Spectroscopic signatures of bond-breaking internal rotation. II. Rotation-vibration level structure and quantum monodromy in HCP

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 1 1 JANUARY 2001 Spectroscopic signatures of bond-breaking internal rotation. II. Rotation-vibration level structure and quantum monodromy in HCP Matthew P. Jacobson and Mark S. Child a) Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom Received 1 September 2000; accepted 16 October 2000 The rotation-vibration level structure of ground electronic state HCP is investigated at vibrational energies approaching and exceeding that of the linear CPH saddle point. With respect to energies above the saddle point, we investigate possible spectroscopic manifestations of strong Coriolis interactions between the hindered, bond-breaking internal rotation of the hydrogen about the CP core and the rotation of the molecule in the space-fixed axis system. With respect to energies below the saddle point, we provide new interpretations, from quantum and semiclassical points of view, of previously observed anomalously large B rotational and g 22 energy dependence on the vibrational angular momentum constants for the large-amplitude pure bending states of HCP referred to elsewhere as isomerization or saddle node states. We also predict similar anomalies in other spectroscopic constants, including the centrifugal distortion constant D and the rotational l-resonance parameter q 2. These changes in the effective spectroscopic rotation-vibration constants are shown to be a direct consequence of the spherical pendulum topology of the HCP bend/internal rotor system, which is associated with a phenomenon called quantum monodromy, defined as the absence of a smoothly valid set of quantum numbers for all states. Our semiempirical model for the HCP bend/internal rotor mode is derived using principles of semiclassical inversion and provides new insights into the breakdown in the ability of rovibrational effective Hamiltonians to model highly vibrationally excited states of HCP American Institute of Physics. DOI: / I. INTRODUCTION a Electronic mail: mark.child@chem.ox.ac.uk This series of papers investigates the changes in frequency domain spectra that accompany the onset of bondbreaking internal rotation in small molecules like HCP, HCN, and acetylene, which can occur when the system has enough vibrational energy for the hydrogens to rotate around the heavy-atom core. In a previous paper, 1 we identified gross changes in the vibrational energy flow and quantum level structure when bond-breaking internal rotation becomes energetically possible, with a particular emphasis on the breakdown of the polyad approximation. Here we turn our attention to rotational dynamics and argue that changes in rotation-vibration structure, associated with the Coriolis coupling between the internal rotation and rotation of the molecule in the space-fixed frame, may provide direct, unambiguous spectroscopic evidence of nascent bond-breaking internal rotation in a number of systems. Experimental spectra recorded at the high vibrational energies necessary for bond-breaking internal rotation to occur will inevitably be congested, but rotationally resolved spectra may contain spectroscopic clues that indicate a transition from vibrational to internal rotor dynamics. That is, even if vibrational assignments are difficult or impossible to make unambiguously, spectroscopic constants associated with rotation and rotation-vibration interaction can still be extracted by recording spectra which access states with different angular momentum J. Except in extremely congested spectra, it is usually possible to follow vibrational levels with increasing J, and local anharmonic or Coriolis perturbations, when they exist, can often be deperturbed. The spectroscopic constants that can be extracted from such rotational progressions include the usual B constant, the centrifugal distortion constant D, Coriolis coupling coefficients such as the rotational l-resonance parameter q, and, in systems with doubly degenerate modes, g constants which parameterize the energy dependence on the vibrational angular momentum. When vibrational assignments can be made, these constants are often given an explicit vibrational dependence; however, we demonstrate here that this vibrational dependence can be highly nonlinear at high vibrational excitation, and we consider each parameter to have an implicit, arbitrarily complicated vibrational dependence. Also, it should be noted that g is generally considered to be an anharmonic vibrational parameter although the effective fitted values may contain contributions from Coriolis interactions. However, we refer to all of the above parameters generically as effective rotational constants because they appear in effective Hamiltonian models used to fit rotationally resolved spectra of HCP and other small molecules. 2,3 In principle, these effective rotational constants together provide a rich source of dynamical information, but actually relating the spectroscopic constants to the underlying rovibrational dynamics is challenging for polyatomic molecules. At relatively low energy, the effective spectroscopic constants can be related, via perturbation theory, to parameters /2001/114(1)/262/14/$ American Institute of Physics

2 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 Bond-breaking internal rotation. II 263 in a normal mode rovibrational Hamiltonian. 4 The derived relationships are generally not simple, e.g., g bb 1 48 bbbb 1 16 s 1 16 b 2 bbb 2 sbb s b s b b (z) b A b b1,b2 b for a symmetric top, 4 where the represent terms in the normal mode expansion of the potential energy surface PES, are the harmonic frequencies of the vibrational modes, and is a Coriolis coupling coefficient. The four terms in this expression represent, respectively, the contributions of the bend mode anharmonicity, stretch bend anharmonic coupling, bend bend anharmonic coupling if there are two or more bend modes, and bend bend Coriolis interactions. Here we refer to bend and stretch modes for concreteness, but b would refer more generally to a doubly degenerate vibrational mode. Thus, even at relatively low vibrational excitation, experimentally measured polyatomic molecule rotation-vibration constants are related in complicated ways to the potential energy surface and thus to the rovibrational dynamics. At high vibrational excitation, the perturbation theory approach of course becomes less useful; certainly at the onset of bond-breaking internal rotation, which is the focus of this paper, it becomes entirely invalid. Certain generic aspects of rovibrational structure in triatomic molecules with bending nonrigidity and internal rotation have been considered previously. 5 This paper explores a specific case, HCP, and provides a relatively simple physical explanation for the evolution of certain rotation-vibration spectroscopic constants with increasing bend excitation, up to and even above the barrier to internal rotation. Rotational constants in HCP have already played a critical role in elucidating the dynamics and spectroscopic signatures of bond-breaking isomerization, albeit in an indirect manner. Much of the initial interest in HCP was generated by stimulated emission pumping SEP spectra which revealed sudden, drastic changes in the rotational constants 2 for a series of states which were initially referred to as the bending progression. That is, these states were assigned as (v 1 0,v 2,v 3 0), and at v 2 36, the rotational constants inferred from the SEP data changed suddenly. Later it was revealed that these states were not in fact pure benders at all. 6 Rather, these states were strongly affected by the Fermi 2:1 resonance and at high energy involved greater excitation in the CP stretch mode than the bend. True pure bending referred to alternately as isomerization states or saddle node states states were predicted theoretically to come into existence at energies higher than that at which the first saddle node bifurcation occurs in the classical mechanics 6 and were subsequently identified experimentally and demonstrated to have anomalously large B and g constants. 6 8 Variational calculations with nonzero J have already been performed on a high quality ab initio potential energy surface, 6,8 and these impressive calculations have reproduced, with reasonable accuracy, the experimentally observed B and g constants at high vibrational excitation (v ). Our contribution is to relate the trends in the spectroscopic constants to the spherical pendulum topology of the HCP bend/internal rotor mode, which possesses a property called quantum monodromy. 9 We do so by performing calculations with only one vibrational degree of freedom, the bend, and we therefore focus on the pure bending states, which have been extensively studied theoretically and experimentally. 6 Whereas the ab initio variational calculations, which require much larger basis sets, could only be performed for J2, with our relatively simple model we are able to calculate rovibrational spectra at any desired vibrational energy and at all experimentally observed values of J. Thus, we can compare our results directly with experiment, and in fact we fit our calculated spectra in exactly the same way as the experimental spectra were fitted. 2 Through this procedure, we extract a wider range of rotational constants than were previously studied theoretically including D and q) and we find that the trends in these are also directly related to the topology of the bend degree of freedom. A final, somewhat tangential contribution of this work is a careful investigation of the limitations of traditional effective Hamiltonians for modeling the highly excited bending states of HCP; this point is explicated in Sec. II, which derives our semiempirical model for the HCP bend/internal rotor mode, using principles of semiclassical inversion. II. THE MODEL This section derives our semiempirical model for the HCP bend/internal rotor mode and contrasts it with the approaches that have been pursued previously, namely full ab initio variational calculations and effective Hamiltonian models. 6 This section may be by-passed on a first reading by anyone interested primarily in our conclusions regarding the HCP rotational constants. The starting point for our model is the exact rotationvibration Hamiltonian for HCP in Jacobi coordinates. The Hamiltonian is taken from the literature, 8,10 but a cursory overview of its construction is highly relevant to much of what follows. The relevant Jacobi coordinates are R, the CP bond length; r, the distance from the CP center-of-mass to the hydrogen; and, the angle between R and r. We denote the angular momentum of the CP core as j and that of the the hydrogen as l with respect to the CP center-of-mass. The simplest expression for the Hamiltonian is Ĥ 2 2 r 2 r r r 2 lˆ R 2 R 2 2 R R 2 ĵ 2 VR,r,. 2 In order to study Coriolis effects related to the hydrogen internal rotation, it is convenient to rewrite the Hamiltonian in terms of the total angular momentum Jj l and the hydrogen internal rotation angular momentum l. Using the identity ĵ 2 Ĵ 2 lˆ2ĵ lˆĵ lˆ2lˆzĵ z, we arrive at 3

3 264 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 M. P. Jacobson and M. S. Child where Ĥ 2 2 r 2 r R 2 R 2 Ĥ rot Ĥ Cor VR,r,, 2 Ĥ rot 2 R R 2 Ĵ 2 2lˆzĴ z is the rotational energy, and 2 Ĥ Cor 2 R R 2 Ĵ lˆĵ lˆ 2 2 r r R R 2 lˆ2 represents the Coriolis coupling. The operators lˆz and Ĵ z represent the projection of the hydrogen atom rotation and total angular momentum, respectively, onto the body-fixed z-axis, which is chosen to be R, the CP bond. The corresponding quantum number for both of these operators is k the vibrational angular momentum. Exact variational calculations have been performed using this Hamiltonian and a high quality ab initio PES. 6,8 Here, we develop an approximate bending Hamiltonian that is computationally less demanding than full variational calculations and provides new insights into the effects of Coriolis coupling. That is, we begin by treating the bend/internal rotor degree of freedom as being adiabatically decoupled from both stretch modes. This is a poor approximation in a global sense, because it is well known that the bend and CP stretch modes are strongly coupled, through Fermi 2:1 resonance, at energies up to at least cm 1. It is possible to explicitly incorporate this coupling, but here we focus more narrowly on the well-studied class of states which can be described as pure bending, in the sense that they follow, to a good approximation, the minimum energy isomerization potential. Also, note that the form of the Coriolis operator above makes it clear that all Coriolis coupling in HCP, in Jacobi coordinates at least, occurs within the manifold of bending states. That is, the Coriolis term mixes bending states with k quantum numbers that differ by one. All eigenstates of the full Hamiltonian except those with J0 and k0) will feel some effects of the Coriolis coupling, due to the anharmonic coupling of the bend mode to the remaining degrees of freedom, but it can be anticipated that the effects of Coriolis will be maximal for states with maximal bend excitation. A straightforward adiabatic treatment of the bend mode would proceed by using the ab initio PES to solve the stretch eigenfunctions of the Hamiltonian at every value of the bend angle. The adiabatic bend Hamiltonian would then take the form Ĥ bend f lˆ2v eff, where V eff () is determined by the lowest stretch eigenvalues E (), and the f () factor in the kinetic energy would be determined by the expectation values where () are the stretching eigenvectors corresponding to the eigenvalues E (). We have performed such a calculation but do not report the results here. Instead, we use a similar model for the bend but derive the effective bend potential using a semiclassical inversion method. We opt for this somewhat more circuitous approach for two major reasons. First, the effective bend potential derived by semiclassical inversion reproduces the experimental rotational constants to significantly better accuracy than the straightforward adiabatic approximation, which provides only qualitative agreement with experiment we elaborate on this point in Note 11. Second, potential energy surfaces with accuracy comparable to that of the HCP surface exist for only a few mostly 3 atom polyatomic molecules. For this reason, we have been interested in exploring approximate, semiempirical models that make maximal use of experimental data and may be particularly useful for studying larger molecules, for which the ab initio approach is substantially more difficult. The most common semiempirical models are spectroscopic effective Hamiltonians, which have yielded numerous insights into the large-amplitude vibrational dynamics of many small molecules, including HCP. 6,12 However, as explored below and in Ref. 1, such models have serious deficiencies at energies near and above that of a saddle point. Our semiclassically inverted effective bending potential remedies several of these problems, but retains computational simplicity. The semiclassical inversion method that we employ is very closely related to the well known RKR method. In the usual RKR approach, which is routinely applied only to diatomic molecules, there are two working equations which together uniquely determine the two classical turning points for the one-dimensional vibrational motion. The first of the working equations determines the difference between the turning points and involves only the vibrational energies. The second equation determines the difference between the inverses of the turning points and depends on both the vibrational energies and the B rotational constants. The HCP bend potential is symmetric about 0, and the symmetrically related turning points are uniquely determined by either one of the equations. In practice, the second equation is not generally useful for polyatomic potentials because, unlike the case of diatomics, the measured rotational constants are not necessarily directly related to the molecular geometry. It will be seen below that Coriolis effects in HCP make very substantial contributions to the effective rotational constants. Ratner and co-workers have successfully employed a semiclassical inversion method, RKR-SCF, to determine a 3D PES for CO 2, using rovibrational energies. 13 They note however that deperturbation is necessary to remove the effects of... Coriolis resonance, which is impossible to fully achieve in the present case, at least at the high vibrational energies of interest. Thus, we utilize the first of the two working equations for the inversion. For k0, the bend Hamiltonian takes the form f r r 2 2 R R 2, 8 f d2 d 2 V eff 9

4 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 Bond-breaking internal rotation. II 265 FIG. 1. Upper left: The effective bending potential determined from the RKR-type semiclassical inversion procedure. The dashed line is the potential using the extrapolated effective Hamiltonian, while the solid line is the corrected effective potential, using knowledge of the saddle point energy. The dashed dotted line is the difference between the inverted potentials using the HCP Ref. 14 and DCP Ref. 20 effective Hamiltonian models. Lower left: The variation of the Jacobi distances R dashed line and r solid line along the HCP minimum energy isomerization pathway, using the potential of Ref. 8. Right panels: The bending frequency vs bend energy of our approximate bend/ internal rotor Hamiltonian, with k0 and k2. The solid line is the classical bend frequency, and the circles are quantum energy level spacings. The dashed line in the k0 case is the classical bend frequency predicted by the effective Hamiltonian model of Ref. 14. The dashed line in the k2 case is the classical azimuthal frequency. and the appropriate RKR-type inversion equation can readily be derived to be t 1 t f d v(u) 1 1/2 UEv dv 10 in which t is the classical turning point, U is a given vibrational energy, and E(v) is the vibrational energy dependence, which in practice is determined by an effective Hamiltonian fit to relevant data. Our estimate for f () will be discussed below. As a practical matter, we obtain the turning points at any energy U by first calculating the righthand side integral using Gauss Jacobi quadrature, and then varying the integration region on the left to obtain the equality using a root finding routine. Several effective Hamiltonian models for HCP have appeared in the literature in the past few years, 6,12,14 all of which have included states with 30 quanta of bend excitation. We have performed calculations using each of these models as input, and found minimal differences among the predicted sets of rotational constants. The results shown below use the most recent effective Hamiltonian, which is reported in Ref. 14. The dashed line in the upper left panel of Fig. 1 represents the effective bending potential determined by the RKR-type semiclassical inversion procedure. The arrow along this potential is positioned at E cm 1, which is the highest bend energy included in the fit of the effective Hamiltonian model. Thus, the inverted bend potential beyond 1.4 is determined by extrapolating the effective Hamiltonian to higher energies than were fit, and must be treated with suspicion. In fact, the inverted potential is clearly in error at higher energies, because the maximum energy at lies at only cm 1, whereas the best electronic structure calculations predict the saddle point to lie at cm 1 once zero-point effects have been taken into account. 8 Thus, although the inverted potential reproduces the bending energies with excellent precision a few cm 1 )uptoe cm 1, the predictions at higher energies must be grossly in error. Although effective Hamiltonians cannot in general be expected to extrapolate far beyond their fitted energies, the drastic failure in this case is intriguing. The upper right panel of Fig. 1 provides some insight into this phenomenon. The dashed line is the classical frequency of the bend mode (k 0) as predicted by the effective Hamiltonian. Quantum mechanical effective Hamiltonians generally express the zero order energy of a vibrational mode as a polynomial expansion in the quantum number; all of the effective Hamiltonian models that have been used to represent the highly excited bend states of HCP have utilized quartic expansions for the bend, i.e., E b n b 1xn b 1 2 yn b 1 3 zn b The classical equivalent of a quantum number is an action, and using standard correspondence principles, the classical energy can be written as E b I b xi 2 b yi 3 4 b zi b 12 with associated classical frequency C de b 2xI di b 3yI 2 b 4zI 3 b. 13 b Thus, is the nominal bend frequency i.e., in the limit of very small bend excitation, while the x, y, and z terms represent the linear, quadratic, and cubic dependence of the frequency on the action/quantum number. z-terms are rarely used in effective Hamiltonians, and y terms are often not well defined. However, as can be seen in the upper right panel of Fig. 1, the bending frequency drops extremely rapidly above cm 1. Within the space of four quantum bending levels, the classical frequency of the bend drops by

5 266 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 M. P. Jacobson and M. S. Child nearly one-half. This drop in the frequency is very closely related to the kink in the PES that occurs when the hydrogen changes its bonding from the carbon to the phosphorus. 6 It is clear that the drop in frequency could not be represented by a model with only an and an x term; in such a model, the frequency can only decrease linearly with the bend quantum number. In practice, both y and z terms are needed. While a comparatively high-order Dunham expansion makes it possible to model the sudden drop in the bend frequency, the model extrapolates poorly. A polynomial expansion for the energy vs action will always cause the frequency to diverge to either positive or negative infinity at large actions/quantum numbers. In this case, the frequency drops very rapidly toward negative infinity, as the z term begins to dominate. This unphysical extrapolation of the effective Hamiltonian in no way undermines its utility for analyzing data at lower energies, but the effective Hamiltonian is incapable of modeling states with more than cm 1 of bend excitation (32 quanta). Poor extrapolation is a general problem of effective Hamiltonians which is made much more severe in this instance by the sharp drop in frequency. Regardless of the details of the effective Hamiltonian, if experimental data were only available with up to 30 quanta of bend excitation, then one would be led to conclude that the barrier height for the isomerization is less than cm 1, because a sudden dip in the bending frequency or in the quantum mechanical level spacing is the most obvious signature of approaching a saddle point see, e.g., Refs. 9 and 15, and the classic work of Dixon 16. In this case, at an energy roughly two-thirds that of the saddle point, a kink in the potential drives down the frequency drastically, but the classical bend frequency will only reach zero at the saddle point itself. Thus, it is inevitable that the bending frequency will partially stabilize after its initial drop. In principle, an even higher order Dunham expansion would be capable of modeling not only the initial drop in the bend frequency but also its stabilization. Such a model would be very difficult to fit to data reliably, although high order effective Hamiltonians have been derived by perturbation theory from potential energy surfaces such as the study of HCP in Ref. 17. However, using a very high order expansion would only exacerbate the problem of poor extrapolation. For example, if a tenth-order polynomial were capable of providing a good fit up to cm 1, it would diverge extremely rapidly at higher energies. To obtain a more robust model for the bending energetics, we cease to employ the RKR potential at c 1.4 and instead extrapolate the bending potential such that it reaches the predicted saddle point energy, which we take to be B cm 1. The detailed behavior of the bending frequency between and cm 1 will of course depend on the precise form of the extrapolation. We could in principle utilize the ab initio PES to obtain a reasonable form for the extrapolation i.e., by inspecting the minimum energy or adiabatic bend potential. However, Ref. 8 notes that the relevant portions of the PES contain slight undulations... which are partly due to the analytic fit. For this reason, we choose a very simple, smooth analytical extrapolation for the bend potential, of the form B1cos, 14 where is a dummy angle related linearly to the Jacobi bend angle, with the proportionality constant chosen such that the value of the RKR potential and the extrapolation potential match at c. The complete extrapolated potential is labeled as Extended RKR in the upper left panel of Fig Although we cannot, of course, guarantee that the extrapolated portion of the potential is numerically precise in its details, we expect to obtain qualitatively reasonable results at energies above the kink, and indeed above the barrier to internal rotation. Our success at reproducing key behaviors of the rotational constants at high vibrational excitation suggests that this expectation is correct. Up to this point we have not specified the precise form of f () in the kinetic energy term of the approximate bending Hamiltonian. We could of course choose the adiabatic approximation in Eq. 8, but we choose instead a simpler form, f 2 2 r r R R 2, 15 which is similar in spirit to a semirigid bender model. 5,19 We specifically choose to estimate the bond-length dependence on as the values along the minimum energy isomerization pathway of the ab initio PES. 8 Thus, although we determine the effective bending potential using semiclassical inversion, electronic structure data remains critical to the success of our model. However, we do wish to point out that 1 the determination of an approximate isomerization pathway is computationally less demanding than the calculation and fitting of a global potential energy surface and 2 a small number of points along the minimum energy pathway is sufficient to estimate the relevant changes in bond lengths with bend angle. In this particular case, the geometry of the saddle point CPH cannot be determined from available experimental data and cannot be estimated to an appropriate accuracy using simple valence bonding concepts. Indeed, in the absence of electronic structure calculations, it would be uncertain whether CPH is indeed a saddle point, rather than possibly a shallow local minimum. In addition to the HCP and CPH stationary points, the other critical region for estimating f () is that of the kink, which occurs at 60 and cm 1 of bend excitation, and which is closely related to the transfer of the hydrogen bonding from the carbon to the phosphorus. Thus, electronic structure calculations at 3 judiciously chosen nuclear geometries could provide an adequate estimate of the isomerization pathway. However, because an excellent surface is available, we have utilized 15 evenly spaced points along the isomerization coordinate, for maximum accuracy. The resultant functional forms of r() and R() are shown in the lower left panel of Fig. 1. The approach that we have taken highlights how experimental data and electronic structure calculations can be used in complementary ways to determine a model with excellent predictive power. In this and other works e.g., Ref. 13,

6 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 Bond-breaking internal rotation. II 267 experimental data have been inverted to determine potentials which reproduce the data with excellent accuracy. However, available experimental data generally does not provide adequate information about all desired regions of a PES, and thus electronic structure calculations can be used, e.g., to extrapolate the inverted potential to higher energies in a reasonable manner. Admittedly, we have used such an approach only for an effective 1D potential and the construction of a global PES by similar means can be expected to be much more difficult. We believe that our approximate model for the HCP bend mode is physically reasonable primarily due to its success at reproducing the rotational constants of highly vibrationally excited (40 quanta) states, as demonstrated below. The predictive power of our model can also be independently assessed in the context of isotopic substitution. Although DCP has not been studied as thoroughly as HCP, either experimentally or theoretically, an effective Hamiltonian model has recently been published for DCP, 20 which can also be used to obtain a semiclassically inverted effective bending potential, in the same manner as for HCP. The effective potentials obtained using the HCP and DCP data are extremely similar; 21 the dashed dotted line in the upper left panel of Fig. 1 is the difference between the two potentials. III. VIBRATIONAL STRUCTURE AND SPHERICAL PENDULUM MONODROMY The central thesis of this article is that unusual variations of the effective rotational constants with increasing bend excitation, some of which have been observed previously 6 and others of which are predicted here, are closely related to the spherical pendulum topology of the HCP bend/internal rotor mode and its associated quantum monodromy. This section introduces the concept of quantum monodromy in the context of a careful quantum and classical study of the approximate bending vibrational Hamiltonian derived in the preceding section. Before proceeding, we first provide a few details on the quantum and classical calculations. For convenience, in both the quantum and classical calculations, we parameterize f () and V eff () as cosine Fourier expansions. The kink in the bend potential, as with any sudden change in a periodic signal, is associated with high frequency components. In practice, we found that a 15 term Fourier expansion was sufficient to quantitatively reproduce the effective bending potential and converge the eigenenergies. With respect to quantum calculations, we obtain the HCP bending vibrational eigenstates in a basis set of spherical harmonics 100 basis functions is sufficient to converge the results up to cm 1 ). The kinetic and potential energy matrix elements in the spherical harmonic basis set were computed both by quadrature and by using analytical expressions obtained by using the Wigner Eckart theorem. The classical vibrational Hamiltonian is H bend f p 2 k2 sin 2 V eff, 16 where kp is a constant of the motion. Trajectories can be obtained in a straightforward manner by numerically integrating Hamilton s equations. The right column of Fig. 1 provides a simple introduction to the monodromy associated with the HCP bend/ internal rotor system. The dots represent quantum energy level spacings, specifically 0.5 (E n1 E n1 ). The solid lines are classical frequencies, which of course are in close agreement with the quantum spacings, once zero-point effects are taken into account. The main point to note here is that the classical frequency for k0 drops to zero at E B. The system has an infinite classical period at this energy because it will come to rest at the saddle point. On the other hand, the classical frequency does not reach zero for k2, or any nonzero value of k, because the system cannot pass through the saddle point due to the infinite centrifugal potential at created by the k 2 /sin 2 term. That is, the system never comes to rest at EB with nonzero angular momentum and instead loops around the saddle point, rather than passing over it. The quantum energy level spacings of course do not go to zero for either k0 ork2, but the level spacings for k0 do fall more rapidly and to a lower minimum value that do those for k2. This difference between the behavior of the k0 and k0 levels is the first of several consequences that we will consider of a restructuring of the quantum energy levels at the saddle point, which is associated with the quantum monodromy. The concept of quantum monodromy provides a framework for understanding certain generic behaviors of systems with cylindrically symmetric potential energy barriers, of which the HCP bending system is an example. The saddle point in this system possesses cylindrical symmetry because it is located at, at which point the variable is indeterminate. The critical property of a system with quantum monodromy is the absence of any smoothly valid set of quantum numbers for the entire spectrum; from the classical point of view, the monodromy is associated with a gross topological obstruction to the global construction of angleaction variables. The existence of quantum monodromy implies a large-scale restructuring of the quantum eigenstates at energies near the energy of the potential barrier responsible for the monodromy, and it is this restructuring that is responsible for the enormous changes in the HCP rotational constants with increasing bend excitation. The specific topology associated with the HCP bending system is that of the spherical pendulum. Imagine a pendulum that is not constrained to move in a plane but instead can move freely on the surface of a sphere; the equilibrium point pointing straight down corresponds to HCP, and the saddle point CPH occurs when the pendulum is pointing straight up. The monodromy in the spherical pendulum is associated with a transition from oscillatory vibrational behavior at EB to rotary internal rotation of the hydrogen behavior for EB. This transition from vibration to rotation distinguishes the topology of the spherical pendulum from that of the champagne bottle, which is associated with quasilinear e.g., Ref. 9 and Renner Teller systems e.g., Ref. 15. Champagne bottle systems also possess quantum monodromy but undergo only bound vibrational motion both

7 268 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 M. P. Jacobson and M. S. Child above and below the potential barrier. The distinction between the two limiting behaviors in the champagne bottle case is that linear molecule quantum numbers are more appropriate at EB, and bent molecule quantum numbers at EB. Due to these differences, comparisons between spherical pendulum and champagne bottle systems must be made cautiously. The mathematical origin of the monodromy of the spherical pendulum has been discussed in detail elsewhere. 9 Here we wish to provide an intuitive feel for the changes in the rovibrational structure that are associated with the monodromy and the transition from oscillatory to rotary motion of the hydrogen about the CP core, primarily from the classical point of view. In the limit E 0, the behavior of H bend of course approaches that of the two-dimensional harmonic oscillator 2DHO. For our purposes, the critical characteristic of the classical 2DHO is that the radial and angular frequencies remain locked in a 2:1 ratio at all energies. That is, the general classical trajectory for the 2DHO is an ellipse, and during one cycle of the ellipse, the radial variable which only takes positive values undergoes two cycles, while the value of undergoes a single cycle. An important consequence of this behavior is that a total action can be defined as I2I I. We designate the equivalent quantum number as n b 2v k, and all eigenstates of the 2DHO with the same value of n b are exactly degenerate. For EB, H bend in our model can be considered an anharmonic 2D oscillator, where is the radial dimension. The critical property of the classical anharmonic 2D oscillator is that the radial and angular azimuthal frequencies are no longer locked into a precise 2:1 ratio. That is, as can be seen in the left column of Fig. 2, dashed line advances by slightly more than during one period of the motion solid line. Thus, while the 2DHO trajectories are perfect ellipses, the trajectories for the anharmonic oscillator are characterized by approximately elliptical motion which precesses over time. This precession can be clearly observed in the right panel of Fig. 2, which projects the trajectories into the (x, y) plane, which is orthogonal to the z-axis represented by the CP bond; more precisely, we define xsin cos and ysin sin. As explored in greater detail in Sec. IV, the key quantum manifestation of the precession is that, although the quantum eigenstates can continue to be labeled by a quantum number n b 2v k, states with the same value of n b are no longer exactly degenerate. This lifted degeneracy can be observed in the top panel of Fig. 3, which plots the eigenenergies of Ĥ bend vs k. The states with n b 10 are connected with a line to aid the eye; a modest, approximately quadratic energy dependence on k can be observed, which is parameterized in an effective Hamiltonian by a g constant; see Sec. IV. The lower right panel of Fig. 1 plots the radial ( ) and azimuthal ( ) frequencies for H bend for the specific case of k2. The definition of for the anharmonic oscillator is not entirely trivial, because the time required for to increase by 2 varies from cycle to cycle, depending on the behavior of during that cycle the converse is not true; the frequency does not depend on the behavior of, only the FIG. 2. Classical trajectories for the HCP bend/internal rotor model, with ki 2. Left column: The values of solid line and dotted line over one period of the motion. Right column: A projection of the classical motion onto the x/y plane, which lies perpendicular to the CP axis. That is, the right column plots ysin sin vs xsin cos. value of k, which is constant. However, can be defined as a long-time average, which is equivalent to 2, 17 where is the change in during one period of the motion. In the context of this broad overview, the key behavior that we wish to point out is the evolution of the ratio /, which is close to 2 at low energy near-harmonic limit, but becomes nearly 1 at high energy. That is, in the limit of free rotation, the and dimensions are in some sense indistinguishable and thus their frequencies are identical. The bottom panels of Fig. 2 depict a classical trajectory at cm 1, which is slightly above EB. Although this is certainly not the high energy limit, the and frequencies are already nearly identical. This has important consequences for the quantum structure. In the high energy limit, the appropriate total action is I j I I and, as can be observed in the quantum eigenenergy plot in Fig. 3, quantum states at EB with the same values of the corresponding quantum number v j v k are nearly degenerate. Clearly, there is a large-scale restructuring of the classical dynamics and quantum structure between the low ( 2 ) and high ( ) energy limits, associated with the monodromy due to the saddle point at EB. The ratio between the classical and frequencies changes most rapidly at EB, as can be understood simply in the following

8 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 Bond-breaking internal rotation. II 269 can clearly be observed, and near this energy, neither states with constant n b nor states with constant v j are nearly degenerate. IV. THE g PARAMETER The variation of the g parameter, which in a rovibrational effective Hamiltonian represents the dependence of the energy levels on the vibrational angular momentum k, with increasing bend excitation is extremely closely related to the spherical pendulum quantum monodromy. The Dunham expansion for an anharmonic 2D oscillator takes the form En b,kn b 1xn b 1 2 gk The g parameter is sometimes given an explicit vibrational dependence, usually of the form, g eff g y n kk n b 1 20 in a multimode system, g eff would be a function of all the vibrational quanta. This type of parameterization is much too simple to reproduce the effects that will be studied here, and we consider g to be an arbitrarily complicated function of n b. Using the Dunham expression, g(n b ) can be estimated as gn b 1 4 E (nb ) 2 E (nb ) FIG. 3. Top: Eigenenergies of the HCP bend/internal rotor model as a function of k. The solid lines connect states with the certain values of n b. Bottom: Effective g values, as estimated from Eq. 21 solid line and as obtained from a rotational effective Hamiltonian fit to the Coriolis-coupled rovibrational eigenstates dashed line. The circles are the g values obtained from variational calculations on the ab initio surface Ref. 6, 8. manner. The variable receives the greatest torque and thus advances most rapidly when 0 or, as can be seen from the application of Hamilton s equations, d H 2 f p dt p sin In fact, as can be seen in Fig. 2, it is a reasonable approximation that advances by each time that approaches one of the two poles, 0 and. Below the barrier, the trajectories can only approach the 0 pole, whereas for EB both poles can be approached. Thus, the ratio of and changes quite rapidly near EB, as the trajectories just begin to approach the pole. The consequences of the corresponding quantum mechanical restructuring will be considered below, in the specific context of effective spectroscopic constants. However, the qualitative features of this restructuring can clearly be observed in the eigenenergy plot in Fig. 3. A coalescence of points about EB and k0 Here and below we use the notation (n b ) k to label the eigenstates of Ĥ bend the sign of k is largely irrelevant to our discussion. Note that, although these labels can be applied to all eigenstates, the v j quantum numbers discussed above are more physically meaningful for the states with EB. The expression for g in Eq. 21 is not quite correct, because we have neglected the requirement that Jk. That is, g can only be properly determined by performing a calculation including the rotational and Coriolis portions of the bending Hamiltonian. However, the rotational contributions to g, at least in this particular case, are relatively small, as will be demonstrated explicitly below. The g constant, as estimated by Eq. 21 and depicted in the bottom panel of Fig. 3, increases relatively gradually up to n b 60, at which point it increases extremely rapidly. The very rapid increase at n b 60 is due, of course, to the quantum monodromy. As can be seen in the upper panel of Fig. 3, the curve connecting the states with n b 70, which lie at E B, is not even approximately parabolic. Rather, it comes to a point at k0. A g value can still be computed using Eq. 21 for states with internal rotor character, but it is no longer meaningful in the sense of describing a quadratic energy dependence on k. Thus, the g constant is guaranteed to increase substantially as E B due to the quantum monodromy, before it ceases to be a meaningful quantity. The variation of the g constant at intermediate bend excitation is somewhat more difficult to explain. As discussed in Sec. III, for a purely harmonic 2D oscillator, g0, and qualitatively it is clear that g will increase as anharmonicity becomes more important. Indeed, as can be seen in the upper panel of Fig. 3, the curve connecting the states with n b 40 demonstrates substantially more curvature than that for

9 270 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 M. P. Jacobson and M. S. Child n b 10; the g values has increased by 50%. However, as can be seen in the bottom panel of Fig. 3, our model predicts that the g constant reaches a local maximum at n b 30 and then decreases slightly before rising steeply again. This behavior is not an artifact of our model. Although g constants have not yet been measured experimentally for the highly excited pure bending levels, they have been calculated for these states by variational calculations on the ab initio PES Refs. 6, 8 circles in the bottom panel of Fig. 3. The results of the two calculations demonstrate exactly the same qualitative behavior, including the local maximum, and in fact are in good quantitative agreement. The behavior of the g constant at intermediate bend excitation can be understood most powerfully from the classical point of view. Specifically, the g constant is closely related to the precession observed in the trajectories in the right column of Fig. 2. To derive the precise relationship, we define a classical analog of the quantum mechanical g as g C lim k 0 E k n This definition simply restates that g encodes the quadratic energy dependence on k, about k0. We evaluate this classical quantity as follows: E 2 k n n 2 k E n E n C 2k n, 23 ke where C is the classical bending frequency. The remaining partial derivative n ke n v E v k E 2 v k E 24 is somewhat more complicated to derive; we follow a similar treatment for the champagne bottle potential in Ref. 9. First, we write the semiclassical action integral v b p d, 25 a where a and b represent the classical turning points and p EV k2 1/2 f sin Thus, v k 1 b E a k p sin 2 d. Now, using Hamilton s equations, 2 f p, 2kf sin 2, we find that FIG. 4. Left: The rate of precession in the azimuthal angle as a function of the quanta of bend excitation. Right: Comparison of the quantum mechanical g circles, as calculated by Eq. 21, with its classical equivalent, g C solid line. In both panels, k1 is used for the classical calculations. v k 1 b E a d 1 2 dt Finally, we arrive at C g C lim k 0 2k. 31 In the right panel of Fig. 4, this g C is compared to the quantum g estimated from Eq. 21. We did not actually take the limit k 0 for the classical calculation; the values of g C change slowly with k, and we simply used k1. The classical and quantum values of g are in very close agreement. The above derivation provides a framework for understanding the origin of the local maximum in the g constants at n b 30. The C is a smooth function without a local maximum, as can be seen in Fig. 1. Therefore, the local maximum arises from the portion of Eq. 31; in other words, the rate of precession in the azimuthal () dimension slows down slightly between n b 30 and n b 40, as can be seen in the left panel of Fig. 4. This strange behavior of the rate of precession is in turn directly related to the kink in the potential that marks the transfer of the hydrogen bonding from the carbon to the phosphorus. In the simplest sense, the anharmonicity of the effective bending potential increases up to the kink, and the g constant increases monotonically as well. However, just beyond the kink, the bending potential briefly rises somewhat more steeply, before again flattening out toward the saddle point. This behavior can also be studied numerically, using the relationship, implied by the above derivation, that b k 2 a p sin 2 d. 32 This integral clearly accumulates most rapidly near the turning points a and b, where p 0. At the inner turning point the dynamics is dominated by the k 2 /sin 2 centrifugal potential, which is unrelated to the details of the potential. In fact, the potential near the inner turning point is nearly indistinguishable from that of the 2DHO potential, which is dominated by a similar centrifugal potential of the form k 2 /(2 2 ). Thus, the precession is due largely to the outer turning point b, and reaches a local maximum when the kink first becomes energetically accessible, because the classical trajectory spends more time near the outer turning point.

10 J. Chem. Phys., Vol. 114, No. 1, 1 January 2001 Bond-breaking internal rotation. II 271 FIG. 5. Overview of Coriolis effects in the bend manifold. Upper panels: The sticks represent the projection-squared of basis states with well-defined n b and k onto the rovibrational eigenstates of the rotating kinked spherical pendulum. The x-axis is the energy in cm 1, scaled to remove the nominal rotational contribution. Lower panels: Close-up views of relevant portions of the HCP bend eigenstate plot in Fig. 3. The n b 10, 40, and 70, k0 zero-order states are marked by larger circles. V. EFFECTS OF CORIOLIS ON THE SPECTROSCOPIC CONSTANTS In order to simulate states with nonzero J, we need to incorporate the effects of Ĥ rot and Ĥ Cor from the Jacobicoordinate Hamiltonian. Consistent with our treatment of the vibrational Hamiltonian, we invoke the substitution R R 2 2 R R 2, 33 where R() is estimated from the minimum energy isomerization path of the ab initio PES and is depicted in the lower left panel of Fig. 1. As a practical matter, we perform the quantum calculations with nonzero J by using a diagonalization/truncation scheme, in which we obtain the eigenstates of the vibrational Hamiltonian Ĥ bend for each relevant value of k i.e., kj), and then couple these states with the Coriolis interaction, which has a selection rule k 1. That is, the full rotational diagonalizations are performed in a basis set defined by (n b ) k, as well as the rigorously conserved quantum number J. The upper portion of Fig. 5 consists of simulated spectra for various k0 states. That is, the sticks represent the projection-squared of various (n b ) 0 states onto the eigenfunctions. The diagrams at the bottom of Fig. 5 are close-up views of the eigenstate diagram in Fig. 3 and help to rationalize the trends in Coriolis coupling and the associated changes in the effective rotational constants. For example, the 10 0 level interacts primarily with 9 1 and 11 1, which lie 600 cm 1 lower and higher in energy, respectively. These perturbing levels gain some intensity from 10 0, increasingly so as J increases due to the scaling of the Coriolis interactions with J). However, the nominal 10 0 level the central peak itself is shifted only slightly in energy by the Coriolis interaction, particularly because the n b 9 lower energy and n b 11 higher energy push the n b 10 level in opposite directions. Below, we will examine in some detail the effects of the Coriolis interactions on the effective rotational constants at relatively low energy. At n b 40, the effects of the Coriolis interactions have become greatly amplified. At low but nonzero J, a 3-line pattern can be observed which is similar to that for n b 10, and again results from the interaction of 40 0 with its n b 1, k1 neighbors. However, as can be seen in lower panels of Fig. 5, the spacing between the levels is smaller (300 cm 1 ), because the bend frequency has dropped by a factor of 2 this is above the kink in the potential. At higher J, the spectrum becomes more complicated, as a greater number of states becomes entangled by the Coriolis interaction. At J12, the additional states that are strongly coupled to 40 0 include directly coupled states with k1 but n b 1, particularly 43 1 and 37 1 ; and indirectly coupled states with k1, including 40 2, 39 3, and Despite the increased complexity of the spectrum, the nominal 40 0 state can be followed up to at least J 12, although its energy is much more strongly perturbed by the Coriolis coupling, relative to n b 10, which will be seen to result in extreme changes in the effective rotational constants. The Coriolis interactions at n b 40 are much stronger than those at n b 10 both because the relevant Coriolis matrix elements are larger and because the coupled states are closer in energy. At n b 70, however, a more fundamental change in the nature of the Coriolis interactions can be observed. The 70 0 level lies slightly above the monodromy point, E cm 1, and the associated restructuring of the bending quantum level structure causes states with k 1 to become nearly degenerate, whereas they are separated by approximately the harmonic frequency at low energy. That is, as discussed in Sec. III, states with the same value of v j are exactly degenerate in the pure internal rotor limit. The 70 0 level lies only slightly above the monodromy point, and deviations from pure rotational behavior can be observed, although they are relatively small. As a result, 70 0 is extensively fractionated through direct, strong Coriolis interaction with a manifold of nearly degenerate states. The associated spectrum consists of a fan of states, which is a signature of the angular momentum coupling between the internal rotation of the hydrogen and the rotation of the CP core. The massive Coriolis effects at EB could in principle be experimentally observable, either directly or through perturbations; the published SEP data extend to cm 1, 2 which is believed to be below the saddle point energy, but ongoing work by Ishikawa and co-workers 22 continues to extend the experimental data set to higher energies.