J. Phys. Chem. 1988, 92, 3217-3225
3217
Classical Trajectory Calculation of the Benzene Overtone Spectrat Da-Hong Lu and William L. Hase* Department of Chemistry, Wayne State University, Detroit, Michigan 48202 (Received: June 11, 1987; In Final Form: October 1, 1987)
A quasi-classical-trajectory method is used to calculate line widths for CH(D) overtone states in benzene. The potential energy function used for the trajectories is derived in part from ab initio calculations. An important potential property is the attenuation of a CCH(D) bend force constant as the CH(D) bond is extended. The probability P(n,t) of populating the initially prepared overtone state In) versus time is calculated for both CH and CD overtone states. All the CD states in C6D6and the high-energy CH states in C6H6decay nonexponentially. Absorption envelopes are determined from the Fourier transform of the square-root of the P(n,t) probabilities. The line widths vary from 30 to 110 cm-' and are somewhat smaller for the CD overtones. Comparisons are made between the calculated and experimental line widths.
I. Introduction Extensive studies have been performed on the visible absorption spectra of molecules containing the X H chromophore (X = C, Si, 0,N, etc.).l+ Transitions are observed which are consistent with initial excitation of a particular X H bond to an overtone state containing n quanta. This state is not an eigenstate of the bound molecular Hamiltonian, but a superposition of the eigenstates. Since the initially prepared state is thought to involve excitation of a particular bond, it is often called a local mode. There is considerable interest in the origin of the line width (structure) of the local mode overtones. The most widely accepted interpretation of the widths is that they result from intramolecular vibrational energy redistribution (IVR) from the local mode overtone state.'&'' A possible hierarchy of states involved in the IVR is depicted in Figure 1, where the overtone state Is) is strongly coupled to the states 11) but is weakly coupled to the b ) states. Theoretical interpretations of the line widths may provide information about these couplings, and, thus, IVR within the molecule. Holme and Hutchinsonl* have studied the effect of the couplings on the excitation process. Recent experiments have also probed the relationship between the excitation and intramolecular proces~es.'~ The C H and C D overtone states in C6H6, C6D6, and C6HD5 have received much s t u d y . ' ~ ~ J ~The - ' ~ line widths for the n = 3-9 states as measured by Berry and co-workers3 are -50-120 cm-' at room temperature. They have argued that the lines are not inhomogeneously broadened by rotational band structure, sequence congestion, or collisions, but instead are broadened by homogeneous intramolecular relaxation proce~ses.~Their analysis is supported by experiments at 1.8 KsZo However, concern was expressed that inhomogeneous effects due to thermal congestion may contribute to the benzene overtone line widths.21 In recent work Page et aLZ2have measured a bandwidth of 10 cm-' for the n = 3 overtone state of rotationally cold benzene. This bandwidth is substantially smaller than that measured by Berry and co-workers, which indicates there are significant inhomogeneous components to the room temperature overtone line widths for the low-level overtone states. Preliminary classical trajectory calculations1z~23 illustrated the sensitivity of IVR in benzene to potential energy surface properties. This work has been amplified by that of Thompson and co-workers who have used classical trajectories and different model potential energy surfaces to study the flow of energy from highly excited ~ ~ ~one ~ model potential energy C H overtones in b e n ~ e n e . *With function they found it is necessary to treat the CH mode anharmonically to obtain appreciable energy flow from the excited C H stretch local mode.24 However, using different potential energy parameters, rapid energy transfer out of an excited C H
-
'Preliminary reports of this research were presented at the 20th Midwest Theoretical Chemistry Conference, University of Pittsburgh, Pittsburgh, PA, May 14-16, 1987; at the American Conference on Theoretical Chemistry, Gull Lake, MN, July 25-31, 1987; and at the 10th International Conference on Molecular Energy Transfer, Emmetten, Switzerland, August 23-28, 1987.
stretch mode occurred for a purely harmonic internal coordinate potential energy function.25 An advance in understanding the relaxation of the benzene overtone states was provided by the work of Sibert et al." They identified a nonlinear Fermi resonance coupling between the excited CH(D) stretch and the contiguous CCH(D) in-plane wag as a term in part responsible for the initial relaxation of the stretch. Since the stretch vibrational frequency decreases as higher overtone states are prepared, this Fermi resonance coupling is mediated by the level of excitation. Though their work suggests the stretchlwag Fermi resonance coupling is very important, it did not rule out other relaxation paths. In the model proposed by Sibert et al. the CH(D) stretch and the CCH(D) wag internal coordinates are coupled via a kinetic energy term. This coupling arises from an increase in the effective mass of the wag as the CH(D) bond is extended. In recent work, it has been suggested that a potential energy coupling term be-
(1) Swofford, R. L.; Long, M. E.; Albrecht, A. C. J . Chem. Phys. 1976, 65, 179. (2) Henry, B. R. Acc. Chem. Res. 1977, 10, 207. (3) Reddy, K. V.;Heller, D. F.; Berry, M. J. J . Chem. Phys. 1982, 76, 2814. (4) Diibal, H.-R.; Quack, M. J . Chem. Phys. 1984, 81, 3779. (5) Baggott, J. E.; Chuang, M.-C.; Zare, R. N.; Diibal, H. R.; Quack, M. J . Chem. Phys. 1985, 82, 1186. (6) Baggott, J. E.; Law, D. W.; Lightfoot, P. D.; Mills, I. M. J . Chem. Phys. 1986, 85, 5414. (7) Jasinski, J. M. Chem. Phys. Lett. 1984, 109, 462. (8) Coy, S. L.; Lehmann, K. K. J . Chem. Phys. 1986,84, 5239. (9) Bernheim, R. A.; Lampe, F. W.; OKeefe, J. F.; Qualey 111, J. R. J . Chem. Phys. 1984, 80, 5906; J . Phys. Chem. 1985,89, 1087. (10) Sage, M. L.; Jortner, J. Chem. Phys. Lett. 1979,62, 451. Sage, M. L. J. Phys. Chem. 1979,83, 1455. (1 1) Sibert 111, E. L.; Reinhardt, W. P.; Hynes, J. T. J . Chem. Phys. 1984, 81, 1115. (12) Sibert 111, E. L.; Hynes, J. T.; Reinhardt, W. P. J . Chem. Phys. 1984, 81, 1135. (13) Stannard, P. R.; Gelbart, W. M. J . Phys. Chem. 1981, 85, 3592. (14) Buch, V.; Gerber, R. B.; Ratner, M. A. J . Chem. Phys. 1984, 81, 3393. (15) Shi, S.;Miller, W. H. Theor. Chim. Acta 1985, 68, 1. (16) Heller, E. J.; Davis, M. J. J . Phys. Chem.. 1980, 84, 1999. (17) Kellman, M. E.; Lynch, E. D. J . Chem. Phys. 1986, 85, 7216. (18) Holme, T. A.; Hutchinson, J. S. J . Chem. Phys. 1986, 84, 5455. Hutchinson, J. S. J . Chem. Phys. 1986, 85, 7087. (19) Mukherjee, P.; Kwok, H. S. J . Chem. Phys. 1986,85, 4912. ( 2 0 ) Perry, J. W.; Zewail, A. H. J . Chem. Phys. 1984,80, 5333. (21) Scherer, G. J.; Lehmann, K. K.; Klemperer, W. J. Chem. Phys. 1983, 85, 2817. (22) Page, R. H.; Shen, Y. R.; Lee, Y. T. J . Chem. Phys, submitted for
publication. (23) Nagy, P. J.; Hase, W. L. Chem. Phys. Lett. 1978, 54, 73; 1978, 58, 482. (24) Blintz, K. L.; Thompson, D. L.; Brady, J. W. J . Chem. Phys. 1986, 85, 1848. Blintz, K. L.; Thompson, D. L.; Brady, J. W. Chem. Phys. Lett. 1986, 131, 398. (25) Blintz, K. L.; Thompson, D. L.; Brady, J. W. J . Chem. Phys. 1987, 86, 4411.
0022-3654/88/2092-3217$01 .SO10 0 1988 American Chemical Society
3218
Lu and Hase
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
the Hamiltonian. If the 4l form an orthonormal basis, the C matrix is orthogonal and S
$ = e4
T
(3)
In time-dependent perturbation theory,36 \k( t ) is expanded in terms of the zero-order product basis 41,which yields \ k ( t ) = Cc,(t)exp(-iE;t/h)$,
(4)
I
Here the coefficients ci are time-dependent in contrast to the time-independent coefficients in eq 1. The cl(r) are obtained from the quantum mechanical equations of motion k,(t) = Figure 1. A possible hierarchy of zero states participating in an overtone absorption envelope. Is) is the initially prepared state and is strongly coupled to the 11) states. The Is) state is weakly coupled to the b).
tween the stretch and wag is also important.26 The origin of this coupling is the attenuation of the wag force constant as the CH(D) bond is extended.27 Preliminary calculations indicate that this potential energy coupling has a pronounced effect on the decay of benzene overtone states.% Apparently, the potential and kinetic coupling are not additive and, instead, tend to cancel.29 In the work presented here classical trajectories are used to calculate the C6H6 and C6D6overtone spectra. The potential energy function used contains recent ab initio results for the stretch/wag attenuation and it is expected to be sufficiently detailed for studying relaxation of overtone states. Thus, if the classical trajectory approach is appropriate, the calculated and experimental overtone spectra are expected to be in good agreement. Classical trajectories are widely used in investigating chemical dynamics. For example, they are used to study gas-phase unimolecular30 and bimolecular reaction^,^' solution kinetics,32 gas-surface interaction^,^^ and the dynamics of macromolecular motion.34 Since trajectories are used so extensively, it is somewhat troublesome that often is not known a priori whether the classical trajectory method will be accurate.35 It has been suggested that trajectories should be an accurate procedure for studying the relaxation of local mode overtone states.12.16Thus, it is of considerable interest to determine whether the trajectory and experimental overtone spectra for benzene agree. 11. Overtone Spectra If the excitation bandwidth of an experiment is not sufficient to resolve a single eigenstate ,$, a time-dependent superposition of the eigenstates is created, which is defined by \k(t) =
Cc, exp(-iE,t/h)$,
(1)
m
To obtain the eigenstates $, and energies E , the molecular Hamiltonian is diagonalized with a zero-order product basis set 4i. The $, and 4l are related by the matrix relationship
4 = C$
(2)
where C is the eigenvector matrix which results from diagonalizing (26) Swamy, K. N.; Hase, W. L. J . Chem. Phys. 1986, 84, 361. (27) Wolf, R. J.; Bhatia, D. S.; Hase, W. L. Chem. Phys. Left.1986, 132, 493. (28) Lu, D.-H.; Hase, W. L.; Wolf, R. J. J . Chem. Phys. 1986, 85, 4422. (29) Ezra, G. S., private communication. (30) Hase, W. L.; Wolf, R. J. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G.,Ed.; Plenum: New York, 1981; p 37. (31) Chapman, S. J . Chem. Phys. 1981, 7 4 , 1001. Whitlock, P. A.; Muckerman, J. T.; Fisher, E. R. J . Chem. Phys. 1982, 76, 4468. (32) Bergsma, J. P.; Gertner, B. J.; Wilson, K. R.:Hynes, J. T. J . Chem. Phys. 1987, 86, 1356. (33) Tully, J. C. Acc. Chem. Res. 1981, 24, 188. Lucchese, R. R.; Tully, J. C. J. Chem. Phys. 1984, 80, 3451. Foley, K. E.; Winograd, N.; Garrison, B. J.; Harrison Jr., D. E. J . Chem. Phys. 1984, 80, 5254. (34) Elber, R.; Karplus, M. Science 1987, 235, 318. Brady, J. W. J . A m . Chem. SOC.1986, 108, 8153. (35) Hase, W. L. J . Phys. Chem. 1986, 90, 365.
-i -Cck(f) hk
yk exp[-i(EkO - E ? ) f / h ]
(5)
where K k is the matrix element of the perturbation between the zero-order states. The term Icl(t)12represents the probability of being in the zero-order state 4l at time t. Experimental overtone spectra may be interpreted and understood through a time-dependent perturbation approach as outlined above. The interpretation of the time evolution of \ k ( t ) is facilitated by using a zero-order product basis set which includes the initially prepared state \k(O). In the analysis presented here it is assumed, following Sibert et al.,]] Q(0) for benzene is an overtone state which contains n quanta of vibrational energy in the CH(D) local mode and zero-point energy in the remaining modes. The basis for this assumption is that a broad-band excitation source overlapping the complete overtone envelope would give rise to initial excitation in a particular CH(D) bond. It should be noted there is still some question about the nature of the zero-order state defined by the absorption envelope. The product basis function dl is equated to e(0). The relationship between the product bases and the states in Figure 1 is straightforward. The initially prepared state Is) is the same as c $ ~ = \k(O). The remaining 11) and states in Figure 1 are the states with i # 1. In the weak-field limit, the overtone absorption spectrum is given by
u)
1
Z(w) = -J-dt
27r --
(\k(0)(9(t))eiW'
For an orthonormal basis, where 'k(0) has been replaced by which can be chosen without restriction ( \ k ( O ) l \ k ( t ) ) = c l ( t ) exp(-iElot/h)
and the absorption spectrum becomes 1 I ( w ) = 27r -- c l ( t ) exp[i(w - E l o / h ) t ]dt
-j
(8)
(9)
where E l ois the overtone transition energy. In general c,(t) is complex, and c l ( t ) c l ( t ) *= [cl(t)I2represents the probability of occupying the overtone state at time t. In the general case the integral in eq 9 will have both real and imaginary components, Le., Re [ I ( w ) ]and Im [ I ( w ) ] so , that the absorption spectrum I(@) is given by (Re [I(w)]*Im [ I ( W ) ] ~ ] I / ~For . the special case where the real part of cl(t) is even with respect to t = 0, while the imaginary part is odd, the spectrum has only a real component. The relationship between c l ( t ) and the nature of the overtone absorption spectrum has been discussed on several occasions."~13s37 If cl(t) is evaluated to infinite time, it is straightforward to show analytically that I(w) in eq 9 becomes a stick spectrum and is given by
+
m
(36) Merzbacher, E. Quanfum Mechanics; Wiley: New York, 1970. (37) Heller, E. J. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 103.
Calculation of the Benzene Overtone Spectra Thus, the spectrum consists of lines at the eigenstate energies E , whose heights are proportional to the magnitude squared of the overlap integral ( r$,I+,) between the CH overtone state and molecular eigenstates. For a benzene overtone state, where the density of eigenstates is immense ( l o 5 states/cm-' or larger), it is extremely difficult, if not impossible, to study individual eigenstates in the absorption envelope. What is of interest in the benzene overtone experiments is the shape of the absorption envelope. It is well-known that, if cl(t) decays exponentially to zero and never recurs, Z(w) consists of one structureless Lorentzian band envelope whose width r (cm-I) is related to the decay rate via the relationship k = 27rcr. In reality the decay of cl(t) is not irreversible, so that it decays with recurrences. However, for many cases, where the density of molecular eigenstates is very large, the recurrences occur on a much longer time scale than for the initial decay. For such a situation the shape of the absorption envelope is determined by the initial decay of c l ( t ) . Recurrences, which occur after the initial decay, give additional fine structure to the band envelope. The determination of low-resolution absorption envelopes from the short time intramolecular dynamics has been emphasized in the work of Heller.37 The density of molecular eigenstates for each benzene overtone state is very large and one expects to be able to determine the overtone absorption envelope from the initial decay of cl(t). Sibert et al.12 found this to be the case in their work and, as shown below, the same result is obtained here.
The Journal of Physical Chemistry, Vola92, No. 11, 1988 3219 is accomplished by adding zero-point energy to each normal mode, with a random phase for the normal mode motion. Normal mode coordinates and momenta are then transformed to Cartesian coordinates and momenta with the normal mode eigenvector. This transformation is not exact and the Cartesian momenta and coordinates are scaled until the energy calculated with the Cartesian Hamiltonian agrees with E,, to within 0.1%. Total angular momentum of benzene is constrained to equal zero during this tran~formation.~~ The addition of the above normal mode zero-point energy puts kinetic and potential energy in each CH(D) bond. Since the normal mode zero-point energy is added with random phases, the total bond kinetic and potential energies differs for each CH(D) bond in this step of the selection of initial conditions. Similarly, the energy in a particular CH(D) bond will vary at this step between initial conditions. A CH(D) bond is then excited to the local mode state In) by adding more energy to the bond so that the bond energy, as defined by
is the same as the Morse eigenvalue E(n) = (n + l/z)we - (n + 1/2)2wrye.This energy is added by either extending or compressing (chosen randomly) the CH(D) bond. In eq 11 p is the CH(D) reduced mass. For the C H bond we = 3074 cm-l and wexe = 61.4 cm-I. Their values are we = 2257 cm-' and wexe= 33.1 cm-' for the CD bond. Since the addition of the zero-point energy puts a varying amount of energy in the CH(D) bond to be excited, the final initial conditions for the trajectories in an ensemble for a 111. Overtone Spectra from Classical Trajectories particular overtone state In) differ in the total molecular energy. A . Accuracy of Classical Trajectories. Before discussing the However, each trajectory has the same bond energy given by eq approach used here to calculate the overtone spectra from classical 11. trajectories, it is useful to consider possible inaccuracies in the Clearly the above is not a unique procedure for choosing the classical trajectory method.35 For example, if the classical trainitial conditions. There are certainly other ways to add the jectories which represent the overtone state are quasi-periodic, molecular zero-point energy and excite a CH(D) bond to an energy may not flow out of the excited CH(D) bond. Thus, though overtone state. The CH(D) bond length is compressed or extended the overtone state is in reality a nonstationary state, it would be in the method used here to model initial conditions for a displaced predicted to be stationary by the classical trajectories. The decay wave p a ~ k e t . ~ Another ~ , ~ ~ , procedure ~~ would be to choose a of the overtone state for such a situation occurs by a nonclassical random phase for the CH(D) bond using the prescription for a tunnelling This type of inaccuracy in classical Morse o ~ c i l l a t o r . ~An ~ ambiguity in this procedure is the lack trajectories has been found for O H overtone states in water,38and of separability between the CH(D) bond and the remaining for triatomic model molecules excited into the dissociative conmolecular Hamiltonian. The decay of several overtone states was tin~um.~' studied by choosing initial conditions with a CH(D) random phase. In contrast to the above situation, trajectories may overestimate For all cases, the results (Le., exponential or nonexponential decay, IVR as a result of incorrectly treating vibrational a d i a b a t i ~ i t y . ~ ~ and line width) are nearly identical with those reported in which The transfer of energy between zero-order states may be slower initial conditions are chosen with an extension or compression of quantum mechanically than classically. However, classical methe CH(D) bond. chanics may treat vibrational adiabaticity correctly for the initial The average energy of the initially excited bond and the decay of the overtone state, so that the trajectories will give the probability the molecule remains in the initial overtone state are proper absorption envelope. If the trajectories become chaotic evaluated versus time by phase averaging over 100 trajectories. as the overtone state decays, irreversible energy transfer is expected The latter quantity, P(n,t), is the quasi-classical trajectory apwith no recurrences at long times. For such a situation the traproximation to the probability Icl(t)I2discussed in section 11. The jectories may correctly predict the absorption envelope, a lowlocal mode is identified as being in state n if the bond energy, eq resolution experimental result, but would not give the fine structure 11, is within the Morse eigenvalue energy interval E(n-'/2) to within the band envelope which could be studied by higher resE(n+'/2). olution experiments. To determine the quasi-classical absorption spectrum for state B. Quasi-Classical Trajectory Method. The quasi-classical n, the assumption is made that P(n,t)'iz = cl(t), so that eq 9 trajectory method proposed by Sibert et al.," and used in our becomes earlier work,2Bwas also used here for studying the decay of benzene 1 overtone states and calculating the overtone absorption spectra. Z(w) = - J m P ( n , t ) ' / z exp[i(w - E , / h ) t ] dt (12) 2 1 -In this method, initial conditions for the classical trajectories are chosen to simulate the excitation of benzene to a local mode state The term c l ( t ) is necessarily real by equating it to P(n,t)'/*.As with n quanta in a particular CH/CD bond. In the first step the discussed above, the quantum mechanical cl(t) may be complex. harmonic zero-point energy E,, is added to the molecule. This The validity of assuming a real cl(t) has been discussed previously by Heller and co-workers.16,44It is also assumed that the real c,(t) = P(n,t)'I2is symmetric about t = 0. This is an assumption that (38) Lawton, R.; Child, M. S . Mol. Phys. 1981, 44, 709. (39) Davis, M. J.; Heller, E. J. J . Chem. Phys. 1981, 75, 246. is widely made in calculating absorption s p e ~ t r a . ~Assuming ~,~~,~~ (40) Hutchinson, J. S.; Sibert 111, E. L.; Hynes, J. T. J. Chem. Phys. 1984,
81, 1314.
(41) Wolf, R. J.; Hase, W. L. J . Chem. Phys. 1980, 73, 3779. Hase, W. L. J . Phys. Chem. 1982.86, 2873. Swamy, K. N.; Hase, W. L.; Garrett, B. C.; McCurdy, C. W.; McNutt, J. F. J. Phys. Chem. 1986, 90,3517. (42) This is a well-known problem in unimolecular dynamics. See for example: Marcus, R. A. Ber. Bunsenges. Phys. Chem. 1977, 81, 190. Hase, W. L.; Buckowski, D. G. J . Compur. Chem. 1982, 3, 335, and ref 35.
(43) The addition of zero-point energy with zero angular momentum is described in: Sloane, C. S.; Hase, W. L. J. Chem. Phys. 1977, 66, 1523. (44) Heller, E. J.; Stechel, E. B.; Davis, M. J. J. Chem. Phys. 1980, 73, 4720. (45) Porter, R. N.; Raff, L. M.; Miller, W. H. J. Chem. Phys. 1975, 63, 2214
3220 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 a real symmetric c l ( t )about t = 0 will give rise to a symmetric [ ( w ) about the overtone transition energy. If the overtone state In) decays exponentially according to P(n,t) = exp(-kt), eq 12 will give a Lorentzian line shape with a line width at half-maximum equal to r (cm-I) = k / 2 ~ c .As shown below, many of the overtone states decay nonexponentially. For these cases the trajectory P(n,t). which are evaluated out to 0.25 ps, are fit to the function P(n,t) =
+ .&,e-kJ
cos
(w,t)
Lu and Hase frequencies. Both the stretch and bend vibrational frequencies are lowered as the CH(D) bond is excited. The diminution of the stretch frequency is adequately represented by the Morse function. The lowering of the bend frequency results from the attenuation of the bending force as the CH(D) bond is extended. From ab initio calculation^,^^ it has been found that the change in the quadratic bend force constant with bond length may be approximated by
(13)
1=2
with a nonlinear least-squares procedure. This fitted P(n,t) is inserted into eq 12 and Z(w) is determined with a fast Fourier transform algorithm.,* In finding Z(w) the fitted P(n,t) is extrapolated to 8 ps, and eq 12 is solved with the integral limits of -8 and 8 ps. The decay of P(n,t) is essentially complete at 8 ps. Also, if this integration interval is combined with a 213 point representation of P(n,t), a satisfactory spectral resolution of approximately 4 cm-I results from the Fourier transform analysis. C . Integrating the Classical Equations of Motion. The Hamiltonian and classical equations of motion for planar benzene are written in terms of Cartesian coordinates. The potential energy function (described below) is first written in curvilinear internal coordinates, and then transformed to a Cartesian coordinate representation by using the relationships between curvilinear internal and Cartesian coordinates. The accuracy of the Cartesian Hamiltonian depends only upon the potential energy since terms are not neglected in the kinetic energy expression. Each trajectory was numerically integrated for 2500 steps with s. The energy in the initially excited CH(D) a step size of bond was calculated every 10 integration steps. The trajectory calculations were performed with the general computer program VENUS, which is an extension of the program MERCURY available through the Quantum Chemistry Program Exchange.49 The potential energy function used in this work and the quasi-classical method are standard options in the computer program.
IV. Potential Energy Function The same analytic potential energy function is used for the work presented here as in our previous study, Le., eq 3 and 4 in ref 28. The potential energy function is quadratic with two modifications. First, the CH(D) and C C stretching potentials are represented by Morse functions. Second, the CCH(D) bend diagonal quadratic force constants are attenuated as the CH(D) bond is extended. This is accomplished via the switching function described below. The quadratic force constants and Morse function parameters are listed in Table I of ref 28. The harmonic frequencies for this potential function are, on the average, only 1.021 times larger than the experimental anharmonic frequencies (Table I1 in ref 28). The experimental normal mode harmonic frequencies are 1.00-1.04 times larger than the anharmonic values.so The quadratic force constants for the analytic potential were taken from set I1 of Pulay et aLsl Very small nondiagonal terms in this force field were neglected to reduce the computational time.2s Set I1 is thought to be a good model of the physical quadratic force field of benzene.52 However, questions still remain concerning the values for several nondiagonal terms,53and the correction for a n h a r m o n i ~ i t y . ~ ~ As Sibert et al. have discussed,” the coupling between the CH(D) stretch and the contiguous CCH(D) bend is a particularly important property of the Hamiltonian. This coupling is influenced by the relationship between the stretch and bend vibrational (46) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. J. Chem. Phys. 1977, 67, 404.
(47) Heller, E. J. J. Chem. Phys. 1978, 68, 3891 (48) Brigham, E. 0. The Fasf Fourier Transform; Prentice-Hall: Englewood Cliffs, NJ, 1974. (49) Hase, W. L. QCPE, 1980, 1 1 , 453. (50) Thakur, S.N.; Goodman, L.; Ozkabek, A. G. J. Chem. Phys. 1986, 84, 6642. (51) Pulay, P.; Fogarasi, G.; Boggs, J. E. J. Chem. Phys. 1981, 74, 3999. (52) Pulay, P. J . Chem. Phys. 1986, 85, 1703. (53) Ozkabak, A. G.; Goodman, L.; Thakur, S. N.; Krogh-Jespersen, K. J . Chem, Phys. 1985, 83, 6047.
where
S ( r ) = 1.0, r
< ro
(14)
In our previous study of IVR in benzene,28three different surfaces with values for the a attenuation parameter of 0,0.825, and 2.000 A-2 were investigated. The surface with a = 0.825 .&-I gave results in the best qualitative agreement with experiment. The attenuation parameter a has not been evaluated for the CCH bend in benzene. However, from recent ab initio calculations for CH,, C2H4, and C3H6at the MP4/6-31G** level of theory it is found that H C H and HCC bends contiguous to simple H C bond ruptures have similar attenuations. The a parameter, in A-2 units, for these molecules is found to be 0.825 (CH,), 0.917 (C2H4), and 0.768 (C3H6)by fitting ab initio force constants for rcHextended to 2.00 A. The fitted a parameter is weakly sensitive to the maximum extension considered. If a maximum extension of 1.75 is considered for C,H, (the approximate maximum C H extension for the benzene overtone states) instead of 2.00 A, the a parameter increases from 0.768 to 0.825 A-2, an 8% increase. The nonterminal vinyl C H bond is extended in the C3H6calculations, and the stretch/bend attenuation coupling is expected to model that for benzene. Thus, the a parameter for the C C H bend attenuation about the nonterminal vinyl C H bond in propylene of 0.825 A-2 is used for the benzene calculations reported here. As discussed above, this is the parameter for C H bond elongation up to 1.75 A. The sensitivity of the trajectory results to minor variations in the a parameter is considered in the calculations reported below.
V. Results A . Sensitivity of Potential Energy Surface Parameters. Previously it was found that both the rate and nature of IVR in benzene are very sensitive to the attenuation of the C C H bend force constant.28 Without attenuation (a = 0) the decay of each overtone state is exponential. Increasing the attenuation parameter to 0.825 A-* has the effect of decreasing the rate of IVR from the excited C H bond. In addition, the decay of the n = 7-1 1 states becomes nonexponential with recurrences. An increase in the attenuation parameter to 2.000 A-2 causes a further lowering of the IVR rate with more prominent recurrences in the nonexponential decay. In the work reported here, calculations were also performed to determine how variations in the quadratic force field affect the decay of the C H overtone states. The C H stretch force constants Cf,) for the unexcited bonds were increased by a factor 2 in one of the studies. In the other two studies the C C H bend force constants (f,) and CC stretch force constants CfR) were increased by factors of 8.5 and 2.0, respectively. The results are listed in Table I. Increasingf, for the five unexcited C H bonds has no effect on the IVR. In contrast increasing either f, or fR has a significant effect. For the normal values off+ and f R , and the a parameter equal to zero, the decay of each C H overtone state is exponential. The decay becomes nonexponential, with recurrences, as a result of the increases in f + and f R . For the larger a value of 2.000 A-2, the decay is nonexponential with either the normal or increased value of fR. As discussed above, ab initio calculations indicate that a value of a = 0.825 is appropriate for attenuation of the CCH bend force constant in benzene. To determine the sensitivity of the results to minor changes in a, calculations were performed where a is
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3221
Calculation of the Benzene Overtone Spectra TABLE I: Effect of Varying Potential Energy Surface Parameters
a overtone surface parameter state@) result increase f. bv a factor of 2 for all five unexcited CH bonds 0. 2.00 n = 6 and 8 P(n,t) are the same as found with the normal f, value increasei l l by a factor of 8.5 0 , 6,and 8 pronounced recurrences in P(n,t); negligible energy transfer from the excited CH bond with 0.25 ps increase all fR by a factor of 2 for all CC bonds 0 pronounced recurrences in P(n,t); energy transfer from the excited CH bond is incomplete and ( E C H )is still decreasing within 0.25 ps 2.00 8 increase allfR by a factor of 2 for all CC bonds decay is nonexponential, with recurrences, and (ECH)is still decreasing within 0.25ps
i,
TABLE II: Dependence of the First-Order Rate Constants for C6H6 Overtone States on the Bend Attenuation Parameter
n=6
"=lo
rate const, ps-I
overtone state n
a = 0.743
a = 0.825
3 4 5 6 '7 '8
15 18 25 18 17 12
16 16 20 19 15 14
a = 0.908 18 16 24 17 20 18
1
.
4
0.2
00
I n=9
"There is a small nonexponential component to the decay of these states.
-I i
TABLE III: Parameters for the c& Nonexponential P ( n , t )
overtone state n=7 0.6575 15.58 0.2650 60.89 119.8 0.06338 10.44 217.9 0.02685 15.58 301.1
n=8 0.6358 16.68 0.1944 32.42 122.1 0.09709 12.92 199.9 0.04486 16.68 267.6
n=9 0.4715 13.02 0.2612 23.97 81.2 0.2076 32.56 191.1 0.05719 13.02 259.8
I
n = 10 0.3130 13.17 0.2282 19.21 67.6 0.5823 77.37 145.3 -0.08232 13.17 142.2
n=8
n=3
"The constraint k4 = k l was made in determining the parameters. The parameter a is unitless. The parameters k and w are in units of ps-1.
increased and decreased by 10%. There are no qualitative changes in the decay of the CH overtone states with these variations in a. States which decay exponentially or nonexponentially with a = 0.825 remain that way as a is changed 10%. However, there are small quantitative changes in the decay. The first-order rate constants for the states which decay exponentially are listed in Table 11. At the 95% confidence limit, there is a 10% or less uncertainty in each rate constant. It is seen that varying the attenuation parameter a by 10% results in a 10-30% change in the first-order rate constant. B. C6H6Overtone Spectra. Decay of the C H overtone states with n = 3-6 is exponential. Previously it was reported that the n = 7 and 8 states also decay exponentially.28 However, a careful inspection of their decay shows that a small nonexponential contribution should be included. This is illustrated in Figure 2 where the P(n,t) for the overtone states are plotted. In these plots, the points are the trajectory results. The solid lines in the n = 7-10 plots are the fits with eq 13. Three cosine terms are needed for these fits, and the parameters are listed in Table 111. The overtone spectra for the n = 7-10 states, evaluated according to eq 12, are shown in Figure 3. Each spectrum, eq 9, is symmetric about the zero-order transition energy, and only one-half of the spectrum is shown. The trajectory line widths for the C H overtone states are listed in Table IV and compared with the experimental line widths. For the states with exponential decay ( n = 3-6) the line width is r = k / 2 m The absorption spectra, Figure 3, are used to determine the line widths (at half-maximum initial absorption intensity) for the nonexponentially decaying states. Also listed in Table IV are
i
Time(ps)
Time(ps)
Figure 2. Probability versus time P(n,t) of populating the initially prepared CH overtone state in C6H6. The solid line is the fit given by eq 13.
::
00
800
WO
Freq. (-1)
2400
3200
4000 00
800
(600
2400
3200
4000
FBC. (m-1)
Figure 3. Absorption spectra for CH overtone states which decay nonexponentially.
the positions of peaks with smaller absorption intensity. It should be noted that the line widths for the n = 7 and 8 states extracted
Lu and Hase
3222 The Journal of Physical Chemistry, Vol. 92, No. 11 1988 ~
1 rdJ
I
n=4
i
I
1 i 1
-‘
-,\i 0.00
005
010
0.15
0.20
0.25
000
+ -
I
*
010
005
015
020
025
Tme(Ps)
Time(ps)
Figure 4. Probability versus time P(n,r) of populating the initially prepared C D overtone state in C6D6. The solid line is the fit given by eq 13.
TABLE I V C a n Overtone Line Width line width“ state 3 4 5 6 7 8 9 10
exptC 43 82 111 95 90 100 56
00
85 85 106 101 85 90 70 71
first
second
566 642 429 359
>lo62 >lo62 1020 896
1600
2400
M
O
Freq (-1)
secondary spectral peaksb
traj
800
lF7
L
mo 00
800
1600
a00
m o
mo
Freq (m-1)
Figure 5. Absorption spectra for CD overtone states, all of which decay
third
nonexponentially.
>I062 >I062
“Line width (fwhm) in cm-’. bSecondary absorption peaks in the trajectory spectra with respect to the zero-order transition energy. ‘Reference 3. Inhomogeneous rotational broadening is not removed from the line width. dThe number in parentheses is the bandwidth measured by Page et al., ref 22, for rotationally cold benzene. It is thought that inhomogeneous contributions are removed from the spectrum.
from the spectra (85 and 90 cm-I) are similar t o those reported previously (80 and 74 cm-I) assuming exponential decay.28 c. c6D6 Overtone Spectra. The most striking feature of the C6D6overtone states is that each state decays nonexponentially. This is illustrated in Figure 4 where the trajectory P(n,t) for n = 3-10 are plotted. Because of the extensive recurrences in the
decay of these states, five cosine terms are required to fit the P(n,t) in contrast to the three cosine terms required for C6H6. The parameters for the fits to the C6D6 P(n,t) are listed in Table v. The C6D6 absorption spectra are plotted in Figure 5 . Considerable structure is present in the spectrum of each CD overtone state. The line width at half-maximum of the principal absorption peak for each state is listed in Table VI and compared with the experimental overtone line width. Also listed, with respect to the principal transition frequency, are the positions of secondary peaks. The trajectory calculations predict narrower lines for C6D6 versus C6H6, as is observed e~perimentally.~ D. Extent ofZVR. The line widths of 50-100 cm-I found in this work for C6H6 yield IVR relaxation times of 0.05-0.10 ps. For C6D6 the relaxation times are approximately 2 times larger. Thus, if all modes (or all regions of classical phase space) participate in the IVR and are strongly coupled, IVR would be complete after 0.25 ps. The average energy in the CHfD) bond at this time would be that for a statistical distribution. Two different statistical models are used to calculate the average energies in the C H and CD bonds. In the classical micrmnonical
TABLE V: Parameters for the CsDsP ( n , f ) 0.5520 7.10 0.4836 8.36 0.3996 9.45 0.2308 6.51 0.1948 8.27 0.1729 11.84 0.1838 17.74 0.0959 12.73
0.0203 3.21 0.1 106 90.05 0.0785 37.92 0.0980 58.89 0.0893 63.18 0.1080 67.28 0.1 117 62.95 0.1805 103.1
283.1 279.6 569.6 509.7 452.6 429.4 432.7 362.1
0.0392 0.0074 0.3182 0.3509 0.1843 0.2156 0.2310 0.3287
7.62 13.07 44.17 64.34 71.70 60.08 64.06 75.82
209.5 -0.0332 4.87 195.2 0.0377 5.17 4.38 194.8 0.0477 191.0 0.0759 6.69 241.8 0.0609 11.29 245.7 0.1348 25.27 232.8 3.1690 32.67 189.6 0.2268 45.56
DTheparameter a is unitless. The parameters k and w are in units of ps-’.
171.4 136.4 125.0 108.5 168.6 165.9 152.7 108.9
0.3831 0.3007 0.1529 0.2368 0.4363 0.2130 0.1187 0.0502
67.45 119.3 0.0682 4.01 91.3 17.02 91.1 0.0033 0.48 47.9 83.1 0.048 1 9.17 39.0 9.62 19.89 74.4 0.0681 10.58 25.1 58.81 93.0 0.1082 12.61 57.7 25.05 99.3 0.2205 15.83 54.3 17.64 100.9 0.2605 18.15 51.7 10.10 87.1 0.1794 16.06 44.0
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3223
Calculation of the Benzene Overtone Spectra TABLE VI: CsDs Overtone Line Widths
state 3 4 5 6 7 8 9 10
line width" expte trai 36 65 50 37 35 65
39 31 51 35 44 61 95 68
secondary spectral peaksb second third fourth fifth
first
437 482 441 391 896 508 529 462
354 254 199 129 308 287 279 233
484 508 662 574 >lo62 887 753 575
520 662 729 >lo62 >I062 862 878
600 724 808 o'2 0.0
>lo62 979
1
'Line width (fwhm) in cm-l. bSecondary absorption peaks in the trajectory spectra with respect to the zero-order transition energy. Reference 3. Inhomogeneous rotational broadening is not removed from the line width. TABLE VII: Comoarison of Average CH(D) Bond Energies
overtone state
class. trai'
statistical models vibrnl adiabb class. microcanone (ECH)
9.5 f 12.5 f 13.2 f 14.4 f 16.6 f 21.4 25.7 f 28.6
* *
1.2 1.7 1.8 1.9 2.8 3.2 3.5 3.6
9.6 f 13.3 f 17.5 f 23.0 f 25.7 f 27.2 f 30.2 f 31.4 f
1.2 1.3 1.6 1.7 1.9 1.7 1.7 1.9
5.5 5.9 6.2 6.5 6.8 7.1 7.4 7.6
3.6 4.0 4.3 4.6 4.9 5.2 5.5 5.8
4.1 4.3 4.6 4.9 5.1 5.3 5.6 5.8
2.9 3.1 3.4 3.7 3.9 4.1 4.4 4.6
(ECD)
aAverage energy in the initially excited bond after 0.25 ps of internal motion. bA statistical model in which the excess energy in the excited bond is distributed randomly between all the planar modes. The zero-point energy is treated vibrationally adiabatic and is, thus, not distributed. The classical microcanonical average bond energy. model, the total energy is assumed to be distributed statistically between all 21 planar degrees of freedom. The total benzene energy is E,, ( E , - Eo),where E, and Eo are the bond overtone and zero-point energies and E,, is the total molecule zero-point energy. The microcanonical CH(D) bond energy is simply 1/21 of the total energy. In the statistical vibrationally adiabatic model, the excess energy in the excited bond is assumed to be distributed statistically between the planar vibrational modes. However, the zero-point energy is treated adiabatically and, thus, does not participate in the redistribution. For this model, the average CH(D) bond energy is Eo (E, - E0)/21. As shown in Table VII, the statistical vibrational adiabatic bond energies are somewhat larger than those of the classical microcanonical model. Comparing the classical trajectory and statistical values of (ECH)and ( E c D )in Table VI1 shows that IVR within benzene is incomplete within 0.25 ps. Thus, IVR with a rate determined from the overtone line width involves incomplete redistribution of the initial excitation energy. This finding is consistent with sequential IVR, with a gradation of IVR rates, as discussed by Sibert et al." The trajectory values of (ECH)and ( E c D )at 0.25 ps may be compared with the initial energy in the bond (section 1II.B). The ( E C Hvalues ) at 0.25 ps are 0.3-0.4 times smaller than the initial energy in the excited C H bond. The smaller fractions are for the n = 5-7 states and the larger for n = 9 and 10. The fraction of the initial energy which remains in the excited CD bond is larger than for the excited C H bond. This is consistent with the narrower line widths (Le., slower relaxation) for the CD overtone states. For the n = 5-10 C D states the fraction of the initial energy
+
+
1
0.0 o.2 0.00
0 05
0.10
0.15
0 20
0.25
Time(ps)
Figure 6. Average energy in the CH(D) bond divided by the initial overtone bond energy plotted versus time. Plots are for the n = 6 and 9 overtone states: (-) CD bond in C6D.4 and (---) CH bond in C6H6.
remaining after 0.25 ps ranges from 0.55 to 0.60. The fraction is slightly lower, 0.50, for n = 3 and 4. To illustrate the rate of average energy transfer from the excited C H and CD bonds, ( E ) relative to the initial bond energy E, is plotted versus time in Figure 6 for the n = 6 and 9 states. There is an initial rapid transfer of energy from the excited bond, but the complete transfer of energy to form a statistical distribution apparently occurs on a much longer time scale.
VI. Discussion In the work presented here the quasi-classical trajectory method is used to calculate low-resolution spectra for benzene overtone transitions. It is thought that the potential energy function used in the calculations accurately describes the interaction of a highly excited CH(D) bond with the remaining benzene moiety. Experimental benzene harmonic frequencies and those for the potential energy function agree to within 5%. Morse functions are used to represent anharmonicity in the CH and CC stretching motions. Anharmonicity in the H C C bends is represented by attenuating the bend quadratic force constant as the H C bond is extended.*' This attenuation lowers the bend frequency as the H C bond is vibrationally excited.26 The stretch to bend frequency ratio is mediated by both the stretch and bend anharmonicity. The quasi-classical trajectory calculations give broad overtone envelopes with widths ranging from 70 to 110 cm-' for C6H6 and 30 to 100 cm-' for C6D6. A nonexponential component is present in the decay of the high-energy C H overtones and all of the CD overtones. Initial IVR is faster for a C H overtone state In) than for the analogous CD state. A comparison between the line widths found here and those of previous classical, semiclassical, and quantum mechanical studies is given in Table VIII. No attempt was made in this work to unravel the complete IVR mechanism for benzene CH(D) overtone states. However, the results do present some clues about the IVR process. There are apparently at least two time scales for IVR, which is consistent with the multitiered IVR model of Sibert et al." The short-time IVR causes the initial overtone relaxation and is related to the overtone line width. Complete IVR, to form a microcanonical ensemble, occurs on a longer time scale. Short-time IVR occurs faster in C6H6 than C6D6, but this does not mean that C6H6 will relax more rapidly to a microcanonical ensemble. Decay of the overtone states is affected by gross changes in the potential energy function. Either increasing the CCH bend force constant by a factor of 8.5, increasing the CC stretch force constant by a factor of 2.0, or significantly enhancing the bend attenuation decreases the IVR rate. Nonexponential decay with recurrences becomes more predominant, and the overtone line widths become narrow. However, it is noteworthy that substantially increasing
Lu and Hase
3224 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 TABLE VIII: Theoretical Calculations of Benzene Overtone Line Widths
overtone line width," cm-l, for n = method
classical classical classical semiclassical statistical quantum quantum
2
3
4
5
6
7
8
9
133 143 101 40 134 79
152 85 60 110 71
156 90 56 114
140 126 70 32 66 33
60 35 >7 37
29 44 >8 50
ref
10
C6H6
80 52
91 85
121 85
66
116
129 143 106 82 176 86
12 b this work 15 14 11
71
C6D6
classical classical classical semiclassical statistical quantum quantum
40
31 39
76 31
75
65
60 73 51 50 28
60 41 61 >6 65 53
95
12 b this work 15 14 11
68
"Line widths are fwhm. bClarke, D. L.; Collins, M. A. J . Chem. Phys. 1987, 86, 6871. Line width for In P(n,t) down to -3.5. the stretching force constant for the five unexcited CH bonds does not affect the decay of the overtone state. Modest changes in the potential energy function have only minor effects on the quasi-classical trajectory results, and, thus, the overtone line widths. For example, varying the bend attenuation parametrer a, eq 14, by 10% only changes the overtone line width by 10-30%. As discussed previously,28a gross change in (I from 0.0 to 2.00 k'decreases the line width by a factor of 2-3. The overtone line widths from the quasi-classical trajectories for C H with n I4 and C D with n I5 are in remarkably good agreement with those reported by Reddy et aL3 Similar trends in the line width versus n are observed in the calculations as in the experiments. Artificial and inaccurate coupling in the trajectories, resulting from zero-point energy flow, is expected to become increasingly more important as the zero-point energy makes a larger contribution to the total energy. This is a possible explanation for the large differences between the trajectory and experimental line widths for the lower overtone states. The experiments by Page et al. for the n = 3 overtone state of rotationally cold benzene give an overtone bandwidth of 10 cm-I. A similar bandwidth has been observed for benzene C C normal mode overtones.54 Thus, there are apparently large inhomogeneous contributions to the line widths reported by Reddy et al. for the low-energy overtone states. Furthermore, the implication is that the Reddy et al. line widths for the higher energy overtone states may also contain inhomogeneous contributions. If this is the case, the quasi-classical trajectory line widths found here may be in error. However, it is possible that the line widths may broaden with increasing n, so that the trajectory line widths are accurate for the large n overtone states. However, for the n = 3 overtone state of C6H6 the trajectory line width is a factor of 8 too large. There are two possible origins of error in the quasi-classical trajectory line widths for the small n overtone states. One is an inaccurate potential energy function. The other results from inherent difficulties with the quasi-classical trajectory app r o a ~ h . ~ Our ~ , suspicion ~ ~ ~ ~ is, that ~ ~ the latter is the principal source of error in the calculated lower n overtone line widths. As discussed above, gross changes in the potential energy function are required to obtain appreciable changes in the calculated line widths. It does not seem that realistic changes in the potential energy function can be made to decrease the calculated n = 3 line width by a factor of 8 to agree with the results of Page et al. As discussed above, varying the a attenuation parameter between 0 and 2.00 A-' only changes the line width by a factor of 2-3. The likely source of the problem with the quasi-classical trajectories, particularly for small n, is the involvement of too many relaxation pathways (channels). Relaxation may be occurring classically at short time by pathways which are not open (or have
small probabilities) quantum mechanically. Thus, as has been discussed on several o c c a s i ~ n sthe , ~ trajectories ~ ~ ~ ~ ~ ~may ~ not be treating vibrational adiabaticity correctly. As an illustration, consider the zero-point level of benzene. Quantum mechanically this is a stationary state. However, the classical intramolecular motion of benzene with zero-point energy (Le., 52.2 kcal/mol) is chaotic. Thus, the average energy of a C H stretch mode, if followed classically, would change from the zero-point value to a lower value corresponding to that for a microcanonical ensemble. Thus, a quasi-classical trajectory study would incorrectly find the zero-point level to be nonstationary. The above problem with zero-point energy is expected to become less important for smaller systems and/or at higher levels of excitation. Thus, the benzene fragment HC,, i.e.
H I
-
(54) Chernoff, D. A,; Myers, J. D.; Pruett, J. G. J. Chem. Phys. 1986. 85, 3132. ( 5 5 ) Schatz, G. C. J . Chem. Phys. 1983, 79, 5386.
t -
/\
C' -C which contains all contiguous couplings with the C H local mode, may be a more accurate model than the complete benzene molecule for a quasi-classical trajectory study of the initial decay of benzene overtone states. Preliminary quasi-classical calculations for the HC3 n = 3 overtone give a line width of approximately 4 A definitive way has not been prescribed to extract the necessary information from the classical trajectories to calculate an absorption spectrum. Heller and Davis have suggested that (*( O ) l \ k ( t ) ) in eq 6 may be represented classically by the average internuclear coordinate autocorrelation function for a swarm of trajectories. The method used in the work presented here is based on the proposal Sibert et al.Iz that the probability of occupying an overtone state with n quanta in a local mode be equated to the quasi-classical trajectory probability P(n,t) of having n quanta of energy in the local mode. The time-dependent expansion coefficient c , ( t ) which is needed to calculate the absorption spectrum, eq 9, is assumed to equal P(n,t)'I2. One of the merits of this ansatz is that the transfer of zero-point energy between the nonexcited modes, an inherent problem with classical traj e c t o r i e ~does , ~ ~ not contribute directly to the absorption spectrum. However, as discussed above, this zero-point energy flow may lead to extraneous relaxation channels. Though for many situations trajectories may be adequate for calculating a low-resolution absorption spectrum, it is not clear that they may be used to determine fine details in the spectrum. To evaluate details in the absorption spectrum with a time-independent formalism requires knowing the energy for each of the ~~
( 5 6 ) Lu, D.-H.; Hase, W. L. Chem. Phys. Letr.,
in
press.
J . Phys. Chem. 1988, 92, 3225-3235 transitions in the absorption envelope; Le., eq 10. Calculating these transition energies quantum mechanically is extremely difficult due to the large number of states which must be handled. However, given the current understanding of classical and semiclassical mechanics, it may be fundamentally impossible to calculate the transition energies from a classical trajectory study. In general, the determination of semiclassical transition energies for polyatomic systems (i.e., more than three dimensions) has required quasi-periodic classical motion. For highly excited overtone states the classical intramolecular motion is usually chaotic, which is the case for all of the benzene CH(D) overtone states. From a time-dependent point of view, recurrences in the probability of occupying the initially prepared state gives rise to
3225
the fine structure in the overtone absorption spectrum. Though rudiments of these recurrences may be present in the short-time trajectory P(n,t), the chaotic classical motion destroys the longer time recurrences, which occur quantum mechanically. It is these latter recurrences which are needed to evaluate fine details in the absorption spectrum. Thus, as applied in this work, the classical trajectory method appears limited to the evaluation of low-resolution absorption spectra. Acknowledgment. This research was supported by the National Science Foundation. We thank Professors G. S. Ezra and Y . T. Lee for very helpful discussions. Registry No. C6H6, 71-43-2; D,,7782-39-0.
On the Relationship between the Classical, Semiclassical, and Quantum Dynamics of a Morse Oscillator Jeffrey R. Reimerst Department of Theoretical Chemistry F l l , University of Sydney, Sydney, New South Wales 2006, Australia
and Eric J. Heller* Physics and Chemistry Departments BG- 10, University of Washington, Seattle, Washington 981 95 (Received: July 31, 1987)
The relationship between the classical, semiclassical, and quantum dynamics of the Morse oscillator is investigated, and the almost classical, frozen Gaussian approximation is shown to give an accurate solution of a differential equation similar to the Schriidinger equation that shares its eigenfunctions with the Schriidinger equation. This result is used to justify a posteriori the success of the De Leon-Heller spectral quantization method for determining eigenfunctions and eigenvalues from frozen Gaussian dynamics, and techniques are described for the optimization of such calculations. A close relationship is shown to exist between the various dynamical schemes, and physical problems that reveal or cloak this relationship are described.
The main argument against the use of Gaussian wave packets refers to their behavior on one-dimensional anharmonic potential
I. Introduction
A large part of the study of chemical physics is concerned with problems that are so complex that accurate quantum mechanical calculations are impractical. Our hope is that many systems might be treated by semiclassical means. The paradigm for the semiclassical correspondence is the harmonic oscillator. Here, the coherent stateI4 undergoes the same dynamics5-' using classical mechanics (Le., the Liouville equation) as using quantum mechanics (Le,, the Schrodinger equation), as exemplified by Ehrenfest's theorem.8 Much e f f ~ r t ~ *has ~ -been ' ~ extended to find analogous coherent states for anharmonic problems, but these states rarely have all of the desired properties. Problems for which such states have been found include the particle in a boxI7 and the two-dimensional rigid-rotor" problems. Another school of thought prefers to use the coherent states, or Gaussian wave packets. Properties of anharmonic quantum systems may be deduced from the classical or semiclassical dynamics of these wave packets. Widely differing views are held as to the merits of this approach, and it is the purpose of this paper to resolve some of these differences. We take a semiclassical approach, using the tools of Gaussian wave packet dynamicsIH1 to relate the classical to the quantum dynamics of the Morse oscillator, and show how it arises that some properties are poorly reproduced while other properties are closely related. This paper asserts that there does exist a strong correspondence and that this correspondence can be exploited to develop computational algorithms: it provides a n ~ t h e r ' ~ - ' *a*posteriori ~* justification of the De Leon-Heller spectral quantization method.31
(1) Glauber, R. J. Phys. Rev. Lett. 1963, 10, 84. (2) Glauber, R. J. Phys. Rev. 1963, 131, 2766. (3) Glauber, R. J. Phys. Rev. 1963, 130, 2529. (4) Schrodinger, E. Natunvissenschaften 1926, 14, 664. (5) Saxon, D. S. In Elementary Quantum Mechanics; Holden Day: San Francisco, 1964; p 158. (6) Nieto, M. M.; Simmons, L. M., Jr. Phys. Reu. A 1979, 19, 438. (7) Nieto, M. M.; Simmons, L. M.. Jr. Phvs. Reu. Lett. 1978. 41. 207. (8) Merzbacher, E. In Quantum Mechanics; Wiley: New York, 1970; p 41. (9) Nieto, M. M.; Simmons, L. M., Jr. Phys. Reu. D 1979, 20, 1321. (10) Nieto, M. M.; Simmons, L. M., Jr. Phys. Reu. D 1979, 20, 1332. (1 1) Nieto, M. M.; Simmons, L. M., Jr. Phys. Reu. D 1979, 20, 1342. (12) Nieto, M. M. Phys. Reu. D 1980, 22, 391. (13) Nieto, M. M.; Gutschick, V. P. Phys. Rev. D 1981, 23, 922. (14) Gutschick, V. P.; Nieto, M. M.; Simmons, L. M., Jr. Phys. Lett. A 1980, 76A, 15. (15) Perelomov, A. M. Commun. Math. Phys. 1972, 26, 222. (16) Barut, A. 0.;Girardello, L. Commun. Math. Phys. 1971, 21, 21. (17) Reimers, J. R.; Heller, E. J. J . Phys. A 1986, 19, 2559. (18) Reimers, J. R.; Heller, E. J. J . Chem. Phys. 1985, 83, 511. (19) Heller, E. J. J . Chem. Phys. 1975, 62, 1544. (20) Heller, E. J. Chem. Phys. Lett. 1975, 34, 321. (21) Heller, E. J. J . Chem. Phys. 1976, 64, 63. (22) Heller, E. J. J . Chem. Phys. 1976, 65, 4979. (23) Heller, E. J. J . Chem. Phys. 1977, 67, 3339. (24) Heller, E. J. J . Chem. Phys. 1978, 68, 3891. (25) Davis, M. J.; Heller, E. J. J . Chem. Phys. 1979, 71, 3383. (26) Heller, E. J.; Stechel, E. B.; Davis, M. J. J . Chem. Phys. 1980, 73, 4720. (27) Heller, E. J. J . Chem. Phys. 1981, 75, 2923. (28) Davis, M. J.; Heller, E. J. J . Chem. Phys. 1981, 75, 3916.
'Queen Elizabeth I1 Fellow.
0022-3654/88/2092-3225$01.50/0 , ~~I
I
1
~
0 1988 American Chemical Society
