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A classical model for time- and frequency-resolved spectroscopy of nonadiabatic excited-state dynamics
Volume 197, number 43
CHEMICAL PHYSICS LETTERS
A classical model for time- and frequency-resolved of nonadiabatic excited-state dynamics Gerhard
18 September 1992
spectroscopy
Stock ’ and William H. Miller
Department of Chemistry, University of California, Berkeley, CA 94720, USA
Received 22 June 1992
Based on the framework of the classical electron analog due to Meyer and Miller, we outline a new classical formulation of the spectroscopy of nonadiabatically coupled electronic states. The timedependent molecular correlation functions are expressed in terms of the classical dipole function that is the classical analog of the quantum-mechanical electronic dipole operator. Explicit expressions for cw absorption and for time- and frequency-resolved pump-probe spectra are derived. The capability of the classical method is illustrated by computational results for a four-mode model of the SI-S, conical intersection in pyrazine. Comparison to exact quantum-mechanical calculations reveals that the classical model reproduces the qualitative features of the cw absorption and the time- and frequency-resolved spectra.
1. Introduction
by a
of classical
variables (i.e.
inverse of the Heisenberg correspondence relation),
Recent experimental and theoretical progress in the description of excited electronic state structure have revealed that nonadiabatic interactions, such as intersections and avoided crossings of potential energy surfaces, are more the rule than the exception. Dynamical computations for nonadiabatically coupled electronic surfaces in general is a difficult numerical task and has thus motivated the development of a number of approximate methods, such as the time-dependent self-consistent field (TDSCF) approximation [ l-31 a variety semiclassical methods, “rigorous” semiclassical [ 41, various versions of the surface hopping model of Tully and Preston [ 7-101, and a variety of classical path models (i.e. classical nuclear motion coupled to time-dependent electronic motion) [ 1l- 141. Some while ago Meyer and Miller proposed a model that extends the appealing simple technique of classical trajectory simulation to the treatment of nonadiabatic dynamics [ 13 1. classical elecanalog (CEA) a set electronic states to: W.H.
University of
’
396
Department of Berkeley, CA USA. fellow of Deutsche
so that one obtains a classical Hamiltonian for the complete set of all heavy particle and electronic degrees of freedom, the dynamics of which is then determined by integrating the classical equations of motion (i.e. Hamilton’s equations). Representing any quantum-mechanical operator by a corresponding classical function, the model is formally appealing, and has been applied successfully to a variety of problems, including the evaluation of nonadiabatic collision processes 14 1, calculation of Oppenheimer dynamics conical intersections [ l&l6 1, and photodissociation to nonadiabatically coupled states both in the gas phase [ 171 and in solution [ 181. In this Letter we use the CEA model to obtain a general description of the spectroscopy of nonadiabatically coupled electronic states. To this end we first derive the CEA dipole correlation function, which is the key quantity for calculating any spectroscopic information within the weak-field limit. In particular, we show how this approach can be used to study the dynamics exhibited in time- and frequency-resolved pump-probe spectroscopy, which has become a focus of increasing experimental (see papers in ref. [ 191, [ 20-25 ] ) and theoretical inter-
0009-2614/92/$ 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
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CHEMICAL PHYSICS LETTERS
est (for recent theoretical reviews see refs. [ 26-28 ] ) . It has been shown experimentally [ 20-231 and theoretically [ 26,27,29-3 1] that time-resolved techniques provide novel information on the ultrafast excited-state dynamics of polyatomic molecules, information that is unobtainable by standard cw techniques, which typically yield only rather diffuse absorption and fluorescence spectra in complex systems. The purpose of the present work is therefore to investigate the extent to which the simple CEA model is capable of providing a quantitative understanding of time-dependent observables and the pump-probe spectra of nonadiabatically coupled systems. Numerical simulations carried out for a four-mode model of the conical S2-SI intersection in pyrazine [ 321 demonstrate the capabilities and limitations of the method.
performed by representing the dipole operator in the Heisenberg picture ji(t)=exp(iZ?t)F,exp( =,&2(t)+P20(t)
2.1. The classical electron analog (CEA) dipole correlation function
As a generic nonadiabatic model problem we consider an electronic three-state system, consisting of the electronic ground state (qo) and the two excited states I al) and I p2). We assume that the excited electronic states are coupled by an intramolecular interaction VIZ, whereas the coupling of the excited states with the well-separated electronic ground state may be neglected. Furthermore, we assume that only the I po) - I (p2) transition is radiatively coupled within the dipole approximation, i.e. we have the common situation of an optically bright excited electronic state ( 1q2) ) coupled to a dark background state ( 1p, ) ). Adopting a diabatic electronic representation we write for the molecular Hamiltonian and the electronic dipole operator (“- ” denotes a quantum-mechanical operator)
+(I~d~n<~~l+h.c.), P=I~2)~2o<~oI+IPo>Po2(~2I~ The
(1) (2)
transition to classical mechanics is most clearly
-i&It)
(3)
3
and writing the expressions for the single- (multi-) photon spectra in terms of one- (multi-) time dipole correlation functions, see, for example, ref. [ 33 1. The simplest and best-known case is the cw absorption spectrum, which is directly given by the Fourier transform of the dipole correlation function C’o,( t), 00
Z,(o)=2Re
s
dtexp(iot)
0
(4)
xexp(-tlT2)CPdt),
CQM(t)=
2. Theory
18 September 1992
< yI
<~Olib2(t)CiZO(O)
lq0)
I y>
,
(5)
where I !P) denotes the initial vibrational wavefunction of the electronic ground state and T2 is the phenomenological total dephasing time of the optical transition. We now want to define the corresponding classical expressions for the eqs. (3) and (5). Within the theoretical framework of the CEA model [ 13 ] the classical function A corresponding to any quantum-mechanical operator A defined on the F-dimensional Hilbert space is given by (fi = 1):
+ ,J,
7.’
=, J(nk+f)(nk’+f)
Xexp[i(a-q~)lAkks,
(6)
where AkkPare the diabatic electronic matrix elements of a and {n,J and {qk} are the classical actionangles variables corresponding to the electronic degrees of freedom. To obtain a dynamically consistent model within the framework of the classical Smatrix theory [ 341, Langer-type modifications have been employed to the off-diagonal terms [ 13 1. For the case R=g,_,, eq. (6) gives the CEA Hamiltonian function, which depends parametrically (through the matrix elements Hk,kF(x) ) on the nuclear coordinates X. The complete classical vi397
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1992
bronic Hamiltonian is then obtained by adding the classical nuclear kinetic energy, giving
and simulate the projection on the initial state 1Y) by sampling over the initial conditions, i.e.
H(p,x,n,q)=P2/2~+noI/o(x)+n,~,/,(x)+n2V2(X)
c,,(t)HCCEA(t)=(~OZ(f)~2O(O)).
(7)
+2J(nO+t)(n2+t)cos(q2-41)~~2(x)~
The time evolution of both heavy-particle degrees of freedom (x, p ) and electronic degrees of freedom ( n, p) are then described consistently by Hamilton’s equations i.(t)I
-
aH api’
d.(f)=-
’
!!!!
8H G
The brackets denote the averaging over the initial positions Xi(0) and momenta pi(O), and over the initial electronic phases qk( 0). Writing the average as a sum over, say, Ntraj trajectories and evaluating the electronic phase difference by formal integration of eqs. (9 ), we obtain for the CEA dipole correlation function (normalized to Cc,,( 0) = 1)
aXi
(i= l,..., NrnOd)>
ilk(t) =
(LO,
3
(14)
(8) 1,2) .
I
(9)
I~2(2(N+~)--1’2~~exp
c,(t)=
As by construction only the two excited electronic states are nonadiabatically coupled, the population of the electronic ground state no and the sum of the excited-state electronic populations N= n, + n2 are constants of the motion, depending only on initial conditions (e.g., for absorption N=O, for emission N= 1). Introducing the canonical transformation [341 n=n2,
4=42-41,
Q=a,
N=n,+n2,
we rewrite the CEA Hamiltonian
(10)
(7) as
H@, z, n, 4) = $
+n,V,(x)+V,(x)+n[V,(x)-VI:(x)]
+2
(N+i-n)(n+f)cosq
V,,(x).
(11)
Due to the canonical transformation ( 10) the only (non-constant) action-angle variables entering the Hamiltonian ( 11) are n(t) and q(t), thus reducing the number of differential equations (9) from six to two. In order to obtain the classical analog for the quantum-mechanical dipole correlation function C,, ( t ) ( 5 ), we replace the electronic dipole operator matrix elements fiZo(t) by the corresponding CEA dipole function ih2(t)t-+POoa(n(t),4(t)) =~2J(no+t)(n+t)exp[i(q2-q0)l,
398
(13)
(12)
x
(
v,-v,+v,,
J~cosq)].
(15)
Eq. ( 15) represents the main theoretical result of this Letter. In connection with eqs. (4) and ( 14) it gives a simple and physically clear picture of the linear response of this classical model of a vibronically coupled system. The resonance condition of the electronic transition is determined by the argument of the exponential function in ( 15 ), where the time integration accounts for the fact that the potentials depend on time via the trajectory x(t). In absence of intramolecular coupling ( l’,:,=O), the excited state population n(t) =const., and the electronic transition frequency is simply given by the vertical electronic energy difference V2- I’,, stating the classical Franck-Condon principle. The nonadiabatic coupling V,, introduces an additional term into the resonance condition reflecting the change of the adiabatic potential surfaces due to the vibronic coupling. Furthermore, the factor Jm accounts for the fact that only a part of the electronic excited-state population is in the optically accessible bright state. So far we have specialized to the linear response of the classical system. Employing time-dependent density-matrix perturbation theory, however, it has been shown that any order of the nonlinear polarization can be expressed in terms of the same classical correlation function ( 15) [ 351. The CEA dipole correlation function ( 15 ) is therefore a rather gen-
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era1 concept, allowing for the calculation of any kind of spectroscopic information within the low-field limit. 2.2. Time-dependent observables and pump-probe spectroscopy Since one computes (numerically) classical trajectories of the molecular system in this approach, it is straightforward to monitor any time-dependent molecular property of interest. Obvious examples are the mean values of the positions (x,(t) ) and the momenta (p,(t)) which describe the vibrational motion of the molecule. Due to recent progress in femtosecond laser technology [ 19-25 ] in some cases these time-resolved information is not only of academical interest but amenable to experimental measurement. A key quantity in dealing with intramolecular dynamics occurring on two coupled potential energy surfaces is the time-dependent population probability P2( t) of the optically bright state I p2), defined as Pz(t)=(‘Y(t)lbD2)(a)2lyl(t))
>
(16)
where ( !P( t) ) is the nonstationary molecular state vector. As P2 (1) = 1 in the absence of intramolecular coupling ( V,, =O), the electronic population probability directly monitors the electronic non-BornOppenheimer dynamics. Besides its important role in the interpretation of nonradiative processes [ 361, it has been shown in a series of recent papers that P2( t) can be measured by a stimulated emission pump-probe experiment in the impulsive limit, i.e. for resonant laser pulses with durations shorter than the characteristical (e.g. vibrational) time scales of the system [ 29,30 1. Employing excited-state absorption with fluorescence detection, Zewail and coworkers have shown a clear experimental example of the real-time observation of electronic population dynamics for sodium iodine [ 23 1. In the CEA formalism the electronic population dynamics is described by the classical action variable n(t) which, by virtue of the “Langer modification”, can take on continuous values from - 4 to $. To make the correspondence to discrete electronic states n = 0 or 1 we invoke the standard histogram method of as-
LETTERS
18 September
1992
signing the system to the electronic state to which n(t) is closest, i.e. P2(t)= kNrx(n,(0), tra,r-l
(17)
x(n)=l,
ifn>t
,
x(n) = 0 ,
otherwise .
(18)
Time-dependent observables as mentioned above are an instructive and useful concept for the interpretation of molecular dynamics, but do not necessarily describe the outcome of a time-resolved experiment with realistic laser fields. These experiments rather measure the nonlinear polarization [2628,301, which may be evaluated quantum mechanically or with a classical model [ 25,27,37]. In this Letter we want to consider a simple but rather generic type of stimulated-emission pumpprobe experiment. Specifically we consider the case of impulsive excitation by a pump pulse at t = 0, followed by a delayed probe pulse which is characterized by the laser frequency o, and the pulse envelope function t2 ( t) centered at t = At. Measuring, as usual, the total differential intensity Zof the probe pulse (i.e. zwith pump-zwithout pump 1 as a function of the delay time At and the probe frequency wp, the pump-probe signal can be written as [30,35] m &,(w,,
At) =o,
Xfm[h.(t2
s -m -h)
cu dt,
s dt, G(t,)tZ(f2) --a,
1
x(~l(nlP2o(tl)~2(t2)IV)Z)I~).
(19)
In the derivation of eq. ( 19) the standard rotatingwave approximation has been employed. Again, replacing the dipole operator by the CEA dipole correlation function and the projection on the initial state by sampling over initial conditions we obtain for the classical pump-probe signal L,(w, =-
At) w?
Nt,
Ntraj r= 4J I
OD
2
dtexp(io,t)t5(t)c,(t>X(n,(t))
,
-m (20)
399
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where x(n) is as above (eq. ( 18) ). Similar to the discussion of the electronic population probability P2(t),the histogram method accounts for the fact that for a single trajectory the electron either is or is not in the optically bright electronic state to couple to the radiation field. It also corrects the effects of the Langer-like modifications of the Hamiltonian ( 11 ), which allows n (2) to range from f to 1, what is clearly unphysical for a population probability and would give rise, e.g., to stimulated emission shifted to the blue of the cw absorption spectrum. Note that, consistent with the quantum-mechanical case, for ultrashort probe pulses the pump-probe signal reduces to the definition of the electronic population probability P2( t ).
3. Computational results Besides providing physical insight, classical and semiclassical models are also meant to be used as a computational tool for treating multidimensional molecular systems that are too complex for a full quantum calculation. We therefore wish to study how accurately the classical observables and spectra introduced above are able to describe the quantummechanical dynamics of a nontrivial model problem. With the aid of ab initio calculations, Domcke and co-workers recently have characterized the conical intersection of the S, (rut*) and S,( XX*) states of pyrazine [ 321. Taking into account three totally symmetric modes (vi, Y6a,~g,) and a Singk COUphg mode (vlOa) within the framework of linear vibrational and vibronic coupling, it has been shown that the model Hamiltonian ( 1) reproduces the available experimental data (absorption and resonance Raman spectra) reasonably well (see ref. [ 321 for detailed information). Furthermore it has been shown that this intersection triggers an ultrafast S,-S, internal conversion process and a dephasing of the vibrational motion, which has been analyzed in terms of detailed quantum wave-packet studies [ 38 ] and by the calculation of “real-time” pump-probe spectra [ 30,321. Here these exact four-mode three-state calculations serve as a reference for testing the reliability of the classical model. Fig. 1 shows the quantum-mechanical (fig. 1a) and the classical (fig. lb) absorption spectra of the 400
18 September 1992
CHEMICAL PHYSICS LETTERS
S, (~a*) state of pyrazine. The stick spectrum in fig. la is obtained by evaluation of eq. (4) for the microscopic model Hamiltonian (1) without additional phenomenological dephasing ( 1/ T2= 0).The spectral envelopes in fig. la and fig. lb are achieved with the dephasing time T2= 30 fs, which has been chosen to reproduce the homogeneous linewidth of the experimental spectrum [ 391. The dense line structure reflects the complete vibronic mixing of the S2 state with the lower-lying S, state, and gives rise to a rather diffuse appearance of the S2 absorption band [ 361. It is seen that the classical model maps the global features of the absorption spectrum, although it is not capable of reproducing the finer structures, which are a consequence of the details of the quantum-mechanical line spectrum. Considering the simplicity of the classical model, however, the agreement with the exact calculation is quite reason-
5.0
4.5
frequency
5.5
[eV]
Fig. 1. The absorption spectrum of the S, state of pyrazine calculated quantum mechanically (a) and with the CEA model (b). The spectral envelopes are obtained with the phenomenological dephasing time i”, = 30 fs.
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CHEMICAL PHYSICS LETTERS
able. For all classical simulations the number of trajectories has been chosen Ntraj= 2500, although the essential structure may already be recognized with a few hundreds of trajectories. Before considering time- and frequency-resolved spectra (i.e. two-dimensional information), it is instructive lirst to look at the time-dependent observables introduced above. Since these quantities are obtained directly from the propagation of Hamilton’s equations and are highly averaged, the classical simulations may be expected to work well. As representative examples of the electronic and vibrational dynamics, fig. 2 shows the electronic popu-
1.01
300
200
time
[fs]
Fig. 2. Time evolution of the mean values of the positions for the models (a) u1and (b) v~, and of the electronic population probability Pz( f) (c), assuming initial preparation of the Sf state. The full lines represent the classical calculations, the dotted lines correspond to exact quantum-mechanical results.
18 September 1992
lation probability Pz ( t ) (fig. 2c) and the mean values of the vibrational modes V, (fig. 2a) and V6a(fig. 2b), respectively. It has been assumed that at t=O the S2 state has been populated by an ideally short laser pulse. The full lines correspond to the classical, the dotted lines to the quantum-mechanical calculation. It is seen that for the first 500 fs the CEA model is in fairly good agreement with the exact results. The oscillations of the higher-frequency mode vl are reproduced almost quantitatively, whereas the vibrational dephasing of the mode V6ais exaggerated by the classical model. This indicates that the effective classical potential, which the heavy particles experience, is more anharmonic than the quantum-mechanical counterpart. The classical model also nicely monitors the quasi-periodical recurrences of the electronic population probabilities P2 ( t) which are a consequence of the coherent wave-packet motion in the totally symmetric modes, driving the electronic population between the two coupled electronic states [ 381. Due to the stronger vibrational dephasing exhibited by the classical simulation, the recurrences of the electronic population also decay more rapidly. Recalling eq. ( 15 ) for the CEA dipole correlation function cr( t), it is clear that the heavy-particle positions x ( t ) (yielding the transition frequency via the potentials Vij(X(t) ) and the electronic population probability P2 (t) (occurring in dm) are essential ingredients in the calculation of c,(t). The good agreement of the time-dependent quantities thus encourages one to consider the nonlinear pumpprobe spectra which are not as highly averaged as the above quantities and therefore require a much higher accuracy of the method. As a simple example of a stimulated-emission pump-probe experiment we consider the case of impulsive excitation and a probe pulse duration of 7= 20 fs (fwhm). Fig. 3 shows the comparison between the quantum-mechanical (fig. 3a) and classical (fig. 3b) pump-probe spectra as a function of the delay time At and the probe carrier frequency w,. As discussed in some detail elsewhere [ 301, the decay of the pump-probe signal as a function of time qualitatively displays the excited-state population dynamics while the frequency distribution reflects the complex vibrational motion on the coupled electronic surfaces. It is seen that the ultrafast initial decay of 401
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jectory moving in an averaged potential instead of a trajectory for each electronic state. The conical intersection causes a splitting and extensive distortion of the initial Gaussian wave packet [ 381, and is therefore a stringent test for a classical model. For many practical purposes, however, one would be rather happy to know the rough qualitative features of the pump-probe signal for a given model system. Simulation of the time- and frequency-resolved spectra employing the CEA model should be capable to give a first idea of important issues as the rough frequency distribution of the emission, the existence of vibrational recurrences, and the lifetime of the excited electronic state.
4. Concluding remarks
Fig. 3. The quantum-mechanical (a) and classical (b) pumpprobe spectrum of the S1 state of pyrazine, assuming impulsive excitation and a probe pulse duration of T= 20 fs.
P2( t) causes a considerable ( x 1.5 eV) red-shift of the emission spectrum, which is for zero delay time located at the cw absorption band. The classical simulation can be seen to catch the global structure of the quantum calculation. In particular, there is a reasonable agreement of the initial decay, the width of the emission spectrum at a given time, and the delay times, where the spectrum is maximally red- and blue-shifted, respectively. The classical model, however, is unable to reproduce the detailed structure of the vibrational motion. This is probably a consequence of the basic Ansatz of the CEA model, that the vibrational motion on two coupled potential surfaces is described by a single tra402
We have outlined a classical formulation of the spectroscopy of nonadiabatically coupled electronic states within the theoretical framework of the CEA model. We have introduced the CEA dipole correlation function, which is the key quantity for the calculation of any spectroscopic information in the weak-field limit. As important examples, explicit expression for the cw absorption spectrum and the stimulated-emission pump-probe spectrum have been derived. Being a classical method, the actual computations (i.e. the propagation of trajectories and Fourier transformations) are readily performed, and the extension to more degrees of freedom is straightforward and conveniently manageable on a nowadays standard workstation. For many degrees of freedom, and consequently increasing averaging, the classical model would be expected to work even better [ 15 1. It should be noted that the complex electronic and vibrational dynamics occurring on intersecting potential energy surfaces is not reproduced by other standard approximation methods. Quantum-mechanical TDSCF methods, for example, neglect the correlations between the separated heavy-particle degrees of freedom, and are thus capable of accounting only for the very short-time dynamics of the system (l-3 vibrational periods) [ 1,401. The CEA model, however, accounts within the basic classical approximation explicitly for the correlations between the heavy-particle degrees of freedom via the
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coupled equations of motion ( 8). Quantum-mechanical TDSCF methods are therefore well suited for the description of coherent processes (e.g. cw absorption, Raman, CARS) [ 31, occurring on the very fast time scale of the total dephasing time T, (for the pyrazine model, e.g. T2= 30 fs). Stimulated emission and fluorescence, on the other hand, happen on the timescale of the excited electronic state lifetime T,, that is in coupled bound-state problems typically a factor 5-50 larger than T, [ 39 1, On this timescale the correlations between the heavy-particle degrees of freedom may become crucial [ 38 1, suggesting the use of classical models. The critical question, concerning all approximation methods, however, is the reliability of the CEA model. For the given bound-state problem it has been seen that the CEA model maps the absorption spectrum and time-dependent quantities rather well and the more detailed pump-probe spectra at least roughly. In order to trust the predictions of the CEA model in cases that are not feasible by quantum calculations, it is certainly necessary to perform numerical studies on a variety of model problems. An obvious and important amplification is the inclusion of a reactive mode to describe dissociation and isomerization processes. Nonadiabatic reactive dynamics is a stringent test for the CEA model because the inherent Ansatz of an averaged potential becomes inadequate for diverging potential energy surfaces. As has been shown recently in a stimulating paper by Benjamin and Wilson [ 181, the CEA model furthermore provides a promising microscopic approach to study nonadiabatic electronic dynamics in condensed phases. Acknowledgement GS would like to thank Wolfgang Domcke and Hans-Dieter Meyer for numerous stimulating and helpful discussions. Research support by the National Science Foundation (Grant CHE-8920690) is also gratefully acknowledged. References [ 1 ] 2. Kotler, A. Nitzan and R. Kosloff, Chem. Phys. Letters 153 (1988) 483; 2. Kotler, E. Neria and A. Nitzan, Computer Phys. Commun. 63 ( 1991) 243.
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(2 ] E. Neria, A. Nitzan, R.N. Bamett and U. Landman, Phys. Rev.Letters67 (1991) 1011. [3] J.R. Waldeck, J. Campos-Martinez and R.D. Coalson, J. Chem. Phys. 94 ( 1991) 2773. [4] P. Pechukas, Phys. Rev. 181 (1969) 166, 174. [5] W.H. Miller and T.F. George, J. Chem .Phys. 56 (1972) 5637. [6] F.J. Webster, P.J. Rossky and R.A. Friesner, Computer Phys. Commun. 63 ( 199 1) 494. [ 71 J.C. Tully and R.K. Preston, J. Chem. Phys. 55 (1971) 562; J.C. Tully, J. Chem. Phys. 93 ( 1990) 106 1. [ 81 NC. Blais and D.G. Truhlar, J. Chem. Phys. 79 (1983) 1334; NC. Blais, D.G. Truhlar and C.A. Mead, J. Chem. Phys. 89 (1988) 6204. [ 9 ] M.F. Herman, J. Chem. Phys. 8 1 ( 1984) 754. [lO]P.J.Kuntz, J.Chem.Phys. 95 (1991) 141, 156. [ 111 D.A. Micha, J. Chem. Phys. 78 (1983) 7138. [ 12 ] D.J. Diestler, J. Chem. Phys. 78 (1983) 2240; L.L. Halcomb and D.J. Diestler, J. Chem. Phys. 84 ( 1986) 3130; M. Amarouche, F.X. Gadea and J. Dump, Chem. Phys. 130 (1989) 145. [ 131 H.-D. Meyer and W.H. Miller, J. Chem. Phys. 70 (1979) 3214. [ 141 H.-D. Meyer and W.H. Miller, J. Chem. Phys. 71 (1979) 2156; S.K. Gray and W.H. Miller, Chem. Phys. Letters 93 ( 1982) 341. [ 151 H.-D. Meyer, Chem. Phys. 82 (1983) 199. [ 16 ] J.W. Zwanziger, E.R. Grant and G.S. Ezra, J. Chem. Phys. 85 (1986) 2089. [ 171 E.M. Goldfield, P.L. Houston and G.S. Ezra, J. Chem. Phys. 84 (1986) 3120. [ 181 I. Benjamin and K.R. Wilson, J. Chem. Phys. 90 (1989) 4176. [ 19 ] C.B. Harris, E.P. Ippen, G.A. Mouvon and A.H. Zewail, eds., Ultrashort phenomena, Vol. 7 (Springer, Berlin, 1990). [20] F.W. Wise, M.J. Rosker and C.L. Tang, J. Chem. Phys. 86 (1987) 2827. [ 2 1 ] J. Chesnoy and A. Mokhtari, Phys. Rev. A 38 ( 1988) 3566; A. Mokhtari, A. Chebira and J. Chesnoy, J. Opt. Sot. Am. B7 (1990) 1551. [22] H.L. Fragnito, J.-Y. Bigot, P.C. Becker and C.V. Shank, Chem. Phys. Letters 160 (1989) 101; S.L. Dexheimer, Q. Wang, L.A. Peteanu, W.T. Pollard, R.A. Mathies and C.V. Shank, Chem. Phys. Letters 188 ( 1992) 61; R.W. Schoenlein, L.A. Peteanu, R.A. Mathies and C.V. Shank, Science 254 (1991) 412. [23] M.J. Rosker, T.S. Rose and A.H. Zewail, Chem. Phys. Letters 146 (1988) 175; P. Cong, A. Mokhtari and A.H. Zewail, Chem. Phys. Letters 172 (1990) 109. [24] M. Dantus, M.J. Rosker and A.H. Zewail, J. Chem. Phys. 89 (1988) 6128; R.M. Bowman, M. Dantus and A.H. Zewail, Chem. Phys. Letters 161 ( 1989) 297;
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M. Gruebele, I.R. Sims, E.D. Potter and A.H. Zewail, J. Chem. Phys. 95 ( 199 1) 7763. [ 251 R.E. Walkup, J.A. Misewich, J.H. Glownia and P.P. Sovokin, J. Phys. Chem. 94 ( 1991) 3389. 1261 W.T. Pollard and R.A. Mathies, Ann. Rev. Phys. Chem. 43 (1992), in press. [27] S. Mukamel, Ann. Rev. Phys. Chem. 41 (1990) 647. [28] S.H. Lin, B. Fain and N. Hamer, Advan. Chem. Phys. 79 (1990) 133. [ 291 W. Domcke and H. Koppel, Chem. Phys. Letters 140 ( 1987) 133; G. Stock and W. Domcke, Chem. Phys. 124 (1988) 227. [ 301 G. Stock, R. Schneider and W. Domcke, J. Chem. Phys. 90 (1989) 7184; G. Stock and W. Domcke, J. Opt. Sot. Am. B 7 (1990) 1970; G. Stock and W. Domcke, Phys. Rev. A 45 ( 1992) 3032.
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[31] W.T. Pollard, S.-Y. Lee and R.A. Mathies, J. Chem. Phys. 92 (1990) 4012; W.T. Pollard, H.L. Fragnito, J.-Y. Bigot, C.V. Shank and R.A. Mathies, Chem. Phys. Letters 168 (1990) 239. [32] L. Seider, G. Stock, A.L. Sobolewski and W. Domcke, J. Phys. Chem. 96 (1992) 5298. [33] R.G. Gordon, Advan. Magn. Res. 3 (1968) 1. [34] W.H. Miller, Advan. Chem. Phys. 25 (1974) 69. [ 35 ] G. Stock and W.H. Miller, to be published. [ 361 H.-D.Meyer and H. Kiippel, J. Chem. Phys. 8 1 ( 1984) 2605; H. Kiippel, W. Domcke and L.S. Cederbaum, Advan. Chem. Phys. 57 (1984) 59. [37] S.-Y. Lee, W.T. Pollard and R.A. Mathies, Chem. Phys. Letters 163 (1989) 11. [ 381 R. Schneider, W. Domcke and H. Kiippel, J. Chem. Phys. 92 (1990) 1045. [ 391 G. Stock and W. Domcke, J. Chem. Phys. 93 (1990) 5496. [40] H.-D. Meyer, personal communication.
CHEMICAL PHYSICS LETTERS
A classical model for time- and frequency-resolved of nonadiabatic excited-state dynamics Gerhard
18 September 1992
spectroscopy
Stock ’ and William H. Miller
Department of Chemistry, University of California, Berkeley, CA 94720, USA
Received 22 June 1992
Based on the framework of the classical electron analog due to Meyer and Miller, we outline a new classical formulation of the spectroscopy of nonadiabatically coupled electronic states. The timedependent molecular correlation functions are expressed in terms of the classical dipole function that is the classical analog of the quantum-mechanical electronic dipole operator. Explicit expressions for cw absorption and for time- and frequency-resolved pump-probe spectra are derived. The capability of the classical method is illustrated by computational results for a four-mode model of the SI-S, conical intersection in pyrazine. Comparison to exact quantum-mechanical calculations reveals that the classical model reproduces the qualitative features of the cw absorption and the time- and frequency-resolved spectra.
1. Introduction
by a
of classical
variables (i.e.
inverse of the Heisenberg correspondence relation),
Recent experimental and theoretical progress in the description of excited electronic state structure have revealed that nonadiabatic interactions, such as intersections and avoided crossings of potential energy surfaces, are more the rule than the exception. Dynamical computations for nonadiabatically coupled electronic surfaces in general is a difficult numerical task and has thus motivated the development of a number of approximate methods, such as the time-dependent self-consistent field (TDSCF) approximation [ l-31 a variety semiclassical methods, “rigorous” semiclassical [ 41, various versions of the surface hopping model of Tully and Preston [ 7-101, and a variety of classical path models (i.e. classical nuclear motion coupled to time-dependent electronic motion) [ 1l- 141. Some while ago Meyer and Miller proposed a model that extends the appealing simple technique of classical trajectory simulation to the treatment of nonadiabatic dynamics [ 13 1. classical elecanalog (CEA) a set electronic states to: W.H.
University of
’
396
Department of Berkeley, CA USA. fellow of Deutsche
so that one obtains a classical Hamiltonian for the complete set of all heavy particle and electronic degrees of freedom, the dynamics of which is then determined by integrating the classical equations of motion (i.e. Hamilton’s equations). Representing any quantum-mechanical operator by a corresponding classical function, the model is formally appealing, and has been applied successfully to a variety of problems, including the evaluation of nonadiabatic collision processes 14 1, calculation of Oppenheimer dynamics conical intersections [ l&l6 1, and photodissociation to nonadiabatically coupled states both in the gas phase [ 171 and in solution [ 181. In this Letter we use the CEA model to obtain a general description of the spectroscopy of nonadiabatically coupled electronic states. To this end we first derive the CEA dipole correlation function, which is the key quantity for calculating any spectroscopic information within the weak-field limit. In particular, we show how this approach can be used to study the dynamics exhibited in time- and frequency-resolved pump-probe spectroscopy, which has become a focus of increasing experimental (see papers in ref. [ 191, [ 20-25 ] ) and theoretical inter-
0009-2614/92/$ 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
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est (for recent theoretical reviews see refs. [ 26-28 ] ) . It has been shown experimentally [ 20-231 and theoretically [ 26,27,29-3 1] that time-resolved techniques provide novel information on the ultrafast excited-state dynamics of polyatomic molecules, information that is unobtainable by standard cw techniques, which typically yield only rather diffuse absorption and fluorescence spectra in complex systems. The purpose of the present work is therefore to investigate the extent to which the simple CEA model is capable of providing a quantitative understanding of time-dependent observables and the pump-probe spectra of nonadiabatically coupled systems. Numerical simulations carried out for a four-mode model of the conical S2-SI intersection in pyrazine [ 321 demonstrate the capabilities and limitations of the method.
performed by representing the dipole operator in the Heisenberg picture ji(t)=exp(iZ?t)F,exp( =,&2(t)+P20(t)
2.1. The classical electron analog (CEA) dipole correlation function
As a generic nonadiabatic model problem we consider an electronic three-state system, consisting of the electronic ground state (qo) and the two excited states I al) and I p2). We assume that the excited electronic states are coupled by an intramolecular interaction VIZ, whereas the coupling of the excited states with the well-separated electronic ground state may be neglected. Furthermore, we assume that only the I po) - I (p2) transition is radiatively coupled within the dipole approximation, i.e. we have the common situation of an optically bright excited electronic state ( 1q2) ) coupled to a dark background state ( 1p, ) ). Adopting a diabatic electronic representation we write for the molecular Hamiltonian and the electronic dipole operator (“- ” denotes a quantum-mechanical operator)
+(I~d~n<~~l+h.c.), P=I~2)~2o<~oI+IPo>Po2(~2I~ The
(1) (2)
transition to classical mechanics is most clearly
-i&It)
(3)
3
and writing the expressions for the single- (multi-) photon spectra in terms of one- (multi-) time dipole correlation functions, see, for example, ref. [ 33 1. The simplest and best-known case is the cw absorption spectrum, which is directly given by the Fourier transform of the dipole correlation function C’o,( t), 00
Z,(o)=2Re
s
dtexp(iot)
0
(4)
xexp(-tlT2)CPdt),
CQM(t)=
2. Theory
18 September 1992
< yI
<~Olib2(t)CiZO(O)
lq0)
I y>
,
(5)
where I !P) denotes the initial vibrational wavefunction of the electronic ground state and T2 is the phenomenological total dephasing time of the optical transition. We now want to define the corresponding classical expressions for the eqs. (3) and (5). Within the theoretical framework of the CEA model [ 13 ] the classical function A corresponding to any quantum-mechanical operator A defined on the F-dimensional Hilbert space is given by (fi = 1):
+ ,J,
7.’
=, J(nk+f)(nk’+f)
Xexp[i(a-q~)lAkks,
(6)
where AkkPare the diabatic electronic matrix elements of a and {n,J and {qk} are the classical actionangles variables corresponding to the electronic degrees of freedom. To obtain a dynamically consistent model within the framework of the classical Smatrix theory [ 341, Langer-type modifications have been employed to the off-diagonal terms [ 13 1. For the case R=g,_,, eq. (6) gives the CEA Hamiltonian function, which depends parametrically (through the matrix elements Hk,kF(x) ) on the nuclear coordinates X. The complete classical vi397
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bronic Hamiltonian is then obtained by adding the classical nuclear kinetic energy, giving
and simulate the projection on the initial state 1Y) by sampling over the initial conditions, i.e.
H(p,x,n,q)=P2/2~+noI/o(x)+n,~,/,(x)+n2V2(X)
c,,(t)HCCEA(t)=(~OZ(f)~2O(O)).
(7)
+2J(nO+t)(n2+t)cos(q2-41)~~2(x)~
The time evolution of both heavy-particle degrees of freedom (x, p ) and electronic degrees of freedom ( n, p) are then described consistently by Hamilton’s equations i.(t)I
-
aH api’
d.(f)=-
’
!!!!
8H G
The brackets denote the averaging over the initial positions Xi(0) and momenta pi(O), and over the initial electronic phases qk( 0). Writing the average as a sum over, say, Ntraj trajectories and evaluating the electronic phase difference by formal integration of eqs. (9 ), we obtain for the CEA dipole correlation function (normalized to Cc,,( 0) = 1)
aXi
(i= l,..., NrnOd)>
ilk(t) =
(LO,
3
(14)
(8) 1,2) .
I
(9)
I~2(2(N+~)--1’2~~exp
c,(t)=
As by construction only the two excited electronic states are nonadiabatically coupled, the population of the electronic ground state no and the sum of the excited-state electronic populations N= n, + n2 are constants of the motion, depending only on initial conditions (e.g., for absorption N=O, for emission N= 1). Introducing the canonical transformation [341 n=n2,
4=42-41,
Q=a,
N=n,+n2,
we rewrite the CEA Hamiltonian
(10)
(7) as
H@, z, n, 4) = $
+n,V,(x)+V,(x)+n[V,(x)-VI:(x)]
+2
(N+i-n)(n+f)cosq
V,,(x).
(11)
Due to the canonical transformation ( 10) the only (non-constant) action-angle variables entering the Hamiltonian ( 11) are n(t) and q(t), thus reducing the number of differential equations (9) from six to two. In order to obtain the classical analog for the quantum-mechanical dipole correlation function C,, ( t ) ( 5 ), we replace the electronic dipole operator matrix elements fiZo(t) by the corresponding CEA dipole function ih2(t)t-+POoa(n(t),4(t)) =~2J(no+t)(n+t)exp[i(q2-q0)l,
398
(13)
(12)
x
(
v,-v,+v,,
J~cosq)].
(15)
Eq. ( 15) represents the main theoretical result of this Letter. In connection with eqs. (4) and ( 14) it gives a simple and physically clear picture of the linear response of this classical model of a vibronically coupled system. The resonance condition of the electronic transition is determined by the argument of the exponential function in ( 15 ), where the time integration accounts for the fact that the potentials depend on time via the trajectory x(t). In absence of intramolecular coupling ( l’,:,=O), the excited state population n(t) =const., and the electronic transition frequency is simply given by the vertical electronic energy difference V2- I’,, stating the classical Franck-Condon principle. The nonadiabatic coupling V,, introduces an additional term into the resonance condition reflecting the change of the adiabatic potential surfaces due to the vibronic coupling. Furthermore, the factor Jm accounts for the fact that only a part of the electronic excited-state population is in the optically accessible bright state. So far we have specialized to the linear response of the classical system. Employing time-dependent density-matrix perturbation theory, however, it has been shown that any order of the nonlinear polarization can be expressed in terms of the same classical correlation function ( 15) [ 351. The CEA dipole correlation function ( 15 ) is therefore a rather gen-
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era1 concept, allowing for the calculation of any kind of spectroscopic information within the low-field limit. 2.2. Time-dependent observables and pump-probe spectroscopy Since one computes (numerically) classical trajectories of the molecular system in this approach, it is straightforward to monitor any time-dependent molecular property of interest. Obvious examples are the mean values of the positions (x,(t) ) and the momenta (p,(t)) which describe the vibrational motion of the molecule. Due to recent progress in femtosecond laser technology [ 19-25 ] in some cases these time-resolved information is not only of academical interest but amenable to experimental measurement. A key quantity in dealing with intramolecular dynamics occurring on two coupled potential energy surfaces is the time-dependent population probability P2( t) of the optically bright state I p2), defined as Pz(t)=(‘Y(t)lbD2)(a)2lyl(t))
>
(16)
where ( !P( t) ) is the nonstationary molecular state vector. As P2 (1) = 1 in the absence of intramolecular coupling ( V,, =O), the electronic population probability directly monitors the electronic non-BornOppenheimer dynamics. Besides its important role in the interpretation of nonradiative processes [ 361, it has been shown in a series of recent papers that P2( t) can be measured by a stimulated emission pump-probe experiment in the impulsive limit, i.e. for resonant laser pulses with durations shorter than the characteristical (e.g. vibrational) time scales of the system [ 29,30 1. Employing excited-state absorption with fluorescence detection, Zewail and coworkers have shown a clear experimental example of the real-time observation of electronic population dynamics for sodium iodine [ 23 1. In the CEA formalism the electronic population dynamics is described by the classical action variable n(t) which, by virtue of the “Langer modification”, can take on continuous values from - 4 to $. To make the correspondence to discrete electronic states n = 0 or 1 we invoke the standard histogram method of as-
LETTERS
18 September
1992
signing the system to the electronic state to which n(t) is closest, i.e. P2(t)= kNrx(n,(0), tra,r-l
(17)
x(n)=l,
ifn>t
,
x(n) = 0 ,
otherwise .
(18)
Time-dependent observables as mentioned above are an instructive and useful concept for the interpretation of molecular dynamics, but do not necessarily describe the outcome of a time-resolved experiment with realistic laser fields. These experiments rather measure the nonlinear polarization [2628,301, which may be evaluated quantum mechanically or with a classical model [ 25,27,37]. In this Letter we want to consider a simple but rather generic type of stimulated-emission pumpprobe experiment. Specifically we consider the case of impulsive excitation by a pump pulse at t = 0, followed by a delayed probe pulse which is characterized by the laser frequency o, and the pulse envelope function t2 ( t) centered at t = At. Measuring, as usual, the total differential intensity Zof the probe pulse (i.e. zwith pump-zwithout pump 1 as a function of the delay time At and the probe frequency wp, the pump-probe signal can be written as [30,35] m &,(w,,
At) =o,
Xfm[h.(t2
s -m -h)
cu dt,
s dt, G(t,)tZ(f2) --a,
1
x(~l(nlP2o(tl)~2(t2)IV)Z)I~).
(19)
In the derivation of eq. ( 19) the standard rotatingwave approximation has been employed. Again, replacing the dipole operator by the CEA dipole correlation function and the projection on the initial state by sampling over initial conditions we obtain for the classical pump-probe signal L,(w, =-
At) w?
Nt,
Ntraj r= 4J I
OD
2
dtexp(io,t)t5(t)c,(t>X(n,(t))
,
-m (20)
399
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where x(n) is as above (eq. ( 18) ). Similar to the discussion of the electronic population probability P2(t),the histogram method accounts for the fact that for a single trajectory the electron either is or is not in the optically bright electronic state to couple to the radiation field. It also corrects the effects of the Langer-like modifications of the Hamiltonian ( 11 ), which allows n (2) to range from f to 1, what is clearly unphysical for a population probability and would give rise, e.g., to stimulated emission shifted to the blue of the cw absorption spectrum. Note that, consistent with the quantum-mechanical case, for ultrashort probe pulses the pump-probe signal reduces to the definition of the electronic population probability P2( t ).
3. Computational results Besides providing physical insight, classical and semiclassical models are also meant to be used as a computational tool for treating multidimensional molecular systems that are too complex for a full quantum calculation. We therefore wish to study how accurately the classical observables and spectra introduced above are able to describe the quantummechanical dynamics of a nontrivial model problem. With the aid of ab initio calculations, Domcke and co-workers recently have characterized the conical intersection of the S, (rut*) and S,( XX*) states of pyrazine [ 321. Taking into account three totally symmetric modes (vi, Y6a,~g,) and a Singk COUphg mode (vlOa) within the framework of linear vibrational and vibronic coupling, it has been shown that the model Hamiltonian ( 1) reproduces the available experimental data (absorption and resonance Raman spectra) reasonably well (see ref. [ 321 for detailed information). Furthermore it has been shown that this intersection triggers an ultrafast S,-S, internal conversion process and a dephasing of the vibrational motion, which has been analyzed in terms of detailed quantum wave-packet studies [ 38 ] and by the calculation of “real-time” pump-probe spectra [ 30,321. Here these exact four-mode three-state calculations serve as a reference for testing the reliability of the classical model. Fig. 1 shows the quantum-mechanical (fig. 1a) and the classical (fig. lb) absorption spectra of the 400
18 September 1992
CHEMICAL PHYSICS LETTERS
S, (~a*) state of pyrazine. The stick spectrum in fig. la is obtained by evaluation of eq. (4) for the microscopic model Hamiltonian (1) without additional phenomenological dephasing ( 1/ T2= 0).The spectral envelopes in fig. la and fig. lb are achieved with the dephasing time T2= 30 fs, which has been chosen to reproduce the homogeneous linewidth of the experimental spectrum [ 391. The dense line structure reflects the complete vibronic mixing of the S2 state with the lower-lying S, state, and gives rise to a rather diffuse appearance of the S2 absorption band [ 361. It is seen that the classical model maps the global features of the absorption spectrum, although it is not capable of reproducing the finer structures, which are a consequence of the details of the quantum-mechanical line spectrum. Considering the simplicity of the classical model, however, the agreement with the exact calculation is quite reason-
5.0
4.5
frequency
5.5
[eV]
Fig. 1. The absorption spectrum of the S, state of pyrazine calculated quantum mechanically (a) and with the CEA model (b). The spectral envelopes are obtained with the phenomenological dephasing time i”, = 30 fs.
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able. For all classical simulations the number of trajectories has been chosen Ntraj= 2500, although the essential structure may already be recognized with a few hundreds of trajectories. Before considering time- and frequency-resolved spectra (i.e. two-dimensional information), it is instructive lirst to look at the time-dependent observables introduced above. Since these quantities are obtained directly from the propagation of Hamilton’s equations and are highly averaged, the classical simulations may be expected to work well. As representative examples of the electronic and vibrational dynamics, fig. 2 shows the electronic popu-
1.01
300
200
time
[fs]
Fig. 2. Time evolution of the mean values of the positions for the models (a) u1and (b) v~, and of the electronic population probability Pz( f) (c), assuming initial preparation of the Sf state. The full lines represent the classical calculations, the dotted lines correspond to exact quantum-mechanical results.
18 September 1992
lation probability Pz ( t ) (fig. 2c) and the mean values of the vibrational modes V, (fig. 2a) and V6a(fig. 2b), respectively. It has been assumed that at t=O the S2 state has been populated by an ideally short laser pulse. The full lines correspond to the classical, the dotted lines to the quantum-mechanical calculation. It is seen that for the first 500 fs the CEA model is in fairly good agreement with the exact results. The oscillations of the higher-frequency mode vl are reproduced almost quantitatively, whereas the vibrational dephasing of the mode V6ais exaggerated by the classical model. This indicates that the effective classical potential, which the heavy particles experience, is more anharmonic than the quantum-mechanical counterpart. The classical model also nicely monitors the quasi-periodical recurrences of the electronic population probabilities P2 ( t) which are a consequence of the coherent wave-packet motion in the totally symmetric modes, driving the electronic population between the two coupled electronic states [ 381. Due to the stronger vibrational dephasing exhibited by the classical simulation, the recurrences of the electronic population also decay more rapidly. Recalling eq. ( 15 ) for the CEA dipole correlation function cr( t), it is clear that the heavy-particle positions x ( t ) (yielding the transition frequency via the potentials Vij(X(t) ) and the electronic population probability P2 (t) (occurring in dm) are essential ingredients in the calculation of c,(t). The good agreement of the time-dependent quantities thus encourages one to consider the nonlinear pumpprobe spectra which are not as highly averaged as the above quantities and therefore require a much higher accuracy of the method. As a simple example of a stimulated-emission pump-probe experiment we consider the case of impulsive excitation and a probe pulse duration of 7= 20 fs (fwhm). Fig. 3 shows the comparison between the quantum-mechanical (fig. 3a) and classical (fig. 3b) pump-probe spectra as a function of the delay time At and the probe carrier frequency w,. As discussed in some detail elsewhere [ 301, the decay of the pump-probe signal as a function of time qualitatively displays the excited-state population dynamics while the frequency distribution reflects the complex vibrational motion on the coupled electronic surfaces. It is seen that the ultrafast initial decay of 401
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jectory moving in an averaged potential instead of a trajectory for each electronic state. The conical intersection causes a splitting and extensive distortion of the initial Gaussian wave packet [ 381, and is therefore a stringent test for a classical model. For many practical purposes, however, one would be rather happy to know the rough qualitative features of the pump-probe signal for a given model system. Simulation of the time- and frequency-resolved spectra employing the CEA model should be capable to give a first idea of important issues as the rough frequency distribution of the emission, the existence of vibrational recurrences, and the lifetime of the excited electronic state.
4. Concluding remarks
Fig. 3. The quantum-mechanical (a) and classical (b) pumpprobe spectrum of the S1 state of pyrazine, assuming impulsive excitation and a probe pulse duration of T= 20 fs.
P2( t) causes a considerable ( x 1.5 eV) red-shift of the emission spectrum, which is for zero delay time located at the cw absorption band. The classical simulation can be seen to catch the global structure of the quantum calculation. In particular, there is a reasonable agreement of the initial decay, the width of the emission spectrum at a given time, and the delay times, where the spectrum is maximally red- and blue-shifted, respectively. The classical model, however, is unable to reproduce the detailed structure of the vibrational motion. This is probably a consequence of the basic Ansatz of the CEA model, that the vibrational motion on two coupled potential surfaces is described by a single tra402
We have outlined a classical formulation of the spectroscopy of nonadiabatically coupled electronic states within the theoretical framework of the CEA model. We have introduced the CEA dipole correlation function, which is the key quantity for the calculation of any spectroscopic information in the weak-field limit. As important examples, explicit expression for the cw absorption spectrum and the stimulated-emission pump-probe spectrum have been derived. Being a classical method, the actual computations (i.e. the propagation of trajectories and Fourier transformations) are readily performed, and the extension to more degrees of freedom is straightforward and conveniently manageable on a nowadays standard workstation. For many degrees of freedom, and consequently increasing averaging, the classical model would be expected to work even better [ 15 1. It should be noted that the complex electronic and vibrational dynamics occurring on intersecting potential energy surfaces is not reproduced by other standard approximation methods. Quantum-mechanical TDSCF methods, for example, neglect the correlations between the separated heavy-particle degrees of freedom, and are thus capable of accounting only for the very short-time dynamics of the system (l-3 vibrational periods) [ 1,401. The CEA model, however, accounts within the basic classical approximation explicitly for the correlations between the heavy-particle degrees of freedom via the
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coupled equations of motion ( 8). Quantum-mechanical TDSCF methods are therefore well suited for the description of coherent processes (e.g. cw absorption, Raman, CARS) [ 31, occurring on the very fast time scale of the total dephasing time T, (for the pyrazine model, e.g. T2= 30 fs). Stimulated emission and fluorescence, on the other hand, happen on the timescale of the excited electronic state lifetime T,, that is in coupled bound-state problems typically a factor 5-50 larger than T, [ 39 1, On this timescale the correlations between the heavy-particle degrees of freedom may become crucial [ 38 1, suggesting the use of classical models. The critical question, concerning all approximation methods, however, is the reliability of the CEA model. For the given bound-state problem it has been seen that the CEA model maps the absorption spectrum and time-dependent quantities rather well and the more detailed pump-probe spectra at least roughly. In order to trust the predictions of the CEA model in cases that are not feasible by quantum calculations, it is certainly necessary to perform numerical studies on a variety of model problems. An obvious and important amplification is the inclusion of a reactive mode to describe dissociation and isomerization processes. Nonadiabatic reactive dynamics is a stringent test for the CEA model because the inherent Ansatz of an averaged potential becomes inadequate for diverging potential energy surfaces. As has been shown recently in a stimulating paper by Benjamin and Wilson [ 181, the CEA model furthermore provides a promising microscopic approach to study nonadiabatic electronic dynamics in condensed phases. Acknowledgement GS would like to thank Wolfgang Domcke and Hans-Dieter Meyer for numerous stimulating and helpful discussions. Research support by the National Science Foundation (Grant CHE-8920690) is also gratefully acknowledged. References [ 1 ] 2. Kotler, A. Nitzan and R. Kosloff, Chem. Phys. Letters 153 (1988) 483; 2. Kotler, E. Neria and A. Nitzan, Computer Phys. Commun. 63 ( 1991) 243.
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(2 ] E. Neria, A. Nitzan, R.N. Bamett and U. Landman, Phys. Rev.Letters67 (1991) 1011. [3] J.R. Waldeck, J. Campos-Martinez and R.D. Coalson, J. Chem. Phys. 94 ( 1991) 2773. [4] P. Pechukas, Phys. Rev. 181 (1969) 166, 174. [5] W.H. Miller and T.F. George, J. Chem .Phys. 56 (1972) 5637. [6] F.J. Webster, P.J. Rossky and R.A. Friesner, Computer Phys. Commun. 63 ( 199 1) 494. [ 71 J.C. Tully and R.K. Preston, J. Chem. Phys. 55 (1971) 562; J.C. Tully, J. Chem. Phys. 93 ( 1990) 106 1. [ 81 NC. Blais and D.G. Truhlar, J. Chem. Phys. 79 (1983) 1334; NC. Blais, D.G. Truhlar and C.A. Mead, J. Chem. Phys. 89 (1988) 6204. [ 9 ] M.F. Herman, J. Chem. Phys. 8 1 ( 1984) 754. [lO]P.J.Kuntz, J.Chem.Phys. 95 (1991) 141, 156. [ 111 D.A. Micha, J. Chem. Phys. 78 (1983) 7138. [ 12 ] D.J. Diestler, J. Chem. Phys. 78 (1983) 2240; L.L. Halcomb and D.J. Diestler, J. Chem. Phys. 84 ( 1986) 3130; M. Amarouche, F.X. Gadea and J. Dump, Chem. Phys. 130 (1989) 145. [ 131 H.-D. Meyer and W.H. Miller, J. Chem. Phys. 70 (1979) 3214. [ 141 H.-D. Meyer and W.H. Miller, J. Chem. Phys. 71 (1979) 2156; S.K. Gray and W.H. Miller, Chem. Phys. Letters 93 ( 1982) 341. [ 151 H.-D. Meyer, Chem. Phys. 82 (1983) 199. [ 16 ] J.W. Zwanziger, E.R. Grant and G.S. Ezra, J. Chem. Phys. 85 (1986) 2089. [ 171 E.M. Goldfield, P.L. Houston and G.S. Ezra, J. Chem. Phys. 84 (1986) 3120. [ 181 I. Benjamin and K.R. Wilson, J. Chem. Phys. 90 (1989) 4176. [ 19 ] C.B. Harris, E.P. Ippen, G.A. Mouvon and A.H. Zewail, eds., Ultrashort phenomena, Vol. 7 (Springer, Berlin, 1990). [20] F.W. Wise, M.J. Rosker and C.L. Tang, J. Chem. Phys. 86 (1987) 2827. [ 2 1 ] J. Chesnoy and A. Mokhtari, Phys. Rev. A 38 ( 1988) 3566; A. Mokhtari, A. Chebira and J. Chesnoy, J. Opt. Sot. Am. B7 (1990) 1551. [22] H.L. Fragnito, J.-Y. Bigot, P.C. Becker and C.V. Shank, Chem. Phys. Letters 160 (1989) 101; S.L. Dexheimer, Q. Wang, L.A. Peteanu, W.T. Pollard, R.A. Mathies and C.V. Shank, Chem. Phys. Letters 188 ( 1992) 61; R.W. Schoenlein, L.A. Peteanu, R.A. Mathies and C.V. Shank, Science 254 (1991) 412. [23] M.J. Rosker, T.S. Rose and A.H. Zewail, Chem. Phys. Letters 146 (1988) 175; P. Cong, A. Mokhtari and A.H. Zewail, Chem. Phys. Letters 172 (1990) 109. [24] M. Dantus, M.J. Rosker and A.H. Zewail, J. Chem. Phys. 89 (1988) 6128; R.M. Bowman, M. Dantus and A.H. Zewail, Chem. Phys. Letters 161 ( 1989) 297;
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M. Gruebele, I.R. Sims, E.D. Potter and A.H. Zewail, J. Chem. Phys. 95 ( 199 1) 7763. [ 251 R.E. Walkup, J.A. Misewich, J.H. Glownia and P.P. Sovokin, J. Phys. Chem. 94 ( 1991) 3389. 1261 W.T. Pollard and R.A. Mathies, Ann. Rev. Phys. Chem. 43 (1992), in press. [27] S. Mukamel, Ann. Rev. Phys. Chem. 41 (1990) 647. [28] S.H. Lin, B. Fain and N. Hamer, Advan. Chem. Phys. 79 (1990) 133. [ 291 W. Domcke and H. Koppel, Chem. Phys. Letters 140 ( 1987) 133; G. Stock and W. Domcke, Chem. Phys. 124 (1988) 227. [ 301 G. Stock, R. Schneider and W. Domcke, J. Chem. Phys. 90 (1989) 7184; G. Stock and W. Domcke, J. Opt. Sot. Am. B 7 (1990) 1970; G. Stock and W. Domcke, Phys. Rev. A 45 ( 1992) 3032.
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[31] W.T. Pollard, S.-Y. Lee and R.A. Mathies, J. Chem. Phys. 92 (1990) 4012; W.T. Pollard, H.L. Fragnito, J.-Y. Bigot, C.V. Shank and R.A. Mathies, Chem. Phys. Letters 168 (1990) 239. [32] L. Seider, G. Stock, A.L. Sobolewski and W. Domcke, J. Phys. Chem. 96 (1992) 5298. [33] R.G. Gordon, Advan. Magn. Res. 3 (1968) 1. [34] W.H. Miller, Advan. Chem. Phys. 25 (1974) 69. [ 35 ] G. Stock and W.H. Miller, to be published. [ 361 H.-D.Meyer and H. Kiippel, J. Chem. Phys. 8 1 ( 1984) 2605; H. Kiippel, W. Domcke and L.S. Cederbaum, Advan. Chem. Phys. 57 (1984) 59. [37] S.-Y. Lee, W.T. Pollard and R.A. Mathies, Chem. Phys. Letters 163 (1989) 11. [ 381 R. Schneider, W. Domcke and H. Kiippel, J. Chem. Phys. 92 (1990) 1045. [ 391 G. Stock and W. Domcke, J. Chem. Phys. 93 (1990) 5496. [40] H.-D. Meyer, personal communication.