same type of molecule in a specific aggregation state, such as molecules that are members of a hydrogen-bonded chain (as defined by a chosen set of criteria defined below). These functions provide a good description of the short-range order present in liquids. 13 They also allow the estimation of single-component aggregate sizes by integrating over the first-neighbor peak: N i ) 1 + 4π F i 0 r mingii (r)r 2 dr (3) where F i is the average number density of species i. By choosing r min as the location of the first minimum in g ii (r), N i becomes the number of molecules comprising an aggregate. Molecular radial distribution functions were generated for solute-solute (g pp (r)), solute-solvent (g pw (r)), and solventsolvent (g ww (r)) using the center of mass of the molecules as their location. These results were compared to g ij (r) data calculated using the R-carbon coordinates of n-propanol (CA) and the oxygen coordinates of water (OW) as the molecular coordinates due to the proximity of these atoms to the center of mass. Additionally, functional radial distribution functions were calculated, including g OH-OH (r), g OH-HO (r), and g OH-OW (r). As described in Section 4.2, both solute and solvent can be sorted into groups according to their hydrogen-bonding state, resulting in the creation of four species: chain n-propanol, free n-propanol, bulk water, and nonbulk water. Chain n-propanol molecules are defined as n-propanol molecules that are hydrogenbonded to other n-propanol molecules. Free n-propanol molecules are not hydrogen-bonded to any other n-propanol molecules. Bulk water is defined as water clusters consisting of 10 or more members, as determined from the coordination data. All remaining water molecules are considered to be a separate species from the bulk. Functional radial distribution functions were calculated for these four species in order to decompose both the molecular and functional radial distribution functions and illuminate the various intermolecular interactions responsible for the observed aggregation Kirkwood-Buff Integrals. Kirkwood-Buff integrals, 14 G ij, are derived from integrals of g ij (r) and can be considered to be a measure of the average excess or deficit of species j around species i. This makes G ij an excellent measure of the degree of aggregation. They are formally defined by G ij ) 0 [gij (r) - 1]4πr 2 dr (4) Kirkwood-Buff integrals were computed from the radial distribution functions using a trapezoid-rule based integration routine. G pp, G pw, G ww, G CA-CA, G CA-OW, G OW-OW, G OH-OH, and G OH-OW were calculated and compared to experiment. While there is no extant direct experimental measurement of g ij (r) for aqueous alcohols, G ij can be experimentally measured by a variety of methods. Since they are easily calculated from the

152 152/ J. Phys. Chem. B, Vol. 108, No. 22, 2004 Roney et al. TABLE 2: Harmonic Oscillator Potential Parameters. Harmonic Oscillator Potential Parameters Used for the Molecular Dynamics Simulations (V HO (r) ) 1 / 2 k f (r - r eq ) 2 ) atom pair k f(k/å 2 ) r eq(å 2 ) n-propanol C-C C-H C-O O-H water O-H H-H TABLE 3: Torsion Potential Parameters. Torsion Potential Parameters for n-propanol Used for the Molecular Dynamics Simulations, in Kelvins (K) (V torsion (O) ) 1 / 2 k p k cos k (O)) atom sequence p 0 p 1 p 2 p 3 X 1-C-O-X X 1-C-C-X TABLE 4: Lennard-Jones Potential Parameters. Lennard-Jones Potential Parameters Used to Model Intermolecular and One-Four Intramolecular Interactions (V LJ (r) ) 4E[(σ/r) 12 - (σ/r) 6 ]) atom σ (Å) ɛ (K) C H aliphatic O propanol O water g ij (r) data obtained from MD simulations, they are used in this paper as a convenient comparison to experiment and serve to validate the MD models used in the present investigations. 3. Molecular Dynamics Methods 3.1. Computational Details. The MD production runs produced trajectories by solving dynamical equations of motion using a modified form of the velocity-verlet algorithm that takes advantage of multiple time-step integration. 15 Modified OPLS parameters were used. 16,17 All molecules were flexible, and harmonic stretching and bending interactions were used. The harmonic potential parameters are presented in Table 2. For n-propanol, torsions were represented by a power series using the coefficients presented in Table 3. Additionally, one-four intramolecular interactions were represented by Lennard-Jones potentials presented in Table 4. Intermolecular interactions were modeled by Lennard-Jones 6-12 potentials and electrostatic interactions. The Lennard-Jones parameters used to model the intermolecular potentials were identical to the parameters used to model intramolecular onefour interactions. The electrostatic potential surfaces were represented by partial-charges located on specified atoms of each molecule. Long-range electrostatic interactions were calculated using Ewald sums. 18 For n-propanol, the hydroxyl oxygen was assigned a charge of e, the hydroxyl hydrogen was assigned a charge of e, and the R-carbon was assigned a charge of e. The remaining atoms bore no charges. This charge distribution produced a dipole moment of 2.36 D. While substantially larger than the gas-phase dipole moment (1.68 D), this value mimics the increase in the dipole moment observed for polar molecules in the condensed phase, and is proportionally similar to the increase in dipole moment for liquid-phase water. 19 For water, the oxygen atom was assigned a charge of e and the hydrogens were assigned charges of e. These model parameters produced aqueous n- propanol solution densities in good agreement with experimental measurements over a broad range of concentrations. 20 Simulations of 16% mole fraction n-propanol in water were performed using a 2523 atom system. This system consisted of 91 n-propanol molecules and 477 SPCE water molecules, with cubic periodic boundary conditions. A Linux Beowulf cluster composed of 1.8 GHz AMD Athlon nodes performed the computations, using the MPI version of a code developed at the Center for Molecular Modeling at the University of Pennsylvania, which uses reversible integration and extended system techniques. 15,21 The processing load was distributed over two nodes, which resulted in a processing rate of 0.90 s per time step with a 1.0 fs time step. The simulation was carried out in three phases. In the first phase, an initial set of atomic coordinates which evenly distributed the n-propanol and water molecules on a simple cubic lattice was assigned random velocities sampled from a Gaussian distribution, and the velocities were scaled so that the initial temperature was 293 K. The system was allowed to equilibrate in the isothermal-isobaric (NPT) ensemble using extended Lagrangian techniques. 15 Multiple time-scale integration was used, with the long-range intermolecular forces calculated every 1.0 fs, forces due to torsions calculated every 0.5 fs, and other intramolecular forces calculated every fs. Quantities of interest, such as the volume, temperature, pressure, and total (extended system) and component energies, were monitored every 0.01 ps. The external pressure was set to 1.0 atm and the barostat frequency was chosen as 1.0 ps ,22 The second phase began once a stable volume had been achieved through a 100 ps simulation. Volume data were collected for 2.0 ns. From this the average volume was calculated, and the periodic boundaries were adjusted from their instantaneous values to the calculated values. This converted the system to an NVT ensemble with an overall density of 0.91 g/cm 3, which is in reasonable agreement with the accepted value of g/cm Each atom in the system was then assigned a new random velocity sampled from a Gaussian distribution. The velocities were initially scaled to give a temperature of 4.0 K, and the system was allowed to equilibrate for 1.0 ps at this temperature in order to resolve any bad contacts resulting from the sudden change of the periodic boundary length without disrupting the aggregate structure. This second equilibration was necessary since the atomic coordinates were not scaled to fit the new periodic boundaries. A further NVT simulation of 9.0 ps at 293 K then ensured that the system dynamics were stable. In the final production phase atomic coordinates were collected for subsequent analysis. First, 100 sets of 10.0 ps each were collected and used to calculate g ij (r). Using time-dependent plots of g ij (r) from the equilibrating system, it was determined that the radial distribution functions had converged to stable values after 200 ps, suggesting that equilibrium had been achieved. To analyze the structure of the solute clusters, a 2 ns simulation was performed to generate the atomic configurations analyzed using the various methods described above. For comparison, control systems of pure n-propanol and water were simulated using identical potential parameters. For n- propanol, a 256-molecule simulation was used equilibrated in a manner identical to the mixed system. This gave a density of 0.86 g/cm 3, a value within 6.4% of the accepted value of g/cm For water, a 512-molecule system was prepared using the same method, resulting in a density of 0.95 g/cm 3. This value is within 5.2% of the experimental density of pure water ( g/cm 3 ). 19 Both systems were used to generate 2 ns of configurations in the NVT ensemble. These sets of atomic coordinates

153 153/224 Aggregation Phenomena in Aqueous n-propanol J. Phys. Chem. B, Vol. 108, No. 22, TABLE 5: Kirkwood-Buff Integral Results. G ij Values Calculated from the g ij (r) Data Presented in Figures 4 and 5 Compared to Results from Small-Angle Neutron Scattering (SANS) 4 and Small-Angle X-ray Scattering (SAXS) 2 Experiments (Note that experimental agreement deteriorates when the site of hydrogen bonding, i.e., the oxygen atom, is not used as the location of the molecule.) molecule pair i j MD SANS SAXS n-propanol n-propanol OH OH 1044 CA CA 1395 n-propanol H 2O OH OW CA OW H 2O H 2O OW OW 1032 were then analyzed using the same methods as the simulated solution to generate comparison data. G ij 4. Results and Discussion 4.1. Model Validation. Kirkwood-Buff Integral values compared with experimental data are presented as Table 5. While the correct way to calculate the integrals is to calculate the distance and angular dependent distribution function and integrate over the angle variables, this is very computationally demanding and the present approximation was sufficient for comparison purposes. Table 5 demonstrates the good agreement that is obtained when the integrals are performed with the hydroxyl oxygen or R-carbon of the propanol. Using the hydroxyl oxygen, the resulting values differ by only 6% and 4%, respectively, providing strong evidence that the system model is reasonable. The overall agreement of G ij values from the MD simulation with experimental results confirms that the model system is a good representation of the aggregation phenomena observed by experiment. From visual inspection of the system animation it can be observed that n-propanol formed exclusive aggregates in agreement with conclusions from the Raman and scattering studies. 1-4 As shown in Figure 1(a), not displaying the water molecules revealed large open spaces occupied by bulk water, and not displaying the n-propanol molecules in Figure 1(c) revealed that the aggregates were largely anhydrous. Figures 1(d-f) show a snapshot of subsets of n-propanol molecules grouped according to chain membership as defined above. It is apparent that the chains themselves self-aggregate in addition to hydrophobic association with the free n-propanol Hydrogen-Bond Coordination Results. Despite the evidence of component structure preservation in the Raman spectra, the coordination data presented in Table 6 indicate that the hydrogen-bonding structures of both n-propanol and water Figure 2. Histogram of the average number of n-propanol chains or water clusters of a given size (number of molecules) observed in an average configuration. are signifigantly affected upon mixing. While only slightly more than 1% of the alcohol molecules do not form hydrogen bonds with other alcohol molecules in the neat simulations, more than half of the alcohol molecules are free in the mixture. Furthermore, the occurrence of triple-coordinated molecules in the neat simulation is nearly seven times the occurrence of triplecoordination in the presence of water. The structure of pure water is similarly disrupted in the presence of n-propanol. Almost five times the number of free water molecules were found in the solution simulation when compared to the control system, and an overall reduction in hydrogen bonding was observed compared to the pure water simulation. The hydrogen-bonding data reveal much about the types of aggregation phenomena present. On average, approximately 46 n-propanol molecules out of 91 (51%) do not form hydrogen bonds with other n-propanol, approximately 32 molecules are singly coordinated with other n-propanol, and approximately 12 n-propanol are doubly coordinated. On average, one triple- TABLE 6: Hydrogen-Bond Coordination Numbers. Percentage of the Central Molecule Having the Specified Number of Hydrogen Bonds with the Coordinating Molecule a molecule pair coordination number central coordinated n-propanol n-propanol 51.26% 35.06% 13.14% 0.55% 0.00% 0.00% n-propanol water 16.67% 34.60% 35.75% 12.79% 0.19% 0.00% water n-propanol 76.83% 19.12% 3.58% 0.45% 0.02% 0.00% water water 1.90% 11.75% 30.06% 36.17% 19.09% 1.03% pure n-propanol 1.07% 13.50% 81.76% 3.66% 0.001% 0.00% pure water 0.41% 5.65% 23.58% 40.39% 28.17% 1.78% a For example, 12.79% of the n-propanol molecules are hydrogen-bonded to three water molecules, while 0.45% of the water molecules are hydrogen-bonded to three n-propanol molecules. Included for comparison in the bottom two rows are the self-coordination data for simulated systems of n-propanol and water, respectively. Insignificant occurences of coordination numbers greater than 5 are not included in this table.

154 154/ J. Phys. Chem. B, Vol. 108, No. 22, 2004 Roney et al. Figure 3. Snapshots of hydrogen-bonded chains of n-propanol representing the various chain sizes present. A branch head is an n-propanol molecule that accepts two hydrogen bonds from n-propanol and donates one to a third n-propanol, resulting in a fork in the chain. Note that the alkyl tails form a large hydrophobic volume that shields the chain backbone from exposure to the solvent. coordinated n-propanol molecule was observed in half of the configurations, resulting in a branched chain. Since the singly coordinated n-propanol molecules terminate the chains, ignoring the branched chains indicates that an average configuration contains 16 chains. If it is assumed that no chains larger than three members in size exist, the coordination data imply that at least four chains (25%) are dimers. If we assume that only one chain with more than two members exists, then a maximum of 94% of the chains can be dimers. The coordination histogram of n-propanol in Figure 2 confirms that in fact more than half of the chain aggregates are dimers, and that chains of up to four n-propanol appear in every atomic configuration sampled. Chains of up to sixteen members were observed, albeit infrequently. Snapshots of several chains taken from various sets of atomic coordinates are included as Figure 3. In these chains, the alkyl tails of n-propanol turn away from the chain backbone of O-H O hydrogen bonds, presumably due to steric effects. In this manner, the larger chains create a structure similar to a highly branched hydrocarbon molecule, which results in a large hydrophobic surface area for the chain. This structure also shields the hydrogen bonds from interaction with the bulk solvent, which may be responsible for the linear independence of the O-H stretching bands for n-propanol and water. 1 The coordination data presented in Table 6 also suggest that the aggregate structure is micellar in nature. Despite the existence of large single-component regions in the liquid, substantial hydrogen bonding between n-propanol and water was observed. Since only 17% of the n-propanol molecules were not hydrogen-bonded to water while slightly less than half of all n-propanol molecules were members of chains, it can be deduced that water forms hydrogen bonds to the O-H O backbone of the chains. If we assert that alcohols participate in a maximum of two hydrogen bonds, then one can imply that these hydrogen bonds occur only at the terminal hydroxyl groups of the chains. 5 This long-held theory is contradicted, however, by the observance of a substantial percentage of n-propanol molecules possessing 3 hydrogen bonds to water. Due to the water-exclusion observed in the system visualizations, it can be assumed for this system that the chain-water interactions occur primarily at the ends of the chains. Further evidence of this hypothesis is presented in Section 4.4. The aggregate structure disrupts the structure of bulk water at the interface to some degree, as determined from data presented in Table 6. Here we find that, on average, 1.03% of water molecules are coordinated with five other waters, 1.90% of water molecules are free of any hydrogen bonds to other water molecules, and most water molecules are forming twofour hydrogen bonds, consistent with a disrupted tetrahedral network reminiscent of neat liquid water. An interesting occurrence is the existence of a significant number of water molecules (3.58%) hydrogen-bonded to two n-propanol molecules, suggesting the existence of composite chains. Given the number of n-propanol molecules doubly coordinated with water, at least one n-propanol-water chain exists. There are possibly up to 15 (n-propanol) 2 (H 2 O) trimers in an average configuration, with the remaining doubly coordinated waters forming bridges between an n-propanol chain and a free n-propanol or chain. This is a possible mechanism to explain the proposed shift to

155 155/224 Aggregation Phenomena in Aqueous n-propanol J. Phys. Chem. B, Vol. 108, No. 22, Figure 4. Plots of g ij(r) vs distance, r, for n-propanol-n-propanol (g pp(r)), n-propanol-water (g pw(r)), and water-water (g ww(r)). Lines labeled with the subscripts p and w used the center of mass of the respective molecules as the location of the molecule. All other lines used the subscripted atom as the location of the molecule. Concentrations are given as mole fraction of n-propanol. Note that for 16% n-propanol, g pp(r) is nearly identical to g pp(r) for 100% n-propanol. Figure 5. Comparison of g OH-OH(r) with g OH-OW(r) and g OH-HO(r). The alternating peaks of g OH-OH(r) and g OH-HO(r) for the 16% n-propanol solution are indicative of hydrogen-bonded chains. This alternating structure is clearly observable in the neat n-propanol data. Also note the similarity of the peak structures of g OH-OH(r) and g OH-OW(r) for the 16% n-propanol solution.

156 156/ J. Phys. Chem. B, Vol. 108, No. 22, 2004 Roney et al. water clusters in bulk n-propanol for concentrations above χ p ) 0.6. The occasional observance of coordination with three or more n-propanol molecules corroborates this idea, which will be the subject of future research. Figure 2 presents a histogram of the sizes of small water aggregates observed. In addition to the one large cluster of bulk water observed in every configuration (not shown), small clusters of up to 23 water molecules were observed. The majority of these small clusters are dimers, and Figure 9 demonstrates that they occur primarily at the interface between n-propanol aggregates and bulk water. This figure also indicates that, surprisingly, the free waters exist primarily in the regions occupied by the bulk solvent. Small clusters of 10 or more waters appear so infrequently that they were considered bulk water for the cluster-state-dependent g(r) data discussed in Section Molecular Distributions. The molecular and functional g(r) plots in Figures 4 and 5 provide further characterization of the types and extent of aggregation phenomena. In the n-propanol-n-propanol case, g pp (r) from Figure 4 is nearly identical to g pp (r) for neat n-propanol. This is strong evidence that the structure of neat n-propanol is preserved in aqueous solution. The first peak of g pp (r) for the aqueous solution, when integrated using eq 3, indicates that the average n-propanol cluster is comprised of 10 members, much larger than any commonly observed chain in the simulation. This discrepancy can be attributed to hydrophobic association. Despite the proximity of CA to the center of mass, g CA-CA (r) differs significantly from g pp (r), revealing that the first peak in g pp (r) actually is the average of at least two types of firstneighbor interactions for the aqueous solution. The sharp first peak at 4.5 Å can be thought of as a chain signature, and the broad shoulder from 5 to 6.5 Å can be attributed to the hydrophobic association of alkyl tails from neighboring n- propanol molecules. This structural feature of g CA-CA (r) also exists in the data for the pure n-propanol system, although the magnitudes of the individual features are modulated. The similar intensities of the first peak result from the higher extent of hydrogen bonding in the neat system. The elevated hydrophobic shoulder on the plot for the solution indicates that the alkyl tails are localized, resulting in the increased ratio of their density at this radius compared to the overall density of n-propanol in the solution. The similarities in the line shapes of the two systems suggest that the aggregates resemble regions of pure n-propanol surrounded by pure water. Figure 5 provides evidence of both chain formation and hydrophobic aggregation of multiple chains. The location of the first-neighbor peak of g OH-OH (r) slightly off center between the first and second peaks of g OH-HO (r) is an unmistakable signature of the O-H O atom sequence of the chains, as is the similarity in the peak spacing between g OH-OH (r) and g OH-OW (r). The existence of slight third- and fourth-neighbor peaks confirms the persistence of chains having at least five members. Identical short-range features exist in the control system data, but the absolute magnitudes of the data for the solution are slightly elevated above the controls due to aggregation. One important difference in the two systems is the broad rise from 6 to 14 Å in g OH-OH (r) for the solution. This feature results from the hydrophobic association of neighboring chains through their alkyl tails. This attribution will be discussed further in Section 4.4. The size and spacing of the aggregates can be discerned by comparing g pw (r) with g pp (r). The large trough from 4.5 to 7.65 Å results from water exclusion inside the n-propanol aggregates, Figure 6. Distributions of the radial distance to the first neighbor for CA-CA, CM-CM, CA-OW, and CM-OW. Note the near unimodality of the distributions. Figure 7. Distributions of the radial distance to the first neighbor for OH-OH and OH-OW. Note the bimodality of the distributions. and also indicates that the aggregates exhibit an average radius of 7.65 Å. Its coincidence with the first peak of g pp (r) shows that the first peak in g pp (r) is indeed representative of aggregation, and that the clusters are spaced an average of 9.6 Å apart. Further evidence of two types of interactions between n-propanol can be observed by comparing g pw (r) with g CA-OW (r). The first-neighbor peak of g pw (r) exhibits bimodality, with a slight peak at 3.5 Å growing from the side of a larger peak at 4.2 Å. The first peak coincides with the first-neighbor peak of g CA-OW (r), indicating that two mechanisms of interaction with water exist for n-propanol. Furthermore, the mechanism that results in a closer association to the center of mass does not occur very often, suggesting that one of the species defined by this interaction is shielded from bulk water. Furthermore, while g CA-OW (r) shows evidence of exposure to bulk water in its regularly spaced humps and troughs, the center-of-mass based g pw (r) lacks this pattern. This is compelling evidence for the existence of a micelle structure. The data presented in Figures 6 and 7 justify the division of n-propanol into two species on the basis of the hydrogenbonding state. Figure 6 shows the first-neighbor distance distributions for the hydrophobic section of the n-propanol molecule, and their unimodality reveals that their mechanism of association is independent of chain state. Furthermore, the broadness of the CA-CA and CM-CM peaks corresponds to the amorphous nature of these nonpolar interactions. The coincidence of the maximum of CA-OW and CM-OW at 3.5 Å indicates that their interaction with the bulk is parallel to the interface, suggesting a cylindrical micelle structure centered around the chain backbone. This is corroborated by the long tail of the CM-OW curve, which indicates that a significant portion of the methyl groups are buried inside the aggregates.

157 157/224 Aggregation Phenomena in Aqueous n-propanol J. Phys. Chem. B, Vol. 108, No. 22, Figure 8. Distributions of the radial distance between water oxygens for the first seven neighbors. Note that up to seven water molecules are found in the first coordination sphere, despite the fact that no more than 5 hydrogen bonds were observed for a single water molecule in the 16% n-propanol solution. Also note that the plateau on the firstneighbor distribution is similar to the first-neighbor distribution for OH-OW. Also worthy of mention is the lack of a shoulder from 5 to 6.5 Å on the CA-CA curve, which suggests that the R-carbons aggregate independently of hydrogen bonding, despite their proximity to the hydroxyl group. Since the chain structure for n-propanol places the R-carbons in such close proximity, it is surprising that two peaks representing the chain and free species were not observed in the CA-CA first-neighbor histogram. This supports the idea that hydrophobic aggregation is independent of hydrogen bonding, and also provides further evidence of composite chains. The histogram of OH-OH distances from Figure 7 indicates that the hydroxyl groups of n-propanol molecules are separated by 2.75 Å in the chains and by 4.6 Å for free n-propanol. The spacing of the free n-propanol from this graph coincides with the second-neighbor peak from g OH-OH (r) from Figure 5. When considered in light of the coordination data and the lack of a second peak in the first-neighbor distribution for CA-CA, this indicates that the contribution of composite n-propanol-water chains is significant. These composite chains are interpreted as comprising the outer layer of a two-species micelle, with the hydroxyl groups from the composite chains forming hydrogen bonds with the bulk water, and suspending the hydrophobic chain structures in the solvent through the interaction of their propyl groups with those of the chain members. Significant water structure disruption can be determined by the existence of up to seven water neighbors at hydrogen-bond distance for water, as demonstrated by coincident peaks at 2.75 Å in all the distributions presented in Figure 8. The firstneighbor distribution also indicates that water can be divided into two species on the basis of cluster size. While most free water exists either in the bulk solvent or at the solventaggregate interface, as observed in Figure 9, the hump in the first-neighbor distribution for OW-OW indicates that occasionally a water molecule is pulled away from the interface by an n-propanol molecule. Cluster-state-dependent g OH-OW (r) data confirm that interfacial extraction does occur for both single water molecules as well as for small clusters, which is discussed further in Section Cluster-State-Dependent Radial Distribution Functions. From the analysis of data presented and discussed above, it was determined that n-propanol formed a chain-centered micelle structure with a layer of disrupted water at the interface. Radial distribution functions were generated which treated chain and free n-propanol as separate species, and used the small water Figure 9. Snapshots of a representative configuration from the molecular dynamics simulation are shown that display the nature of solvation for a single aqueous n-propanol cluster with the surrounding n-propanol molecules hidden for visualization. The solvating water is characterized as free (not hydrogen-bonded to other water molecules), clustered, and bulklike. Note that the clustered and free water molecules form a curved interface between the bulk water region and the region occupied by the n-propanol aggregates.

158 158/ J. Phys. Chem. B, Vol. 108, No. 22, 2004 Roney et al. Figure 10. Cluster-state-dependent g(r) plots showing the interactions of the hydrophilic end of n-propanol with bulk water. Note that g uoh-cow(r) is quite similar in structure to g OW-OW(r) for neat water from Figure 4. Figure 11. Cluster-state-dependent g OH-uOW(r). Note that only g coh-uow(r) has a significant second-neighbor peak for nonbulk waters, indicating that the chain ends are more likely to extract small clusters from the solvent-micelle interface. clusters as an approximate way to divide water into bulk solvent and disrupted water in order to probe the water structure disruption at the n-propanol aggregate-bulk water interface. In Figure 10, the first-neighbor peak intensity ratios for both g uoh-cow (r)/g coh-cow (r) and g uca-cow (r)/g cca-cow (r) are 3: 1. This reveals that the free n-propanol molecules are three times more likely than chain n-propanol to be hydrogen-bonded to bulk water. This strengthens the idea that free n-propanol molecules form a surrounding layer that shields the chains from exposure to bulk water. The broadening of the second peak in g uoh-cow (r) may indicate that water structure disruption does occur at the aggregate interface, but that this disruption does not usually result in the complete extraction of a small solvent cluster. Despite the tendency of the n-propanol chains to be surrounded by free n-propanol, the signature of the water structure in g coh-cow (r) indicates that they are exposed to the bulk solvent to a significant degree, and that their exposure does not result in the same type of structure disruption that is observed for the free n-propanol. This also agrees with the idea of an oblong spherical micelle structure, since it indicates that the chain ends are not always shielded from the bulk. Figure 11 provides evidence that the extraction of small water clusters occurs primarily at the interface between the ends of the chains and the bulk solvent. The existence of a secondneighbor peak in g coh-uow (r) that is hardly discernible in g uoh-uow (r) indicates that the free n-propanol is less likely to pull a small cluster into the aggregate region. Free n-propanol is more likely to disrupt the water structure, however, as indicated by the lack of broadening of the second-neighbor peak in g coh-cow (r). The tendency of the free n-propanol to turn alkyl groups away from the bulk solvent and toward other alkyl groups is described by Figure 12. The 3:2 ratio of intensities for the slight firstneighbor peaks of g ucm-cow (r)/ g ccm-cow (r) supports the idea of a micelle structure since it indicates that free n-propanol molecules are exposed to a higher number of water molecules. Both g ccm-cow (r) and g ucm-cow (r) show a marked deficit in the density of water in close coordination with the hydrophobic portion of n-propanol. They also lack the 2.75 Å peak spacing signature associated with exposure to the solvent. The lack of a sharp first-neighbor peak for both g ccm-cow (r) and g ucm-cow (r) indicates that the small amount of close association of the methyl group and water occurs without a well-defined structure. Both functions also exhibit a region of depressed intensity extending to 7.5 Å which coicides with the water-exclusion trough observed in g pw (r). The independence of the hydrophobic association in cluster formation can also be seen in Figure 12. An overall similarity in the structure of CM-CM interactions, as indicated by coincident peaks at 4 Å, 5.25 Å, and 8-9 Å, as well as a similar minimum at 7 Å, indicates that the alkyl tail interaction results in the same type of hydrophobic association regardless of chain state. The overall elevation of intensity of g ccm-ccm (r) can be attributed to the large hydrophobic surface area presented by the chains, which results in a greater tendency to aggregate, but its similarity in structure to g ccm-ucm (r) and g ucm-ucm (r) asserts that the hydrophobic interaction mechanism is unaffected by hydrogen bonding at the hydroxyl group. The chain-chain hydrophobic aggregation interpreted from g OH-OH (r) is confirmed by g coh-coh (r) from Figure 13. Distinct neighbor peaks can be observed out to the fourth neighbor, indicating five-member chains. The third- and fourth-neighbor peaks lie on top of a broad region of increased intensity from 6 to 13.5 Å which coincides with the second peak in g pp (r). This chain-chain aggregation results from the hydrophobic association of the alkyl tails, which are turned out and away from the hydrogen-bond backbone. In this manner, the chains aggregate with themselves as if each chain were a highly branched hydrocarbon, since the hydrophilic groups are only exposed to the bulk solvent at the ends of the chains, and are possibly further shielded from bulk water by micellar n-propanol molecules and water-n-propanol composite chains attached to the chain terminus. Further evidence of the existence of the composite chains can be found in Figure 13. The regular peak structure of g OH-OH (r) appears in all three g(r) plots. Since the chain structure is defined by hydrogen bonding, g coh-coh (r) exhibits a very strong peak at 2.75 Å, and the lack of intensity for

159 159/224 Aggregation Phenomena in Aqueous n-propanol J. Phys. Chem. B, Vol. 108, No. 22, Figure 12. Cluster-state-dependent g(r) plots showing the interactions of the hydrophobic end of n-propanol. The similarities in the line shapes indicate that the alkyl tail interactions are independent of the hydrogen-bonding status of n-propanol. composite chains tend to be excluded from the water structure much like a hydrocarbon, and are suspended in the solution by the micellar n-propanol molecules. These micellar n-propanol molecules form an interface between the aggregates and the solvent by donating hydrogen bonds to the water structure, and turning their alkyl groups toward the aliphatic shell surrounding the chain backbones. 5. Conclusion and Future Direction Figure 13. Cluster-state-dependent g OH-OH(r). Note that the firstneighbor peaks for interactions involving free n-propanol coincide with the second-neighbor peak for the n-propanol chains, neglecting the remnants of chain formation observed between 0 and 3.5 Å. This is strong evidence for the existence of composite chains of the form (npropanol)-(h 2O) x-(n-propanol). g coh-uoh (r) and g uoh-uoh (r) in the region between 2.5 and 3.25 Å indicates that unless n-propanol molecules are hydrogenbonded together, their hydroxyl groups do not associate closely. Free n-propanol instead tends to associate with both free and chain n-propanol at a spacing of 5 Å, which coincides with the second-neighbor peaks of g OH-OH (r), g OH-OW (r), g OW-OW (r), and g coh-coh (r). This indicates that both species participate in composite chains, where n-propanol molecules are displaced from the chain backbone and substituted by a water molecule. The existence of first- and second-neighbor peaks at 5 Å and 7 Å, repectively, in g uoh-uoh (r) indicates that both (n-propanol)- (H 2 O)-(n-propanol) and (n-propanol)-(h 2 O) 2 -(n-propanol) structures form, with the single-water structure preferred. Similar structures also join chains and composite chains, as determined by the similar peak structures of g coh-uoh (r). Furthermore, these chain-composite chain structures participate in chain-chain hydrophobic aggregation, as the general rise in g coh-uoh (r) from 6.5 to Å demonstrates. Up to this point, little has been said about the micelle structure other than the observation that the methyl groups tend to bury themselves in an oblong spherical aggregate structure. The micelles appear to be largely amorphous, with no well-defined structures observed other than the pure and composite chains. Specifically, g ucm-ccm (r) from Figure 12 exhibits the same firstneighbor peak location as both g ccm-ccm (r) and g ucm-ucm (r), which rules out the existence of micelle structures where the free n-propanol molecules insert into the spaces between the alternating pendant alkyl groups of the chains. The chains and The analysis of the molecular dynamics indicates that at 16% mole fraction, n-propanol forms an amorphous micelle centered around a hydrogen-bonded chain structure resembling the pure n-propanol structure. These structures consist of both pure n-propanol chains, as well as chains that include water molecules in the hydrogen-bond backbone. These chains turn their pendant alkyl groups out and away from the backbone, and the resulting structure resembles a highly branched aliphatic hydrocarbon with hydrophilic sites at the ends of the chains. The nonpolar aliphatic regions of the clusters then aggregate together in a manner similar to branched-chain hydrocarbons in water, and are suspended in the polar solvent by n-propanol molecules which turn their alkyl tails inward toward the chains and form hydrogen bonds with the solvent, disrupting the water structure out to the second hydration sphere, as determined by peak broadening in g OH-OW (r). This structure is best described as small droplets of neat n-propanol emulsified by the free n-propanol molecules. A simplified schematic representation of the aggregate structure is included as Figure 14. The current method of sorting water into bulk solvent and interfacial water needs refinement. Since the present method uses only the hydrogen-bonding data to sort the water molecules, a highly disrupted cluster of water needs only one hydrogen bond to the bulk solvent to be considered a part of the bulk. A more comprehensive method of water structure analysis should be implemented. One possibility is the use of order parameters such as those used by Fidler and Rodger, 8 which take into account the near tetrahedral bonding angles of water. The ability of the chain terminal hydroxyl groups to extract small clusters of water from the bulk solvent is significant in that it provides a mechanism for the transition from separate clusters of n-propanol and water at χ p e 60% to water clusters in n-propanol solvent at χ p > 60%. It is suspected that as the concentration of n-propanol is increased, both the number and size of the small water clusters will increase, and these clusters will be cut off from the each other by surrounding n-propanol at the highest concentrations. Additionally, as the mole fraction

160 160/ J. Phys. Chem. B, Vol. 108, No. 22, 2004 Roney et al. Figure 14. Simplified two-dimensional diagram of the types of structures observed in the solution. Oxygen atoms are highlighted in red. Molecular bonds are indicated by solid black lines, and hydrogen bonds are represented by dotted lines. The alkyl hydrogens are not shown. Starting from the top and proceeding counterclockwise, the hydrogen-bonded chains which dominate the structure of pure n-propanol are shortened and terminated by hydrogen bonds to water molecules. Adjacent chains form composite chains via bridging water molecules, which can extract and isolate small water clusters from the solute-solvent interface. Due to the large hydrophobic volume created by the pendant propyl groups (see Figure 3), individual and composite chains tend to aggregate to minimize solvent exposure. The outer layers of free n-propanol (not shown) form micelles centered around the structures presented here. is reduced to χ p < 10%, the transition to n-propanol clusters in bulk solvent should also be observed by a reduction in the size and number of small water clusters formed. These hypotheses are being investigated in ongoing studies, and the results will be reported in future publications. Another topic of interest is the percentage of structure disruption for both n-propanol and water, as investigated by Raman spectroscopy. 1 Simulations of the low-frequency Raman spectroscopy will be performed for various mole fractions of n-propanol, in an attempt to replicate the linear combination of spectra observed by experiment. Additionally, the structure disruption observed in this study should result in an increase in the amount of free O-H stretching signal present in the IR spectrum for water, and should produce peak components that are similar to the IR spectrum of either gas-phase or interfacial water. This is currently an active area of research, and the results will be presented in future publications. Acknowledgment. The authors acknowledge the use of the services provided by the Research Oriented Computing Center, University of South Florida. The research at USF was supported by an NSF Grant (No. CHE ) and a grant from the Petroleum Research Foundation to Brian Space. References and Notes (1) Yoshida, K.; Yamaguchi, T. Low-frequency Raman spectroscopy of aqueous solutions of aliphatic alcohols. Z. Naturforsch. 2001, 56 a, (2) Hayashi, H.; Nishikawa, K.; Iijima, T. Small-angle X-ray scattering study of fluctuations in 1-propanol-water and 2-propanol-water systems. J. Phys. Chem. 1990, 94, (3) Shulgin, I.; Ruckenstein, E. Kirkwood-Buff integrals in aqueous alcohol systems: Comparison between thermodynamic calculations and X-ray scattering experiments. J. Phys. Chem. B 1999, 103, (4) Almásy, L.; Jancsó, G.; Cser, L. Application of SANS to the determination of Kirkwood-Buff integrals in liquid mixtures. Appl. Phys. A, 74[Suppl.] 2002, S1376-S1378. (5) Franks, F.; Ives, D. J. G. The structural properties of alcoholwater mixtures. Q. ReV. Chem. Soc. 1966, 20, (6) Matteoli, E.; Lepori, L. Solute-solute interactions in water. ii. An analysis through the Kirkwood-Buff integrals for 14 organic solutes. J. Chem. Phys. 1984, 80 (6), (7) Takamuku, T.; Yamaguchi, T.; Asato, M.; Matsumoto, M.; Nishi, N. Structure of clusters in methanol-water binary solutions studied by mass spectrometry and X-ray diffraction. Z. Naturforsch. 2000, 55 a, (8) Fidler, J.; Rodger, P. M. Solvation structure around aqueous alcohols. J. Phys. Chem. B 1999, 103, (9) Debye, P. Angular dissymmetry of the critical opalescence in liquid mixtures. J. Chem. Phys. 1959, 31, (10) Frank, H. S.; Evans, M. W. Free volume and entropy in condensed systems. iii. Entropy in binary liquid mixtures; partial molal entropy in dilute solutions; structure and thermodynamics in aqueous electrolytes. J. Chem. Phys. 1945, 13,

161 161/224 Aggregation Phenomena in Aqueous n-propanol J. Phys. Chem. B, Vol. 108, No. 22, (11) Jeffrey, G. A. An Introduction to Hydrogen Bonding; Oxford: New York, (12) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, (13) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, (14) Kirkwood, J. G.; Buff, F. P. The statistical mechanical theory of solutions. I. J. Chem. Phys. 1951, 19, (15) Tuckerman, M.; Berne, B. J.; Martyna, G. J. Reversible multiple time scale molecular dynamics. J. Chem. Phys. 1992, 97, (16) Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J. Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids. J. Am. Chem. Soc. 1996, 118, (17) Cornell, W. D.; Cieplak, P.; Bayly, C. I.; Gould, I. R.; Merz, K. M.; Ferguson, D. M.; Spellmeyer, D. C.; Fox, T.; Caldwell, J. W.; Kollman, P. A. A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J. Am. Chem. Soc. 1995, 117 (19), (18) Ewald, P. Die berechnung optischer und elektrostatischer gitterpotentiale. Ann. Phys. 1921, 64, (19) Lide, D. R. Handbook of Chemistry and Physics, 84th ed.; CRC Press LLC, New York, (20) Roney, A. B.; Space, B.; Moore, P. B.; Castner, E. W. A molecular dynamics study of the concentration dependence of aggregation phenomena in aqueous n-propanol. To be submitted. (21) Moore, P. B.; Zhong, Q.; Husslein, T.; Klein, M. L. Simulation of the hiv-1 vpu transmembrane domain as a pentameric bundle. FEBS Lett. 1998, 431, (22) Paci, E.; Marchi, M. Constant-pressure molecular dynamics techniques applied to complex molecular systems and solvated proteins. J. Phys. Chem. 1996, 100,

162 JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 8 22 AUGUST 2004 A time correlation function theory of two-dimensional infrared spectroscopy with applications to liquid water Russell DeVane, Brian Space, a) Angela Perry, Christine Neipert, and Christina Ridley Department of Chemistry, University of South Florida, SCA400 Tampa, Florida T. Keyes Department of Chemistry, Boston University, Boston, Massachusetts Received 12 March 2004; accepted 2 June 2004 A theory describing the third-order response function R (3) (t 1,t 2,t 3 ), which is associated with two-dimensional infrared 2DIR spectroscopy, has been developed. R (3) can be written as sums and differences of four distinct quantum mechanical dipole multi time correlation functions TCF s, each with the same classical limit; the combination of TCF s has a leading contribution of order 3 and thus there is no obvious classical limit that can be written in terms of a TCF. In order to calculate the response function in a form amenable to classical mechanical simulation techniques, it is rewritten approximately in terms of a single classical TCF, B R (t 1,t 2,t 3 ) j (t 2 t 1 ) i (t 3 t 2 t 1 ) k (t 1 ) (0), where the subscripts denote the Cartesian dipole directions. The response function is then given, in the frequency domain, as the Fourier transform of a classical TCF multiplied by frequency factors. This classical expression can then further be quantum corrected to approximate the true response function, although for low frequency spectroscopy no correction is needed. In the classical limit, R (3) becomes the sum of multidimensional time derivatives of B R (t 1,t 2,t 3 ). To construct the theory, the response function s four TCF s are rewritten in terms of a single TCF: first, two TCF s are eliminated from R (3) using frequency domain detailed balance relationships, and next, two more are removed by relating the remaining TCF s to each other within a harmonic oscillator approximation; the theory invokes a harmonic approximation only in relating the TCF s and applications of theory involve fully anharmonic, atomistically detailed molecular dynamics MD. Writing the response function as a single TCF thus yields a form amenable to calculation using classical MD methods along with a suitable spectroscopic model. To demonstrate the theory, the response function is obtained for liquid water with emphasis on the OH stretching portion of the spectrum. This approach to evaluating R (3) can easily be applied to chemically interesting systems currently being explored experimentally by 2DIR and to help understand the information content of the emerging multidimensional spectroscopy American Institute of Physics. DOI: / I. INTRODUCTION Revealing the three-dimensional molecular structure of transient species in condensed phases is experimentally challenging, but necessary to understand the nature of timedependent processes, such as protein folding. The congested spectra produced under these conditions often cannot be resolved using linear infrared IR spectroscopy or other traditional techniques. X-ray diffraction and multidimensional nuclear magnetic resonance NMR effectively reveal timeaveraged three-dimensional structures, but fail to accurately describe short-lived species, such as peptides in solution. Two-dimensional infrared 2DIR spectroscopy, recently the subject of extensive theoretical and experimental study, has shown promise for providing new information about these time evolving structures. 1 5 While multidimensional NMR only captures processes on a millisecond time scale, 2DIR a Author to whom correspondence should be addressed. Electronic mail: space@cas.usf.edu can approach a time scale on the order of picoseconds or even femtoseconds Recently, 2DIR techniques have been used to investigate the coupled carbonyl stretches of Rh CO) 2 (C 5 H 7 O 2 ), 7,11 14 the nuclear potential energy surface of coupled molecular vibrations, 15 vibrational relaxation, 16,17 interactions between solvent and solute, conformational fluctuations in peptides, 6,9,10,21 the threedimensional structure of peptides and small proteins, 8,22 28 and the coupling of cytidine and guanosine in DNA. 29 Using 2DIR, couplings and projection angles between two coupled anharmonic vibrations can also be inferred. 15 2DIR uses a sequence of four time ordered femtosecond length IR pulses to create vibrational coherences, couple them, and detect the final polarization state of the system being investigated. In the experiment, a third-order nonlinear polarization is thus generated and subsequently detected. This polarization P (3) i at time t is simply the convolution of the third-order response function R (3) with the three input electric fields: /2004/121(8)/3688/14/$ American Institute of Physics 162/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

163 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 Two-dimensional infrared spectroscopy 3689 P 3 i 1, 2,t Rijk t 1,t 2,t 3 E 3 j t t 3 0 E 2k t 2 t 3 t 2 E 1 t 1 2 t 3 t 2 t 1 dt 1 dt 2 dt 3. 1 The E represent the pulsed laser fields and t is the time elapsed after the final laser pulse. The variable 1 is the time delay between the first and second pulses and 2 is the delay between the second and third pulses. In the vibrational echo 2DIR experiment ,33 Also note that in the idealized limit of function pulses the polarization becomes equivalent to the response function in the time domain, which becomes the experimental observable. Because the waiting times are by definition positive, in the frequency domain the Fourier-Laplace transform of the response function is often reported, giving a real and an imaginary part that contain both the Fourier transform of the response function and principal part integrals over the frequency domain response function. With the theoretical expressions to be developed we have no difficulty taking the full Fourier transform, which contains all time-domain information. The numerical results on water presented below will focus on the case where 2 t 2 0 although a more general theory is developed. The t n are the time intervals between the field-matter interactions, and equal the n if the pulse lengths are substantially shorter than the time scale of the dynamics. The third-order response function R (3) is a fourth rank tensor and depends on the polarization components i, j, k, XYZ of the incident laser fields. For a resonant 2DIR experiment, R (3) is given by 13 R 3 ijk t 1,t 2,t 3 i 3 i t 3 t 2 t 1, j t 2 t 1, k t 1, 0. 2 In Eq. 2, the represent the dipole moment operators in the subscripted laboratory Cartesian direction and the time-dependent dipole is the Heisenberg representation of the operator, (t) e iht/ e iht/. The square and the angle brackets represent commutators and quantum mechanical averages, respectively, in a standard notation. 34 Evaluation of the R (3) expression is computationally demanding because it requires the evaluation of several quantum mechanical TCF s that result from expanding the commutators. Even in the classical limit, the commutators become Poisson brackets of the dynamic variables at various times that are prohibitively difficult to evaluate computationally. 35,36 Theoretical methods that approximately relate quantum mechanical and classical TCF s have successfully been applied to one-dimensional spectroscopies that have an exact classical limit in terms of time derivatives of a classical TCF and to describing high frequency spectroscopy using quantum correction schemes to approximate the relevant quantum mechanical TCF. 31,37 45 Recently, we have developed a TCF theory of the fifth-order Raman response function and associated two-dimensional Raman spectroscopy technique 46 that produced results in excellent agreement with numerically exact molecular dynamics MD simulations 35,46 and experiments. 46,47,48 A similar method will prove useful here to simplify the R (3) expression and obtain an approximate TCF theory of the third-order response function and 2DIR spectroscopy that is amenable to evaluation using classical MD and TCF techniques. The R (3) expression contains a trace of nested commutators, which must be imaginary and have a leading contribution of order 3 to cancel the prefactor of i/ 3 in the classical limit; this is a useful way to examine the expression because a theory in terms of a single TCF must have a classical limit of order 0. It should be noted that we are interested in a result that is appropriate for high frequencies, and examining the low frequency expansion is used primarily as analytic tool in seeking an expression in terms of a single TCF, although low frequency 2DIR spectroscopy is also possible. Expansion of the commutators reveals four distinct quantum mechanical time correlation functions. In Sec. II, it will be demonstrated that R (3) can be written approximately in terms of only one of these TCF s. It will also be demonstrated that in the classical limit, 0, a contribution of order 3, is obtained from the linear combination of TCF s that cancels the prefactor. Without resorting to approximations, one prefactor is eliminated exactly, and R (3) can be rewritten in terms of only two TCF s. This is accomplished using frequency domain detailed balance relationships between two pairs of quantum mechanical TCF s appearing in R (3). The remaining two are not eliminated exactly, but by using exact results for a harmonic system with a linearly varying dipole to relate the TCF s. The second prefactor is then canceled by deriving a general relationship between two remaining TCF s for the harmonic system and rewriting R (3) in terms of the real and imaginary parts of only one TCF. The final result is obtained by finding the relationship between the real and imaginary parts of the last remaining TCF for the harmonic system. The resulting expression is exact for the harmonic system and can be applied to fully anharmonic dynamics derived using classical MD. The use of the results for the harmonic reference system only serves to weight the different dynamical events that contribute to the response function as they would be for a harmonic system. Consequently, our theory has the essential feature that a harmonic oscillator with a linearly varying dipole no electrical anharmonicity gives no 2DIR signal; the approximation serves to filter out the less interesting harmonic dynamics and enhances the anharmonic couplings in the resulting signal. 13 A similar approach was very successful in developing a TCF theory of the fifth-order Raman response function. 46 For example, the earlier theory nearly quantitatively reproduced exact MD calculations for liquid xenon. 35,46 Anharmonic dynamics and couplings are a main focus of 2DIR spectroscopy and they are critical in extracting physical insight. 13,28 It is possible to use a different reference system to relate the TCF s, and a cubic and quartic anharmonic oscillator system treated perturbatively may be another good, albeit more complicated, choice for the present methods and this avenue is being pursued. 49 The remainder of the paper is organized as follows: The derivation of the TCF theory of the third-order response function, R (3), is presented in Sec. II. A description of the 163/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

164 3690 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 DeVane et al. MD and spectroscopic model is given in Sec. III. The application of the TCF theory to the 2DIR spectrum of liquid water is discussed in Sec. IV. Potential quantum correction schemes are presented and the paper is concluded in Sec. V. A t 1,t 2,t 3 1 Q abcd e E i a ad j dc k cb ba e ie ab t 1 / e ie ac t 2 / e ie ad t 3 /, E ab E a E b, II. MODELS AND METHODS The commutators in Eq. 2 can be expanded to give four TCF s and their complex conjugates. In terms of these time correlation functions, R (3) can now be written as R 3 ijk t 1,t 2,t 3 i 3 A t 1,t 2,t 3 B t 1,t 2,t 3 C* t 1,t 2,t 3 D* t 1,t 2,t 3 D t 1,t 2,t 3 C t 1,t 2,t 3 B* t 1,t 2,t 3 A* t 1,t 2,t 3. 3 In Eq. 3 the star superscript represents the complex conjugate of the complex time domain TCF and the expression in square brackets can be seen to be the sum and difference of the imaginary parts of the four TCF s, namely, 2(A I B I C I D I ). The subscripts I and R denote the imaginary and real parts of a time-domain TCF or of its Fourier transform, respectively. At this point evaluation of the R (3) expression requires the calculation of four different quantum mechanical time correlation functions, a formidable task. To minimize computational effort, it is desirable to rewrite R (3) in terms of only one TCF that can be approximated as its classical counterpart. The four TCF s that appear in Eq. 3 are A t 1,t 2,t 3 i t 3 t 2 t 1 j t 2 t 1 k t 1 0, B t 1,t 2,t 3 j t 2 t 1 i t 3 t 2 t 1 k t 1 0, C t 1,t 2,t 3 0 j t 2 t 1 i t 3 t 2 t 1 k t 1, D t 1,t 2,t 3 0 i t 3 t 2 t 1 j t 2 t 1 k t 1. Because the dipole moment operators commute classically, the four TCF s have the same classical limit. To proceed and to verify the classical limit of a TCF, it is helpful to rewrite it in the energy representation, for example, A t 1,t 2,t 3 i t 3 t 2 t 1 j t 2 t 1 k t 1, A t 1,t 2,t 3 1 Q a a e H e ih t 3 t 2 t 1 / i e ih t 3 t 2 t 1 / e ih t 2 t 1 / j e ih t 2 t 1 / e iht 1 / k e iht 1 / a. In Eq. 5 the dipole moment operators are multiplied by e H /Q and the trace is taken. Q is the partition function and a (t) e iht/ a e iht/. Next, it is possible to insert four complete sets of energy eigenstates, a a a with H a E a a, operate, and simplify 46 to get expressions of the form 4 5 ab a b. Next, consider the complex conjugate of A(t 1,t 2,t 3 ); note that the dipole moment operator matrix elements are Hermitian, i.e., ij * ji. The matrix elements can be chosen and are taken as real, it is then clear that A*(t 1,t 2,t 3 ) A( t 1, t 2, t 3 ): A* t 1,t 2,t 3 0 k t 1 j t 2 t 1 i t 3 t 2 t 1, 6 A* t 1,t 2,t 3 1 Q abcd e E i a ad j dc k cb ba e ie ba t 1 / e ie ca t 2 / e ie da t 3 /. The triple Fourier transform of A(t 1,t 2,t 3 ) gives the frequency domain function A( 1, 2, 3 ). Similar manipulations give the frequency domain functions for B, C, and D. Index switching exchanging summed over dummy indicies, which is equivalent to taking cyclic permutations of the trace, have been used to maximize the similarity between the four expressions. 5,46 A 1, 2, 3 1 Q abcd B 1, 2, 3 1 Q abcd C 1, 2, 3 1 Q abcd D 1, 2, 3 1 Q abcd e E i a ad j dc k cb ba 1 E ab / 2 E ac / 3 E ad /, e E i a dc j ad k cb ba 1 E ab / 2 E ac / 3 E dc /, e E i b dc j ad k cb ba 1 E ab / 2 E ac / 3 E dc /, e E i b ad j dc k cb ba 1 E ab / 2 E ac / 3 E ad /. Using the fact that the Fourier transforms of the TCF s are real, 46 it is simple to find frequency domain expressions for the complex conjugates of the TCF s. Taking the triple Fourier transform FT of the complex conjugates gives the same frequency domain functions evaluated at negative frequency, i.e., FT f *(t 1,t 2,t 3 ) f ( 1, 2, 3 ): 7 164/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

165 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 Two-dimensional infrared spectroscopy 3691 A 1, 2, 3 1 Q abcd B 1, 2, 3 1 Q abcd C 1, 2, 3 1 Q abcd e E i a da 1 E ba / 2 j cd k bc ab E ca / 3 E da /, e E i a cd 1 E ba / 2 j da k bc ab E ca / 3 E cd /, e E i b cd 1 E ba / 2 j da k bc ab E ca / 3 E cd /, 8 D 1, 2, 3 A 1, 2, 3 tanh 1 /2 D 1, 2, 3 A 1, 2, 3, C 1, 2, 3 B 1, 2, 3 tanh 1 /2 C 1, 2, 3 B 1, 2, 3, C 1, 2, 3 B 1, 2, 3 tanh 1 /2 C 1, 2, 3 B 1, 2, Fourier transforming R (3) into the frequency domain and directly substituting the relationships from Eqs. 9 and 10 into it, the third order response function can be written exactly in terms of only two-time correlation functions A and B, D 1, 2, 3 1 Q abcd e E i b da 1 E ba / 2 j cd k bc ab E ca / 3 E da /. At this point, if the frequency domain expressions only differed by the Boltzmann factor weighting them, it may be possible to derive detailed balance relationships between all of them. This is only the case for the pairs of TCF s, (A,D) and (B,C). Using the Boltzmann factors to relate the members of each pair and then enforcing the functions, four detailed balance relationships arise. Simple relationships between members of different pairs and between positive and negative frequency TCF s do not generally exist. 46 This is true even though the time-domain TCF s, e.g., A and A*, have the same real part, and their imaginary parts are the same function of opposite signs; both the real and imaginary portions of the time domain TCF contribute to the frequency domain TCF s: D 1, 2, 3 e 1A 1, 2, 3, D 1, 2, 3 e 1A 1, 2, 3, 9 C 1, 2, 3 e 1B 1, 2, 3, C 1, 2, 3 e 1B 1, 2, 3. If the classical limit of each TCF is taken as e E 1, an unexpected problem arises. The members of each pair can clearly be seen to have the same classical limit, i.e., (A,D) and (B,C), but the equivalence between the pairs is not obvious. This point will be reexamined later. Along with the relationships shown in Eq. 9, taking the ratios between sums and differences of time correlation functions will provide four relationships useful for simplifying the R (3) expression: 46 D 1, 2, 3 A 1, 2, 3 tanh 1 /2 D 1, 2, 3 A 1, 2, 3, R 3 1, 2, 3 i tanh e 1 B A 1 e 1 B A. 11 In Eq. 11 the notations f f ( 1, 2, 3 ) and f f ( 1, 2, 3 ) are introduced for clarity and f represents any of the functions A, B, C, D. This equation is exact for all frequencies and implies that the 2DIR signal is zero along the 1 0 frequency axis. At this point it is insightful to examine the classical limit of R (3). The exponentials are expanded and terms of order are retained. The result is that one prefactor is eliminated exactly from R (3) in the classical limit: R 3 1, 2, 3 i B A 2 1 B A. 12 It is easily proven that f f 2 f R, where f R ( 1, 2, 3 ) denotes the Fourier transform of the real part of f (t 1,t 2,t 3 ), and f f 2 f I, where f I ( 1, 2, 3 ) is the Fourier transform of the imaginary part of f (t 1,t 2,t 3 ), both of which are themselves real functions. These relationships allow R (3) to be written as R 3 1, 2, 3 i 1 2 B 2 R A R 1 B I A I. 13 Equation 13 makes it clear that the Fourier transform of the imaginary parts of A and B must have a leading difference order and the real parts order 2 for a meaningful classical limit to exist. At this point, back Fourier transforming to the time domain gives R (3) written in terms of time derivatives. In general f t 1,t 2,t 3 e i 1 t 1e i 2 t 2e i 3 t 3f 1, 2, 3 d 1 d 2 d 3, 165/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

166 3692 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 DeVane et al. d dt 1 f t 1,t 2,t 3 i 1 e i 1 t 1e i 2 t 2e i 3 t 3f 1, 2, 3 d 1 d 2 d 3, d 2 dt 1 2 f t 1,t 2,t 3 i 1 2 e i 1 t 1e i 2 t 2e i 3 t 3f 1, 2, 3 d 1 d 2 d 3. Then, in the time domain, R (3) can be written in terms of t 1 derivatives as shown. This expression is exact in the classical limit: R 3 t 1,t 2,t 3 2 d B 2 dt R t 1,t 2,t 3 d A 1 dt R t 1,t 2,t 3 1 i 2 d2 dt 1 2 A I t 1,t 2,t 3 d2 dt 1 2 B I t 1,t 2,t In order to further simplify R (3), it is necessary at this point to make approximations. By evaluating the TCF s invoking a harmonic approximation it is possible to find a relationship between A and B. A harmonic potential V m 2 q 2 /2 is assumed with partition function Q e /2 /(1 e ). Additionally, the dipole moment functions appearing in the time correlation functions are expanded out to first order in the harmonic coordinate, giving the required matrix elements: 46 ij 0 ij q ij, 1/2 q ij 2m i, j 1 j 1 1/2 i, j 1 j 1/2. 15 In Eq. 15 the primes represent derivatives with respect to the harmonic coordinate q. In both A and B, expansion of the four s gives the sum of 16 terms, each distinct in the powers of coordinates used to evaluate the dipole moment matrix elements. Many of these terms can be neglected because their functions force them to equal zero. Additionally, from examining Eq. 14 it is clear that to contribute to R (3), a term in the time domain must have a nonzero derivative with respect to t 1 although Eq. 14 only considers the classical limit the neglected higher order terms would also involve higher order t 1 derivatives. Under these conditions, only four nonzero terms remain for each TCF: three terms with two and two 0 denoted 0110, 1100, and 1010 to denote the order of coordinate that is used in evaluating the dipole matrix elements in the order presented above in Eqs. 7 and 8, then one term with four denoted The 0110, 1100, and 1010 terms are identical in TCF s A and B. Because R (3) is now expressed in terms of differences between TCF s A and B, these three terms vanish. The 1111 term is the only one left to consider. In A(t 1,t 2,t 3 ) the 1111 term is the sum of six distinct parts, labeled a through f. The six terms from A 1111 are A 1111a t 1,t 2,t 3 i j k A 1111b t 1,t 2,t 3 i j k 2 2m 1 1 e 2 2e 2 e i t 1 2t 2 t 3, 2 2m 1 1 e 2 2e i t 1 2t 2 t 3, e A 1111c t 1,t 2,t 3 i j k 2m 2 1 e e 2 e i t 1 t 3, A 1111d t 1,t 2,t 3 i j k 2m 1 e e i t 1 t 3, 2 A 1111e t 1,t 2,t 3 i j k 2m A 1111f t 1,t 2,t 3 i j k 1 e e 2 2e e i t 3 t 2, 2 2m 1 1 e 2 2e e i t 3 t Similarly, for TCF B, the 1111 term can be written as the sum of terms a through f. Terms a and b are identical to those found for TCF A, while terms c through f have different prefactors. The four terms, unique B 1111 terms, are B 1111c t 1,t 2,t 3 i j k B 1111d t 1,t 2,t 3 i j k 2 2m 1 1 e 2 2e e i t 1 t 3, 2 2m 1 1 e 2 2e e i t 1 t 3, e B 1111e t 1,t 2,t 3 i j k 2m e 1 e i t 3 t 1, 166/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

167 B 1111f t 1,t 2,t 3 i j k 2m 2 1 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 Two-dimensional infrared spectroscopy 1 e 2 e 2 e e i t 3 t Considering only the 1111 terms, it is now possible to define an exact relationship between the two TCF s A and B and use it to simplify R (3). For terms a through f, the relationship between A 1111 and B 1111 is easily found by comparing their cofactors. The result is a set of six relationships between the two TCF s. These relationships hold in both the time and frequency domains, since they only depend on the terms coefficients: A 1111a B 1111a, A 1111b B 1111b, A 1111c e B 1111c, A 1111d e B 1111d, 18 exponentials out to first order in, g( 1, 2, 3 ) and g( 1, 2, 3 ) can be simplified as shown below: g 1, 2, /4 3 /2, g 1, 2, /4 3 /2. 20 Next, g( 1, 2, 3 ) and g( 1, 2, 3 ) can be incorporated into Eq. 12 to simplify the R (3) expression even further. TCF A has been completely eliminated from R (3), leaving behind only TCF B. In the classical limit R 3 1, 2, 3 i B R 1, 2, 3 4B I 1, 2, A e B 1111e, 1 e A f B 1111f. 1 e Carefully considering the six relationships shown above, it is possible to define a function g( 1, 2, 3 ) and its negative frequency counterpart g( 1, 2, 3 ) that relates A and B in positive and negative frequencies in terms of the dynamical variables 1, 2, and 3 ; this function is not necessarily unique but is the simplest functional form that was obtained. The frequency domain TCF functions allow the harmonic frequency to be written in terms of 1, 2, and 3 in the usual fashion: A 1, 2, 3 g 1, 2, 3 B 1, 2, 3, g 1, 2, 3 1 e /2 1 e 1 3 /2, A 1, 2, 3 g 1, 2, 3 B 1, 2, 3, 19 g 1, 2, 3 1 e /2. 1 e 3 1 /2 To demonstrate this result, the explicit expressions for A and B Eqs. 16 and 17 can be substituted into Eq. 18 and then enforcing the functions immediately leads to the results in Eq. 19 this type of manipulation is demonstrated in detail in a previous paper. 46 At this point, it is interesting to revisit the question of the classical limits of TCF s A and B. Above, it was not obvious that these two TCF s shared the same classical limit. It is worth noting that as 0, all the six relationships in Eq. 18 approach A B. Additionally, g( 1, 2, 3 ) 1 and g( 1, 2, 3 ) 1. It is clear that the two TCF s have identical classical limits for the harmonic system with a linear dipole. If the classical limit is taken by expanding the Two of the three prefactors have been eliminated completely and R (3) is written in terms of only B R ( 1, 2, 3 ) and B I ( 1, 2, 3 ), the Fourier transforms of the real and imaginary parts of B(t 1,t 2,t 3 ); in this limit B R becomes the classical TCF that can be computed using MD. Finding a relationship between B R and B I will allow the removal of the last prefactor in the classical limit and also give an expression in which the frequency factors are not Taylor expanded entirely in terms of B R that is valid for all frequencies the primary goal of this paper. Considering a one-time correlation function f (t), f R ( ) and f I ( ) have a simple functional relationship, f I ( ) tanh( /2) f R ( ). 46,50 If a similar relationship exists between B I ( 1, 2, 3 ) and B R ( 1, 2, 3 ), R (3) can then be written in terms of only B R. However, such a relationship is not immediately obvious. To attempt to find one each nonzero term of B(t 1,t 2,t 3 ) terms 0011, 0101, 1001 and the six 1111 terms was separated into its real and imaginary components, and the two parts were Fourier transformed separately. The ratio between B I ( 1, 2, 3 ) and B R ( 1, 2, 3 ) was then determined for each term. Comparing the nine ratios, an exact relationship for the terms considered, similar in form to the above relationship for the one-time correlation function, becomes apparent, B I 1, 2, 3 tanh B R 1, 2, Equation 22 is exact for the harmonic system. It should be noted that we are only considering the 1111 terms for the harmonic system in constructing this relationship the lowest order terms that contribute to R (3) within the harmonic approximation. With the ability to relate B R ( 1, 2, 3 ) and B I ( 1, 2, 3 ), it is possible to make one final simplification, which removes the final remaining prefactor in the classical R (3) expression. In the classical limit, the final expression for R (3) is given by 167/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

168 3694 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 DeVane et al. R 3 1, 2, R 3 t 1,t 2,t d B R 1, 2, 3, 23 B R dt 1 2 dt 2 2 d3 B R dt 1 dt d3 B R dt 1 2 dt 3 2 d3 B R dt 1 dt d3 B R dt 1 dt 2 dt 3. The derivatives appearing in the time-domain expressions result from taking time derivatives of the Fourier transform of B( 1, 2, 3 ). To examine high frequency phenomena, taking the classical limit of the R (3) is not desirable. In this case, in the frequency domain expression takes the following form: R 3 1, 2, 3 i 3 tanh 1 /2 1 e 1 1 g 1, 2, 3 1 tanh /4 1 e 1 1 g 1, 2, 3 1 tanh /4 approximation to the real experiment, we will evaluate the 2D response, specifically, R 3 1, 3 R 3 1,t 2 0, dt 1 e i 1 t 1 dt 2 e i 2 t 2 dt 3 e i 3 t 3R 3 t 1,t 2,t 3 t Equation 25 implies that we need to evaluate the response function given by the Fourier transform of Eq. 24 at t 2 0. Within our theory it is computationally far more convenient to evaluate the TCF B R (t 1,t 2,t 3 ) using the condition t 2 0. This is complicated by the form of Eq. 24 because the TCF is multiplied by a complicated function of frequency and the product would need to be back Fourier transformed to implement the condition in time and in general this does not lead to evaluating B R (t 1,t 2 0,t 3 ). However, in the limit that all three frequencies are high, 1,2,3 1, an approximate simplification is possible and this is the relevant experimental frequency regime. Defining everything on the righthand side of Eq. 24 that multiplies B R as h( 1, 2, 3 ) gives R (3) within our theory as R 3 t 1,t 2 0,t 3 d 1 e i 1 t 1 d 2 e i 2 t 2 d 3 B R 1, 2, e i 3 t 3h 1, 2, 3 B R 1, 2, t 2 0 Equation 24 is now in a form that can be evaluated using classical MD and TCF techniques. B R (t 1,t 2,t 3 ) is replaced by its classical counterpart and Fourier transformed into the frequency domain. The frequency factors are then taken into account and the resulting function can be back transformed to the time domain to produce the desired response function that can then be used in Eq. 1 ; Eq. 24 is the central result of the paper and will be used to examine the 2DIR spectrum of ambient water in Sec. IV. As noted in the Introduction, Eq. 24 would give zero signal for a harmonic system with a linear dipole. This result can be verified by substituting the Equations in Eq. 17 into Eq. 24. Thus the use of the harmonic reference system makes the expression sensitive to anharmonic couplings while filtering out the harmonic contributions. We have presented a fully three-dimensional theory. While expressing the response in terms of a classical correlation represents an enormous simplification, a three-time classical correlation remains a formidable challenge. Because of that, and because of the available experiments, we aim to discuss a two-dimensional 2D theory in detail. There are two questions: do the experiments probe a 2D response function, and is the 2D response determined by a 2D classical correlation? The IR echo experiment nominally has t 2 0. For finite pulse lengths this condition is not literally enforced. However, as the ideal limiting case, and as a good Equation 26 only implies that the TCF only needs to be evaluated at t 2 0ifhis not a function of 2. In the limit of high frequency this is the case, although at 3000 cm 1 it is still weakly 2 dependent and the limiting value is not reached until 6000 cm 1 when cm 1 at 300 K. Thus, h hardly changes over the width of an intramolecular resonance and the resulting line shapes will be little changed. Therefore, for the purpose of demonstrating the theory we will assume the high frequency limit and take h as only a function of 1 and 3, h( 1, 3 ). In this case Eq. 26 can be evaluated to give B R ( 1,t 2 0, 3 ) and when Eq. 24 is evaluated a large 2 is chosen, in this case cm 1. To test the choice a variety of smaller 2 values were tried and the line shapes were very similar. This 2D theory involves further approximation but is not a limitation of the general theory presented but rather a computational convenience to avoid calculating the three-time TCF that is generally required in Eqs. 1 and 24. Equating the TCF B R (t 1,t 2,t 3 ) with its classical counterpart is a further approximation and this is frequently done in the case of one-dimensional TCF s and tends to change the magnitude of spectroscopic signals more than the actual line shapes themselves. 43 A better approach is to quantum correct the Fourier transform of the classical TCF based on the relationship between the quantum and classical TCF for an 168/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

169 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 Two-dimensional infrared spectroscopy 3695 exactly solvable model system, e.g., a harmonic system. In the multidimensional case this is somewhat more difficult. One proceeds by forming the ratio of the terms in the B function, Eq. 17, with their classical limit. That ratio is then used to quantum correct the frequency domain classical TCF. The problem in the multidimensional case is that 1, 2, and 3 dependence is underdetermined and only the harmonic frequency dependence on is known; in the onedimensional case there is a one to one correspondence between the harmonic frequency and the spectroscopic observation frequency conjugate to t and no ambiguity results. In Sec. IV two quantum correction schemes that are both exact for the harmonic model system results are presented and it is demonstrated that the resulting line shapes are not greatly changed. Further, the lack of a unique quantum correction scheme is not a major limitation given that challenging 2DIR experiments do not focus on absolute intensities and the present approach represents a computationally tractable theory of 2DIR spectroscopy that can be applied to chemically interesting systems simulated in atomistic detail. III. APPLICATIONS TO AMBIENT WATER: MODELS AND COMPUTATIONAL DETAILS The present theory of the third-order response function and 2DIR spectroscopy was applied to ambient liquid water. Recent experiments have examined the OH stretching region for dilute solutions of HOD in D 2 O; there are experimental challenges associated with examining neat H 2 O due to its strong IR absorbance. 51,52 Applying the present theory to HOD in D 2 O is the subject of an ongoing investigation. To obtain time ordered water configurations, classical MD simulations were performed using a code developed at the Center for Molecular Modeling at the University of Pennsylvania, which uses reversible integration and extended system techniques. 53,54 A flexible SPC model was used to perform the MD simulations. Microcanonical simulations were performed on a system consisting of 64 H 2 O molecules with a density of 0.99 g/cm 3 and a temperature of 295 K; these conditions produce a pressure of 1.0 atm for this water model as determined by constant temperature and pressure (N- P-T) simulations. Previous studies demonstrated that a system size of 64 water molecules is adequate to reproduce the IR spectrum of water. 55 The water intramolecular potential includes a harmonic bending potential, linear cross terms, and a Morse OH stretching potential, and the model parameters are presented in previous works. 44,50 MD simulations with a length of 1.4 million 1.0 fs time steps were performed and configurations were stored every 4.0 fs producing a total of configurations that are subsequently used to calculate two-time TCF s; this time spacing gives a Nyquist frequency of 4167 cm The MD was performed without explicit polarization forces. However, a spectroscopic model is used to calculate the time dependent dipole of the liquid water that explicitly includes many-body polarizability. The model is parametrized to produce accurate dipoles and polarizability tensors and their coordinate derivatives in the gas phase. Specifically a point atomic polarizability PAPA model is used that has been successfully applied in quantitatively reproducing the linear IR spectrum of water 44,55 and the SFG spectroscopy of the water/vapor interface. 50 The PAPA model accurately accounts for induced dipoles and their derivatives in the condensed phase and this is evidenced by the success of the model in reproducing condensed phase spectra. In addition, a permanent dipole model, fit to ab initio calculations used in another study, 57 was also adopted and checked successfully against experimental IR gas phase intensities. 58 Induced dipole derivatives are responsible for most of the observed liquid state IR intensity in the OH stretching region while the bending intensity is largely determined by the permanent dipole derivative. To evaluate the present theory in the specific case of 2DIR spectroscopy with no time delay between the second and third laser pulses it is necessary to calculate the classical two-time TCF. B R (t 1,t 2,t 3 ) j (t 1 ) i (t 3 t 1 ) k (t 1 ) (0) and subsequently Fourier transform the result. This implicitly involves calculating the TCF at both positive and negative times in performing the integral: B R 1, dt1 e i 1 t 1 dt3 e i 3 t 3 j t 1 i t 3 t 1 k t To calculate the correlation function it is convenient to only work with positive times and thus it is necessary to consider the behavior of the two-time TCF in four different cases corresponding to quadrants in the two-dimensional plane determined by t 1 and t 3. It is easy to show that in the diagonal quadrants both the quantum and classical TCF s are identical; the quadrants are denoted below by a superscript for t 1 0 t 3 0 and by when t 1 0 t 3 0. Thus they can be evaluated for positive values of the time arguments as B j (t 1 ) i (t 3 t 1 ) k (t 1 ) (0). In the off-diagonal quadrants, and, the two functions are also identical and can be evaluated for positive values of the time arguments as B j (t 1 t 3 ) i (t 1 ) k (t 1 t 3 ) (t 3 ), and here time stationarity properties have been used to rewrite the TCF in terms of positive times; 34 the order of the dipole evaluation is irrelevant classically and is presented in the quantum mechanically correct form. Thus two different classical TCF s need to be calculated and both contribute to the 2DIR signal at a given pair of frequencies. Note that this can be avoided in calculating R (3) (t 1,t 2 ) in the classical limit, Eq. 23, because the result is given in that case by time derivatives that can be calculated via finite difference using the data local in time and Fourier transforming the TCF can be avoided. 46 The scheme employed here, where high frequency vibrations are of interest, is to perform a series of one-dimensional FFT s to evaluate Eq. 27 and use the B data in the diagonal quadrants and B data in the off-diagonal quadrants. The nature of the contribution made by the two TCF s can be understood by rewriting Eq. 27 as 169/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

170 3696 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 DeVane et al. B R 1, 3 1 dt1 cos 1 t dt3 cos 3 t 3 B B dt1 sin 1 t 1 0 dt3 sin 3 t 3 B B. 28 Therefore, the frequency contributions to the 2DIR spectrum arise from adding the double cosine transform of the sum of the two TCF s to the double sine transform of their difference. Therefore, B and B were calculated for liquid water using the model described above. The two-dimensional TCF s decayed slowly 59 and were calculated using a maximum correlation time of 20 ps in each time dimension. Over this duration each TCF decays 90% from its initial value but has not quite reached the asymptotic value of zero and the final value at 20 ps is slightly different for distinct time slices. Therefore, different base line values are subtracted off for each slice in performing the series of one-dimensional Fourier transforms. This does not significantly affect the line shape, and this was checked by performing the identical protocol on the linear IR absorption line shape 55 and the relevant TCF, (0) (t), is similar to the 2DIR time slices and it was unaffected as compared to a Fourier transformed TCF that decayed fully to zero. Ideally, longer correlation times would be employed, allowing the signal to decay to zero. However, the computational demands become large as the time is increased due to the small time step needed ( t 4.0 fs) to produce a Nyquist frequency that allows the resolution the O-H stretch frequency of water 56 and the twodimensional nature of the data. For example, to allow the TCF to get 99% of the way to the zero base line would require a maximum correlation time of 50 ps that represent the well averaged portion of a calculation correlated out to about 100 ps and this would produce several gigabytes of data to be stored and processed. IV. THE 2DIR SPECTRUM OF AMBIENT WATER A. Properties of the frequency domain two-time TCF Figure 1 shows the magnitude of the Fourier transformed fully polarized two-time TCF i.e., the dipole components are all the same, B R ( 1, 3 ), for the intramolecular vibrations; multiplying this result by the appropriate frequency factors in Eq. 24 would give our approximation to the third-order response function. For the purpose of demonstrating the theory only the fully polarized signal will be presented. Note that B R (t 1,t 2,t 3 ) and its Fourier transforms are real functions while R (3) (t 1,t 3 ) is real and its Fourier transform is purely imaginary. They both have positive and negative contributions in time and frequency. As expected, a strong diagonal signal dominates the frequency domain landscape indicating strong self-coupling of vibrational FIG. 1. The magnitude of the Fourier transform, in arbitrary units, of the two-time correlation function of the system dipole, B R ( 1, 3 ), is shown for the intramolecular stretching region. modes 28 dominant peaks are present in the water bending 1800 cm 1 and stretching 3300 cm 1 regions. Also, there is a continuous nonzero signal along the diagonal analogous to the nonzero signal present in the linear IR spectrum in this region. Aside from the dominant diagonal signal, slowly decaying ridges are present that run along 3. These ridges are positioned at and 3300 cm 1 indicating coupling between water bending and stretching in the condensed phase revealed via the two-time correlation function. Also, note that while water has a distinct antisymmetric and symmetric stretch in the gas phase, the condensed phase normal modes are nearly pure local O-H stretching modes and the mixing of the two gas phase vibrational resonances in the liquid leads to the broad O-R absorption One-dimensional frequency slices of B R ( 1, 3 ) are shown in Figs. 2 a 2 c to reveal the detailed line shapes. Figures 2 a and 2 b are edge slices along 3 with 1 0 cm 1 and 1 with 3 0cm 1, respectively. Figure 2 a reveals a signal with a different phase for the bend and stretch. Figure 2 b shows a signal similar to the linear IR experiment although the bend is relatively more intense. Figure 2 c shows a diagonal slice, where 1 3, of B R ( 1, 3 ) and here the phases of the bend and stretch are reversed from Fig. 2 a and the intensity of the bend is diminished with respect to the O-H stretching. The relative intensities along the diagonal are reminiscent of the linear IR experiment. Seeing differing phases for the resonances is interesting and similar phase information from the 2DIR experiment has been used to extract structural information about the system being probed, for example, in determining the relative orientation of transition dipoles of coupled vibrational modes. 30 To display the data in a different format, Figs. 3 a and 3 b display a series of single time Fourier transforms of the TCF, B R (t 1, 3 ) and B R ( 1,t 3 ), respectively. In both cases the signal magnitudes decay monotonically with time and the widths stay about the same. It remains to be seen what way of examining the spectra proves most useful. For example, the question arises in this mixed time frequency representation what physical insight can be obtained by understanding the rate of decay of the signal in time at a given frequency? To quantitate the signal diminution, the decay of 170/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

171 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 Two-dimensional infrared spectroscopy 3697 FIG. 2. Three one-dimensional frequency slices of B R ( 1, 3 ) are shown: a along 3 with 1 0, b along 1 with 3 0, and c along 1 3. While the spectra are shown in absolute units, the diagonal slice is shown smoothed using a simple multipoint average in both frequency directions to eliminate the oscillations near and along the diagonal apparent in Fig. 1 and this leads to a smaller signal magnitude. the OH stretching band maximum amplitude vs time was fit to both single and double exponentials and the twoexponential fit was much better at capturing an apparent fast and slower decay process. The OH stretching peak had a fast and a slow time decay constant of 42 fs 2.4 ps, respectively. These numbers compare favorably to recent spectroscopic measurements of vibrational relaxation of OH stretching in D 2 O that also find two decay time scales. 30,60,61 The rapid, resonant decay time is faster in our case because the OH oscillator is strongly coupled to other OH stretches via hydrogen bonding and in their case the OD stretch is only nonresonantly coupled. The slower relaxation time is due to structural rearrangements that would be expected to be similar for both pure and isoptopically mixed water, and the experiments obtain similar values for this relaxation time. Displaying the time-dependent frequency data highlights the rich information content of 2DIR spectroscopy and one of the goals of using short pulses in time is to resolve timedependent structural and dynamical features. B. Application of the TCF theory of 2DIR to liquid water Figure 4 a shows the theoretical linear IR spectrum that results from Fourier transforming the dipole-dipole autocorrelation function derived from the water model described above. 45 In the figure, the location of the major vibrational resonances is clear and it is presented to aid in interpreting the more complicated 2D spectrum presented in Fig. 4 b. Application of the present theory of the fully polarized third-order response function is presented as a contour plot of the nonquantum corrected signal R (3) ( 1, 3 ) in Fig. 4 b. As explained in Sec. II, Eq. 24 is evaluated using B R ( 1,t 2 0, 3 ) and setting cm 1 to mimic the 2DIR experiment. R (3) (t 1,t 3 ) becomes equivalent to the third-order polarization P (3) (t 1,t 2 0,t 3 ) in the limit of function pulses although this limit is not generally experimentally feasible and in that case Eq. 1 needs to be evaluated to calculate the observed polarization; this limiting case is a good approximation to the real experiment. 13 It should be noted that the back Fourier transform of R 3 1, 3 is the time-domain response function R (3) (t 1,t 3 ) that is measured in the ideal short pulse limit experiment. However, a frequency domain display of experimental data requires use of the Fourier-Laplace transform of the time-domain response function that includes both the frequency domain response function full Fourier transform itself and principal part integrals over the function. For simplicity, we show only R (3) ( 1, 3 ). It contains same information as the experimental observable, and is susceptible to the same physical interpretation, but does not correspond exactly to the real or imaginary parts of the Fourier-Laplace transform. 171/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

172 3698 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 DeVane et al. FIG. 4. Theoretical spectra for ambient water are presented. a The linear IR spectrum of water is shown in the top panel. b A contour plot of the application of the present theory of the third-order response function, R (3) ( 1, 3 ), is shown in the bottom panel. FIG. 3. 1D Fourier transforms of the TCF are shown: a B R (t 1, 3 ), b B R ( 1,t 3 ). In each case the times shown are slices at 0, 1, 2, 3, 4, 5, and 6ps. Figure 4 b reveals a strong echo signal along the diagonal with major peaks at about 1850 bending and 3300 cm 1 stretching. The frequency factors that multiply the frequency domain TCF in Eq. 24 significantly modify the relative intensities on and off the diagonal. The diagonal peak located at 3300 cm 1 is elongated parallel to the diagonal indicating inhomogeneous broadening. It is also elongated at an angle to the diagonal, parallel to 3 axis, indicating lifetime broadening. 11 The angle of the elongation of peaks relative to the diagonal yields information on the degree of correlation of the broadening. 11 Although the magnitude of the signal in the off-diagonal coupling regions is low compared to the diagonal, significant signal is present and experimental pulse sequences can be constructed that suppress the often less interesting strong diagonal features; 28 a principal goal of 2DIR spectroscopy is to uncover couplings between vibrational modes. The broad and extensive couplings present in the water 2DIR spectrum are consistent with the fast vibrational energy redistribution known to take place in water. 60,62 Figures 5 a 5 d show one-dimensional frequency slices of R (3) ( 1, 3 ). Figure 5 a displays a slice along 1 with 3 0cm 1. The intense water bending and stretching peaks located in the regions of 1850 and 3300 cm 1, respectively, dominate the frequency slice line shape. Figure 5 b shows the diagonal, 1 3, slice of R (3) ( 1, 3 ). The intense diagonal water stretching peak shows up in the 3300 cm 1 region and the less intense water bending peak shows up in the 1850 cm 1 region. Figure 5 c presents a slice along 1 with cm 1. The water bending peak, located in the region of cm 1, corresponds to a diagonal peak. The inset provides details of the weaker water OH stretch peak, with a negative amplitude, located in the cm 1 region. Also of interest are the relative intensities of the diagonal and off-diagonal peaks. Figure 5 d shows a slice along 1 with cm 1. The dominant peak is located in the region of cm 1 and represents the dominant peak of the diagonal. Off-diagonal peaks are also located approximately in the region of cm 1 as can be seen in the inset. V. QUANTUM CORRECTIONS AND CONCLUSIONS As was pointed out in Sec. II, equating the TCF B R (t 1,t 2,t 3 ) with its classical counterpart is a further approximation and this is a common practice in the case of one-dimensional TCF s. A better approach is to quantum correct the Fourier transform of the classical TCF based on the relationship between the quantum and classical TCF for an exactly solvable model system, e.g., a harmonic system. In the multidimensional case this is somewhat more difficult and this will be discussed below. Two quantum correction schemes for our theory of R (3) were constructed and tested for plausibility. These schemes provide two alternatives that relate the real part of the quantum mechanical two-time TCF to the corresponding classical TCF. The quantum corrections are chosen to exactly relate the real part of the six lowest order contributing terms for a harmonic system B function, Eq. 17, with their classical limits. That ratio of the quantum and classical TCF s is then used to quantum correct the frequency domain classical TCF: /224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

173 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 Two-dimensional infrared spectroscopy 3699 FIG. 5. Four frequency slices of R (3) ( 1, 3 ) are shown. Insets provide detailed structure of line shape. a Along 1 with 3 0cm 1, b along 1 3, c along 1 with cm 1, d along 1 with cm 1. B R Q 1, 2, coth csch 1 2 B C 1, 2, 3, 29 2 B Q R 1, 2, coth csch 3 2 B C 1, 2, B Q R represents the real part of the quantum time correlation function and B C represents its classical limit. Note that only the harmonic frequency dependence on is known from Eq. 17 and not the 1, 2, and 3 dependence. It might be possible to unambiguously determine this dependence for a more complex reference system, perhaps by considering a stepwise time-dependent harmonic oscillator model system. Here, the two possibilities given in Eqs. 29 and 30 were chosen as the simplest functional forms relating B Q R and B C exactly for a harmonic system. Equation 29 may be more reasonable than Eq. 30 because the latter equation predicts no signal along the 1 axis with 3 0, although Eq. 24 approaches zero for high frequencies even uncorrected. Equation 29 gives the third-order response as zero along the 3 axis with 1 0 and this is consistent with the exact result, Eq. 11. Figures 6 a 6 c show slices of the quantum corrected R (3) ( 1, 3 ) signal in order to compare the two quantum correction schemes considered. Figures 6 a and 6 b show the 1 with 3 0cm 1 slice and the 1 3 slice, respectively, using the quantum correction scheme in Eq. 29. Figure 6 c shows 1 3 for the quantum correction scheme in Eq. 30 and it would give no signal along 1 with 3 0 cm 1. The changes to the presented signals for both schemes are largely only in magnitude and there is no significant difference in line shape with and without quantum correction as is noted by comparing the three 1 3 slices, Figs. 5 b, 6 b, and 6 c. This result is consistent with quantum correction schemes for one-dimensional spectroscopies that change the magnitude of the signal more than the line shape because quantum corrections are nearly flat functions over the width of a vibrational resonance. 39,43 This implies that the present classical MD based TCF theory may be able to capture the essential features of 2DIR spectroscopy even without quantum correction. Given the present computationally tractable theory for R (3), the 2DIR technique can now be theoretically examined 173/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

174 3700 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 DeVane et al. FIG. 6. Three frequency slices of the quantum corrected R (3) ( 1, 3 ) signal are shown. Insets provide detailed structure of line shape: a along 1 with 3 0 with quantum correction scheme shown in Eq. 29, b along 1 3 with quantum correction scheme shown in Eq. 29, and c along 1 3 with quantum correction scheme shown in Eq. 30. for complex molecular systems and other polarization conditions to better understand the information content of the nonlinear spectroscopy. The present theory is well suited as both a predictive and interpretive tool and retains a fully molecular detailed description of 2DIR spectroscopy. In constructing our TCF theory it was necessary to invoke certain relationships that are only exact for a harmonic systems. 63 The apparent success of an analogous theory of the fifth-order Raman response function suggests that such an approximation may be reasonable. 46 For example, the earlier theory nearly quantitatively reproduced exact MD calculations for liquid xenon. 35,46 It also remains to be seen how accurate a classical MD based description of an inherently quantum mechanical, high frequency spectroscopy can be. However the success of analogous theories of linear spectroscopy also suggests the present theory may prove effective. ACKNOWLEDGMENTS The research at USF was supported by NSF Grant No. CHE and a grant from the Petroleum Research Foundation to B.S. The authors would like to acknowledge the use of the services provided by the Research Oriented Computing center at USF. The authors also thank the Space Basic and Applied Research Foundation for partial support. The research at BU was supported by NSF Grant No. CHE to T.K. 1 N.-H. Ge, M. T. Zanni, and R. M. Hochstrasser, J. Phys. Chem. A 106, S. Gnanakaran and R. M. Hochstrasser, J. Am. Chem. Soc. 123, C. Scheurer, A. Piryatinski, and S. Mukamel, J. Am. Chem. Soc. 123, K. Kwac, H. Lee, and M. Cho, J. Chem. Phys. 120, M. Cho, J. Chem. Phys. 115, J. Bredenbeck, J. Helbing, R. Behrendt, C. Renner, L. Moroder, J. Wachtveitl, and P. Hamm, J. Phys. Chem. B 107, M. Khalil, N. Demirdoven, and A. Tokmakoff, Phys. Rev. Lett. 90, S. Woutersen and P. Hamm, J. Phys. Chem. B 104, S. Woutersen, Y. Mu, G. Stock, and P. Hamm, Proc. Natl. Acad. Sci. U.S.A. 98, M. T. Zanni and R. M. Hochstrasser, Curr. Opin. Struct. Biol. 11, N. Demirdoven, M. Khalil, and A. Tokmakoff, Phys. Rev. Lett. 89, N. Demirdoven, M. Khalil, O. Golonzka, and A. Tokmakoff, J. Phys. Chem. A 105, O. Golonzka, M. Khalil, N. Demirdoven, and A. Tokmakoff, J. Chem. Phys. 115, K. A. Merchant, D. E. Thompson, and M. D. Fayer, Phys. Rev. Lett. 86, O. Golonzka, M. Khalil, N. Demirdoven, and A. Tokmakoff, Phys. Rev. Lett. 86, P. Hamm, M. Lim, and R. M. Hochstrasser, J. Phys. Chem. B 102, I. V. Rubtsov and R. M. Hochstrasser, J. Phys. Chem. B 106, I. V. Rubtsov, J. Wang, and R. M. Hochstrasser, Proc. Natl. Acad. Sci. U.S.A. 100, /224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

175 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 Two-dimensional infrared spectroscopy D. E. Thompson, K. A. Merchant, and M. D. Fayer, J. Chem. Phys. 115, D. E. Thompson, K. A. Merchant, and M. D. Fayer, Chem. Phys. Lett. 340, S. Woutersen and P. Hamm, J. Chem. Phys. 115, M. C. Asplund, M. T. Zanni, and R. M. Hochstrasser, Proc. Natl. Acad. Sci. U.S.A. 97, J. Bredenbeck and P. Hamm, J. Chem. Phys. 119, P. Hamm, M. Lim, W. F. DeGrado, and R. M. Hochstrasser, J. Chem. Phys. 112, P. Hamm, M. Lim, W. F. DeGrado, and R. M. Hochstrasser, Proc. Natl. Acad. Sci. U.S.A. 96, S. Woutersen and P. Hamm, J. Chem. Phys. 114, M. T. Zanni, M. C. Asplund, and R. M. Hochstrasser, J. Chem. Phys. 114, M. T. Zanni, N.-H. Ge, Y. S. Kim, and R. M. Hochstrasser, Proc. Natl. Acad. Sci. U.S.A. 98, A. T. Krummel, P. Mukherjee, and M. T. Zanni, J. Phys. Chem. B 107, O. Golonzka and A. Tokmakoff, J. Chem. Phys. 115, S. Mukamel, Principles of Nonlinear Optical Spectroscopy Oxford University Press, Oxford, A. Tokmakoff, J. Chem. Phys. 105, S. Mukamel, Principles of Nonlinear Optical Spectroscopy Oxford University Press, Oxford, 1995, Chap D. A. McQuarrie, Statistical Mechanics Harper & Row, New York, A. Ma and R. M. Stratt, J. Chem. Phys. 116, S. Saito and I. Ohmine, Phys. Rev. Lett. 88, H. Kim and P. J. Rossky, J. Phys. Chem. B 106, B. J. Berne, J. Jortner, and R. Gordon, J. Chem. Phys. 47, J. Borysow, M. Moraldi, and L. Frommhold, Mol. Phys. 56, E. W. Castner, Jr., Y. J. Chang, Y. C. Chu, and G. E. Walrafen, J. Chem. Phys. 102, X. Ji, H. Ahlborn, P. Moore, and B. Space, J. Chem. Phys. 113, X. Ji, H. Ahlborn, B. Space, P. Moore, Y. Zhou, S. Constantine, and L. D. Ziegler, J. Chem. Phys. 112, P. Moore, H. Ahlborn, and B. Space, in Liquid Dynamics Experiment, Simulation and Theory, edited by M. D. Fayer and J. T. Fourkas ACS, New York, H. Ahlborn, X. Ji, B. Space, and P. B. Moore, J. Chem. Phys. 111, H. Ahlborn, X. Ji, B. Space, and P. B. Moore, J. Chem. Phys. 112, R. DeVane, C. Ridley, T. Keyes, and B. Space, J. Chem. Phys. 119, V. Astinov, K. Kubarych, C. Milne, and R. D. Miller, Chem. Phys. Lett. 327, R. DeVane, C. Ridley, T. Keyes, and B. Space, J. Chem. Phys. to be published.. 49 G. Herzberg, Molecular Spectra & Molecular Structure: Volume I Spectra of Diatomic Molecules Van Nostrand Reinhold, New York, A. Perry, H. Ahlborn, P. Moore, and B. Space, J. Chem. Phys. 118, J. E. Bertie and Z. Lan, Appl. Spectrosc. 50, C. J. Fecko, J. D. Eaves, J. J. Loparo, A. Tokmakoff, and P. L. Geissler, Science 301, P. Moore and M. Klein unpublished. 54 M. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys. 97, H. Ahlborn, B. Space, and P. B. Moore, J. Chem. Phys. 112, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes Cambridge University Press, Cambridge, A. Morita and J. T. Hynes, J. Phys. Chem. B 106, R. D. Amos, Adv. Chem. Phys. 67, R. van Zon and J. Schofield, Phys. Rev. E 65, A. Pakoulev, Z. Wang, Y. Pang, and D. D. Dlott, Chem. Phys. Lett. 380, T. Steinel, J. B. Asbury, S. A. Corcelli, C. P. Lawrence, J. L. Skinner, and M. D. Fayer, Chem. Phys. Lett. 386, Z. H. Wang, A. Pakoulev, Y. Pang, and D. D. Dlott, Chem. Phys. Lett. 378, S. Mukamel, V. Khidckel, and V. Chernyak, Phys. Rev. E 53, R /224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

176 JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER SEPTEMBER 2003 A time correlation function theory for the fifth order Raman response function with applications to liquid CS 2 Russell DeVane, Christina Ridley, and Brian Space a) Department of Chemistry, University of South Florida, Tampa, Florida T. Keyes Department of Chemistry, Boston University, Boston, Massachusetts Received 15 May 2003; accepted 25 June 2003 A new theory for the fifth order Raman response function, R (5) (t 1,t 2 ), is presented. Using this result, R (5) (t 1,t 2 ) is shown to have a classical limit given by a combination of time derivatives of the real and imaginary parts of a two time correlation function TCF of the polarizability. In contrast with one time correlation functions, no exact analytic relationship exists between the real and imaginary parts of the quantum mechanical TCF that would allow the classical limit to be written in terms of classical TCF s. Writing the nonlinear response function in terms of classical TCF s would allow R (5) (t 1,t 2 ) to be calculated with minimal computational effort, in contrast to existing exact classical formulations. However, a simple approximate relationship is shown to exist between the real and imaginary parts of the two time TCF for a harmonic system with nonlinear polarizability. In the spirit of quantum correction, this relationship is used to write the exact classical response function in terms of classical TCF s. The resulting TCF expression is then calculated from fully anharmonic molecular dynamics calculations supplemented by a suitable spectroscopic polarizability model. The approximate expression is demonstrated to have correct limiting behaviors and leads to a two-dimensional spectrum for ambient carbon disulfide in excellent agreement with existing experimental and theoretical work. The proposed approach makes the calculation of fifth order response functions practical for a wide variety of chemically interesting systems American Institute of Physics. DOI: / I. INTRODUCTION Fifth order Raman experiments have recently been the subject of active investigation, both experimentally 1 8 and theoretically, 9 21 in the hope of obtaining unique insights into intermolecular dynamics. 20 However, obtaining unambiguous results has been difficult. Experimental measurements are complicated by the difficulties in separating third order cascade signals from the desired fifth order spectrum, although recent heterodyne detection schemes have demonstrated promise and some success in liquid CS Analytic theories and computer simulations must deal with commutators, or Poisson brackets in the classical limit, of dynamical variables at different times. The tools to manipulate such objects are virtually nonexistent and they are far more complex than the familiar time correlation functions TCF s that determine conventional spectroscopy. Numerically exact simulations have been performed in liquid xenon, but are computationally demanding and are made possible by approximations that are inappropriate for polarizable molecular liquids. 17 Other theoretical investigations on several systems have relied on either certain normal mode, 9 11,14,16,18 mode coupling, 15 generalized Langevin equation, 9 or finite field approximations. 12,22 Here, an approximate TCF method is developed that is In Eq. 1.1, e H /Q, for a system with Hamiltonian H and partition function Q at reciprocal temperature 1/kT, and k is Boltzmann s constant; Tr represents the trace of the operators, and is the system polarizability tensor and the Greek superscripts denote the elements being considered. The square brackets represent the commutator of the operators. The trace gives a classical limit that is an order 2 contribution that results from a combination of four two time correlation functions that are themselves equivalent classically. This is similar to the third order spectroscopies resulting from an order contribution of the trace in the third order response function, R (3) (t) (i/ )Tr (t), (0),. 20,26,27 The trace can be written as the differa Author to whom correspondence should be addressed. Electronic mail: space@cas.usf.edu computationally tractable and includes the essential physical elements needed to describe the fifth order response function for liquid CS 2. The work takes the approach similar to that which was successful in relating third order spectroscopies to classical TCF s. 20,23 25 While it is not possible to obtain a simple TCF expression that is exact in this case, 19 significant progress along these lines is possible. The quantum mechanical expression for the electronically nonresonant fifth order polarization response is given by 14,20,26,27 R (5) t 1,t 2 i 2 Tr t 1 t 2 t 1, 0, /2003/119(12)/6073/10/$ American Institute of Physics 176/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

177 6074 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 DeVane et al. ence between two one time correlation functions that are equivalent classically, e.g., (0) (t) (t) (0) ; the angle brackets are the trace of the operators divided by the partition function in the standard notation. 28 In that case the difference between the quantum mechanical TCF s is the imaginary part of the TCF that is exactly related to the real part in frequency space, C I ( ) tanh( /2)C R ( ), where the subscripts denote real functions that are the Fourier transform of the real and imaginary parts of C(t) (0) (t). In the classical limit C R ( ) is the Fourier transform of the classical TCF, 28 making the third order response function and the linear spectroscopies simply related to conventional TCF s. For R (5), the trace of the nested commutators must be of order 2 in the classical limit, to cancel the prefactor of 2. In Sec. II it will be demonstrated that a multiplicative factor of leading order can be obtained exactly using frequency domain detailed balance relationships between the two time quantum mechanical TCF s that constitute R (5) (t 1,t 2 ). A combination of real and imaginary parts of a two time TCF remain, and we require their O( ) contribution for the classical limit. If a relationship between the real and imaginary parts of the two time correlation function existed, the explicit -dependence could be determined as in the third order case, and the classical fifth order response could be written as second derivatives in time of a classical TCF. However, no exact analytic relationship is discernible. To proceed further, a simple approximate relationship is shown to exist between the real and imaginary parts of the two time TCF for a harmonic system with nonlinear polarizability, that is required to produce a fifth order response for a harmonic system. 10,11,29 In the spirit of quantum correction, this relationship is used to write the exact quantum mechanical response function approximately in terms of classical TCF s. The resulting TCF expression is then calculated from fully anharmonic molecular dynamics calculations supplemented by a suitable spectroscopic polarizability model. The approximate expression is demonstrated to have correct limiting behaviors and leads to a two-dimensional spectrum for ambient carbon disulfide in excellent agreement with existing experimental and theoretical work. The proposed approach makes the calculation of fifth order response functions practical for a wide variety of chemically interesting systems. The remainder of the paper is organized as follows: The formal development is carried out in Sec. II. The application of our TCF theory of the fifth order response to liquid CS 2 is compared with prior theories and experiments in Sec. III. The paper is concluded in Sec. IV. II. MODELS AND METHODS Expanding the commutators in the trace in Eq. 1.1 gives the fifth order response function as R (5) t 1,t 2 1/ 2 g t 1,t 2 f * t 1,t 2 f t 1,t 2 g* t 1,t In Eq. 2.1, f (t 1,t 2 ) *(t 1 ) (t 2 ), g(t 1,t 2 ) (t 2 ) *(t 1 ), f *(t 1,t 2 ) (t 2 ) *(t 1 ), and g*(t 1,t 2 ) *(t 1 ) (t 2 ) ; the superscript star is the complex conjugate. The superscript notation on denoting the polarization directions is suppressed and the results apply to all possible polarizations. The expression inside the square brackets in Eq. 2.1 is the difference between the real part of the functions f and g, g R (t 1,t 2 ) f R (t 1,t 2 ), with a leading term order 2 in the classical limit. This also demonstrates that R (5) is real in time, while the correlation functions are complex functions of time. It is convenient to evaluate f and g explicitly in the energy representation, f t 1,t 2 1 Q i i e H e iht 1 / e iht 1 / e iht 2 / e iht 2 / i. Inserting three complete sets of energy eigenstates,, with H E, operating and simplifying gives f t 1,t 2 1 Q i j k e E i ik kj ji E E E,. e it 1 E ki / e it 2 E kj /, As noted above, the functions all obey, f (t 1,t 2 ) f *( t 1, t 2 ), and this also implies that the matrix elements can be chosen real, and will be taken as real. In the frequency domain, f 1, 2 1 dt e i 1 t dte i 2 t 2 f t 1,t 2, f 1, 2 1 Q i j k e E i ik kj ji 1 E ki / 2 E kj /. 2.2 Similar manipulations lead to the frequency domain functions below. To obtain them in the form presented, the identity that the double Fourier transform of the complex conjugate of the TCF s gives the complex conjugate of the frequency domain function evaluated at negative the frequency argument, e.g., FT f *(t 1,t 2 ) f *( w 1, w 2 ), is used. FT represents the Fourier transform process shown in Eq. 2.2, and the fact that the frequency domain functions are all real is also used. The functions can be seen to be real because, e.g., FT f (t 1,t 2 ) f ( 1, 2 ) FT f *( t 1, t 2 ) f *( 1, 2 ), f * t 1,t 2 t 2 * t 1, /224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

178 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 Fifth order Raman response 6075 f 1, 2 1 Q i j k e E i ik kj ji g t 1,t 2 t 2 * t 1, 1 E ik / 2 E jk /, 2.4 e 1g 1, 2 f 1, 2, e 2g 1, 2 h 1, 2, e ˆ f 1, 2 h 1, 2, e 1f 1, 2 g 1, g 1, 2 1 Q i j k e E k ik kj ji 1 E ki / 2 E kj /, g* t 1,t 2 * t 1 t 2, g 1, 2 1 Q i j k e E k ik kj ji 1 E ik / 2 E jk /. 2.5 Also, another correlation function h(t 1,t 2 ) is presented that does not appear in the response function, but has the same classical limit as f and g and will prove useful later; this set of two time correlation functions represents the entire set that can be created with the relevant operators for the two time TCF s, h t 1,t 2 * t 1 t 2, h 1, 2 1 Q i j k e E j ik kj ki 1 E ki / 2 E kj /, h* t 1,t 2 t 2 * t 1, h 1, 2 1 Q i j k e E j ik kj ji 1 E ik / 2 E jk / In Eqs the forms presented are a result of using index switching to maximize the similarity of the delta functions. For example, another form of Eq. 2.7 can be obtained by switching the i and j indicies. Index switching is equivalent to taking advantage of cyclic permutations of the trace to obtain different expressions. The Hermiticity of the polarizability was also used to equate, e.g., ik ki, which is true for real matrix elements. It is clear from Eqs that the set of functions f ( 1, 2 ), g( 1, 2 ), and h( 1, 2 ) differ only in the Boltzmann factor weighting the expressions, e.g., e E i in f ( 1, 2 ) and e E k in g( 1, 2 ). The same applies to the set of function of negative frequency f ( 1, 2 ), g( 1, 2 ), and h( 1, 2 ). This implies the following relationships between the frequency domain functions: ˆ 1 2, e 1g 1, 2 f 1, 2, e 2g 1, 2 h 1, 2, e ˆ f 1, 2 h 1, 2, e 1f 1, 2 g 1, 2, In Eqs enforcing the relevant delta functions allows the frequency factor to be taken outside the summations. There is, however, no direct relationship between the functions of positive and negative frequency. It is now useful to take sums and differences of the functions, and use the relationships in Eqs , e.g., f ( 1, 2 ) g( 1, 2 ) (e 1 1)g( 1, 2 ). Taking the ratio of the sum and differences leads to the following relationships: f 1, 2 h 1, 2 tanh ˆ /2 f 1, 2 h 1, 2, f 1, 2 g 1, 2 tanh 1 /2 f 1, 2 g 1, 2, h 1, 2 g 1, 2 tanh 2 /2 h 1, 2 g 1, 2, f 1, 2 h 1, tanh ˆ /2 f 1, 2 h 1, 2, 2.20 f 1, 2 g 1, 2 tanh 1 /2 f 1, 2 g 1, 2, 2.21 h 1, 2 g 1, 2 tanh 2 /2 h 1, 2 g 1, This allows the fifth order response function to be rewritten as R (5) 1, 2 1/ 2 tanh 1 /2 f 1, 2 f 1, 2 g 1, 2 g 1, In the classical limit, Eq has a prefactor of 1 /2, and to obtain an expression in terms of classical TCF s a term order needs to be obtained from the functions in the square brackets. The expression inside the square brackets in Eq is the difference between the Fourier transform of the imaginary parts of the functions f (t 1,t 2 ) and g(t 1,t 2 ), that will be denoted as g I ( 1, 2 ) f I ( 1, 2 ), where the subscripts denote the Fourier transform of the real (R) or imaginary (I) parts of the TCF s, both of which are themselves real functions of frequency. It is now useful to write Eq in terms of only one function, g( 1, 2 ), using Eqs. 2.9 and 2.13, 178/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

179 6076 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 DeVane et al. R (5) 1, 2 1/ 2 tanh 1 /2 1 e 1 g 1, 2 1 e 1 g 1, To obtain the classical limit, kt, the exponentials are expanded and terms order are retained, R (5) 1, 2 1/ 2 tanh 1 /2 2 1 g 1, g 1, 2, R (5) 1, 2 1/ 2 tanh 1 /2 2 g 1, g 1, 2 1 g 1, 2 g 1, 2, R (5) 1, 2 1/ 2 1 /2 4g I 1, g R 1, 2, where it is understood that the leading contributions in to the TCF are to be kept, zero-order for g R and first-order g I. In the time domain Eq takes the form, R (5) t 1,t / t 2 1 g R t 1,t 2 2 i / t 1g I t 1,t It is worth noting that Eq can be obtained from the frequency domain version of Eq. 2.1 by writing R 5,at that point, in terms of the g( 1, 2 ) using the frequency domain relationships between f and g and Taylor expanding the resulting exponentials. Equation 2.24 is, however, exact for all frequencies and is a starting point to develop an approximation that is valid for intramolecular or high frequency intermolecular spectroscopy for which kt. Equation 2.28 is exact in the classical limit. When kt the function, g R ( 1, 2 ) becomes the Fourier transform of the classical TCF, (0) ( t 1 ) (t 2 ) (0) (t 1 ) (t 1 t 2 ), and the times have been reordered because they commute in the classical limit. For a one time correlation function, C(t), a simple relationship exists between the real and imaginary parts of the function, C I ( ) tanh( /2)C R ( ). While it is clear that g I makes a contribution that is order, there is no general relationship between the real and imaginary parts of g. Therefore, the g R term in Eq can be calculated from classical TCF s but g I has no apparent classical limit. Nevertheless an approximate connection between g R and g I can be established for a harmonic system with a nonlinear polarizability, and this is the simplest harmonic system that produces a fifth order response. 10,11,29 The goal is to determine the relationship between the real and imaginary parts of g( 1, 2 ) for a harmonic system with potential energy, V 1/2m 2 Q 2. Such a result leads to an approximate correlation function expression for R (5). To proceed, the polarizability is expanded to second order in the harmonic coordinate in order to evaluate the matrix elements in Eq. 2.4, 0 Q 1/2 Q In Eq the primes represent derivatives with respect to the harmonic coordinate, Q. When Eq. 2.4 is evaluated for the harmonic system, the lowest order nonzero contributions to R (5) involve two and one ; the correlation functions have contributions from one 0 and two, that can be denoted as g 011 ( 1, 2 ), g 101 ( 1, 2 ), and g 110 ( 1, 2 ), but these do not contribute to the response function when treated exactly; this point will be revisited below. The superscripts on g represent the power of the coordinate that is used to evaluate the polarizability matrix elements appearing as ik kj ji, respectively. The total contribution to the first non-vanishing order, is then written as g( 1, 2 ) g 211 ( 1, 2 ) g 121 ( 1, 2 ) g 112 ( 1, 2 ). The evaluation involves standard results for the harmonic oscillator and evaluating several geometric series, 30 and gives, in the frequency domain, g 211 1, 2 1 Q 2m k e k k 1 k k 1 2k k 2k k k Fourier transforming to the time domain, 2 g 211 t 1,t 2 2m 2 1 e 2 e i2 t 1e i2 t 2 3e 1 1 e 2 e i t 2 3e e 2 1 e 2 e i t 2 2e 2 1 e 2 e i2 t 1e i at Adding the complex conjugate to yield the real part in the time, g 211 R t 1,t 2 2m e 2 1 e 2 cos 2 t 1 t 2 e 2 6e 1 1 e 2 cos t Subtracting the complex conjugate to yields the imaginary part in the time, 179/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

180 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 Fifth order Raman response 6077 TABLE I. The success of the relationship g I ( 1, 2 ) tanh( ( 1 /4 2 /2))g R ( 1, 2 ) for the harmonic system is demonstrated. In the classical limit, for the proposed relationship between the real and imaginary parts of g to work, multiplying the coefficient of g R column 2 by the value of ( 1 /4 2 /2) column 6 should give the coefficient of g I column 3 for each term. The proposed relationship works exactly for three of the terms and very nearly for the other three g R g I 1 2 ( 1 /4 2 /2) g 211 term 1 4 Eq Eq g 211 term 2 8 Eq Eq /2 g 121 term 1 4 Eq Eq /4 g 121 term 2 8 Eq Eq /4 g 112 term 1 4 Eq /4 g 112 term 2 8 Eq Eq /4 g 211 I t 1,t 2 2m e e 2 sin 2 t 1 t 2 e e 2 sin t Similar manipulations lead to the following expressions in the other two cases: g 121 R t 1,t 2 2m e 2 1 e 2 cos t 1 2 t 2 e 2 6e 1 1 e 2 cos t 1, g 121 I t 1,t 2 2m e e 2 sin t 1 2 t 2 e e 2 sin t 1, g 112 R t 1,t 2 2m e 1 e 2 cos t 1 t 2 3e 2 2e 3 1 e 2 cos t 1 t 2, g 112 I t 1,t 2 2m e e 2 sin t 1 t The three functions just discussed are composed of two terms with distinct t or dependence for a total of six contributions to g. Each has an order relationship between the real and imaginary parts. However, to get a classical result, the relationship for all the terms must be of the same form because the different contributions cannot be distinguished in performing a classical MD and TCF calculation; this point will be revisited below. While no expression works perfectly for all the terms, we will now demonstrate that g I ( 1, 2 ) tanh( ( 1 /4 2 /2))g R ( 1, 2 ) is an excellent approximation in the classical limit, and leads to a TCF approximation for R (5) that has reasonable limiting behaviors. In the classical limit, to leading order in, tanh( ( 1 /4 2 /2)) ( 1 /4 2 /2). Table I shows the limiting form of the frequency prefactors and the value of ( 1 /4 2 /2) for the six terms listed above, and each contribution to g consists of two terms. In the classical limit, for the proposed relationship between the real and imaginary parts of g to work, multiplying the coefficient of g R column 2 by the value of ( 1 /4 2 /2) column 6 should give the coefficient of g I column 3 for each term. The proposed relationship works exactly for three of the terms and very nearly for the other three. For the terms that are not handled quite correctly, namely the g 211 term 2 and g 112 term 1, this leads to errors in filtering the dynamics according to the harmonic system dynamics. While the present theory is not a harmonic theory it relies on fully anharmonic dynamics to calculate the relevant TCF derivatives the harmonic system serves to weight the different phonon processes as they would contribute in such a system. 10 Thus the errors associated with this approximation are to relatively over weight the processes associated with the stated terms. It a subsequent paper it will be shown that this approximation leads to results in liquid xenon, for which the fifth order response can be calculated numerically exactly, 16 in nearly quantitative agreement /224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

181 6078 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 DeVane et al. This analysis focuses on the frequency prefactors and the value of each frequency argument, 1 and 2 for each term. This is sufficient because odd functions of frequency, such as tanh( ), are able to convert the real part of such a function to the imaginary part by converting the sum of delta functions to a difference. The delta function combinations are themselves the result of Fourier transforming the sin and cos functions associated with the imaginary and real parts of g, respectively that are referenced explicitly in Table I. For example, in one time dimension, FT cos t 1/2, FT sin t 1/2i When tanh( ) multiplies Eq the frequency argument is replaced by the value of the harmonic frequency,. has the same magnitude but opposite sign for each delta function in Eq The resulting function of the harmonic frequency can then be factored out and a difference of delta functions remain, i.e., those in Eq Thus the relationship g I ( 1, 2 ) tanh( ( 1 /4 2 /2)) g R ( 1, 2 ) is nearly exact for the harmonic system in the classical limit. The small errors associated with this approximation serve to slightly overweight the physical processes associated with the terms that are not accounted for exactly. In the spirit of quantum correction schemes, 24,32,33 in which an exact result for a relatively simple system is used to make quantum mechanical expressions amenable to classical calculation, the relationship g I ( 1, 2 ) tanh( ( 1 /4 2 /2))g R ( 1, 2 ) is used in Eq to give in the classical limit R (5) 1, 2 1/ 2 1 /2 2 2 g R 1, 2 1 g R 1, 2. In the time domain, Eq takes the form, R (5) t 1,t 2 2 /2 2 / t 1 2 g R t 1,t / t 1 t 2 g R t 1,t Equation 2.41 is a principle result of this paper and represents an approximation to the fifth order response function that only requires calculating a classical TCF, albeit a somewhat novel correlation of three variables with two time arguments, identifying g R (t 1,t 2 ) (0) (t 1 ) (t 1 t 2 ) in the classical limit with,. Here we have identified g R (t 1,t 2 ) as the correlation function of the polarizability fluctuations; this does not affect the response function or the formal development of the TCF theory, but is a necessary step. There is a remaining problem with the result g I ( 1, 2 ) tanh( ( 1 /4 2 /2))g R ( 1, 2 ) in that it works well for the higher order terms g 211, g 121, and g 112, but is wrong for the lower terms, g 011, g 101, and g 110, that have exact relationships between the real and imaginary parts that differ from the choice made here; the exact relationships lead to the cancellation between the two terms in Eq and thus no contribution to the fifth order response. Physically, calculating R (5) (t 1,t 2 ) using Eq and g R (t 1,t 2 ) (0) (t 1 ) (t 1 t 2 ) leads to contamination from the lower order terms involving 0, the static polarizability, that do not contribute to the fifth order experiment. To correct this problem it is sufficient to use the correlation function of the polarizability fluctuations, g R (t 1,t 2 ) (0) (t 1 ) (t 1 t 2 ). To see that this eliminates these contributions consider a harmonic system in the energy representation with the polarizabilty expanded via Eq. 2.29, 0 i i Q i 1/2 i i Q 2 i. The second term is clearly zero and the third term is order and would not contribute classically. Thus correlating the polarizability fluctuations eliminates contributions from the static polarizability by construction and thus does not allow the lower order contaminating terms, g 011, g 101 and g 110 to contribute to the correlation function. This particular demonstration relies on a harmonic system but a similar argument, based on orders of is clearly possible; all terms but the first are proportional to some power of. Further, it is easy to show that the original R (5) (t 1,t 2 ) expression, Eq. 1.1, is unaffected by the substitution of for and thus the TCF theory of the fifth order response is established. Equation 2.41 can also be obtained more directly by starting with the the fifth order response function directly, Eq below. To obtain a classical limit, one of the commutators of the polarizability is replaced by a Poisson bracket, its classical limit, A,B i A,B ; the curly brackets represent the Poisson bracket of operators A and B. 14 The resulting expression contains a classical TCF piece and Poisson bracket contribution, but here the quantum version is retained. It is worth noting that the present TCF theory of the fifth order Raman response function results in a TCF approximation for the Poisson bracket contribution to the classical response function. Rewriting Eq. 1.1 gives R (5) t 1,t 2 i 2 t 1 t 2, t 1, 0, 2.42 R (5) t 1,t / t / t 1 t 2 0 t 1 t 1 t 2 i / t 1 t 2 0 t 1 t 2 t The first term, Eq is already a TCF contribution. The second term can be identified, using Eqs. 2.4 and 2.7 as g(t 1,t 2 ) h*(t 1,t 2 ). Using Eq. 2.14, this can be rewritten in frequency as g( 1, 2 ) e 2g( 1, 2 ). Taylor expanding the exponential and keeping terms order gives 2g I ( 1, 2 ) 2 g R ( 1, 2 ). Using g I ( 1, 2 ) tanh( ( 1 /4 2 /2))g R ( 1, 2 ) gives the result in Eqs and This approach does, however, immediately invoke the classical limit, and the generality of Eq is lost. Next the behavior of our theory of the fifth order response function, Eq is assessed. It is easy to show that at the time origin, t 1 t 2 0, R (5) (0,0) 0. This implies that ( 2 / t 2 1 (0) (t 1 ) (t 1 t 2 ) ) 0,0 2( 2 / t 1 t 2 181/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

182 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 Fifth order Raman response 6079 (0) (t 1 ) (t 1 t 2 ) ) 0,0, and this can be easily verified; 28 the vertical line denotes evaluating a function at the specified time points. Perhaps a more demanding test is that R (5) (t 1,t 2 0) 0. This implies that ( 2 2 / t 1 (0) (t 1 ) (t 1 t 2 ) ) t1,0 2( 2 / t 1 t 2 (0) (t 1 ) (t 1 t 2 ) ) t1,0, which is also true. It is encouraging that we find these correct limiting behaviors. To test Eq against experimental and other theoretical approaches, the two time TCF was calculated for ambient CS 2. Classical MD simulations were performed using a code developed at the Center for Molecular Modeling at the University of Pennsylvania, which uses reversible integration and extended system techniques. 34,35 Microcanonical MD simulations were performed on ambient CS 2 with an appropriate density and an average temperature of 298 K. 7.0 million four femtosecond MD time steps were performed, and TCF data was correlated every 8.0 fs femtoseconds for a total of 3.5 million configurations; the fifth order response functions derived from these computations were smoothed by averaging every five points together to better visualize the overall shape of the resulting functions. A shorter run of a million configurations was performed correlating the polarizability data every 40 fs and this will be discussed in Sec. III. The results were tested and found to be system size independent, and the results that are presented were generated from 50 molecule simulations. A simple, flexible CS 2 model was employed; details are summarized in earlier publications ,41,42 Briefly, the model includes harmonic stretching and bending potentials, as well as linear cross terms derived to reproduce the gas phase vibrational frequencies. 43 The electrostatics are represented by partial charges on the atoms, which are chosen to reproduce the average condensed phase quadrapole moment. 42 Polarization forces are not included in the dynamics, and are not essential to reproducing the spectroscopy. To model the polarizability of CS 2, when calculating spectroscopic quantities, many body polarization equations are solved exactly by matrix inversion to calculate the system polarizability matrix. 37,41,44,45 A PAPA polarizability model is employed that is fit to reproduce the polarizability tensor of an isolated molecule. 44,45,49 The potential and polarizability parameters were summarized previously The fifth order signal is also sensitive to the polarizability derivatives with respect to the vibrations. PAPA polarizability models naturally incorporate parameters that determine the polarizability derivatives. To implement this, it is sufficient to make the point polarizabilities on the atomic centers carbon and sulfur bond length dependent. 46,47,50 The point polarizabilities then change as (r) 0 (r) r, where r is displacement from the equilibrium bond length. The parameters for carbon and sulfur ( C Å 2, S Å 2 ) were fit to reproduce density functional calculations at the b3lyp/aug-cc-pvdz level. FIG. 1. The two time correlation function of the polarizability, (0) (t 1 ) (t 1 t 2 ), is shown. As an initial test of the TCF theory the fully polarized response of ambient CS 2 is considered. Figure 1 shows the two time TCF, (0) (t 1 ) (t 1 t 2 ) for the fully polarized geometry with all of the polarizability elements are the same Cartesian direction. The TCF decays to zero at long times with only limited structure at intermediate times, in contrast to its highly structured derivatives. The function decays most rapidly between 0 and 500 fs and shows some oscillatory behavior along t 2, with t 1 0. Even after significant averaging the TCF shows some noise at longer times, although the short time information between 0 and 1 ps, that is significant in calculating the fifth order response function is well averaged. Figures 2 a and 2 b present the magnitude of the fully polarized fifth order response function, R (5) (t 1,t 2 ) calculated using Eq with two different TCF time steps. The limiting behaviors, with R (5) (t 1,t 2 ) going to zero at the origin and everywhere along t 2 0 are apparent. III. RESULTS FOR AMBIENT CS 2 FIG. 2. The application of the present theory of R (5) (t 1,t 2 ) is shown for ambient CS 2. The TCF used to generate the fifth order response function is calculated with a time step of a dt ps and b dt 0.04 ps. 182/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

183 6080 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 DeVane et al. Further, along t 1 0 the response exhibits an intermediate-time plateau. This behavior has been observed experimentally for ambient CS 2. 1,2,26 The ridge along t 1 0 has also been observed in numerically exact calculations of R (5) performed in liquid xenon 17 and for model systems. 9,20,22,29 It is encouraging that the present TCF theory captures this distinctive feature of the fifth order response. The fifth order response function does exhibit oscillatory behavior with a frequency corresponding to symmetric stretching ( 658 cm 1 ). To verify the origin of the feature a CS 2 model was run with a frequency 20 cm 1 larger and the oscillations in the signal appeared with the new frequency. The symmetric stretch is the only vibrational mode that has a nonzero first polarizability derivative and all vibrational coordinates have zero second polarizability derivatives in the gas phase. The physical mechanism permitting the intramolecular mode signal appearing in the fifth order response function is currently under investigation and may be interaction induced with intermolecular couplings leading to nonzero second polarizability derivatives for a particular mode. Nonzero second polarizability derivatives would be required for at least an independent harmonic mode to contribute as can be seen in Eq Other theoretical investigations in CS 2 have used rigid models 51,52 or do not state/control the polarizability derivatives to match experimental values. 26 Figure 2 b shows the response that results from calculating the TCF using a larger time step that naturally filters out most of the high frequency oscillations and makes the overall shape of the surface easier to discern. Overall, in the fifth order response, a single large peak is found at short times for both time arguments, and this is similar to the experimental results that are available. Recent work by Jansen et al. has also described the fully polarized fifth order response of ambient CS 2 using a nonequilibrium simulation/finite field approach that can be shown to reproduce the third order polarization response calculated for the same model using TCF methods. 51,52 The results are difficult to compare because the polarizabilty models used are different. The present model was employed because it quantitatively reproduced the OKE spectrum of liquid CS 2 from boiling to freezing at atmospheric pressure. In any case, Jansen obtained results for four different polarizabilty models all of which show the signal decaying over like time scales and exhibiting limiting behaviors similar to the results presented here. Trying our theory with different polarization models is planned for future investigations. Figures 3 a 3 d highlight time slices of R (5), t 1 0, t 2 0, and t 1 t 2, and Fig. 3 d compares the theoretical time slices on an absolute scale. Figures 3 a 3 c also show recently published experimental data for the same time slices; the experimental measurements are not in absolute units and are scaled to match the theoretical data. In contrast to the theoretical results, the experimental measurements 1 are not simply the nuclear contribution to R (5). They also include an electronic response at short times and perhaps some contamination from other processes. 1 Nonetheless, the agreement, especially the rate and location of decay of the functions is very similar outside of 100 fs although it appears somewhat longer for t 1 0). Up to 100 fs the electronic con- FIG. 3. Three time slices of R (5), a along t 2 with t 1 0, b along t 1 with t 2 0, and c along t 1 t 2 are shown. The theoretical results lines with symbols are compared with recent experimental measurements lines with no symbols Ref. 1. d The theoretical times slices are compared with an absolute scale (t1 0, lines with circles; t1 t2, lines with triangles; t2 0 no symbols. In this and subsequent figures the theoeretical results are given in the stated absolute units with in Eq taken as unity. 183/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

184 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 Fifth order Raman response 6081 tribution to the signal is dominant experimentally. A measure of this is the signal along t 2 0 where the nuclear fifth order response vanishes and the principle contribution to the experimental signal is from electronic response. Our theoretical expression, Eq. 2.41, also gives zero along t 2 0 and the theoretical data demonstrate this result. The first few points are small nonzero values that are statistically equivalent to zero; taking derivatives at points close to the axis requires numerical algorithms that are correct to lower order, and the signal results from the cancellation of positive and negative contributions. Along t 1 0 both sets of data show a rapid decay and a plateau at about 500 fs that can be seen as a ridge in Figs. 2 a 2 b, and the agreement between the two sets of data is excellent. The theoretical data show an oscillation with a period characteristic of the symmetric stretch. This oscillation, in phase and magnitude, is also hinted at in the experiments, but it is unclear how noisy the experimental measurements are. Fourier transforming both the theoretical and experimental one-dimensional data sets along t 1 0) does give a peak of similar width in the symmetric stretching spectral region. At longer times, after a few picoseconds, the response function approaches zero everywhere. Along t 1 t 2 there is a small or no signal and no significant echo that has been the subject of discussion concerning R (5). 17 The experimental data along t 1 t 2 are very similar to t 2 0 again reflecting the small nuclear response that is observed. A large contribution along t 1 t 2 could be interpreted as an echo, indicating the existence of long lived intermolecular modes but this appears not to be the case. This absence of a significant response along t 1 t 2 suggests that intermolecular modes are highly damped. From Eq. 2.41, the contribution along t t 1 t 2 can be shown to result from the difference between two TCF s: (0) (t) (2t) (0) (t) (2t), in which represents a time derivative. Evidently the TCF s are very similar. IV. CONCLUSIONS One way to take the classical limit is to replace commutators with Poisson brackets. Then, R (5) is seen to contain brackets of variables at different times, which in principle requires the exceedingly difficult task of calculating the dependence of a many-body dynamical variable on its initial conditions. Correspondingly, one might hope 14 that R (5) contains some fundamentally new information about liquids, absent from conventional TCF s. Here we have reduced the problem to the calculation of a two-time TCF. This represents a considerable decrease in difficulty, even though such TCF are not currently well understood. One might also conclude that novel information is less likely to be found in R (5), although the question remains open. By invoking a two-time TCF our theory stands, on a scale of difficulty, in between calculation of a two-time bracket and the GLE Ref. 9 theory, which expresses R (5) with conventional TCF s, as does the mode coupling theory 15 of Denny and Reichman. Given the present computationally tractable theory for R (5), examining the temperature dependence of the signal for CS 2 and for other liquids will help establish the nature of the R (5) measurement and its variability. Further, using the current theory to examine other polarizations can provide valuable insights into the information content of fifth order Raman responses in molecular liquids. Also, the theoretical results for the harmonic system, Eqs can be used to formulate a quantum mechanically derived INM theory of R (5). 53,54 Constructing such a theory, that would include only harmonic dynamics, would permit comparison with the present TCF theory that is a hybrid of harmonic and fully anharmonic dynamical behaviors. Work along these lines is being pursued. ACKNOWLEDGMENTS The research at USF was supported by an NSF Grant No. CHE and a grant from the Petroleum Research Foundation to B.S. The authors would like to acknowledge the use of the services provided by the Research Oriented Computing center at USF. The authors also thank the Space Foundation for partial support. The research at BU was supported by NSF Grant No. CHE to T.K. 1 K. Kubarych, C. Milne, S. Lin, V. Astinov, and R. Miller, J. Chem. Phys. 116, V. Astinov, K. Kubarych, C. Milne, and R. D. Miller, Chem. Phys. Lett. 327, O. Golonzka, N. Demirdoven, M. Khalil, and A. Tokmakoff, J. Chem. Phys. 113, K. 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Mukamel, Principles of Nonlinear Optical Spectroscopy Oxford University Press, Oxford, J. Cao, S. Yang, and J. Wu, J. Chem. Phys. 116, T. Steffen, Ph.D. thesis, The University of Sydney, B. J. Berne, J. Jortner, and R. Gordon, J. Chem. Phys. 47, J. Borysow, M. Moraldi, and L. Frommhold, Mol. Phys. 56, E. W. Castner, Jr., Y. J. Chang, Y. C. Chu, and G. E. Walrafen, J. Chem. Phys. 102, S. Saito and I. Ohmine, Phys. Rev. Lett. 88, S. Saito and I. Ohmine, J. Chem. Phys. 106, D. A. McQuarrie, Statistical Mechanics Harper and Row, New York, K. Okumura and Y. Tanimura, J. Chem. Phys. 107, J. E. B. Wilson, J. Decius, and P. Cross, Molecular Vibrations Dover, New York, /224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

185 6082 J. Chem. Phys., Vol. 119, No. 12, 22 September 2003 DeVane et al. 31 R. DeVane, C. Ridley, T. Keyes, and B. Space, J. Chem. Phys. to be published. 32 H. Ahlborn, X. Ji, B. Space, and P. B. Moore, J. Chem. Phys. 112, J. L. Skinner, J. Chem. Phys. 107, P. B. Moore, Q. Zhong, T. Husslein, and M. L. Klein, FEBS Lett. 431, M. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys. 97, R. C. Weast, CRC Handbook of Chemistry and Physics CRC, West Palm Beach, FL, P. B. Moore, X. Ji, H. Ahlborn, and B. Space, Chem. Phys. Lett. 296, X. Ji, H. Ahlborn, P. Moore, and B. Space, J. Chem. Phys. 113, X. Ji, H. Ahlborn, B. Space, P. Moore, Y. Zhou, S. Constantine, and L. D. Ziegler, J. Chem. Phys. 112, S. Constantine, A. Gardecki, Y. Zhou, X. Ji, B. Space, and L. Ziegler, J. Phys. Chem. A 105, P. Moore and B. Space, J. Chem. Phys. 107, Y. Fujita and S. Ikawa, J. Chem. Phys. 103, G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules Van Nostrand, New York, T. Keyes, J. Chem. Phys. 104, T. Keyes, J. Chem. Phys. 106, K. A. Bode and J. Applequist, J. Phys. Chem. 100, J. Applequist, J. R. Carl, and K.-K. Fung, J. Am. Chem. Soc. 94, K. A. Bode and J. Applequist, J. Phys. Chem. 100, B. M. Ladanyi, T. Keyes, D. Tildesley, and W. Street, Mol. Phys. 39, J. Applequist and C. O. Quicksall, J. Chem. Phys. 66, T. I. Jansen, K. Duppen, and J. Snijders, Phys. Rev. B 67, T. I. Jansen, Ph.D. thesis, The University Groningen, B. Space, H. Rabitz, A. Lörincz, and P. Moore, J. Chem. Phys. 105, P. Moore, H. Ahlborn, and B. Space, in Liquid Dynamics Experiment, Simulation, and Theory, ACS Symposium Series, edited by M. D. Fayer and J. T. Fourkas American Chemical Society, New York, /224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

186 JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 18 8 MAY 2003 A combined time correlation function and instantaneous normal mode study of the sum frequency generation spectroscopy of the waterõvapor interface Angela Perry, Heather Ahlborn, and Brian Space a) Department of Chemistry, University of South Florida, Tampa, Florida Preston B. Moore Department of Chemistry and Biochemistry, University of the Sciences in Philadelphia, Philadelphia, Pennsylvania Received 27 November 2002; accepted 17 February 2003 Theoretical approximations to the interface specific sum frequency generation SFG spectrum of O H stretching at the water/vapor interface are constructed using time correlation function TCF and instantaneous normal mode INM methods. Both approaches lead to a SSP polarization geometry signal in excellent agreement with experimental measurements; the SFG spectrum of the entire water spectrum, both intermolecular and intramolecular, is reported. The observation that the INM spectrum is in agreement with the TCF result implies that motional narrowing effects play no role in the interfacial line shapes, in contrast to the O H stretching dynamics in the bulk that leads to a narrowed line shape. This implies that SSP SFG spectroscopy is a probe of structure with dynamics not represented in the signal. The INM approach permits the elucidation of the molecular basis for the observed signal, and the motions responsible for the SFG line shape are well approximated as local O H stretching modes. The complexity of the broad structured SFG signal is due to O H stretching motions facing toward the bulk or vacuum environments that are characteristic of the interface. The success of both approaches suggests that theory can play a crucial role in interpreting SFG spectroscopy at more complex interfaces. It is also found that many-body polarization effects account for most of the observed signal intensity American Institute of Physics. DOI: / I. INTRODUCTION a Author to whom correspondence should be addressed. Electronic mail: space@cas.usf.edu Liquid water interfaces are ubiquitous in chemistry and in the environment. Understanding their properties is of obvious importance. Thus, with the advent of interface specific spectroscopies, water interfaces have been intensely studied Sum frequency generation SFG spectroscopy is a powerful experimental method for probing interfacial structure. SFG is a second-order, electronically nonresonant, polarization experiment and is therefore dipole forbidden in isotropic media such as a bulk liquid. 27,28 It should be noted that bulk electric quadrapoles can contribute to the signal but appear to be negligible, and a demonstration of this fact will be presented in Sec. III below. 29 Interfaces, however, serve to break the symmetry and produce a signal. The SFG experiment is a frequency domain technique that employs both a visible and tunable infrared laser field overlapping in time and space at the interface. In the absence of any vibrational resonance at the instantaneous infrared laser frequency, a structureless signal due to the static hyperpolarizability of the interface is obtained. 1,6,24 When the infrared laser frequency is in tune with a vibration at the interface a resonant line shape is obtained with a characteristic shape. Thus this signal contains conformational information about the molecular species that are resident at the interface. The SFG signal would also be expected to include dynamical effects but it will be demonstrated here that it appears to be dominated by structural contributions, at least for aqueous interfaces. A limiting factor in fully exploiting SFG spectroscopy in the laboratory has been the ability to extract molecularly detailed information from the complicated spectra that are obtained. In this paper, classical molecular-dynamics MD methods are used to model the water/vapor interface. Two complementary theoretical approaches quantum corrected time-correlation function TCF and instantaneous normal mode INM methods are employed to describe the SFG spectrum of the interface and to ascertain the molecular origin of the SFG signal. This dual approach was demonstrated to be highly effective in understanding condensed phase spectroscopy of water and other liquids In particular, TCF methods provided a quantitative description of the O H stretching line shape in ambient liquid normal and heavy water and INM methods served to identify the molecular motions that result in the observed signal; this suggests that these complementary techniques are appropriate for modeling water interfacial spectroscopy. 30,32 34,36 The O H stretching absorption is intense a factor of 18 greater than the gas phase intensity; this is unusual because condensed phase resonances typically integrate to the gas phase value. Because it is such a strong resonance, many SFG studies /2003/118(18)/8411/9/$ American Institute of Physics 186/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

187 8412 J. Chem. Phys., Vol. 118, No. 18, 8 May 2003 Perry et al. have concentrated on its interfacial signature. While SFG is interface specific, there are nonetheless few absorbers present at an interface, and the signal is relatively weak. The central limitation in applying TCF and INM methods to SFG spectroscopy is the quantum-mechanical nature of the intramolecular vibrations that are probed experimentally. Due to the many-body nature of an interface one is limited to a classical description of the nuclear dynamics and a semiclassical representation of the spectroscopy. TCF and INM approximations both overcome this limitation in different ways. Classical mechanical TCF-derived spectra are formally exact, for a given Hamiltonian, in the low-frequency regime ( kt). But they suffer severely from the need for quantum correction at higher frequencies, typically including the entire intramolecular region of a spectrum and significant portions of the intermolecular spectrum of hydrogen bonded liquids. The quantum correction is typically done by multiplying the frequency domain spectrum the Fourier transform of the TCF 47 by a function chosen to produce an exact result for a model system. For example, a harmonic correction factor converts the classical harmonic TCF spectrum to its quantum-mechanical counterpart. 48 A key insight from our earlier work is that, while the commonly used factors are very different in their high-frequency form, they are often smooth, structureless functions over the relevant frequency range This implies that the classical line shape is reasonable, even if the dynamics from which it results seems distant from the true quantum-mechanical, often ground-state dominated, oscillators extant in the liquid. Further, it was found in the bulk that the harmonic correction factor led to quantitative agreement in both line shape and intensity with experiment for the O H stretch and this approach is adopted in the present work. An INM approximation to SFG spectroscopy is quantum mechanical by construction but is a limited dynamical description. As a result, in bulk water and other liquid state intramolecular line shapes INM intramolecular resonances are broader than their TCF counterparts, but with the same central frequency and integrated intensity. This suggests that the intramolecular INM spectra represent an underlying spectral density that is dynamically motionally narrowed in the actual line shape. 57 However, in applying INM methods to the SSP SFG spectroscopy of the water/vapor interface, INM and SFG spectra are in agreement. This suggests that dynamical motional narrowing effects are not important at the interface, and that the dynamics can be described in the slow modulation limit of motional narrowing. In that limit all frequency fluctuations of the oscillator are represented in the line shape. 35,57,58 While dynamical effects are apparently not well represented in the common SSP polarization geometry, a recent study in the Shen group suggests that motional narrowing effects may well be significant in the SPS geometry, and in that case the free O H stretching peak is greatly diminished. 59 That work also argues that motional effects are not important in the SSP and PPP configurations, and this supports our hypothesis. This result is consistent with an earlier theoretical description of SFG spectroscopy at the water/vapor interface. 23 Thus, as a prelude to more complex interfaces, this joint TCF/INM approach is applied to the water/vapor interface producing excellent agreement with experimental SFG measurements. The molecular dynamics, dipole, and many-body polarization methods and associated parameters are summarized along with a quantum correction scheme for the resonant contribution to the SFG signal in Sec. II. The theoretical results and their comparison to experiment are discussed in Sec. III. The paper is concluded in Sec. IV. II. MODELS AND METHODS The SFG signal consists of a nonresonant due to the static hyperpolarizability and resonant contribution that is important when the infrared laser frequency is tuned to a vibrational transition at the interface. The signal intensity is proportional to the square of their sum: I SFG ( ) R ( ) NR ( ) 2, where the subscripts respectively denote the resonant and nonresonant contributions. The molecular information is contained in the resonant signal, and this signal has been recently determined via measuring the nonresonant contribution in D 2 O for the O H stretching region it is unaffected by isotopic substitution. 1 Independently measuring this contribution permits the deconvolution of the full H 2 O signal to extract R ( ) for the O H stretching region in H 2 O. R ( ) is given by 27,28 R i dte i t Tr, i jk t In Eq. 2.1 e H /Q for a system with Hamiltonian H and partition function Q at reciprocal temperature 1/kT, and k is Boltzmann s constant; is the system dipole and its polarizability tensor where the subscripts represent the vector and tensor components of interest, respectively. The operator evaluated at time t is the Heisenberg representation of the operator jk (t) e iht/ jk e iht/ ; Tr represents the trace of the operators. It is convenient to proceed by rewriting the Fourier-Laplace transform in Eq. 2.1 as the Fourier transform of a correlation function that can then be interpreted in the classical limit and quantum corrected. Evaluating the commutator in Eq. 2.1 gives Tr, i jk t C t C* t 2i Im C t, 2.2 C t i jk t. In Eq. 2.2 the superscript star is the complex conjugate and Im denotes the imaginary part; angle brackets are the trace of the operators divided by the partition function in the standard notation. 47 The imaginary part of the one time-correlation function is related in frequency space exactly to the real part: C I ( ) tanh( /2)C R ( ), where the subscripts denote the Fourier transform of the real and imaginary parts of C(t), both of which are themselves real functions of frequency. Using this result for the trace in Eq. 2.1 gives R i dte i t D R t, 0 D R t 2i d e i t tanh /2 C R /224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

188 J. Chem. Phys., Vol. 118, No. 18, 8 May 2003 Study of sum frequency generation spectra 8413 After changing the order of integration in Eq. 2.3 R ( ) takes the form R 2 tanh /2 C R, C R 1 dte i t C 2 R t. 2.4 In the classical limit, C R (t) becomes the classical cross correlation function of the system dipole and polarizability elements, i jk (t). Unfortunately, SFG experiments focus on high-frequency spectra where ( kt) and classical mechanics is clearly invalid. Building on our previous work, the classical correlation function result, that is amenable to calculation using MD and TCF methods, is quantum corrected using a harmonic correction factor: C R ( ) C Cl ( ) /(1 e ). This correction factor is exact in relating the classical harmonic coordinate correlation function to its quantum-mechanical counterpart. Here we are using it to correct functions, the dipole and polarizability, that contain higher orders of the coordinates, and exact corrections for harmonic systems of this type are still possible but unneeded here. 49 Using this result, the TCF approximation to the resonant part of the SFG spectrum, R TCF, takes the form TCF R 2 1 e C Cl, C Cl t i 0 jk t. 2.5 In Eq. 2.5, the angle brackets represent a classical TCF that can be computed using MD and a suitable spectroscopic model. 60 A similar TCF approach was adopted earlier for modeling the SFG spectrum of both solid 28 and liquid interfaces, 24 although quantum corrections were not included in those cases. To construct an INM approximation to Eq. 2.1 it is sufficient to evaluate the trace in Eq. 2.1 for a harmonic system. It is convenient to invoke both the Placzek and linear dipole approximation to evaluate the resulting matrix elements, although higher-order contributions can be included and simple analytic expressions result for these contributions. An equivalent approach is to evaluate C Cl (t) for classical harmonic oscillators and quantum correct the resulting expression using the harmonic correction factor. 42 The resulting expression for the INM SFG spectrum, INM R considering the contribution of a single INM with coordinate Q l : INM R 2 1 e C Cl, C Cl i / Q l jk / Q l l kt In Eq. 2.6, l is the frequency of mode Q l and the angle brackets here represent averaging over classical configurations of the system generated. Below it will be demonstrated that the TCF approach, that does not invoke the Placzek and linear dipole approximation, give results in close agreement with the INM results and Eq. 2.6 is therefore sufficient. Molecular-dynamics MD simulations were performed using a code developed at the Center for Molecular Modeling at the University of Pennsylvania, which uses reversible integration and extended system techniques. 61 Microcanonical MD simulations were performed on ambient H 2 O with a density of 1.0 g/cm 3 and an average temperature of 298 K. To create an interface, a cubic simulation box of equilibrated liquid water was extended doubled along the z axis and the system was allowed to equilibrate creating two water/vapor interfaces. The interfaces are sufficiently far apart not to interact strongly and Ewald summation was included in three dimensions. 26 The density profile of the system was monitored to verify equilibration. 26 In all cases the results were tested and found to be system size independent. Most results were generated from 512 molecule simulations, and smaller system sizes down to 108 molecules were tried and did not alter the results. MD simulations were conducted using a flexible simple point charge SPC model that includes a harmonic bending potential, linear cross terms, and Morse O H stretching potentials as in our previous work. 62 In performing the MD, partial point charges are placed on the atoms that were chosen to reproduce the condensed phase dipole moment. At the water/vapor interface, the true water dipole falls from its condensed phase value, about 2.4 D, to that in the gas phase, 1.8 D, over a distance of only a few molecular layers. 63 It would seem that polarizable dynamics would be essential to model the dynamics of aqueous interfaces. However, as will be demonstrated below, the SFG spectrum is not being significantly influenced by the dynamics, and the use of nonpolarizable MD seems to adequately represent the structure of the water/vapor interface. Evaluating the TCF in Eq. 2.5 presents a problem for interfacial systems. The interface was constructed using the standard MD geometry with vacuum/vapor above and below the water. 23,25 Unfortunately this produces two interfaces with average net dipoles in opposite directions. Calculating the SFG spectrum of the entire system would lead to partial cancellation of the SFG signal and meaningless results. Therefore the system is split into two pieces through the center of mass of the interfacial system along the direction normal to the interface. Each of the resulting subsystems is then handled separately, and each molecule is assigned to one half or the other for the entire length of the calculation. Still, a problem arises in that molecules at one interface can diffuse to the other interface over time. This is a problem in calculating the TCF spectra where different molecular contributions are all added to form a single net dipole and polarizability at each step that are then correlated to form C Cl (t). 60 Using the bulk diffusion constant for water it was estimated that it would take about ps for a molecule to diffuse from one interface to the other for the system sizes considered here. It is necessary to therefore limit the length of a TCF correlation time of 15 ps 108 particles or 30 ps 512 particles, and many correlations of this length are performed in calculating an averaged TCF. 188/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

189 8414 J. Chem. Phys., Vol. 118, No. 18, 8 May 2003 Perry et al. The need to do this can be understood from a single molecule perspective. If cross terms between the dipoles and polarizability elements could be neglected C Cl (t) could be written in terms of single molecule contributions: C Cl (t) N i M (0) jk M (t), where N is the number of molecules and the superscript M is the molecule index. 47 Given that bulk isotropic molecular motions give no SFG signal, it is the anisotropic dynamics of molecules at the interface that generates a signal. Both the polarizability tensor elements and dipole moment are independent of translational origin, thus it is only necessary to distinguish between molecules exhibiting bulk and interfacial dynamics to understand the contribution of a molecule to the TCF. If a molecule were to reside at both interfaces during the duration of the TCF calculation, invalid results would be obtained; this is clearly undesirable. For example, in the common SSP SFG geometry 5,6 it is the interface normal component of the dipole that is correlated. Further, if molecules were reassigned to a specific half of the system at each calculation time point, an asymmetry would be introduced at the dividing surface by including the dynamics of a molecule at only certain steps as it appears in and disappears from a given half. This would introduce an artificial inhomogeneity in truly bulklike isotropic dynamics that might generate an SFG signal. The fact the correlation function calculations are limited to short times and that the SFG TCF, a cross correlation between the system dipole and polarizability elements not invariants like in traditional Raman and infrared experiments, is long lived leads to poor averaging at longer times. This makes the signal difficult to accurately Fourier transform, even though the focus is on extracting the short time, high-frequency behaviors. This issue will be discussed further in Sec. III. This problem does not affect the INM calculation because it is local in time and the interface is split for each interfacial configuration. In fact, the INM calculation also directly accesses all frequency scales and the SFG spectrum was calculated for both intramolecular and intermolecular modes. The MD was performed without explicit polarization forces, but when the SFG TCF or INM spectrum is calculated, polarizability is included in our calculations. Two models are employed: the first model includes full manybody polarization effects included explicitly via a point atomic polarizability approximation PAPA polarizability model Another model was also employed with only single body isotropic scalar polarizability using the same point polarizabilities on the atoms ( O Å 3, H Å 3 ). 67 The latter model was used for computational efficiency because, like the experiment, the SFG theoretical calculations exhibit a low signal-to-noise ratio due to the relatively small number of molecules residing at the interface. Including many-body polarization resulted in a large increase in intensity but the line shape was little changed. This increase is due to the large dipole derivative that results in water when many-body polarization is included. 32,34 The SFG signal is sensitive to both dipole and polarizability derivatives. PAPA polarizability models naturally incorporate parameters that determine the polarizability derivatives. To implement this, it is sufficient to make the point polarizabilities on the atomic centers O H bond-length TABLE I. Cartesian components of the derivatives of the molecular polarizability of H 2 O with respect to the symmetric (Q ss ) and antisymmetric (Q as ) stretching modes are given. Only nonzero contributions are shown. The results of our model are shown in the first column and experimentally derived results are in the second column. The values are taken from Ref. 68. The data are in units of Å 2 1/2 m KpsÅ ; the mass units are those derived from a system with K as energy, ps as time, and Å as length. dependent The point polarizabilities then change as: (r) 0 (r) r, where r is displacement from the equilibrium bond length. The parameters for hydrogen and oxygen ( O.442 Å 2, H Å 2 ) were fit to both reproduce ab initio calculations and checked for a single gas phase molecule against experimental Raman cross sections and Cartesian polarizability derivatives derived from experiment; 68,69 excellent agreement was achieved. The results for our polarizability model compared to previous work are presented in Table I. In practice, other polarizability derivative models were tried and the resulting spectra were not very sensitive to this choice. The choice of the polarizability model itself, which determines the induced dipole derivatives, is the most important factor. This is consistent with the results that were obtained in bulk water, where induced dipole derivatives were responsible for 90% of the observed IR absorption and the permanent charge or gas phase dipole derivative portion contributes little and with a different line shape. 34 The PAPA model accounts very accurately for induced dipole derivatives, and this is evidenced by the success of the model in the condensed phase. In addition, a permanent dipole mode, fit to ab initio calculations used in another SFG study, 24 was also adopted and checked successfully against experimental IR intensities. 69 This was done due to concerns that free O H oscillators extending beyond the interface into the vacuum may have a dipole derivative that is not dominated by induced effects, but using a very accurate model did not significantly alter the resulting spectra. III. DISCUSSION Model Experiment xx / Q ss yy / Q ss zz / Q ss xz / Q as Figure 1 presents a comparison of experimental 1 and both TCF and INM derived theoretical descriptions of the resonant contribution, R ( ) 2, for the water/vapor interface. It should be noted that the nonresonant contribution can be well approximated using ab initio methods and is typically structureless with significant amplitude; 24 the focus here is on the resonant portion to compare with the recent experimental measurements of the resonant contribution directly. The spectrum is in the SSP geometry that correlates the dipole moment component normal to the interface with diagonal polarizability matrix elements in the plane of the interface e.g., z (0) xx (t), with the z axis taken as the surface normal direction. This geometry leads to the most intense spectrum due to a relatively sizable net normal dipole 189/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

190 J. Chem. Phys., Vol. 118, No. 18, 8 May 2003 Study of sum frequency generation spectra 8415 FIG. 1. SFG spectra calculated using TCF thick gray line and INM thin solid line methods and reported experimentally dashed line Ref. 1 for the water/vapor interface. The TCF and INM SFG spectra are generated from MD runs with 512 molecules and the scalar polarizability model is used. The TCF correlation time is an average of several 15-ps runs. The INM result is the average of molecule configurations. moment at the interface and the relatively large diagonal polarizability elements; water has a nearly diagonal polarizability matrix with nearly equal elements in both the gas phase and bulk. In this initial paper, we will focus exclusively on the SSP geometry. The experimental data are a fit to a deconvolution of the measured SFG spectrum in which the nonresonant signal is removed. Because they are a fit to analytic functions the experimental data do not include statistical noise. 1,13 The agreement with experiment is outstanding in both cases and is within the statistical error over most of the frequency range. There are certain differences between the theoretical and experimental spectra. When compared with the experimental spectrum, the INM free O H peak is slightly blueshifted while the corresponding TCF line shape displays some asymmetry. At the highest frequencies presented the experimental fit goes to zero, although the TCF and INM results are still showing intensity. The origin of these differences is difficult to determine and is the subject of ongoing investigation. Both the TCF and INM results in Fig. 1 are the result of using the scalar polarizability model detailed in Sec. II. The comparison between the INM and TCF spectra are in absolute units, and no parameters are adjusted to bring them into agreement. However, the experimental measurements are in arbitrary units, thus a reasonable magnitude was chosen for comparison. Figure 2 compares a TCF result using full many-body polarization and scalar polarizability. It is clear that many-body polarization leads to a much more intense signal, and it is this added intensity that many-body polarizability provides that makes the water O H stretch a desirable experimental probe. The fact that the data in Fig. 1 agree well with experiment is further evidence that many-body polarizability modifies the intensity more than the line shapes. In Fig. 2 a the data are highly smoothed for comparison, and the unsmoothed Fourier transform of the TCF is presented in Fig. 2 b. The need to smooth the spectra arise FIG. 2. a Smoothed TCF SFG spectra generated from several 15-ps runs with 125 molecules using full many-body polarization top, larger magnitude line and scalar polarizability bottom line are shown. b Unsmoothed TCF SFG spectra are shown. from two primary sources. First, low signal to noise is inherent in SFG spectroscopy due to the small number of absorbers at an interface. Second, as discussed in Sec. II, our TCF s are not well averaged at long times, and it was found that the TCF s had not reached their final plateau values at 15 ps, the longest times that could be safely accessed. This makes Fourier transforming difficult and noisy and leads to highfrequency contamination of the data that requires smoothing. 70 It may be desirable to simulate larger systems to access longer times, but this requires more computational effort, and these calculations are already reasonably challenging. It is also possible to fit the TCF, assuming a reasonable long term behavior that will not affect the highfrequency O H stretching region in any case and subsequently transforming the fit function; this is an avenue that is being pursued. The INM calculations do not suffer from the second limitation and can directly access both the high- and lowfrequency information. The limitation in INM calculations is the large matrix manipulations that are involved that are of order N 3. This limitation can be overcome by only calculating INM s at the interface, and not including most of the bulk molecules in the force-constant matrix. As long as the modes are sufficiently localized as they are in water in the spectral region of interest this approach should not perturb the interfacial modes that contribute to the SFG signal, and this approach is being implemented to investigate more complex interfaces. The close agreement between the INM and TCF spectra in Fig. 1 is striking. INM approximations to spectroscopy offer only a limited dynamical description and describe an 190/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

191 8416 J. Chem. Phys., Vol. 118, No. 18, 8 May 2003 Perry et al. underlying spectral density that is typically broader than the observed line shape when considering intramolecular modes. Dynamical effects can serve to narrow the underlying spectral density to give the observed line shape. This can be understood from a motional narrowing perspective, in which an harmonic oscillator with a time-dependent frequency leads to a line shape that is narrowed compared to the frequency fluctuations inherent in its dynamics. 57 Indeed, all INM spectroscopy calculations to date have led to line shapes either broader or equivalent to analogous TCF calculations, in which motional narrowing is naturally included. Here INM and TCF calculations that are a result of the same Hamiltonian and spectral model are being considered. Motional narrowing does have a limit in which all frequency fluctuations of a system are exhibited in the line shape, the slow modulation limit. In bulk liquid systems this limit was apparently achieved in intermolecular spectra, and TCF and INM spectra are nearly the same in that case ,36,40,46 However, it is surprising that the high-frequency O H stretching motions can be described in the slow modulation limit, as is suggested by the close agreement between the INM and TCF calculations. The physical origin of this observation is the subject of ongoing investigations, but the observations suggest the nature of the modal frequency fluctuations at the interface are significantly different than in the bulk. The near correspondence of the INM and TCF spectra also implies that the linear dipole and Placzek approximation are sufficient to understand the SFG spectroscopy of the water/vapor interface. In Eq. 2.6, these linear approximations were invoked and proved effective. Further, it has been suggested that bulk quadrapoler contributions may be present in the experimental signal. While these contributions could be explicitly in a TCF calculation the present water model has a realistic quadrapole moment, the close agreement between theory and experiment strongly suggests that this contribution is negligible, as has been demonstrated experimentally. 29 Figure 3 a displays the INM SFG spectrum using the scalar polarizability model over the full water frequency range. Figure 3 b concentrates on the intermolecular spectrum. Clearly the intensity of the intermolecular spectrum is much lower than that of the bend and O H stretching region. In calculating the INM intermolecular spectrum the imaginary frequency modes are included as contributing like the real modes at the magnitude of their imaginary frequency, as in previous work. 34,35 The bend itself is not nearly as intense as the O H stretching region, and the relative intensities presented here are a lower limit. The INM results presented do not include many body polarization effects, and the bend intensity is not significantly enhanced by these effects but, as demonstrated above, the O H stretch signature is enhanced. The intermolecular spectrum is similar in breadth to that observed experimentally for bulk IR spectroscopy but is relatively less intense compared to the intramolecular resonances. Given that the range of the intermolecular spectrum is similar to the bulk IR and Raman spectrum 71 suggests that the intermolecular spectrum can also be described in the slow modulation limit and that the INM spectrum would be a FIG. 3. a The INM SFG spectra generated from Fig. 1 is shown for an expanded frequency range. The O H stretching region is off scale in the graph. b The INM SFG intermolecular spectrum is shown. good approximation to the true spectrum. The SFG signal does show characteristic features at about 200 and 500 cm 1 that are also seen in the bulk. 34 A more careful analysis of the intermolecular spectrum, including many-body polarization effects that significantly alter the shape in the bulk, is needed. However, being able to easily access without long time information the low-frequency SFG spectrum using INM methods demonstrates the strength of this approach. INM methods, especially when in near agreement with TCF calculations and experimental measurements, permit the detailed elucidation of the molecular motions giving rise to the observed line shape. When the TCF and INM spectrum are similar, it is clear that a mode is contributing to the spectrum at its frequency, and it is not necessary to know how the INM frequency is affected by motional narrowing of the underlying spectral density. Figure 4 shows an INM spectrum and its projections onto gas phase normal modes, the symmetric and antisymmetric stretch. The projections are performed as in our previous work. 62 In the condensed phase IR spectrum, it was found that the symmetric and antisymmetric stretching motions were roughly equally contributing to the line shape. This equal contribution amounts to taking a linear combination of the symmetric and antisymmetric stretches that produce a simple O H stretching motion; the bulk normal modes are local O H stretching modes. At the interface, Fig. 4 demonstrates that the antisymmetric stretch dominates at higher frequencies and the symmetric stretch provides most of the intensity at lower frequencies. The interface induces an asymmetry in the symmetric and antisymmetric contributions, although both are still making a significant contribution at all frequencies. As a result, the interfacial normal motions are also es- 191/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

192 J. Chem. Phys., Vol. 118, No. 18, 8 May 2003 Study of sum frequency generation spectra 8417 FIG. 4. An INM SFG spectrum solid line from a few configurations is shown with its projection onto gas phase normal modes. The symmetric stretch contribution is represented by the solid gray line. The antisymmetric stretch projection is represented by the dashed line. sentially O H local stretching modes, as was hypothesized in an earlier SFG study. 23 The asymmetry is a result of the fact that some of the O H modes are spanning the interface, oscillating in free space. These so-called free O H modes produce the high-frequency feature at about 3700 cm 1 that is a characteristic signature of aqueous interfaces. Figure 5 highlights a typical mode at 3700 cm 1 from the free O H spectral region and it is clear that the oxygen atoms are anchored in the interface and the O H is oscillating freely above the interface. Clearly the mode has some antisymmetric stretching character but is closer to an O H local mode. Another point of interest has been the spectral location of the other O H on a water molecule with a free O H. 6,13,23 The INM projected spectrum implies that while the free O H appears as the antisymmetric stretch component the other O H is largely the symmetric stretching contribution; water molecules spanning the interfaces are the dominant species contributing to the SFG spectrum. Figure 5 demonstrates this point by also showing the mode of the O H for which the other stretch is a free O H. This complimentary mode is a local O H stretching motion with some symmetric stretching character and it contributes at 3415 cm 1 in the region dominated by symmetric stretching contributions. These results demonstrate how the INM approach does not require a priori assumptions about the nature of interfacial modes but does reveal their physical characteristics and how different molecular motions contribute to the spectrum. To further demonstrate that the dominant motions are simple O H stretching modes a simple model density of states was constructed, 32 and is shown in Fig. 6. First, the second derivative of the O H Morse potential the force constant at a given local O H stretching mode s instantaneous bond length is evaluated. The reduced mass of the O H bond is nearly that of a hydrogen atom and is a taken as a constant fit to reproduce the INM density of states 72,73 that handles the reduced mass exactly. Then the effective frequency for each oscillator can be calculated as the square root of the force constant over the reduced mass. This density of states DOS is constructed as a function of the molecular layers at the interface. The top layer has a shape characteristic of the SFG FIG. 5. Color A snapshot of a water/ vapor interface containing 512 molecules. A typical INM is shown that is localized on a water molecule that left has one O H stretch that is a free O H at 3700 cm 1 and right the other O H stretch at 3415 cm 1 for the same molecule. 192/224 Downloaded 21 Jul 2005 to Redistribution subject to AIP license or copyright, see

193 8418 J. Chem. Phys., Vol. 118, No. 18, 8 May 2003 Perry et al. FIG. 6. The model O H DOS based on the distribution of O H bond lengths is represented as a function of the number of molecular layers. The bulk DOS is represented by the top thick-dashed line. The first layer thindashed line shows a free O H peak, and the second layer thick solid gray line is already bulklike. The experimental spectrum filled triangles Ref. 1 is also shown for comparison in arbitrary units. spectrum that is shown for comparison. While the SFG spectrum is not directly comparable to the density of states because it lacks weighting by the polarizability and dipole derivatives, the SFG spectrum serves to locate the characteristic features of the line shape. The density of states quickly converges to its bulk value reinforcing the interface specificity of SFG spectroscopy. If even the second molecular layer was making a significant contribution, the SFG spectrum would be similar to that of the bulk. The figure also serves to demonstrate that the DOS itself reflects the molecularly thin nature of the interface. Knowing the INM s and the intensity of their contribution to the INM spectrum allows for quantification of the signal as a function of molecular layers and such an analysis confirms that the signal is primarily a result of the top molecular layer. IV. CONCLUSION The combined use of INM and TCF approximations to SFG spectroscopy represent a powerful complementary approach. The TCF SFG spectrum is an excellent approximation of the line shape measured experimentally. Further, achieving agreement with experimental measurements engenders confidence in the MD and spectroscopic models used to produce the theoretical spectrum. SFG offers an additional observable to calibrate the MD model used to conduct the interfacial simulations. Thus other properties that can be derived from the simulation, but might be difficult to measure experimentally, can be extracted with more confidence from the MD. The INM SFG spectrum is in close agreement with the TCF results. This agreement for an intramolecular mode is surprising and suggests that the interfacial dynamics is qualitatively different than the bulk and that the dynamics probed by the SSP geometry SFG spectrum can be understood as being in the slow modulation limit of motional narrowing. The INM SFG spectrum also predicts an intermolecular SFG signal while it is difficult to obtain low-frequency long time information from the TCF approach. A significant strength of the INM approach is the ability to decompose the spectrum into individual molecular contributions. This was especially simple at the water/vapor interface because the modes are similar at least in their molecular identity if not dynamics to the bulk, essentially O H local stretching modes. It is anticipated that other aqueous interfaces will be somewhat similar because their SFG spectra are themselves similar. They are dominated by a narrow free O H and broader bulk facing O H peaks, although the relative intensities change. 16 The agreement between the INM and TCF derived spectra suggests that SSP SFG spectroscopy is probing primarily structural information; INM methods do not account for detailed dynamical behaviors. Further, a recent study suggests that another polarization geometry SPS may be quite sensitive to the dynamics at interfaces while the SSP and PPP experiments are not. 59 Thus SFG spectroscopy may be capable of giving a complete picture of the interface, including structure and dynamics. Realizing this promise depends critically on the spectra being reliably interpreted and the methods employed in this study are designed to unambiguously characterize the nature of SFG spectroscopy. This paper focused on the relatively simple water/vapor interface. It is planned to now investigate the behavior of the other polarizations and more complex and interesting interfaces using our combined INM/TCF approach. ACKNOWLEDGMENTS B.S. and P.M. would like to thank Professor M. L. Klein for encouragement in their collaboration. The research at USF was supported by an NSF Career Grant No. CHE and a grant from the Petroleum Research Foundation to Brian Space. The authors also thank the Space Foundation for partial support. 1 E. A. Raymond, T. L. Tarbuck, and G. L. Richmond, J. Phys. Chem. B 106, Y. Shen, Solid State Commun. 108, L. F. Scatena, M. G. Brown, and G. Richmond, Annu. Rev. Phys. Chem. 52, D. E. Gragson and G. L. Richmond, J. Phys. Chem. B 102, Y. Shen, Principles of Nonlinear Optics Wiley, New York, Q. Du, R. Superfine, E. Freysz, and Y. Shen, Phys. Rev. Lett. 70, C. Raduge, V. Pflumio, and Y. Shen, Chem. Phys. Lett. 274, Y. R. Shen, Proc. Natl. Acad. Sci. U.S.A. 93, E. Freysz, Q. Du, and Y. R. Shen, Ann. Phys. Paris 19, Q. Du, E. 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195 195/224 Biophysical Journal Volume 85 November A Molecular Dynamics Method for Calculating Molecular Volume Changes Appropriate for Biomolecular Simulation Russell DeVane,* Christina Ridley,* Randy W. Larsen,* Brian Space,* Preston B. Moore, y and Sunney I. Chan z *Department of Chemistry, University of South Florida, Tampa, Florida; y Department of Chemistry and Biochemistry, University of the Sciences in Philadelphia, Philadelphia, Pennsylvania; and z Academia Sinica, Taipei, Taiwan ABSTRACT Photothermal methods permit measurement of molecular volume changes of solvated molecules over nanosecond timescales. Such experiments are an important tool in investigating complex biophysical phenomena including identifying transient species in solution. Developing a microscopic understanding of the origin of volume changes in the condensed phase is needed to complement the experimental measurements. A molecular dynamics (MD) method exploiting available simulation methodology is demonstrated here that both mimics experimental measurements and provides microscopic resolution to the thermodynamic measurements. To calculate thermodynamic volume changes over time, isothermal-isobaric (NPT) MD is performed on a solution for a chosen length of time and the volume of the system is thus established. A further simulation is then performed by plucking out a solute molecule of interest to determine the volume of the system in its absence. The difference between these volumes is the thermodynamic volume of the solute molecule. NPT MD allows the volume of the system to fluctuate over time and this results in a statistical uncertainty in volumes that are calculated. It is found in the systems investigated here that simulations lasting a few nanoseconds can discern volume changes of ;1.0 ml/ mole. This precision is comparable to that achieved empirically, making the experimental and theoretical techniques synergistic. The technique is demonstrated here on model systems including neat water, both charged and neutral aqueous methane, and an aqueous b-sheet peptide. BACKGROUND A novel molecular dynamics (MD) method to determine both thermodynamic volumes of solvated molecules and time-dependent volume changes in the condensed phase, the pluck method, is presented here. The method utilizes contemporary MD methods (Martyna et al., 1996; Tuckerman and Martyna, 2000) specifically, extended system, isothermal-isobaric (NPT) MD to determine the volume of a system while simultaneously exploring the dynamics of the system. The thermodynamic volume of a solution is obtained directly from the volume coordinate in NPT MD. Consider, e.g., a single solute in a solvent. Solute volumes can be calculated by plucking the solvated species from the system, i.e., simulating the remaining system in the absence of solute-solvent forces. After re-equilibration, the volume of the remaining system, in this case pure solvent, is determined using NPT MD. The difference between the solution and solvent volumes determines the solute molecular volume. Plucking the molecule from an otherwise equilibrated system often produces an initial condition for a system configurationally near the new equilibrium and thus aids in equilibration; this is especially important in simulating complex systems. Also, in simple solutions consisting of a single yet variable solute, the volume of the neat solvent need only be determined once, simplifying the computation. Submitted November 11, 2002, and accepted for publication May 6, Address reprint requests to Brian Space, Dept. of Chemistry, University of South Florida, 4202 E. Fowler Ave., SCA400, Tampa, FL space@cas.usf.edu. Ó 2003 by the Biophysical Society /03/11/2801/07 $2.00 The method is, however, not limited to simple solutions and can be used in very complex biological simulations to determine molecular volumes of any of the components of an assembly of biomolecules and solvents. NPT dynamics produces a fluctuating volume coordinate and the thermodynamic volume is the average value over time. This leads to an uncertainty in the thermodynamic volume calculation that must be assessed. It is demonstrated in Results from a Simple Model System that the volume fluctuations are Gaussian and thus the standard deviation of the volume fluctuation is a useful measure of the uncertainty. However, as a consequence of the dynamical nature of the MD, successive volume values are not statistically independent. Following earlier work, the correlation time of the volume coordinate is calculated and only uncorrelated values are sampled. (Allen and Tildesley, 1989; Friedberg and Cameron, 1970; Jacucci and Rahman, 1984) This is equivalent to sampling more frequently and correcting for the correlation between volumes. For the aqueous systems investigated here solute volume uncertainties of ;1.0 ml/ mole are obtained from a few nanoseconds of dynamics and all the volume values in the article are given with respect to the change in solute volume. It should be noted that NPT MD algorithms are not strictly equivalent to microcanonical dynamics, although the method employed here samples the NPT ensemble exactly (Martyna et al., 1996). These methods couple the real system variables to fictitious variables that regulate the thermodynamic properties of interest (e.g., thermostat, the temperature; and barostat, the pressure) such that they fluctuate around the desired, preset average values. The methods for calculating thermodynamic volumes is therefore exact for a given MD potential energy model, and the only issue is whether dynamical events observed are physically relevant. NPT

196 196/ DeVane et al. dynamics does closely mimic true microcanonical (NVE) dynamics p (e.g., the perturbation of the dynamics is order 1= ffiffiffiffiffiffi 3N, where N is the number of atoms in the system), and has even been suggested as the method of choice for biophysical systems. (Hansson et al., 2002) The motivation in using NPT dynamics to calculate volume changes via this approach is to mimic photothermal experiments on biological systems that also determine molecular volume changes on nanosecond timescales with similar precision (Hansen et al., 2000; Larsen and Langley, 1999; Larsen et al., 1998). For example, photothermal experiments can identify protein/peptide intermediates with characteristic volumes that have lifetimes of several nanoseconds. The present theoretical methods can be employed in the same fashion. Transient species with characteristic volumes can be identified by statistically significant changes in the volume coordinate over time, indicative of metastable equilibrium between the solute and solvent. MD can then provide microscopic resolution to the observation by identifying the structures of any intermediates. Further, while the NPT MD is not dynamically exact, it is always possible to verify configurational events by repeating simulations microcanonically to verify the veracity of the dynamics. The dynamic interpretation of the proposed methods is therefore a computational convenience. Computational efficiency is, however, clearly desirable given the inherent challenge of simulating interesting biological systems for hundreds of nanoseconds, a relatively short timescale over which to look for large conformational changes in biopolymers. To access longer timescales and to sample volume efficiently, multiple timescale integration techniques are employed and permit the use of larger MD time steps (Tuckerman and Martyna, 2000). It is notable that photothermal methods can also map out enthalpy profiles over similar timescales, and MD directly complements these measurements by providing a molecular interpretation of the energetics. Other effective methods exist to calculate molecular volumes and related excess compressibilities (Matubayasi and Levy, 2000; Lockwood and Rossky, 1999; Lockwood et al., 2000; Dadarlat and Post, 2001), but the pluck method is ideally suited to modeling biological systems and their time evolution. The flexibility in dissecting the physical origin of volume changes using the pluck method will also be demonstrated by the examples that will be presented. For example, by varying potential energy interactions between a solute and solvent the volume can be dissected into different contributions in a thermodynamically consistent manner (Imai et al., 2001). It is important to note that the precision of measuring volume changes will be affected by the duration of the simulation that is (computationally) possible for larger systems. For example, when modeling large solvated proteins (e.g., [100 residues), simulations are currently limited to durations on the order of 100 ns. It is demonstrated below that 10 ns of MD would permit the determination of the volume to within ;4.0 ml/mole. Also, the protein volume is larger in this case and the relative error in the volume measurement is reduced. Further, volume changes associated with folding may be greater, on a perresidue basis, for larger proteins (Foygel et al., 1995). In Methods and Applications to a Model System, the molecular volume of a b-sheet peptide, which has been investigated experimentally using photothermal methods (Hansen et al., 2000), is calculated to demonstrate the analysis involved in the pluck method. The technique is then applied on model systems including neat water, both neutral and (fictitious) charged aqueous methane in Results from a Simple Model System. A series of molecules was chosen to highlight the methods ability to probe the relative volume changes associated with electrostriction and ionic solvation. The present methods can also be extended to calculate excess compressibilities by simulating at different pressures and calculating the compressibility via finite difference (Lockwood and Rossky, 1999; Lockwood et al., 2000; Dadarlat and Post, 2001). METHODS AND APPLICATIONS TO A MODEL SYSTEM Fig. 1 presents a snapshot of a solvated b-sheet peptide including a panel with the solvent removed for better visualization. The peptide was chosen because it is currently the subject of experimental investigation using FIGURE 1 A snapshot of the b-sheet peptide is shown (right) with solvent and (left) with water removed for better visualization and in a different orientation to better display the three-dimensional structure of the peptide. The colors represent the atom types as follows: C, green; O, red; N, blue; and H, white. Biophysical Journal 85(5)

197 197/224 An MD Method for Molecular Volumes 2803 photothermal methods (Hansen et al., 2000). Chan and co-workers synthesized a caged unfolded version of the peptide that can be photolysed in neat water solution to initiate peptide folding (Hansen et al., 2000; Chen et al., 2001). Using photothermal methods it is then possible to map out volume and enthalpy profiles during the roughly 1.0-ms folding process. The folded structure was created based on an NMR structure (Chen et al., 2001), and two separate folded systems were prepared and their equilibrium structures were compared and found to be essentially indistinguishable. To demonstrate the pluck method, Fig. 2 shows a time trace of the solvated peptide volume coordinate. For comparison, the relaxation of the system volume after the peptide is plucked from solutions is also shown. The inset of Fig. 2 shows the short time dynamics of the system to highlight the transient volume change. The system quickly approaches a new equilibrium and has a system volume characteristic of pure water 50.0 ps after removal of the peptide. Equilibrium volume fluctuations are then followed for several nanoseconds to determine the volume of the systems to a desired precision; 2.0 ns of the dynamics are shown in Fig. 2. Fluctuations of observable quantities from their means, obtained from MD simulations, are typically Gaussian. They can thus be characterized by their standard deviation, s. The inset of Fig. 3 presents a histogram of the volume fluctuations of the solvated peptide system. Clearly the distribution is well-described as Gaussian. If the successive values of the volume were uncorrelated, one could calculate the uncertainty of the volume simply as p DV ¼ s= : (1) In Eq. is the total number of samples. However, closely spaced values of an observable quantity (in this case the volume is of interest) are not statistically independent because they are connected to each other implicitly by the dynamical equations of motion. It is possible to define a correlation time, t c ¼ s Dt, during which the volumes are not independent, and s is the statistical inefficiency or number of correlated data measurements (Allen and Tildesley, 1989; Jacucci and Rahman, 1984). Multiplying s by Dt, the time length between successive measurements, gives the correlation time, t c. The s-value is calculated by performing volume averages over blocks of time of successively longer lengths ending with the entire length of the MD run. The parameter s is formally defined by Friedberg and Cameron (1970) as FIGURE 2 The curve with the larger average value shows a time trace of the volume fluctuations for the folded aqueous b-sheet peptide. The lower curve displays the relaxation of the neat water system volume to its equilibrium value after the peptide is plucked from solution. The inset presents the short time volume fluctuations to highlight the relaxation to equilibrium. Note that the volume of the plucked system transiently spikes to a value greater than the previous equilibrium fluctuations after the peptide is removed. The horizontal lines represent the average system volume. FIGURE 3 The uncertainty in the volume of the b-sheet peptide system as a function of the length of the MD simulation. Volume uncertainties for specific times of interest are also identified. The inset displays a histogram of the system volume fluctuations and a superimposed Gaussian function with a standard deviation calculated from the volume fluctuations. s ¼ lim t B s 2 B =s2 s 2 ¼ 1 N B B + ðhvi t B! N B ÿhviþ 2 : (2) B B¼1 In Eq. 2, N B is the number of blocks of length t B such that the product, N B t B ¼@, is the total number of samples. Thepinherent correlations then modify the uncertainty in the volume as DV ¼ ffiffiffiffiffiffiffi s=@ s. This modified formula reduces to Eq. 1 if the time between successive volume samples is longer than t c, and in that case s ¼ 1. For long MD runs it is computationally convenient to save a minimal amount of data and therefore choose a sampling time slightly greater than t c, and this approach was adopted; s and t c are determined initially via trial MD runs. Fig. 3 also displays a curve demonstrating how the DV values decrease over time for the solvated b-sheet peptide. The graph demonstrates that 50.0 ns of dynamics results in an uncertainty of 0.68 ml/mole. This severalns timescale corresponds to a typical photothermal experimental time resolution for identifying dynamical intermediates. The volume resolution over this timescale is sufficient to identify relatively modest conformational changes in a peptide/protein or other biomolecule. In fact, one study estimated that volume changes of ;3.0 ml/mole/residue are to be expected for a helix-to-coil transition in a protein (Imai et al., 2001). Because the uncertainty in the volume diminishes only as the square root of the number of volume measurements, 50-ns timescales are needed to reduce the error to the 0.68-ml/mole level. However, it is encouraging that meaningful volume differences were obtained from only a nanosecond of dynamics where the error is 4.2 ml/mole. The figure also gives a volume of ml/mole for the folded peptide from the 7.5 ns of dynamics that were performed in this study. The molar volume of the neat water system was determined once with sufficient precision to not effect the net volume uncertainty, and this will be discussed further in Results from a Simple Model System. For the folded peptide, the value of the statistical correlation time was found to be t c ¼ 2.4 ps. Preliminary investigations involving 5.8 ns of MD for an unfolded configuration of the peptide resulted in a volume of ml/mole. It is interesting that the folded and unfolded state, to within the present statistical certainty, have the same volume. This is somewhat surprising due to the difference in solvation structure associated with each state. Longer MD simulations are required to distinguish between the volumes of these two distinct conformational states. Further, this result does not necessarily imply that dynamical intermediates with significantly different volumes are not present during folding. Further experimental and theoretical investigations are required to determine the volume changes during folding. The Biophysical Journal 85(5)

198 198/ DeVane et al. determination and structural and dynamical origin (Dadarlat and Post, 2001) of any volume change upon folding is the subject of ongoing investigation. It is notable that larger proteins may exhibit larger (per-residue) volume changes making them amenable to investigation using the present technique (Foygel et al., 1995). The simulations were performed using a code originally developed by the Klein group at the Center for Molecular Modeling at the University of Pennsylvania, and the Space group is currently a co-developer and user of the code. It is a fast code that includes parallel execution, extended system, particle mesh Ewald, and multiple timescale integration algorithms. The code has been employed in a number of biological MD simulations (Moore et al., 2001, 1998; Zhong et al., 1998; Tarek et al., 1999). The extended system MD methods employed require a coupling between the real system and extended system variables and this could alter the effectiveness of the volume sampling. A variety of coupling constants (representing the coupling of the barostat to the system) between the volume coordinate and the molecular coordinates were tried in preliminary simulations. Physically acceptable values of the barostat mass (Martyna et al., 1996; Tuckerman and Martyna, 2000) led to only a weak dependence on the sampling efficiency for the solvated peptide system. In the simulations, the water model was a flexible SPC model described elsewhere (Moore et al., 2002; Ahlborn et al., 1999, 2000). The force field includes partial charges on the hydrogen and oxygen atoms representing the condensed phase permanent dipole. The protein force field was AMBER ff99. All the aqueous methane systems employed 62 water molecules. The peptide (Chen et al., 2001) had no net charge and was solvated with 810 water molecules using cubic periodic boundary conditions. An all-atom methane model was used, including a flexible force-field fit, to reproduce experimental infrared frequencies with harmonic C-H bonds (Herzberg, 1946). Lennard-Jones interactions were introduced only between the methane carbon and water oxygen with s ¼ 3.33 Å and e ¼ 51.0 K. The equilibrium bond length for the C-H interaction is 1.09 Å, and its geometry is tetrahedral. When simulating ionic systems the net charge on the system is canceled by employing a neutralizing background in the standard fashion (Allen and Tildesley, 1989). In all cases, the temperature was 298 K and the pressure was 1.0 atmosphere. The multiple timescale integration methods allowed the stable use of 4.0 fs time steps performing NPT dynamics and 8.0 fs time steps when simulating at constant NVE. This system serves to introduce the method and its properties. Using NPT MD in this fashion permits both the calculation of equilibrium molar volume changes and identification of metastable dynamic intermediate species exhibiting distinct molar volumes. The method will also be ideally suited to decomposing volume changes into differing physical origins such as attributing a volume change to peptide or solvent rearrangements and assessing the role of, e.g., sterics and electrostatics; this will be discussed further in the next section. differing partial charges on the CH 4 atoms and for an uncharged model. When there are no partial charges on the hydrogens, the methane-water potential energy interaction becomes equivalent to an united atom description of methane. The partial charges in the first model were fit to the electrostatic potential surface calculated via ab initio electronic structure methods that reproduce the octupole moment of gas phase methane (the resulting charges are ÿ0.52 e ÿ on C and e ÿ on H; see Sigfridsson, 1998). With these partial charges, the methane volume is ml/mole; and without them, it is ml/mole. The volume correlation time was found to be t c ¼ 1.6 ps for these systems. The volume of a methane is calculated as the difference between the system volume of the aqueous methane and a neat water system with the same number of H 2 O molecules as the solvated methane, i.e., the volume of the original aqueous system after the methane is plucked out. These uncertainties were obtained from 10.0 ns of dynamics. The slight volume change obtained is not statistically significant; the electrostriction effects associated with the solvated highly symmetric methane, which lacks a permanent dipole and quadrupole moment but has an octupole moment, are expected to be small. Fig. 4 shows the time evolution of the solvated uncharged methane system volume. Fig. 5 shows the time-dependent error estimates for the uncharged methane model, and the quadrupolar methane error estimate curve is very similar. These volume uncertainties are similar to that obtained in the case of the solvated b-sheet peptide, suggesting that the observed behavior would be similar in other aqueous systems. Fig. 5 also demonstrates that slightly over 100 ns of dynamics would be required to resolve any difference between these methane models. The inset of Fig. 5 demonstrates the Gaussian nature of the system volume fluctuations. To further test the effects of electrostatic moments on solvation and to assess the associated volume changes, both RESULTS FROM A SIMPLE MODEL SYSTEM To test the approach and demonstrate its flexibility, molar volumes were calculated for neat water and a solvated methane model. These systems were the subject of earlier investigations, although somewhat different potentials were used in that work (Lockwood and Rossky, 1999). First, the volume of the flexible SPC water was calculated to high precision to minimize the error in calculating volumes of aqueous solvated species, and was found to be ml/mole. The state point considered in all these studies is a pressure of 1.0 atmosphere and a temperature of 298 K. To assess the effects of electrostatic forces in solvation, aqueous methane was simulated for a variety of models with FIGURE 4 The curve shows a time trace of the volume fluctuations for the aqueous uncharged methane system. The average system volume value and its uncertainty are also represented in the figure. Biophysical Journal 85(5)

199 199/224 An MD Method for Molecular Volumes 2805 FIGURE 5 The uncertainty in the uncharged aqueous methane system as a function of the length of the MD simulation is shown. Volume uncertainties for specific times of interest are also identified. The inset displays a histogram of the system volume fluctuations and a superimposed Gaussian function with a standard deviation calculated from the volume fluctuations. fictitious monopolar (charged) and dipolar methane was simulated. The dipolar methane consisted of again placing a partial charge of ÿ0.52 e ÿ on C and a e ÿ on only one of the hydrogens, whereas the other three were uncharged. These charges result in a permanent dipole of 2.7 Debye, comparable to that of liquid water that has an average dipole of 2.4 Debye for the model used here. The permanent dipolar methane exhibits a molecular volume change from the uncharged methane via electrostriction with net volume decrease of ml/mole compared to the uncharged methane model. The relatively small volume change associated with dipolar solvation is consistent with the lack of volume change observed upon solvation of the octupolar model. Next, a charge of 1e ÿ and ÿe ÿ was placed on the methane carbon and the hydrogens were left uncharged. In both cases the aqueous-charged models produced dramatic electrostriction effects, and the solvated anion had the largest volume change. The volume change was ÿ ml/ mole for anionic solvation and ÿ ml/mole for cationic solvation. The errors are the result of 10.0 ns and 12.0 ns of dynamics, respectively. Notice that this implies a negative solution volume for the anion. The larger anionic electrostriction effect is due to the nature of its solvation. This result highlights the importance of properly accounting for electrostatic interactions in calculating molecular volumes, and this effect may well be important in volume changes associated with the dynamics of biomolecules. Fig. 6 shows the radial distribution function between the methane carbon atom and (Fig. 6 A) the water oxygen atoms and (Fig. 6 B) the water hydrogen atoms. The radial distribution functions are presented for both solvated ion systems along with the uncharged methane system. It is clear that the solvated anion permits the water hydrogen to FIGURE 6 (A) The radial distribution function between the methane carbon atom and the water oxygen atoms is shown. (B) The radial distribution function between the methane carbon atom and the water hydrogen atoms is displayed. The solid line represents the anion, the dashed line is the cation, and the dotted line is the uncharged molecule. penetrate effectively into the carbon van der Waals sphere, thus maximizing the interaction between the positive partial charge on the hydrogen and the negative ionic charge. The cation is also tightly solvated compared to the neutral, and both ions display a far more structured solvation shell than the neutral. The cation hydrogen first-neighbor peak is in approximately the same location as the neutral but is sharper, indicative of more ordering; the anion second-neighbor peak is shifted slightly inward from the neutral s first peak. Electrostriction effects are essentially screened out by ;5.5 Å. The radial distribution functions in Fig. 6 are clearly consistent with the observed volume changes and demonstrate the physical mechanism giving rise to electrostriction. Solvating the cation draws the oxygen atoms in more tightly to the methane and causes some solvent ordering, but does not dramatically disrupt the solvent structure. Solvating the anion causes the water to preferentially point a hydrogen in toward the negative charge, creating a new species of coordinated hydrogen atoms appearing between 1.5 and 2.2 Å in Fig. 6 B. Biophysical Journal 85(5)

200 200/ DeVane et al. the experimental and theoretical methods permit the determination of time-dependent molar volume changes, and the identification of metastable intermediate species with lifetimes lasting tens of nanoseconds. The method is also useful in accounting for the molecularly detailed origin of observed molar volume changes, including dissecting such changes into physically meaningful partial contributions from different potential energy interactions. Brian Space and Preston Moore thank Professor M. L. Klein for continuing encouragement in their collaboration. The research was supported by National Science Foundation grants (CHE to B.S., to R.W.L), and the Petroleum Research Foundation to B.S. and the American Heart Association to R.W.L. The authors gratefully acknowledge the University of South Florida Research Oriented Computing Center for a generous allocation of computer time on the University of South Florida Beowulf Cluster. REFERENCES FIGURE 7 Representative snapshots of solvated methane are shown: (A) the anion; (B) the cation. A strength of MD is in providing detailed molecular mechanisms for observed structure and dynamics. Fig. 7 shows a snapshot of both the solvated anion and cation. Fig. 7 A shows the anion solvated such that the water aligns itself with one of the hydrogens effectively penetrating in toward the negative carbon atom whereas the second hydrogen is held at a distance. This is consistent with Fig. 6 B where we see high ordering with the first sharp hydrogen peak representing the hydrogen that effectively penetrates and a second sharp peak representing the second hydrogen being held at a distance from the negative carbon. Fig. 7 B also demonstrates that the water solvates the cation, with the oxygen approaching close to the methane molecule and the hydrogens pushed back; however, the bulkier oxygen cannot penetrate as effectively. The MD models employed here serve to demonstrate the power of the methods presented to assess molar volume changes associated with solvation in the presence of different electrostatic fields. To make the approach more realistic it would be necessary to more carefully simulate a system of interest, including possibly simulating larger systems and including effects such as polarization forces. Nonetheless the observation that anionic solvation leads to larger molar volume changes than cationic solvation is in agreement with experimentally measured trends and the observed volume changes are on the same order of magnitude (van Eldik, 1989). In summary, we have presented an approach to calculating molar volume changes that is especially useful for comparing with photothermal experimental results. Both Ahlborn, H., X. Ji, B. Space, and P. B. Moore A combined instantaneous normal mode and time correlation function description of the infrared vibrational spectrum of water. J. Chem. Phys. 111: Ahlborn, H., X. Ji, B. Space, and P. B. Moore The effect of molecular geometry and detailed balance on the infrared spectroscopy of water (H 2 O and D 2 O): a combined time correlation function and instantaneous normal mode analysis. J. Chem. Phys. 112:8083. Allen, M. P., and D. J. Tildesley Computer Simulation of Liquids. Clarendon Press, Oxford, UK. Chen, P., C. Lin, H. Jan, and S. I. Chan Effects of turn residues in directing the formation of the b-sheet and in the stability of the b-sheet. Prot. Sci. 10: Dadarlat, V. M., and C. B. Post Insights into protein compressibility from molecular dynamics simulations. J. Chem. Phys. B. 105: Foygel, K., S. Spector, S. Chatterjee, and P. C. Kahn Volume changes of the molten globule transition of horse heart ferricytochrome c: a thermodynamic cycle. Prot. Sci. 4: Friedberg, R., and J. E. Cameron Test of the Monte Carlo method: fast simulation of a small Ising lattice. J. Chem. Phys. 52: Hansen, K. C., R. S. Rock, R. W. Larsen, and S. I. Chan A method for photo-initiating protein folding in a non-denaturing environment. J. Am. Chem. Soc. 122: Hansson, T., C. Oostenbrink, and W. F. van Gunsteren Molecular dynamics simulations. Curr. Op. Struct. Biol. 12: Herzberg, G Infrared and Raman Spectra of Polyatomic Molecules. D. Van Nostrand Company, Inc., New York, NY. Imai, T., Y. Harano, A. Kovalenko, and F. Hirata Theoretical study for volume changes associated with the helix coil transition of peptides. Biopolymers. 59: Jacucci, G., and A. Rahman Comparing the efficiency of metropolis Monte Carlo and molecular dynamics methods for configuration space sampling. Nuovo Cimento. D4: Larsen, R. W., and T. Langley Volume changes associated with CO photolysis from fully reduced bovine heart Cytaa3. J. Am. Chem. Soc. 121: Larsen, R. W., J. Osborne, T. Langley, and R. B. Gennis Volume changes associated with CO photodissociation from fully reduced cytochrome. J. Am. Chem. Soc. 120: Lockwood, L. M., and P. J. Rossky Evaluation of functional group contributions to excess volumetric properties of solvated molecules. J. Phys. Chem. B. 103: Biophysical Journal 85(5)

201 201/224 An MD Method for Molecular Volumes 2807 Lockwood, L. M., P. J. Rossky, and R. M. Levy Functional group contributions to partial molar compressibilities of alcohols in water. J. Phys. Chem. B. 104: Martyna, G. J., M. E. Tuckerman, D. J. Tobias, and M. L. Klein Explicit reversible integrators for extended systems dynamics. Mol. Phys. 87:1117. Matubayasi, N., and R. M. Levy Thermodynamics of the hydration shell. 2. Excess volume and compressibility of a hydrophobic solute. J. Phys. Chem. B. 104: Moore, P. B., Q. Zhong, T. Husslein, and M. L. Klein Simulation of the HIV-1 VPU transmembrane domain as a pentameric bundle. FEBS Lett. 431: Moore, P. B., C. F. Lopez, and M. L. Klein Dynamical properties of a dimiristoylphosphatidylcholine fully hydrated bilayer from a multinanosecond molecular dynamics simulation. Biophys. J. 81: Moore, P., H. Ahlborn, and B. Space A combined time correlation function and instantaneous normal mode investigation of liquid state vibrational spectroscopy. In Liquid Dynamics Experiment, Simulation and Theory. M. D. Fayer., and John T. Fourkas, editors. ACS Symposium Series, New York. Sigfridsson, E A comparison of methods for deriving atomic charges from the electrostatic potential and its moments. J. Comp. Chem. 19: Tarek, M., K. Tu, M. L. Klein, and D. J. Tobias Molecular dynamics simulations of supported phospholipid/alkanethiol bilayers on a gold(111) surface. Biophys. J. 77: Tuckerman, M. E., and G. J. Martyna Understanding modern molecular dynamics: techniques and applications. J. Phys. Chem. B. 104: van Eldik, R Activation and reaction volumes in solution II. Chem. Rev. 89:549. Zhong, Q., P. B. Moore, and M. L. Klein Molecular dynamics study of the LS3 voltage-gated ion channel. FEBS Lett. 427: Biophysical Journal 85(5)

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217 217/224 J. Phys. Chem. A 2001, 105, A Novel Technique for the Measurement of Polarization-Specific Ultrafast Raman Responses S. Constantine, J. A. Gardecki, Y. Zhou, and L. D. Ziegler* Department of Chemistry, Boston UniVersity, Boston, Massachusetts Xingdong Ji and Brian Space Department of Chemistry, UniVersity of South Florida, Tampa, Florida ReceiVed: NoVember 22, 2000; In Final Form: June 27, 2001 A simple time domain method for the observation of polarization-specific Raman responses in electronically nonresonant materials is demonstrated. When a cutoff filter is placed in the probe beam path before the detector in the conventional pump-probe configuration, the in-phase dichroic optical heterodyne-detected (OHD) response is enhanced as compared to the usual putative corresponding dichroic response observed when the probe is not dispersed. The ultrafast excited OHD responses of CS 2 obtained by this method are reported for parallel, perpendicular, and magic angle relative orientations of pump and probe pulse polarizations. The observed dispersed dichroic signal can be derived from the real part alone of the third-order nuclear response function. The decay of the CS 2 isotropic response is found to be dominated by a 500 fs decay process for times longer than 0.7 ps. This relaxation time scale matches the nondiffusive exponential decay seen in the birefringent and dichroic anisotropic responses of CS 2. Calculated instantaneous normal mode (INM) isotropic and anisotropic nuclear response functions are found to exhibit exponential decays in this same fs time scale, suggesting that this decay component may be predominantly determined by the distribution of Raman-weighted density of states. I. Introduction To fully exploit the capability of Raman measurements to characterize the nature of inter- and intramolecular nuclear degrees of freedom, all unique elements of the molecular transition polarizability tensor should be obtained. For isotropic media, electronically nonresonant Raman cross sections can be completely described by the isotropic and anisotropic components of the polarizability tensor. 1,2 Different scattering mechanisms, selection rules, and relaxation processes can characterize the spectral densities associated with each of these two scattering components. 1-5 During the past decade, the optical heterodynedetected (OHD) transient birefringence of transparent materials has been shown to be a convenient probe of the low-frequency intermolecular Raman spectral density of liquids due to the impulsive excitation provided by femtosecond pulses In the conventional two-beam OHD configuration, first introduced by Levenson and co-workers, 21 the heterodyning of a phasecontrolled portion of the probe pulse itself limits the measurement to observations of the anisotropic or depolarized portion of the Raman scattering tensor. Previous experimental techniques have been reported for the observation of polarization-specific OHD impulsive Raman measurements. Vohringer and Scherer 22 demonstrated that polarization-specific OHD responses could be observed in a transient grating geometry experiment due to the π/2 phase heterodyning of the probe scattered in the signal direction from a thermal grating. However, without additional phase-locking pulses, partial or incomplete heterodyning of the third-order * Corresponding author. lziegler@chem.bu.edu. response of the sample may be obtained due to this source of local oscillator. Simon and co-workers 23,24 employed a relatively simple two-beam ultrafast technique for the observation of isotropic and anisotropic responses of liquids called positionsensitive Kerr lens spectroscopy (PSKLS). PSKLS is due to the effects of beam distortion on the probe in the far field, resulting from the nonlinear index of the sample. The signalto-noise ratios of these reported responses, even after long data acquisition times, were still relatively poor compared with standard OHD responses. Subsequent techniques, employing three beams with various schemes to achieve phase-locking of the local oscillator field have been demonstrated. Matsuo and Tahara 25 used a three-pulse, active phase-locking scheme in order to obtain polarization-specific responses. Aside from the added experimental complexity of this method, the relative intensities of different polarization conditions are difficult to compare due to the polarization dependence of the transmittance/ reflectivity of optical elements. Most recently, Tokmakoff and co-workers 26 have demonstrated a polarization-specific OHD method that employs diffractive optic beam splitters, to generate a passively stabilized local oscillator field for the observation of phase and polarization-specific, impulsive Raman responses. At the expense of a more complicated experimental setup than the standard OHD impulsive pump-probe arrangement, this configuration provides a measure of polarization-specific thirdorder responses. Subsequently, Tokmakoff and co-workers 27 have reported polarization-specific Raman responses of liquids obtained by essentially a variation of PSKLS. We have discussed 28 how these simple two beam measurements are related to the so-called Z-scan measurements and the standard OHD experimental observables /jp004277x CCC: $ American Chemical Society Published on Web 10/04/2001

218 218/ J. Phys. Chem. A, Vol. 105, No. 43, 2001 Constantine et al. The technique reported here allows the observation of polarization-specific Raman responses in the conventional pump-probe two-beam configuration when just a single optical element, an optical cutoff filter, is added to the probe beam path. These effects are demonstrated for the ultrafast Raman responses of CS 2, which has been a benchmark material for both polarization-specific experimental and theoretical studies. 8,17-19,23-26,31-37 After approximately 1 ps the anisotropic OHD birefringent response of CS 2 is well fit by exponential decays of 1.7 and 0.5 ps at room temperature, as first noted by McMorrow, Lotshaw, and co-workers. 6,8 Recent OHD studies 38 have reported temperature-dependent biexponential decay character in the anisotropic ultrafast response of a number of organic liquids including CS 2. While the longer time decay component is attributed to rotational diffusion, the origin of the less evident, faster ( 500 fs) exponential decay has not been established. 8,38 Possible dynamical mechanisms for this second decay include pure dephasing, energy relaxation, or motional narrowing. Additionally, such an exponential feature can also arise from an appropriately shaped inhomogeneous distribution of Raman active modes. 6,8,17 The novel polarization technique described here is used to measure the corresponding relaxation times of the ultrafast isotropic response of liquid CS 2. These experimental results are compared with the polarization-specific nuclear response functions determined by instantaneous normal mode (INM) calculation. II. Theory a. Dispersed Responses. The dispersed optical heterodynedetected (OHD) birefringent or dichroic response as a function of time delay between pump and probe pulses, τ, and the selected probe pulse frequency, D, is given by S ijkl ( D,τ) )-2Im[Ẽ /LO i ( D )P (3) ijkl ( D,τ)] (1) D is the detuning of the detected probe pulse frequency, ω D, from the carrier frequency of the probe pulse (Ω), i.e., D ) Ω - ω D. P ijkl (3) ( D,τ) is the Fourier transform of the third-order polarization response of the material polarized along the ith direction generated by temporally sequenced pump/probe field interactions with electric vectors along the j, k, l directions. For pump/probe pulses with identical carrier frequencies, this probefrequency-dependent response is described by P (3) ijkl ( D,τ) ) (1/ 2π) - dt exp(-i D t)ê pr j (t-τ) dτ1 0 Ê *pu k (t-τ 1 ) Ê pu l (t-τ 1 ) R (3) ijkl (τ 1 ) (2) R (3) ijkl (t) is the third-order impulse response function of the material, and Ê R j (t) is the pulse envelop of the pump or probe (R )pu, pr) fields. The complex quantity, Ẽ /LO i ( D ) (eq 1), is proportional to the spectrum (Fourier transform) of the probe field, Ẽ /pr i ( D ): Ẽ /LO i ( D ) ) (1/ 2π) exp(-iθ) - dt exp(i D t) Ê /pr j (t-τ) ) (1/ 2π) exp[i( D τ - θ)] Ẽ /pr j ( D ) (3) Dichroic and birefringent OHD responses are obtained with inphase (θ ) 0) and π/2 phase shifted local oscillator fields (θ ) π/2), respectively. In the usual ultrafast OHD investigation of nonresonant materials, changes in the energy of the total, nondispersed probe beam as a function of interpulse delay (τ) are reported The corresponding familiar expressions for OHD birefringence and dichroism 10,39 are recovered when eq 1 is integrated over all probe pulse frequencies ( D ). 40,41 As expected for electronically nonresonant materials, observed integrated OHD birefringent (θ ) π/2) responses are generally more than 1 order of magnitude larger than the corresponding OHD dichroic (θ ) 0) responses. 12,42-44 Formally, this difference is due to the larger magnitude of the real part as compared to the imaginary part of this third-order nuclear impulse response function. These relative magnitudes may be seen as resulting, respectively, from constructive and destructive interfering CARS and CSRS-like density matrix pathways contributing to this nuclear response function. 12,43-45 Within a Born-Oppenheimer description of molecular states, the third-order impulse response function, R (3) ijkl (t), which in general is a complex function, may be written as a sum of contributions from electronic and nuclear degrees of freedom: (3) (3e) R ijkl ) R ijkl + R (3n) ijkl. When the incident frequencies are not coincident with any regions of one- or two-photon electronic absorption, the corresponding nonresonant electronic response can be taken to be instantaneous and proportional to a real constant. 40,41,48 The nuclear response is formally described in terms of the two-time correlation function of the transition polarizability by R (3n) ijkl (t) ) i/h [R ij (t),r kl (0)]. 39 b. Polarization Considerations. In an isotropic medium, there are only three linearly independent susceptibility elements, or response functions, that obey the well-known relationship: (3) (3) (3) (3) R ZZZZ ) R ZZYY + R ZYZY + R ZYYZ. 46 By Kleinman s symmetry, (3e) (3e) (3e) R ZZYY ) R ZYZY ) R ZYYZ ) R (3e) ZZZZ /3 and thus the nonresonant electronic contribution can be described in terms of just a single susceptibility element. When the relative polarization directions of the pump and probe beams have parallel, perpendicular, or magic angle (54.7 ) orientations, the corresponding relative magnitudes of the nonresonant electronic responses, R (3e) ZZZZ : (3e), are in the ratio 3:1: 5 / For nonresonant Ra- R (3e) ZZYY :R Mag.Ang (3n) man transitions, only R ZYZY ) R (3n) ZYYZ, and hence two polarization observations, e.g., pump/probe parallel polarized and pump/ probe perpendicularly polarized, are necessary for the complete characterization of the nonresonant Raman scattering response (time or frequency domain). These responses can be recast in terms of the anisotropic and isotropic contributions to the Raman (nuclear) response by R Aniso ) (R ZZZZ - R ZZYY )/2 and (3) (3) (3) R Iso ) (R ZZZZ + 2R ZZYY )/3. This combination of observed responses is generally taken to allow the contribution of orientational motion to be separated from other sources of Raman response relaxation. 10,44 When the relative angle between the linearly polarized pump and probe pulses is set to the socalled magic angle (54.7 ), the observed signal is directly (3) proportional to the isotropic response, i.e., R Mag.Ang ) R (3) Iso. c. Calculated Spectrogram. A calculated dichroic spectrogram, i.e., the OHD dichroic response as a function of both interpulse delay and probe pulse frequency, is shown in Figure 1 due to a pair of transform-limited 45 fs pulses incident on a material with an impulse response function given by (3) R ZZYY (t) ) aδ(t) - F(ω i ) sin ω i t - b exp(-t/τ d )(1 - exp(-t/τ r )) (4) i where F(ω i ) ) ω i exp(-ω i /ω c ) and ω c ) 20 cm -1, τ d ) 1.7 ps, τ r ) 0.15 ps, a ) 1200, and b ) 140. A response function of (3) this form models a typical depolarized intermolecular R ZZYY response. 24 The δ function component represents the instantaneous nonresonant electronic response. The intramolecular librational or collision-induced and orientational diffusive (3) (3) (3)

219 219/224 Polarization-Specific Ultrafast Raman Responses J. Phys. Chem. A, Vol. 105, No. 43, Figure 1. Calculated OHD dichroic spectrogram (upper panel) for an all-real third-order impulse response function due to an instantaneous electronic and intermolecular and diffusive nuclear contributions (see text for more details) and 45 fs pulses. The corresponding OHD signals due to the totally and partially integrated spectrogram are plotted in the lower panel. nuclear contributions are simulated by the second and third terms, respectively, in eq 4. As seen in Figure 1, which results when eqs 2-4 are substituted in eq 1, a dispersive-like shape with respect to both τ and D is observed in the region where pump and probe overlap, τ 0. Such a shape characterizes nonresonant electronic dichroic spectrograms. 40,41 In the τ < 0 region, the response is dominated by the instantaneous electronic response. Both electronic and nuclear contributions overlap in the 0 < τ < 100 fs region. Antiphased contributions, with respect to the detuning frequency D ) 0, are found at τ > 100 fs due to the intermolecular nuclear responses and exhibit a maximum/minimum at 300 fs (see Figure 1). When this dichroic OHD spectrogram is integrated over all probe frequencies, the response vanishes, as shown in the lower panel of Figure 1 (dashed line). This is a general result for transform limited pulses when the response function is all real as modeled here (eq 4). This is probably a good approximation for lowfrequency, intermolecular modes of interest here. 12,39 However, when the dichroism is observed at a selected frequency, or integrated over only a portion of the probe pulse frequencies, the exact cancellation from the red and blue sides of this dichroic spectrogram is spoiled and a nonzero dichroic response is obtained. The dichroic signal resulting from an integration over the spectrogram responses only on the red side of the probe carrier frequency illustrating this effect is displayed in the lower panel of Figure 1 (solid line). Furthermore, since the dichroic measurement is inherently an in-phase (θ ) 0) local oscillator result, there are no polarization restrictions on the relative orientation of the pump and probe beam for this dispersed technique. Thus, nonputative dichroic responses for electronically nonresonant materials may be obtained for any relative polarization orientation of pump and probe pulses when the interpulse delay dependence of only a selected portion of the probe pulse spectrum is observed, as demonstrated by the results of the calculations presented in Figure 1. We also briefly note here that convolution of the nuclear impluse response function with the frequency selected nonresonant electronic response yields the corresponding frequency selected Raman response (see Appendix). 56 III. Experimental Section The OHD responses are observed in the standard two-beam OKE pump-probe configuration Near transform limited (<1.2 TL) 45 fs pulses centered at 595 nm are used to obtain the data reported here. Ultrafast pulses are produced by an OPA pumped by the second harmonic of a regeneratively amplified Ti:sapphire oscillator operating at 250 khz. In contrast to the usual experimental observation of nonresonant OHD responses, a low-pass red filter (Schott RG610) is placed in the probe beam just before the detector (photomultiplier tube). This filter has a 50% transmission at 605 nm. Thus, only a portion of the dichroic signal due to third-order polarization components on the red side of the probe carrier frequency contributes to the responses reported here. The use of dispersed OHD dichroic responses for polarization selectivity was first qualitatively demonstrated via monochromator dispersion. 43 However, the use of a lowpass filter for this purpose is simpler and cheaper to implement, avoids potential phase front distortion effects 28 at the monochromator entrance slits, and provides a polarization-independent detection scheme. Spectroscopic grade CS 2 is used without further purification and flowed througha1mmquartz sample cell. The pulses were focused with a 175 mm lens, and pulse energies at the sample were of the order of 10 nj/pulse. The intensity dependence of these responses was measured in order to ensure that homodyne contamination was minimal (< 5% at signal maximum). Time constants are determined by fits to the observed experimental decays from an initial delay time such that the determined time constants are independent of this choice (typically from τ 0.7 ps).