Three-pulse photon echoes for model reactive systems

Transcription

1 hree-pulse photon echoes for model reactive systems Mino Yang, Kaoru Ohta, and Graham R. Fleming Citation: he Journal of Chemical Physics 11, 1243 (1999); doi: 1.163/ View online: View able of Contents: Published by the AIP Publishing On: hu, 7 Nov :17:12

2 JOURNAL OF CHEMICAL PHYSICS VOLUME 11, NUMBER 21 1 JUNE 1999 hree-pulse photon echoes for model reactive systems Mino Yang, Kaoru Ohta, and Graham R. Fleming Department of Chemistry, University of California, Berkeley, California and Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, California Received 14 December 1998; accepted 5 March 1999 A theoretical description of the three-pulse photon echo peak shift for model reaction systems is presented. An electronic two-state system with a finite upper-state lifetime and a three-state system in which electronic transitions can occur are considered. A probabilistic argument is employed to incorporate the incoherent transitions. New pathways describing the transition of electronic population are introduced and the nuclear propagator in the electronic population state is written by a convolution integral between those of the nonreactive two-state system weighted by some factors for the electronic transition. he response functions are given by multitime correlation functions and are analyzed by the cumulant expansion method. Some numerical calculations are presented and the influence of incoherent reactions on the peak shift is discussed. Comparison with experimental data confirms the existence of the effects predicted here American Institute of Physics. S I. INRODUCION Photon echo spectroscopy has been developed to a sophisticated level for the study of chromophore bath interactions in dilute samples, where the electronic transition can be modeled as a two-state system interacting with a complex set of intra- and intermolecular nuclear motions. 1 8 In particular we and others 9 22 have made extensive use of the three-pulse photon echo peak shift 3PEPS method to characterize solvent solute interactions in liquids, 9 16 glasses, and proteins It was established that the solvation dynamics influencing the electronic energy gap fluctuation via the electron phonon coupling can be revealed by this kind of experiment. he experiment has an intrinsically high dynamic range over which the transition frequency fluctuation may be followed. A notable feature of this experiment for dilute two-level systems coupled to a bath is that the presence of a finite long-time peak shift is direct evidence for the presence of static on the experimental time scale inhomogeneity in the system. 3 We note that the experiment has very high subpulsewidth intrinsic time resolution. he experimental results are usually analyzed by a theory based on the linearly coupled harmonic bath model and the cumulant expansion method. 23 his model provides a good description of experiments on two-state systems over a wide range of temperature It thus seems appropriate to extend such studies to more complex systems in which electronic population transfer chemical reaction can occur. 3PEPS experiments have been carried out on some reactive systems such as the purple bacterial light harvesting complexes LH1 and LH2 Refs. 2,21 and on the bacterial reaction center. 22 he results showed that the energy excited population transfer process within the biological complexes qualitatively influences the form of the echo signal and that the essential parameters controlling the evolution of the system can be extracted from measurements of the peak shift. hese experiments were analyzed by a theory based on the two-state system incorporating the reactive events intuitively Although this theory explains the experimental results quite well, its domain of applicability needs to be explored via a more formal approach. he theory presented in this paper is extended to the energy transfer system elsewhere. 24 3PEPS experiments on reactive systems 25,26 are being currently performed in our group to investigate the effect of solvation dynamics on chemical reactions. For example, in some polar solvents, the excited state of phenol blue contains a large amount of charge transfer character. Charge recombination leads to very rapid decay to the ground state. 25 Rhodamine6G, oxazine 1, and oxazine 75 in the solvents DMA and DEA comprise another interesting set of systems in which ultrafast photo-induced electron transfer from the solvent to the solute occurs. 26 In order to properly interpret the experimental 3PEPS data on reactive systems, a theoretical model is required. his is the purpose of the present paper. As the simplest examples of reactive systems, we consider a two-state system with a finite lifetime in the upper state such as phenol blue and a three-state system which models an excited state electron proton transfer system. By doing this we shall show how the effect of reaction can be systematically incorporated into a theory for the third-order signal and we will discuss a new kind of phenomenon in the 3PEPS signal specific to these types of systems. In most reactive systems in the condensed phase, the electronic transitions occur on time scales long compared with the electronic dephasing time. In this case, the information of electronic coherence is lost before the electronic transition occurs. So the consideration of population transfer dynamics is sufficient for our purpose. Leegwater 27 showed that incoherent population transfer is quite a good description even for a system with strong electronic coupling if the homogeneous broadening width is larger than the electronic coupling constant. When the population transfer time scale is comparable to that of the degrees of freedom of the bath, the /99/11(21)/1243/1/$ American Institute of Physics On: hu, 7 Nov :17:12

3 1244 J. Chem. Phys., Vol. 11, No. 21, 1 June 1999 Yang, Ohta, and Fleming population transfer kinetics should be described by a rate kernel equation in which the rate kernel contains the information of the bath dynamics Understanding the dynamics of such systems is a topic of much current interest. 32 However, Markovian reactive systems in which the rate kernel is approximated by a time-independent rate constant provide a good starting point of our discussion and should give us an intuitive understanding of the influence of reactive events on the 3PEPS experiment. In this paper, the electronic transitions are assumed to be Markovian. We expect that an extension of the present work to incorporate the non- Markovian effects could be made. II. BASIC HEORY FOR HE SIMPLE WO-SAE SYSEM hird order spectroscopic signals can be predicted by a response function which describes the third order process occurring in the system. In terms of the response function, the three-pulse photon echo peak shift for a simple two-state system is well established and described in detail elsewhere. 8 1 Before proceeding to the reactive systems, we will therefore briefly discuss the basic elements of the 3PEPS for a nonreactive two-state system. In the condensed phase, there are many identical chromophores contained in a region with dimension of the wavelength of light. When these chromophores interact with light, a macroscopic on the wavelength scale nonlinear polarization is induced and this gives the nonlinear signal. If the chromophores are independent of each other, the timedependence of the polarization can be described by the density matrix of a single chromophore. he pulse sequence for a 3PEPS experiment is shown in Fig. 7. he time periods and are experimentally controlled. In the impulsive field limit, the first time period during which the system is in an electronic coherence state is scanned. During the second time period, the system is in a diagonal state population state of density matrix. he third pulse creates the second coherence state which leads to rephasing and echo formation. In a conventional photon echo experiment, the final time period t is integrated over to record the echo intensity as a function of the first time period. he experimental signal should be interpreted, when pulse duration is very short, in terms of the response function R(t,,) by the relation S, dtrt,, Under the rotating wave approximation RWA and an impulsive pulse shape, only two kinds of third-order processes contribute to the signal with the direction of k 3 k 2 k he effects of finite pulsewidth are easily developed by considering two additional third-order processes. 1 R gg and R ee are the response functions which describe the ground and excited population evolution, respectively, during the interval between the second and the third pulses. In this paper, the subscript gg(ee) on the response function denotes that the population state begins at the ground excited state and ends in the ground excited state. he superscript denotes reaction-free response functions. We can write these response functions in terms of the nuclear propagators in Liouville space as 23 t,,g eg,eg R gg tg gg,gg G ge,ge g, R ee t,,g eg,eg tg ee,ee G ge,ge g, where g is the nuclear density matrix in the ground electronic state and denotes the thermal equilibrium average. he nuclear propagator G, (t) for an operator A in the Liouville space is defined by G, tae ih t Ae ih t, 2.4 with H the nuclear Hamiltonian in the electronic state. he response functions, Eqs and 2.3.2, describe the time evolution of the nuclear density matrix accompanying the change of the electronic state. Making use of the cumulant expansion method, one can derive expressions for the response functions 1,11,23 where R gg t,,r re t,,expiq gg t,,, 2.5 R ee t,,r re t,,expiq ee t,,, 2.6 R re t,,exppptpp PtPt, Q gg t,,qqtqqqt Qt, 2.7 Q ee t,,qqtqqqt Qt. Here, the superscript re denotes the modulus of the response function which is determined by the real part of the line broadening function, and P and Q are the real and imaginary he observable of interest is the location of the echo maximum with respect to zero delay of the first time interval for parts of the line broadening function g(t), respectively. he imaginary part of the line broadening function controls the different fixed values of the second period. he shift from phase angle of the response function associated with each zero delay, *(), we refer to as the peak shift. From Eq. pathway. he line broadening function characterizes the 2.1, the peak shift is obtained by solving the equation spectral distribution of the fluctuations, 23,33,34 in the presence S, of static inhomogeneity, in,. 2.2 * gt in t 2 /2 d1costcoth/2 A plot of *() vsconstitutes a 3PEPS data set. he key features of the 3PEPS experiment can be found iti d sint, 2.8 elsewhere On: hu, 7 Nov :17:12

4 J. Chem. Phys., Vol. 11, No. 21, 1 June 1999 Yang, Ohta, and Fleming 1245 where is the Boltzmann factor and is a temperatureindependent spectral density representing the bath density of states. he reorganization energy is defined in terms of the spectral density as d. 2.9 Note that the real part of g(t) is temperature-dependent while the imaginary part is not. For convenience, we set 1. Once and in are determined, the response functions at any temperature can be calculated by Eqs Although the spectral density is quite a general quantity containing the information of bath fluctuation dynamics, it is convenient to introduce a temperature-independent transition energy fluctuation correlation function M(t). he expression linking M(t) and is 23 Mt 1 dcost. 2.1 In this paper, we will assume a simple form of M(t) to simulate the bath dynamics and from this we calculate the spectral density via the inverse Fourier transform of Eq he resultant spectral density will be inserted into Eq. 2.8 to model the line broadening function g(t) at an arbitrary temperature. he total response function is given by the sum of R gg and R ee, R t,,r re t,,expiq gg t,, expiq ee t,, If the imaginary part of g(t) is neglected, the response functions arising from the gg and ee pathways are exactly the same. Since no spectral diffusion time-dependent Stokes shift occurs in this case, the displacement of the excited state potential curve from that of the ground state is zero and so the nuclear dynamics in both states is not different. his is the reason why the gg and ee pathways give identical response functions when Q. Otherwise, some interference between the gg- and ee-pathways will play a role in the signal proportional to the modulus squared of the total response function. FIG. 1. a Energy level diagram for two-state system with a decay constant. b Feynman diagrams for the three kinds of third order processes occurring in the two-state system with decay to ground state. he dotted line on R ge indicates an electronic transition due to the decay process. We can easily see that the peak shift behavior predicted by solving Eq. 2.2 along with Eq. 3.1 via Eq. 2.1 is exactly the same as that for a system with an infinite lifetime. So this kind of response function leads us to the conclusion that there would be no effects of lifetime on the peak shift when pulse duration is impulsively short. In this section, we will show that this may not be always true. We assume the lifetime is much longer than the dephasing time and so we can neglect decay to the ground state during the coherence time when the dephasing dynamics is dominant. In this case, the decay to ground state can occur only during the population time period and the third order process can be represented by the diagrams in Fig. 1b. In the presence of decay from the excited state to the ground state, there are three pathways of third order response. he first one denoted by R gg is for the process in which the system is in the ground state for the entire population period and this process is not affected by the decay process. he expression for this is the same as in the normal two-state system, that is, R gg R gg. he second response function denoted by R ee t,,g eg,eg tg ee,ee G ge,ge g 3.2 is for the process in which the system is in the excited state for the entire population period. he nuclear dynamics for this process must be identical to that for the process described by R ee in the normal two-state system. However, due to the decay to the ground state with the decay constant k d, the population density undergoing this process decreases to give G ee,ee e k d G ee,ee 3.3 and consequently we obtain III. RESPONSE FUNCIONS FOR REACIVE SYSEMS A. wo-state system with finite lifetime We consider an electronic two-state system with a finite lifetime of the excited state. he energy diagram is shown in Fig. 1a. raditionally, it has been accepted that a finite lifetime does not affect the peak shift but only reduces the intensity of the signal. In this case, the response function for a two-state system with an upper state lifetime k 1 d is given by 8 Rt,,e kd R t,,. 3.1 his form is an extension of a standard method introduced in linear spectroscopy to incorporate the lifetime broadening On: hu, 7 Nov :17:12

5 1246 J. Chem. Phys., Vol. 11, No. 21, 1 June 1999 Yang, Ohta, and Fleming R ee t,,e k d R ee t,,. 3.4 his simple form of exponential factor for the decay process means the decay dynamics is separated from the nuclear dynamics. he third response function is denoted by R ge t,,g eg,eg tg gg,ee G ge,ge g 3.5 and describes the process in which the system starts from the excited state at the initial time of the population period and ends in the ground state at the final time of the population period. Now, the electronic state where the population resides changes during the population period due to the decay process and so the nuclear wave packet experiences a different potential energy as the electronic transition occurs. Because, in our model, the decay rate is assumed to be separable from the nuclear dynamics, the nuclear propagator in this case can be written by G gg,ee k d dsggg,gg sg ee,ee se k d s. 3.6 Here, G ee,ee (s)e k d s is the nuclear propagator for time s while the electronic state is in the excited population state. he exponential factor reflects the probability that the population is still in the excited state. k d ds is the decay probability recovering probability of the ground state population during the time interval ds and G gg,gg (s) describes the nuclear propagation in the ground electronic state during the time s since the decay from the excited state to the ground state occurred. he decay process can occur at any time of the population period and so we obtain the convolution integral Eq his relation is exact within the present model. Inserting Eq. 3.6 into Eq. 3.5, weget R ge t,,k d dse k d s G eg,eg G ee,ee sg ge,ge tg gg,gg g. s 3.7 By a straightforward application of the cumulant expansion method, we can easily analyze the four-time correlation function in Eq. 3.7 to obtain an expression for R ge, R ge t,,r gg t,,e 2iQtQ k d dse k d s e 2iQtsQs. 3.8 he total response function is given by a sum of the above three terms Rt,,R gg R ee R ge R gg R ee e k d R gg e 2iQtQ dse 2iQtsQs k d e k d s. 3.9 Since, in the pathway for R ge, the system makes an odd number of interactions with the field on the bra and ket sides, respectively, R ge contributes to the total response function with a minus sign 23 which reflects the recovery of the hole in the ground state. By the time integral of the last term of Eq. 3.9, the population kinetics and the phase shift of the response function associated with the recovered ground state are described. he peak shift behavior can be predicted by a numerical calculation of Eq. 3.9 along with Eqs. 2.1 and 2.2. Some illustrative calculations will be presented in Sec. IV. By employing the linear electron phonon coupling approximation, we implicitly assumed the potential energy surface of the ground and excited states to be harmonic with the oscillator strength for the electronic transition independent of solvation coordinate. his means that only a displacement of the potential minimum position of the excited state from that of the ground state can make the nuclear dynamics in the two states differ. his displacement inducing the spectral diffusion and the Stokes shift in linear spectroscopy is controlled by the imaginary part of g(t). So if we neglect this, we get a response function for a system in which the decay process does not alter the nuclear dynamics but decreases the amplitudes of the population and hole created during the first coherence dynamics in the excited and ground states, respectively. In this case, we can reduce Eq. 3.9 to the conventional form of the response function for the system with a finite lifetime like Eq. 3.1, Rt,,2R re t,,e k d. 3.1 It is evident from Eq. 2.2 that this kind of response function cannot reflect the finite lifetime effects on the peak shift when the pulse shape is impulsive. In the general case, however, there is a finite Stokes shift in the fluorescence spectrum and thus the imaginary part of g(t) must not be neglected. In this case interference among the above three kinds of response functions in Eq. 3.9 can occur reflecting the lifetime effects on the peak shift. he role of the imaginary part of g(t) in the third order response function becomes magnified in low temperature 17 and we expect this kind of interference effect to be most evident in low temperature experiments. B. hree-state system with incoherent reaction incoherent electron transfer As an example of the kind of system we have in mind consider the excited state electron transfer system DA D*A D A DA. he scheme for such a system is shown in Fig. 2a. his model could also be applied to a photo-induced proton transfer system. If the reaction rates are much slower than the dephasing time, we can safely ignore the reaction processes during the coherence time interval. In this paper, we only consider the case that the state e is nonresonant with the laser and so the state does not interact with the light. For the purpose of simplicity, we assume a Markovian process for the incoherent reaction from the state e to the state e. In this case, the electronic transition dynamics is separated from the nuclear dynamics and we can employ a similar procedure as above to understand the effect of incoherent reaction on the peak shift. Moreover, we assume the nuclear coordinates associated with the optical transitions of e and e states should be statistically independent of each other and satisfy the same fluctuation dynamics On: hu, 7 Nov :17:12

6 J. Chem. Phys., Vol. 11, No. 21, 1 June 1999 Yang, Ohta, and Fleming 1247 R ge t,,r gg t,,e 2iQtQ k d dse k r k d s e 2iQtsQs FIG. 2. a Energy level diagram for a three-state system. he reaction rate constants k d, k d, and k r are assumed to be independent of the nuclear motion. b Feynman diagrams for the four kinds of third order processes occurring in the three-state system. he dotted lines on R ge and R ge e indicate electronic transitions due to the reactions. Except that the decay rate of the excited state population is given by the sum of two decay constants k r k d, this response function is the same as that for the process in Sec. III A. he last process represented by the response function R ge e occurs via the third electronic state. Initially the population is created in the excited state e. During the population period, the ee-population moves to the other excited state e by the incoherent transfer reaction with rate constant k r. he newly attained ee-population state does not contribute to the signal because that state is out of the laser window. However, after a while, the ee-population decays to the ground state with the decay constant k d to fade out the ground state grating. he response function for this sequential process is given by R ge et,,g eg,eg tg gg,e e,eeg ge,ge g, 3.17 he four kinds of third order processes occurring in this system are represented by the Feynman diagrams in Fig. 2b. As usual, there is a pathway in which the population state is the ground state. his pathway is common for every reactive system and its expression is the same for all cases. hus we will not write this expression again. For the process in which the population stays in the excited state during the entire population period, the response function is given by R ee t,,g eg,eg tg ee,ee G ge,ge g, 3.11 where the nuclear propagator during the population time is written, similarly to Eq. 3.3, as G ee,ee e k r k d G ee,ee In this system, the decay rate of the excited population is given by the sum k r k d. Consequently, the response function for the ee-pathway is obtained as R ee t,,e k r k d R ee t,, he population of the third process begins at the excited state and ends in the ground state and the response function representing this process is given by R ge t,,g eg,eg tg gg,ee G ge,ge g he nuclear propagator during the population period is also given similarly to Eq. 3.6 as where the nuclear propagator in the population state is G gg,e e,eek d ds1 G gg,gg s 1 k r s 1ds2 G e s e,ee 1 s 2 e k d s 1 s 2 G ee,ee s 2 e k r k d s 2. he physical meaning described by Eq is clear. G ee,ee (s 2 )e (k r k d )s 2 is the nuclear propagator for time s 2 while the population is in the excited state e. he exponential factor reflects the probability that the population is still in the excited state. k r ds 2 is the transfer probability from e to e by the incoherent reaction during the time interval ds 2. hus, k r ds 2 G ee,ee (s 2 )e (k r k d )s 2 is the nuclear density matrix newly transferred to the e state at time s 2. he nuclear propagation afterwards in the state e during the time interval s 1 s 2 is described by the propagator G e (s e,ee 1 s 2 )e k d (s 1 s 2 ) while the exponential factor reflects the depletion of the ee-population due to the decay to the ground state with decay constant k d. k d ds 1 is the decay probability from the state e to the ground state during the time interval ds 1. G gg.gg (s 1 ) describes the nuclear propagation in the ground electronic state during the time G gg,ee k d s 1 since the decay from the excited state e to the ground dsggg,gg sg ee,ee se k r k d s. state occurred. he two kinds of transition process in this pathway can occur at any time during the population period 3.15 and so we get the convolution integral Employing a similar procedure to the previous section, we get the response function for the decay process as By employing the cumulant expansion method, we finally obtain the expression of the last response function as On: hu, 7 Nov :17:

7 1248 J. Chem. Phys., Vol. 11, No. 21, 1 June 1999 Yang, Ohta, and Fleming R ge et,,r gg t,,e 2iQtQ k r dse k r k d s 1e k d s e 2iQtsQs Actually, this response function has a two-dimensional integral with respect to time because there are two types of sequential reaction events. We can easily reduce it to the onedimensional integral given in Eq by simple algebraic manipulation. Referring to the Feynman diagrams shown in Fig. 2b to determine the sign of each pathway contributing to the total response function, we get the total response function as Rt,,R gg R ee R ge R ge e R gg R ee e k r k d R gg e 2iQtQ dse 2iQtsQs k d k r 1e k d s e k r k d s ]. 3.2 he last term of Eq. 3.2 reflects the destruction of the hole, because of the two reaction pathways moving the ee-population to the gg-population, in the ground state created by the first coherence dynamics. he difference in the history of the nuclear trajectories between the original gg-population having existed from the initial population time and the other having decayed from ee-population is incorporated into the response function via the integrand including Q(t). he difference in the nuclear trajectory history will have an effect on the peak shift. If there is no displacement of the potential curve (Q ) of the excited state e from that of the ground state, then the nuclear dynamics is not influenced by the change of electronic state. In this case, Eq. 3.2 simplifies to Rt,,R re t,,f, 3.21 where f () is a parametric function of the rate constants which describes just the time dependence of the population. So the amplitude of the third-order signal will decrease with f () but the peak shift is not affected by the reactions with an impulsive pulse shape. With no decay to the ground state k d k d, an incoherent reaction between two excited states can occur. In this case, Eq. 3.2 becomes Rt,,R gg R ee e k r, 3.22 which describes just the decrease of the ee-population due to population transfer to a state which is out of the laser window. IV. ILLUSRAIVE CALCULAION AND DISCUSSION IN HE IMPULSIVE LIMI In this section, we present some illustrative calculations of the peak shift behavior in the presence of reactive events in the impulsive limit. he influence of using finite pulses will be discussed in the following section. We model the FIG. 3. Model spectral density of Eq he reorganization energy is 2 cm 1. nuclear fluctuation dynamics in terms of the electronic energy gap fluctuation function M(t) defined by Eq We assume that M(t) is given by the sum of a Gaussian and an exponential function 2Mtexpt/ g 2 expt/ e 4.1 to simulate the bimodal character of the fast and slow solvation processes. It is well known that a Gaussian process associated with the inertial solvent motions is responsible for the short time fast dynamics. he exponential part represents a diffusive solvation dynamics. With this form, the temperature independent spectral density is given by the inverse cosine transform of Eq. 2.1 as 2 2 g exp g e 1 e Figure 3 shows the frequency dependence of this spectral density when g 6 fs and e 1 ps. he reorganization energy is set to be 2 cm 1 in all calculations. Figure 4a shows the 3PEPS behavior of a two-state system with a finite lifetime of the excited state predicted by Eq. 3.9 at room temperature 3 K and with zero inhomogeneity. he peak shift for the two-state system with k d is also shown for comparison. Except for population times longer than the lifetime of the excited state, the 3PEPS for the system with a finite lifetime is almost identical with the k d system dotted line. In other words, the finite lifetime effect does not appear until most of the ee(gg)-population disappears due to the decay. his kind of behavior can be understood by looking at Eq Asthe last term of Eq. 3.9 shows, the lifetime effect is brought about by the difference in the nuclear motion history which yields some phase difference between the response functions associated with the original and restored gg-populations. At early times, the portion of the restored gg-population is minor and thus the lifetime effect does not appear at that time region. After the original grating nearly disappears due to the recovery of the ground state population depletion of the population grating, the lifetime effect arising from the difference in history between the restored gg-population and the original one becomes apparent. Because of this, the echo signal intensity in the time region when the lifetime effect On: hu, 7 Nov :17:12

8 J. Chem. Phys., Vol. 11, No. 21, 1 June 1999 Yang, Ohta, and Fleming 1249 FIG. 4. 3PEPS for a two-state system with a finite upper state lifetime a at 3 K and b 3 K. he lifetimes are indicated for each curve. in. he peak shift when k d is shown for the comparison dotted line. FIG. 6. 3PEPS for a three-state system a at 3 K and b 3 K. he three solid lines are for the reaction rates of 3, 6, and 9 fs from top to bottom. he peak shift when k d is shown for the comparison dotted line. appear should be very small and a high dynamic range experiments will be required to observe this effect. In Fig. 4b, we present the peak shift at very low temperature. he other parameter values are the same as those in Fig. 4a. Equation 2.8 shows the relative contribution of the imaginary part of g(t) is enhanced in low temperature and thus the lifetime effect reflected through the imaginary part of g(t) is more pronounced than in Fig. 4a. Even so, the time scale that the lifetime effect begins to appear does not seem to be much altered by the temperature. At low temperature, the rise in the peak shift starting around 1 fs appears even for the normal two-state system. his FIG. 5. 3PEPS for a three-state system with the direct decay time (k d 1 6 fs) at 3 K. in. he time scales (k r 1 ) of population transfer to a state out of laser window are indicated for each curve. he peak shift when k d k r k d is shown for the comparison dotted line. is ascribed to the interference effect between the imaginary parts of the response functions for the gg- and ee-pathways. If some portion of the excited population does not directly go back to the ground state which we expect to be the case most for real systems, a population grating will still persist for a long time. As an example consider a three-state system with k d k r and k d. he direct decay process with rate constant k d decreases the amplitude of the population grating both in the ground and excited states and makes the signal intensity very weak. In contrast to the case of a pure two-level system, this loss of population grating is not complete due to the flow of excited state population to the e state. In this case, the total signal will be dominated by the population grating retained in the ground state. he finite lifetime effect arising from the phase difference in the response functions discussed above will not appear because this phase difference effect only appears after the whole grating has nearly decayed as shown in Fig. 4. As a result the peak shift is mainly determined by the population grating to give similar behavior to that in Fig. 6 below without direct decay to the ground state. he peak shift when k 1 d 6 fs and k d is plotted in Fig. 5 for some values of k r. When 1 the population transferred to the e state is finite (k r 9 fs), no rise in the peak shift is shown as expected. he reason for the small deviation, even in this case, from the peak shift for a nonreactive system does not come from the history effect but from population transfer to the e state as discussed below. As k r becomes small, the history effect arising from the direct decay process becomes comparable to the effect from the ground population grating associated with k r process to give a rise in the peak shift On: hu, 7 Nov :17:12

9 125 J. Chem. Phys., Vol. 11, No. 21, 1 June 1999 Yang, Ohta, and Fleming Now, we discuss the three-state system only incorporating the effect of the incoherent reaction from the donor state to acceptor state without any decay to the ground state. hen the total response function is given by Eq which describes the exponential decay of the ee-population. At long transient population time, the decrease of the role of the ee-population due to the reaction makes the interference between the gg- and ee-pathways different from that in the normal two-level system where both pathways contribute to the signal equally. As a result, the peak shift for the threestate system is affected by the incoherent reaction. When the population time becomes large, the contribution from the ee-pathway becomes minor and the interference effect disappears, i.e., the response comes only from the gg-pathway. hen, because the signal is quadratically proportional to the magnitude of the response function, only the modulus of the response function R re in Eq. 2.5 controls the peak shift. Consequently the peak shift for times longer than the population transfer is higher than that from a normal two-state system where the interference still occurs. A similar effect occurs in the work of Nagasawa et al., 17,19 where the peak shift calculated without the imaginary contribution to the line broadening function is larger than that where both contributions are included. However it is generally the case the interference effect is small at room temperature. his means an incoherent reaction which removes the excited population from the laser window has rather small influence on the peak shift at room temperature. Figure 6a shows this behavior. In fact, the peak shift is not only insensitive to the incoherent reaction rate but also is essentially identical to that with no reaction. At low temperature Fig. 6b, however, the role of the imaginary part of g(t) becomes non-negligible and the interference effect is pronounced to give an apparent rise in the peak shift on the time scale of the reaction. V. EFFEC OF FINIE PULSE SHAPE When the duration of the laser pulse is finite, nonrephasing pathways, in which the second electronic coherence state is the Hermitian conjugate of that in the rephasing pathway, should be considered with a time integral over the pulse envelopes. his kind of effect in an isolated two-level system was discussed in detail in Ref. 1. he nonrephasing pathways yield free induction decay FID while the rephasing ones have echo characteristics. Since only the population states are relevant to reactive events in the simplified reaction model, the nonrephasing response functions (R II ) for the present reactive systems are obtained from the rephasing ones (R I ) given in Eqs. 3.9 or 3.2 by a simple substitution of the rephasing reaction-free response functions R gg and R ee in R I with the nonrephasing ones. he expressions for the nonrephasing reaction-free response functions can be found elsewhere. 1 When the pulse sequence is as given in Fig. 7, R I and R II contribute as follows to a given phase-matched signal: i t 1 t 2 t 3 t:r I (tt 3,t 3 t 2,t 2 t 1 ) Echo ii t 2 t 1 t 3 t:r II (tt 3,t 3 t 1,t 1 t 2 ) FID iii t 3 t 1 t 2 t:r I (tt 2,t 2 t 1,t 1 t 3 ) Echo FIG. 7. ime ordering of pulse sequence. and denote the center to center distances between the pulses which are experimentally controllable. t i denotes the interaction points within the ith pulse. he dashed pulse denotes the signal resulting from the three interactions, respectively, at t 1, t 2, and t 3. iv t 2 t 3 t 1 t:r II (tt 1,t 1 t 3,t 3 t 2 ) FID. he other sequences of interactions yield no signal in the given phase-matched direction under the RWA. he total response function is given by the combination of the four cases weighted by the pulse envelopes. Figure 8 shows the peak shift of a two-state system at room temperature when the finite pulse shape is incorporated. he pulse shape is assumed to be Gaussian with a width of 37 fs FWHM. In addition to the usual overall increase in the magnitude of the peak shift of the nonreactive system, 1 the short time behavior is quite different than in the impulsive case Fig. 4. Non-negligible decay to the ground state during the pulse duration decreases the peak shift compared with that of nonreactive system. his does not imply a change of rephasing capability but reflects the fact that, due to the ground state recovery, the rephasable signal makes a progressively smaller contribution as the interval between the first and second pulses becomes longer. As a result, the effective rephasing capability appears to be decreased. he rise at long times can be explained similarly to that observed in the impulsive case. In contrast with the two-level system described above, the finite pulse shape does not produce any significant effect on the signal from the three-state system in the absence of FIG. 8. 3PEPS for a two-state system with a finite upper state lifetime at 3 K when the laser pulse shape is Gaussian of FWHM 37 fs. he lifetimes are indicated for each curve. in. he peak shift when k d is shown for the comparison dotted line On: hu, 7 Nov :17:12

10 J. Chem. Phys., Vol. 11, No. 21, 1 June 1999 Yang, Ohta, and Fleming 1251 FIG. 9. 3PEPS for a three-state system in the absence of the ground state recovery at 3 K. he pulse shape is the same as in Fig. 8. he lifetimes are indicated for each curve. in. he peak shift when k r is shown for the comparison dotted line. FIG. 11. Experimental data for the peak shift on the phenol blue in methanol. he laser wavelength was 58 nm and the pulse duration was 47 fs FWHM. VI. SUMMARY ground state recovery Fig. 9 since the ground state grating still persists because population transfer occurs from the excited state. A combination of both pathways, ground state recovery and population transfer to a state absorbing out of laser window, shows interesting behavior in the peak shift when the pulse duration is finite see Fig. 1. Initially, the ground state recovery makes gratings on both the excited and ground states disappear and produces the rapid decay of the peak shift as shown in Fig. 8. However, since the quantum yield of ground state recovery is less than one, some portion of the ground state grating still remains. At long time, this ground state grating controls the peak shift behavior and thus a slightly higher peak shift is predicted than in the nonreactive system as discussed in Fig. 6. Surprisingly the overall shape of the peak shift in this case, using the simple spectral density given by Eq. 4.2, is quite similar to that of the phenol blue system experimentally measured in our laboratory Fig. 11. A detailed analysis of the experimental data will be carried out elsewhere. 25 In the 3PEPS experiment, a molecular system successively evolves in coherence, population, and a second coherence electronic states. Because the time period when the system is in coherence states is very small, we can neglect electronic transitions during this period. Under this assumption, a population transition can only occur incoherently in the course of experiments. he population kinetics were incorporated by employing simple probabilistic arguments. As an example, we considered a two-state system with a finite lifetime and a three-state system with two excited electronic states. In the three-state system, only one of the excited states is resonant with the laser frequency. In these population transfer systems, the evolution of the nuclear wave packets depends on the particular electronic state. For example, the shift of the excited potential energy surface from that of the ground state produces different forces acting on the wave packet between the ground and excited electronic states. his is the reason why differences in the nuclear history emerge from the population transfer. he potential difference experienced by the nuclear motion as the electronic transition happens makes additional contributions to the response function and this kind of effect should be reflected, within the linear electron phonon coupling approximation, via the imaginary part of the line broadening function as shown in this paper. Preliminary results of 3PEPS experiments on phenol blue 25 and electron transfer systems Rhodamine6G in DMA and DEA Ref. 26 show qualitatively similar behavior to the predictions of the present approach. We expect that detailed analyses of the experiments with the present theory will provide us with useful information on reactive systems. FIG. 1. 3PEPS for a three-state system in the presence of ground state recovery at 3 K. he pulse shape is the same as in Fig. 8. he recovery time is set to be 6 fs. he time scales of population transfer to the third state are indicated for each curve, in. he peak shift when k d k r k d is shown for comparison dotted line. ACKNOWLEDGMENS his work was supported by a grant from the NSF and in part by the donors of the Petroleum Research Fund of the American Chemical Society. M.Y. wishes to acknowledge the financial support of the Korea Research Foundation made in the program year 1997 and K.O. a Research Fellowship of On: hu, 7 Nov :17:12

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