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A semiclassical self-consistent-field approach to dissipative dynamics.II. Internal conversion processes

Gerhard StockInstitute of Physical and Theoretical Chemistry, Technical University of Munich,D-85748 Garching, Germany

~Received 6 March 1995; accepted 19 May 1995!

A semiclassical time-dependent self-consistent-field~TDSCF! formulation is developed for thedescription of internal conversion~IC! processes in polyatomic molecules. The total densityoperator is approximated by a semiclassical ansatz, which couples the electronic degrees of freedomto the nuclear degrees of freedom in a self-consistent manner, whereby the vibrational densityoperator is described in terms of Gaussian wave packets. The resulting TDSCF formulationrepresents a generalization of familiar classical-path theories, and is particularly useful to makecontact to quantum-mechanical formulations. To avoid problems associated with spurious phasefactors, we assume rapid randomization of the nuclear phases and a single vibrational densityoperator for all electronic states. Classically, the latter approximation corresponds to a singletrajectory propagating along a ‘‘mean path’’ instead of several state-specific trajectories, which maybecome a critical assumption for the description of IC processes. The validity and the limitations ofthe mean-path approximation are discussed in detail, including both theoretical as well as numericalstudies. It is shown that for constant diabatic coupling elementsVkk8 the mean-path approximationshould be appropriate in many cases, whereas in the case of coordinate-dependent couplingVkk8(x) the approximation is found to lead to an underestimation of the overall relaxation rate. Asa remedy for this inadequacy of the mean-path approximation, we employ dynamical corrections tothe off-diagonal elements of the electronic density operator, as has been suggested by Meyer andMiller @J. Chem. Phys.70, 3214~1979!#. We present detailed numerical studies, adopting~i! atwo-state three-mode model of theS12S2 conical intersection in pyrazine, and~ii! a three-statefive-mode and a five-state sixteen-mode model of theC→B→X IC process in the benzene cation.The comparison with exact basis-set calculations for the two smaller model systems and the possiblepredictions for larger systems demonstrate the capability of the semiclassical model for thedescription of ultrafast IC processes. ©1995 American Institute of Physics.

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I. INTRODUCTION

Novel experimental evidence as provided by femtosond spectroscopy has prompted considerable interest intheoretical investigation of photoinduced relaxation dynaics such as ultrafast internal conversion and electron andton transfer processes.1 There has been significant progrein the development of quantum-mechanical techniques inder to describe these dissipative phenomena, including pintegral methods,2–5 time-dependent self-consistent-fie~TDSCF! schemes,6–12 and the reduced density-matrapproach.13–18 Most of the quantum-mechanical methodhowever, are in practice restricted to the description of mosystems consisting of a one-dimensional ‘‘system’’ coornate which is coupled to a harmonic ‘‘bath.’’ Being interestin the study of more realistic models for dissipative dynaics ~including, e.g., multidimensional reaction coordinatand realistic solvent potentials!, classical and semiclassicaapproaches such as surface-hopping19–24 and classical-pathtechniques8,25–32are therefore still the methods of choice.

The surface-hopping approach is the most popuquantum-classical scheme to describe molecular dynamon coupled electronic potential-energy surfaces~PESs!. Theidea is to propagate classical trajectories on a single adiabPES until, according to some ‘‘hopping criterion,’’ the trajetory ‘‘hops’’ to another adiabatic surface. This way it is e

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sured that the system always propagates on asingle PES,which is an important requirement for the description of reactive scattering processes. Owing to the hopping proceshowever, the coherence of the wave function is destroyafter a relatively short time, which may represent a serioshortcoming for the description of bound-state relaxation dnamics involving several transitions between electronPESs. Classical-path methods, on the other hand, propagthe classical degrees of freedom self-consistently in anaver-agedpotential, the value of which is determined by the instantaneous populations of the different quantum states. Tway the coherence of the wave function is conserved, bone has to deal with problems arising from the concept ththe trajectory propagates on a ‘‘mean path,’’ which may leato unphysical results in applications to reactive scatterinwhere the system is asymptotically in a single electronstate.33

The present work has been motivated by the finding ththe simple classical-path concept describes surprisinglycurately the ultrafast electronic and vibrational relaxation dnamics of bound-state systems. In particular it has beshown to reproduce almost quantitatively complex structurof time-dependent observables such as electronic populaprobabilities and mean positions and momenta of vibrationmodes. Examples include electron-transfer dynamics of tdissipative two-level system~often referred to as spin-boson

5/103(8)/2888/15/$6.00 © 1995 American Institute of Physicst¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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2889Gerhard Stock: Semiclassical dissipative dynamics. II©

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model!,34 ultrafast electronic decay processes triggeredconically intersecting PESs,35–37and simple photoisomerization processes.38

In a preceding paper,34 henceforth referred to as paperwe have proposed a semiclassical TDSCF ansatz wcouples the electronic degrees of freedom to the nucleargrees of freedom in a self-consistent manner. The resulTDSCF formulation represents a generalization of existclassical-path theories,8,25–32 and is particularly useful tomake contact to quantum-mechanical formulations, thusploring the validity of the semiclassical ansatz. The methhas been applied to the spin-boson model with an Ohbath, which is one of the best-studied model problemsdissipative quantum dynamics.39,40 Employing the semiclassical TDSCF ansatz, we have derived the approximatileading to the classical-path equations of motion, thus reving the criteria for the validity and applicability of this concept. It has been shown that the assumption of comprandomization of nuclear phases represents the centraproximation of the approach. This condition is fulfilled, foexample, if the electronic relaxation dynamics is stroenough to ensure that the nuclear phases get randomizedidly enough to be unimportant for the overall time evolutiof the system. In particular, the randomization assumpmakes it possible to perform the trace over the nuclear vables through quasiclassical sampling of the nuclear inconditions, which has been found to be the main reasonthe success of the semiclassical approach in the descripof dissipative dynamics.

In this work we want to extend the formulation deveoped in paper I to the description of internal conversion~IC!processes in polyatomic molecules. Unlike the idealizproperties of the spin-boson Hamiltonian, the PESs perting to the individual electronic states of a more realistic mlecular model system may differ substantially. Thereforeassumption of a mean path, i.e., the propagation of a sitrajectory in an averaged potential instead of several stspecific trajectories, may become critical. As has been shin paper I, however, the assumption of a mean path is estial in order to perform the classical sampling of the badegrees of freedom. A major point of the present papetherefore to elucidate the limitations and the effects ofmean-path approximation for the description of IC process

To this end we adopt a semiclassical TDSCF formulatthat describes the vibrational wave function of the systemterms of Gaussian wave packets,6,8,10,41,42and compare theequations of motion for a ‘‘multiconfiguration ansatz’’~i.e.,with individual vibrational wave functions for each eletronic state!to the corresponding equations of motion resuing from a ‘‘single configuration ansatz’’~i.e., with a singlevibrational wave function for all electronic states!. Thanalysis elucidates the nature of the mean-path approxtion, and therefore allows to define criteria for its validity.is shown that the TDSCF formulation can be improvewhen dynamical corrections~‘‘Langer-like modifications’’!are employed to the electronic equations of motion asbeen suggested by Meyer and Miller.28 The effects of thesecorrections are discussed in some detail, and are showdepend on the form of the nonadiabatic coupling, i

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whether the coupling is a constant or depends on a nuclcoordinate.

The theoretical considerations are accompanied by dtailed numerical studies which are compared to exaquantum-mechanical basis-set propagations. As model stems we consider~i! a two-state three-mode model of theS12S2 conical intersection in pyrazine,43 which has beenadopted by several authors as a standard example of etronic relaxation,36,44–46and~ii! a three-state five-mode and afive-state sixteen-mode model of theC→B→X IC processin the benzene cation.37,47–49We introduce time-dependentobservables that characterize the electronic and vibratiorelaxation dynamics of molecular systems, including, in paticular, electronic state-specific observables, which represa stringent test of the mean-path approximation. We giconditions to determine the value of the dynamical corretions, and discuss criteria which indicate the reliability of thmethod. The comparison with reference quantum calcutions for smaller systems~with up to five vibrational modes!and the possible predictions for larger systems clearly deonstrate that the semiclassical model is a powerful methfor the description of ultrafast IC processes.

II. THEORETICAL FORMULATION

A. Model system

In a diabatic electronic representation,50 the Hamiltonianfor a molecular system comprisingNel coupled electronicstates can be written as

H5 (k51

Nel

uwk&hk^wku1 (kÞk851

Nel

uwk&Vkk8^wk8u, ~2.1a!

wherehk5T1Vk denotes the vibrational Hamiltonian of thediabatic electronic basis stateuwk& andVkk8 represents thediabatic coupling between the electronic statesuwk& anduwk8&. Most of the theoretical analysis in this section is independent of a specific form of the diabatic potential elmentsVk , Vkk8 . For later reference and in order to introducsome notation, however, it is convenient to specify a simpmodel for the Hamiltonian~2.1!. As is common practice invibronic coupling theory,51,52we thus approximate the diaba-tic potential matrix elementsVk and the diabatic couplingsVkk8 by a Taylor expansion in~dimensionless!normal coor-dinatesxj , yielding in lowest order

hk5Ek1h01 (j51

Nmod

k j~k!xj , ~2.1b!

h05 (j51

Nmod12v j~pj

21xj2!. ~2.1c!

Hereh0 represents the harmonic oscillator Hamiltonian comprising Nmod nuclear degrees of freedom,v j , xj , and pjdenoting the vibrational frequency, the position, and the mmentum of thej th oscillator.Ek is the vertical transitionenergy of the diabatic stateuwk&, andk j

(k) denotes the gradi-ent of the excitation energy with respect to the totally symmetric coordinatesxj .

, No. 8, 22 August 1995t¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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In molecular physics, two cases of the diabatic coupliVkk8 are of particular interest:~i! the coupling is constant

Vkk85gkk8 , ~2.2a!

leading to anavoided crossing53 of the adiabatic PESs, and~ii! the coupling is linear in one~or several!coordinatesxc~the so-called coupling modes!

Vkk85lkk8xc , ~2.2b!

leading to aconical intersection53 of the adiabatic PESs. FoNel52, the first case is often referred to as ‘‘spin-boson prolem,’’ as the Hamiltonian~2.1! with ~2.2a! is equivalent tothe Hamiltonian of the dissipative two-level system.39 Thelatter case, which we will refer to as ‘‘vibronic-couplingproblem,’’ represents the standard case of IC through a ptochemical funnel.54 Interestingly, it has been found that thdifferent topology of the PESs in the two cases leadsqualitatively different electronic and vibrational relaxatiodynamics.55 Furthermore it should be stressed that thebronic couplingVkk8 converts an otherwise separable hamonic system into a nonseparable multidimensional prolem.

The time evolution of the wave functionuC(t)& underthe model Hamiltonian~2.1! is described by the time-dependent Schro¨dinger equation

i]

]tuC~ t !&5HuC~ t !&. ~2.3!

We assume that at the timet50 the system is exclusively inthe upper electronic state~e.g.,uw2&) and in the ground stateof the vibrational Hamiltonianh0 . This initial condition cor-responds to a photoexcitation of the electron from an initelectronic state~e.g.,uw0&) to the upper (uw2&) state.

B. TDSCF ansatz and equations of motion

As motivated in the Introduction, we wish to compathe equations of motion for a semiclassical TDSCF answith individual vibrational wave functions for each electronic state @henceforth referred to as multiconfiguratio~MC! ansatz#to the equations of motion resulting fromTDSCF ansatz with a single vibrational wave functio@henceforth referred to as single-configuration~SC! ansatz#.In both cases, the derivation of the equations of motion hbeen performed by several workers,8,10,41,42and is thereforeonly sketched briefly. For the sake of notational convenienwe will in the following refer to a single nuclear coordinatex~instead of( j xj ). Furthermore we want to specialize oncoupled electronic two-state system; problems associawith the general case ofNel.2 are considered in Sec. III C

The MC ansatz for the wave functionuC(t)& can bewritten as8,10

uC~ t !&5 (k51,2

xk~ t !uwk&uFk~ t !&, ~2.4!

where thexk(t) are complex numbers, describing the dnamics of the electronic degrees of freedom, anduFk(t)&denotes the vibrational wave function, accounting for the dnamics of the nuclear degrees of freedom in the diaba


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electronic stateuwk&. To make the ansatz unambiguous, weimpose that the vibrational wave functionsuFk(t)& are nor-malized, i.e.,

^Fk~ t !uFk~ t !&51, ~2.5a!

and that

^Fk~ t !uFk~ t !&50. ~2.5b!

Inserting the MC ansatz~2.4! into the time-dependent Schro¨-dinger equation~2.3!, we obtain

i xk5^FkuhkuFk&xk1^FkuVkk8uFk8&xk8, ~2.6!

i uFk&5~hk2 i xk /xk8!uFk&1xk8 /xkVkk8uFk8&, ~2.7!

wherek51,2 andk Þ k8. The vibrational wave functionsuFk(t)& are assumed to be Gaussian wave packets56

Fk~x,t !5p21/4exp$2 12@x2xk~ t !#

2

1 ipk~ t !@x2xk~ t !#1 igk~ t !%, ~2.8!

which are completely described by the trajectoryxk(t), thecorresponding momentumpk(t), and the action integralgk(t). Employing the Dirac–Frenkel–McLachlan time-dependent variational principle57

^dCu i ]/]t2HuC&50, ~2.9!

it is straightforward to show that the equations of motion forthe trajectoryxk(t) and the momentumpk(t) are given by

8,10

xk5vpk12Im xk8 /xk^Fku~x2xk!Vkk8uFk8&, ~2.10!

pk52]Vk~xk!

]xk22Re xk8 /xk^Fku~x2xk!Vkk8uFk8&.

~2.11!

Analogously, the SC ansatz for the wave functionuC(t)& iswritten as8,10

uC~ t !&5 (k51,2

eihkxk~ t !uwk&uF~ t !&. ~2.12!

Contrary to the MC ansatz~2.4!, the vibrational dynamics isdescribed by a single vibrational wave functionuF(t)&,which, in analogy to~2.8!, depends on the trajectoryx(t) andmomentump(t). For convenience, we have furthermore in-troduced a phase functionhk ,

9,10 which is not of furtherinterest for the evaluation of the observables introduceabove. Insertion of the SC ansatz into the time-dependenSchrodinger equation~2.3!yields the equations of motion forthe electronic and vibrational wave function

i xk5^FuhkuF&xk1^FuVkk8uF&xk8, ~2.13!

i uF&5~n1h11n2h212Re x1* x2V12!uF&. ~2.14!

It is seen that within the TDSCF approximation the elec-tronic equations of motion reduce to a coupled two-levesystem with explicitly time-dependent coefficients. Thenuclear dynamics is described by the single vibrational wavfunction uF(t)& propagating in an averaged potential,weighted by the electronic population probabilities

nk~ t !5uxk~ t !u2. ~2.15!


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Employing again the variational principle~2.9!, the nuclearequations of motion pertaining to the SC ansatz are derias

x5vp, ~2.16!

p52^C~ t !u]H

]xuC~ t !&,

52 (k51,2

nk]Vk~x!

]x22Re xk* xk8

]Vkk8~x!

]x. ~2.17!

Equations ~2.16! and ~2.17! are recognized as the welknown equations of motion of a classical-path fomulation.8,25–32Contrary to the MC ansatz leading to a fomulation with state-specific trajectoriesxk(t), the SC ansatzresults in a formulation with a mean trajectoryx(t) driven bythe ‘‘Ehrenfest force’’mp(t).

C. Mean-path approximation

In this section we want to discuss the mean path appromation, that is, the underlying assumption leading from tMC ansatz~2.4! to the SC ansatz~2.12!. Intuitively, it isclear that in the limit

Dx[x2~ t !2x1~ t !→0, ~2.18a!

Dp[p2~ t !2p1~ t !→0, ~2.18b!

the MC ansatz becomes equivalent to the SC ansatz. Letry to assess the validity of condition~2.18!. Due to the ini-tial preparation of the system, Eq.~2.18! is exact at the timet50, and should therefore be a good approximation for shtimes, too. In a dissipative system themean valuesof themomenta fluctuate around zero for long times.44 Condition~2.18b! is therefore also valid for long times and should aproximately hold for intermediate times as long as the vibtional relaxation is fast enough. The validity of Eq.~2.18b!for longer times is not easy to foresee as it depends on hdifferent the geometries of the two electronic statesuw1& anduw2& are. If the overall coupling of the system is distributeover many weakly coupled vibrational modes~and thereforethe geometry change in each individual mode is small!, theapproximation should be justified. If the overall couplingthe system is contained in a few strongly coupled vibrationmodes, however, the approximation may break downlonger times.

To consider the nuclear equations of motion, it is useto recall the relation between the mean path variablesx(t)andp(t) and the state-specific variablesxk(t) andpk(t)

x5n1x11n2x2 , ~2.19a!

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p5n1p11n2p2 , ~2.19b!

where the time dependency of the variables has beedropped for notational convenience. Taking the time derivative of ~2.19!and inserting the MC equations of motions forxk(t) ~2.11! andpk(t) ~2.11!, we obtain the MC equations ofmotion for the mean positionx(t) and the mean momentump(t)

x5vp, ~2.20!

p52 (k51,2

nk]Vk~xk!

]xk, V125g, ~2.21!

52 (k51,2

nk]Vk~xk!

]xk22Re lx1* x2^F1uF2&

1~p22p1!@ n22Re lx1* x2^F1uF2&~p22p1!#,

V125lx. ~2.22!

It is interesting to note that for constant diabatic coupling (V125g) the MC equations of motion forx(t) andp(t) are identical to the SC equations. In the case ocoordinate-dependent coupling (V125lx), the equation forthe mean positionx(t) is still identical, whereas the equationfor the mean momentump(t) differs considerably from~2.17!. In this case we obtain a correction term to the SCequation that vanishes for smallDp, and the off-diagonalterm is augmented by the vibrational overlap integral

^F2uF1&5exp$2 14~x22x1!

22 14~p22p1!

2

1 i /2~p21p1!~x12x2!1 i ~g22g1!%. ~2.23!

It is seen that the SC ansatz neglects the nuclear phase dference acquired by the trajectories. Furthermore the strengof the diabatic coupling is somewhat overestimated. In thlimit ~2.18!, the MC equations, of course, reduce to the Sequations.

Let us next consider the electronic equations of motio~2.6!and~2.13!. These equations describe a driven two-levesystem, that can be characterized by the two time-dependeobservables

DE~ t !5^C~ t !$uw2&h2^w2u2uw1&h1^w1u%C~ t !&, ~2.24!

Vc~ t !52Re^C~ t !uw1&V12 w2uC~ t !&, ~2.25!

whereDE(t) represents the mean value of the diabatic electronic energy gap andVc(t) denotes the mean diabatic cou-pling. Inserting the MC and SC ansatz, respectively, and asumingV125g1lx we obtain

DE~ t !5n2E22n1E11H 12v@n2~p2

21x22!2n1~p1

21x12!#1n2k2x22n1k1x1 ~MC!,

12v@~n22n1!~p

21x2!#1~n2k22n1k1!x ~SC!,~2.26!

Vc~ t !52Rex1* x2H ~g1l/2~x11x21 i ~p22p1!!^F1uF2& ~MC!,

g1lx ~SC!.~2.27!

No. 8, 22 August 1995t¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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Note that for ‘‘symmetric’’ coordinate shifts~k252k1!as assumed in the spin-boson model, the tuning of the etronic energy gapDE(t) owing to the linearly coupledmodes is described correctly by the SC ansatz. BecauseSC ansatz uses the averaged quantitiesx(t) andp(t) for theevaluation of the 1

2v@ ...# term, however, it is expected tunderestimate the magnitude of the electronic energy ga

The accuracy of the SC approximation for the meanabatic couplingVc(t) is seen to depend strongly on the forof the coupling V12. In the case of constant couplinV125g, Vc(t) misses the overlap integral^F1uF2&, whichmainly corresponds to the neglect of nuclear phases. Incase of coordinate dependent couplingV125lx, the situationis more complicated. Besides the neglect of the overlaptegral and a term that vanishes withDp, the SC ansatz replacesl/2(x11x2) by l(n1x11n2x2) , which represents asomewhat problematic approximation. This is because,to the weight by the electronic populationsnk , the bilinearexpressionl(n1x11n2x2) amounts to a dynamically rathedifferent coupling thanl/2(x11x2). For this reason and because of the missingDp term, one may expect that the Sansatz underestimates the mean diabatic coupling.

To summarize, it has been shown that the mean-papproximation should be an appropriate assumption forspin-boson problem~i.e., for V125g andk252k1), whichis in accordance with the finding of paper I. In this caseequations of motion for the nuclear degrees of freedomidentical in the SC and MC formulations, and the tuningthe electronic energy gapDE(t) due to linearly coupledmodes is described correctly by the SC ansatz. The negof nuclear phases inVc(t) and the differences in the12v@...#term inDE(t) should not be a critical approximation as lonasDx,Dp do not become too large. In the case of coordindependent couplingV125lx, however, the SC approximation of the mean diabatic coupling was found to becoproblematic.

D. Dynamical corrections

In order to improve the SC ansatz in the casecoordinate-dependent diabatic coupling, we introduce ‘‘dnamical corrections’’ into the SC equations of motion, as hbeen suggested by Meyer and Miller in the theoretical framwork of the classical electron analog model.28 The basic ideaof the classical electron analog model is to replace theNel

complex electronic variablesxk by 2Nel real classical action-angle variables

xk~ t !5Ank~ t !e2 iqk~ t !. ~2.28!

Here the action variablesnk are identical to the electronipopulations defined in~2.15!, and the angle variablesqk rep-resent the electronic phases. Introducing the classical Hatonian function

H~p,x,n,q!5(knkhk~p,x!1 (

kÞk8Anknk8 e

i ~qk2qk8!Vkk8~x!,

~2.29!

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the equations of motion for the electronic and nuclear dgrees of freedom~2.13! and ~2.16!, ~2.17! can be rewrittenas27,28

nk~ t !52]H

]qk, qk~ t !5

]H

]nk, ~k51,...,Nel!, ~2.30!

p j~ t !52]H

]xj, x j~ t !5

]H

]pj, ~ j51,...,Nmod!, ~2.31!

i.e., both electronic and nuclear degrees of freedom formaobey Hamilton’s equations.

It should be noted that the classical electron analmodel is not the only way to obtain canonical equationsmotion for both electronic and nuclear degrees of freedoDefining xk(t)5ak(t)1 ibk(t) and introducing a similarHamiltonian function as~2.29!, it has been shown by Strocchi that the time evolution of the ‘‘Cartesian electronic varables’’ak ,bk is also given by Hamilton’s equations.58

The formulation of the equations of motion in terms oelectronic action-angle variables, however, allows to intrduce ‘‘Langer-like modifications’’ into the coupling matrixelements of the Hamiltonian function~2.29!, i.e., one makesthe replacement28

Anknk8 Vkk8→A~nk1g!~nk81g! Vkk8 , ~2.32!

whereg is a constant. To understand the effects of the corection termg it is instructive to again consider an electronitwo-state system. Introducing furthermore the canonictransformation59

n5n2 , q5q22q1 ,

N5n11n251, Q5q1 , ~2.33!

the electronic equations of motion read

n~ t !52Im ^F2uV21uF1&A~11g2n!~n1g! eiq,~2.34!

q~ t !5^F2uV2uF2&2^F1uV1uF1&

1Re ^F2uV21uF1&122n

A~11g2n!~n1g!eiq.

~2.35!

Equations~2.34! and ~2.35! are for g50 equivalent tothe MC expression~2.6!; the corresponding SC expressio~2.13! is easily obtained by settinguFk&5uF&. Note that thedifferential equation~2.35! has singularities atn52g andn511g, reflecting the fact that these values ofn are branchpoints for the angle variableq. As has been discussed indetail by Herman and co-workers,32,60 these singularitiesmay result in abrupt phase changes of the semiclassical wfunction, and need therefore be taken care of, if one intento pursue a ‘‘rigorous’’ semiclassical description~i.e., accord-ing to classical S-matrix theory61!. In the present work, how-ever, we restrict ourselves to a more approximate treatmei.e., we perform a quasiclassical average over initial condtions and neglect all interferences between individual trajetories @see Eq.~2.39!below#. In this case, possible discontinuities of the electronic phaseq do not affect the evaluatedquantities.

, No. 8, 22 August 1995ct¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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From ~2.34! it is clear thatn→0 whenevern approaches2g and 11g, i.e., the electronic action variablen(t) isrestricted to the interval2g,n,11g. Note, however, thatn(t) corresponds to a populationprobability, which shouldbe restricted to 0,n,1. The g-modification therefore re-sults in the curious fact that an electronic state can be ‘‘ovpopulated’’~or ‘‘under-populated’’! by the value ofg. Gen-eralizing to a model withNel electronic states, a analogouanalysis reveals that the action variablesnk(t) ~k51,...,Nel!can take on values from2g to 11g(Nel21!. Forg Þ 0 theevaluation of physically meaningful observables thusquires a mapping of the continuous variablen(t) onto dis-crete quantum numbers~e.g., through a histogram methosee Sec. III A!, which ensures that the mean electronic poplation probability is restricted to@0,1#.

Let us consider the consequences of theg corrections onthe electronic dynamics. Firstly note that the mean diabcouplingVc(t) is increased due to the correction. As a rouestimate of this effect one may consider the quantity

Veff52/pE2g

11g

dnA~11g2n!~n1g!5~1/41g1g2!,

~2.36!

the value of which, for example, is doubled forg'0.21. Thesecond main consequence of the modifications is an incrof the magnitude of the diabatic electronic energy gapDE(t)defined in Eq. ~2.24!. Writing DE(t)5E11n(E1 1E2!1~k22k1!x, it is clear that an enlarged range of possibvalues ofn also results in a enlarged range ofDE(t). It hasbeen shown above that in the case of coordinate-depencoupling V12 the SC ansatz underestimates both the etronic energy gapDE(t) and the mean diabatic couplinVc(t). Theg modifications therefore may correct this inaequacy of the SC ansatz to a certain extent. In the casconstant couplingV12, however, the modifications woullead to an overestimation of the mean diabatic couplVc(t), and should therefore be omitted.

So far we have not addressed the question aboutvalue ofg. Because of2g,n,11g, g should be positive,otherwise it would not be possible that the system is copletely in a single electronic state as it requires tn50,1. In accordance with the well known modificatiodue to Langer,62 Meyer and Miller have suggested a valueg5 1

2.28 In order to obtain the correct semiclassical eigenv

ues of a one-dimensional system that is restricted tosemi-infinite interval @0,`#, Langer has shown that onneeds to add a ‘‘correction term’’\2(8mr2)21 to the poten-tial, which results in the replacementn→n1 1

2 in the expres-sion for the WKB eigenvalues. In the same spirit, Adams aMiller have derived an expression for semiclassical eigenues for potential functions defined on a arbitrary fininterval.63

The situation is more involved in the case of the semclassical TDSCF formulation, where the electronic equatiof motion ~2.34! and ~2.35! describe a nonlinearly drivenoscillator. Similarly to Langer’s case, the correspondingsition xel and momentumpel are restricted to a finite intervathrough the relation2g,n5xel

21pel2,11g. Note, how-

ever, that in the case of constanthk ,Vkk8 ~i.e., no nuclear


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motion!, the electronic equations of motion areexact forg50, i.e., they are equivalent to the time-dependent Sch¨-dinger equation for a two-level system.28 The need for dy-namical corrections therefore does not arise from the fininterval the variablesxel ,pel are restricted to, but is rather aconsequence of the self-consistent-field approximation ofinteraction of electronic and nuclear degrees of freedom. Ufortunately, we have not succeeded in finding a relatiwhich would allow to calculateg for a given system. Exten-sive numerical studies on a variety of model systems~seebelow! suggest a value ofg51

4 in the case of coordinate-dependent couplingV12. Note that this value corresponds tan increase ofVeff by a factor of 2.5, compared to a factor o4.0 for g50.5. Furthermore, it is shown in Sec. III A that iis possible to define a consistency criterion@Eq. ~3.17!#,which determines an upper bound forg. As pointed outabove, in the case of constant couplingV12 the modificationslead to an overestimation of the mean diabatic coupliVc(t), and should therefore be omitted.

E. Quasiclassical sampling procedure

It has been shown by several authors that the semicsical approximation of a wave function can be considerabimproved if, instead of a single Gaussian wave-packet,expansion into many Gaussians is employed, the expancoefficients of which are determined by evaluating the semclassical Greens function.64–67These ‘‘rigorous’’ semiclassi-cal methods, however, are known to become quite cumbsome for treating problems with many degrees of freedoIn paper I we have therefore proposed a simpler and mpractical approach, which combines a semiclassicalTDSCF ansatz with a quasiclassical sampling scheme.34 Ithas been shown that the combination of these two techniqrequires~i! to employ the mean-path approximation, and~ii!to assume rapid randomization of nuclear phases.34 The lattercondition allows to perform the trace over nuclear variablin terms of an incoherent sampling of nuclear initial condtions, which has been found to represent a much betterproximation to describe quantum dissipative dynamics ththe propagation of single or double trajectories as requifor the SC~2.12! or MC ~2.4! ansatz, respectively. In whafollows, we briefly summarize the semiclassical TDSCF aproach introduced in paper I and generalized it to incorporthe dynamical corrections discussed above.

To derive the semiclassical TDSCF approximation of pper I, it is convenient to change from the wave-function dscription adopted above to the more general density-madescription. The generalization of the SC ansatz~2.12! to anexpansion into Gaussian wave-packets can then be wrias34

r~ t !5E dx0E dp0E dx08E dp08 cx0 ,p0~ t !cx08 ,p08* ~ t !

3sx0 ,p0 ,x08 ,p08~ t ! ufx0 ,p0

~ t !&^fx08 ,p08

~ t !u, ~2.37!

where

sx0 ,p0 ,x08 ,p08~ t !5(

k,k8xx0 ,p0

~k! ~ t !xx08 ,p08

~k8!*~ t !uwk&^wk8u ~2.38!

, No. 8, 22 August 1995ct¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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represents the electronic density operator depending onnuclear initial conditionsx05x(t50), p05p(t50). Thefunctionscx0 ,p0(t)5Awx0 ,p0

Ax0 ,p0(t) are the semiclassica

expansion coefficients, wherewx0 ,p0denotes the Wigner

distribution,68 describing the initial distribution of thenuclear variablesx0 , p0 , andAx0 ,p0

(t) represents the semiclassical weight function derived by Herman and Kluk.65

Assuming that the nuclear phases become randomirapidly, we may neglect the nondiagonal contributions~2.37!, thus yielding

r~ t !5E dx0E dp0 wx0 ,p0sx0 ,p0

~ t ! ufx0 ,p0~ t !&^fx0 ,p0

~ t !u,

~2.39!

sx0 ,p0~ t !5(

k,k8xx0 ,p0

~k! ~ t !xx0 ,p0~k8!* ~ t !uwk&^wk8u. ~2.40!

Owing to the approximation~2.39!, each individual contri-butionsx0 ,p0

(t) andufx0 ,p0(t)&^fx0 ,p0

(t)u evolves indepen-dently from all other contributions~i.e., with differentx0 ,p0). The time evolution of the total density operator(t) is therefore described by the incoherent additionmany trajectories, each one obeying the SC equations oftion for the electronic~2.13! and nuclear~2.14! degrees offreedom, respectively. Furthermore, it is easy to show tAx0 ,p0

(t)51 for potentials which are at most quadratic inx.For the model system~2.1! under consideration, the individual trajectories in~2.39! are thus solely weighted by theWigner distributionwx0 ,p0

.To introduce the dynamical corrections described abo

it is advantageous to change to electronic action-angle vables defined in~2.28!. The structure of the modification~2.32! then suggest to rewrite the electronic density opera~2.40!as

sx0 ,p0~ t !5(

knkuck&^cku1 (

kÞk8A~nk1g!~nk81g!

3exp i ~qk2qk8!uck&^cku, ~2.41!

where for notational convenience the argumentst,x0 ,p0have been suppressed on the right hand side of~2.41!. Theresulting TDSCF ansatz for the total density operator@Eqs.~2.39! and ~2.41!# combined with the corresponding equations of motion for the nuclear and electronic degreesfreedom@Eqs.~2.16!, ~2.17!and ~2.30!# represent the work-ing equations of the semiclassical TDSCF approach, andbe used for the computational studies in Sec. III.

To illustrate the formal considerations above by a cocrete example, let us consider the evaluation of timdependent mean values. The time-dependent mean valuan observableA is given by

A~ t !5Tr$Ar~ t !%,

5Tre Trv$Ar~ t !%, ~2.42!

where

Tre$...%5(k

^wku...uwk&, ~2.43!


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Trv$...%5E dx ^xu...ux& ~2.44!

denote the trace over the electronic and nuclear degreesfreedom, respectively. Thinking of the electronic and nucleavariables as ‘‘system’’ and ‘‘bath’’ degrees of freedom, re-spectively, we can define a reduced density operators(t)

s~ t ![Trv$r~ t !%,

5E dx0E dp0 wx0 ,p0sx0 ,p0

~ t !, ~2.45!

which describes the dynamics of the electronic ‘‘systemthat is interacting with the nuclear ‘‘bath.’’ Equation~2.45!clearly states the correspondence between the trace over bvariables (Trv) and the integration over initial conditions.

III. COMPUTATIONAL RESULTS

In this section we undertake detailed numerical studieto ~i! investigate the applicability and limitations of themean-path approximation,~ii! discuss the effects of the dy-namical corrections introduced above, and~iii! demonstratethe capability of the semiclassical TDSCF approach for thdescription of ultrafast IC processes. To this end, we introduce time-dependent observables that characterize the eltronic and vibrational relaxation dynamics of molecular systems, and consider as representative examples theprocesses in pyrazine and the benzene cation.

A. Time-dependent observables

To characterize the IC dynamics of a molecular systemit is useful to define time-dependent electronic and vibrational observables which illuminate this relaxation processIn the discussion of radiationless processes~as well as of anyreactive process!the main quantity of interest is given by therate of the reaction. Employing time-dependent wave-packpropagations on coupled electronic PESs, one directly evalates the time-dependent population probabilitiesPk(t) of thekth electronic state. In the case thatPk(t) may be fitted insome sensible way to an exponential function, the rate of thIC process is then given byPk /Pk .

The population of an electronic state can be defined ithe diabatic or the adiabatic representation. In the casesurface crossings, as considered here, these two populatioare not the same.45 Transitions betweendiabatic electronicstates are important for the interpretation of spectroscopdata,47,52,69,70as in the vicinity of a conical intersection theelectronic dipole transition operator is only smooth in thediabatic representation.50 Regarding IC as a chemical reac-tion, however, one is rather interested in the population probabilities of theadiabaticelectronic states.

The time-dependent population probability of the diabatic electronic stateuwk& is defined as the expectation value ofthe diabatic projectorPk

Pk~ t !5Tr$Pkr~ t !%, ~3.1!

Pk5uwk&^wku. ~3.2!

Within the TDSCF formulation~2.39!, the diabatic popula-tion probabilitiesPk(t) can be written as


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Pk~ t !5(rwrnk

~r !~ t !, ~3.3!

where the Monte-Carlo integration over the nuclear initconditionsx0 ,p0 @cf. Eq. ~2.45!# has been expressed bysum over the collective indexr5$x0 ,p0%. Employing thegmodifications introduced above, however, the classical actvariablesnk

(r )(t) no longer directly correspond to the population probability of stateuwk&, asnk

(r )(t) can take on con-tinuous values from2g to 11g(Nel21!. To make corre-spondence to discrete electronic states~i.e., nk50 or 1!, weinvoke a histogram method71 of assigning the system to thediabatic electronic state with the largestnk(t), i.e.

Pk~ t !5(rwrx„nk

~r !~ t !…, ~3.4!

x~nk!5H 1 if nk.nk80 otherwise .

~3.5!

If the electronic state under consideration is degenerate~e.g.,in case of theX state of the benzene cation!, the total popu-lation of the X electronic state is evaluated, i.e., the suof the corresponding action variables is histogramm{ x@nXa

(t) 1 nXb(t)]}.

In a diabatic representation, the projector onto thekth

adiabatic electronic state is defined by

Pkad5S~x!P

kS†~x!, ~3.6!

where

S~x!5(k,k8

uwk&Skk8~x!^wk8u ~3.7!

denotes the unitary transformation between the diabaticadiabatic wave functions.52,45 Quantum mechanically, theevaluation of the adiabatic population probabilities via E~3.6! is a rather time-consuming task,45 because it requiresmanipulations in the full electronic-vibrational producspace. The semiclassical evaluation of the adiabatic protions ~3.6!, on the other hand, requires only manipulationsNel3Nel matrices, as the transformation matrixS(x) isreadily obtained by diagonalizing the classical diabatic ptential matrix. The population probability of thekth adiabaticstate is given by

Pkad~ t !5Tr$Pk

adr~ t !%,

5(rwrnadk

~r !~ t !, ~3.8!

where

nadk~ t !5(inipii

~k!

1(iÞ j

A~ni1g!~nj1g! ei ~qi2qj !pi j~k! , ~3.9!

and pi j(k)(x)5^w i uPk

aduw j& denote the electronic matrix elements of the adiabatic projectorPk

ad. If we allow for g.0,


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.

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-

we again need to map the electronic action variable nadk(t)onto discrete electronic states, i.e., Eq.~3.8! is augmented bythe histogram functionx.

In the discussion of the mean-path approximation whave introduced the mean value of the diabatic electroenergy gapDE(t) ~2.24! and the mean diabatic couplingVc(t) ~2.25!. A closely related quantity is given by the meavibrational energyHk(t) contained in the diabatic stateuwk&, which is given by72

Hk~ t !5Tr$Hkr~ t !%,

Hk5Pk~hk2Ek1ek!, ~3.10!

where ek512( jk j

(k)2/v j denotes the stabilization energy othe oscillator potential in the diabatic stateuwk&. Assuminge15e2 , E15E2 , the diabatic electronic energy gap~2.24! isgiven by DE(t)5H2(t)2H1(t). Similarly, the time-dependent energy content of the diabatic coupling is definas

Vc~ t !5Tr$Vcr~ t !%,

Vc5 (kÞk8

uwk&Vkk8^wk8u. ~3.11!

According to Eq. ~2.42!, the semiclassical evaluation oDE(t) andVc(t) yields

Hk~ t !5(rwrnk

~r !^f r uhk2Ek1ekuf r&, ~3.12!

Vc~ t !5(rwr (

kÞk8A~nk1g!~nk81g! ei ~qk2qk8!

3^f r uVkk8uf r&. ~3.13!

The electronic populationsPk(t) andPk(ad)(t) and the state-

specific energiesHk(t) andVc(t) are examples for electronicobservables, i.e., the are obtained by averaging over allbrational degrees of freedom. The simplest observablesflecting the vibrational dynamics of the system are the meposition and mean momentum of a specific vibrational moxj

xj~ t !5Tr$xjr~ t !%5(rwrxj

~r !~ t !, ~3.14!

pj~ t !5Tr$pjr~ t !%5(rwrpj

~r !~ t !. ~3.15!

The vibrational relaxation dynamics can also be analyzedterms of the time-dependent energy content of a individunuclear degrees of freedom44,72

Ej~ t !5Tr$Ejr~ t !%,

5(rwr(

knk

~r !v j

2$pj

~r !2~ t !1~xj~r !~ t !1k j

~k!/v j !2%.

~3.16!

The energy content of coupling modes is obtained by repling thek j

(k) in ~3.16! by lkk8,73 wherelkk8 is the vibronic

coupling constant defined in~2.2b!.


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Note that the definitions of time-dependent energieschosen such that the conservation of energy can be writte

Esum~ t ![(kHk~ t !1Vc~ t !5(

jEj~ t !, ~3.17a!

5E2(k

Pk~ t !~Ek2ek![Etot~ t !, ~3.17b!

whereE5E21ezp is the total energy of the system, consising of the vertical excitation energy of the initially exciteelectronic state~say,uw2&) and the vibrational zero-point energy ezp. Esum(t) represents the time-dependent total vibtional energy acquired by the system. It can be expreseither as sum over the state-specific energiesHk(t) andVc(t), or as sum over the mode-specific energiesEj (t).Equation ~3.17! describes the transformation of electronenergyEtot(t), that has been put into the system via phoexcitation, into vibrational energyEsum(t), i.e., the electronicrelaxation in general results in an increase of vibrationalergy. In an exact calculation as well as in a standclassical-path simulation, we therefore clearly haEsum(t)5Etot(t).

Employing theg modifications, however, Eq.~3.17!canno longer be taken for granted. It has been pointed out abthat the diabatic population probabilityPk(t) should beevaluated using a histogram technique~3.4!. It is not mean-ingful, however, to histogram the action variablenk(t) in Eq.~3.10!as well, becauseHk(t) represents the total vibrationaenergy of the diabatic stateuwk&, which is correctly given byEq. ~3.10!. As a consequence, the total vibrational eneEsum(t) may not longer be equal toEtot(t). This is to saythat, owing to theg modifications, there is more vibrationaenergyEsum(t) in the system as is provided throughEtot(t)by the electronic relaxation process! Asnk(t) can take oncontinuous values from2g to 11g(Nel21!, this is espe-cially a problem for models including many electronic statas we will find in the case of the benzene cation wNel55. To avoid this dynamical inconsistency, one haschooseg small enough, that Eq.~3.17! is fulfilled, i.e., con-dition ~3.17! determines an upper bound for theg correc-tions.

B. The S12S2 conical intersection in pyrazine

As a first example we consider a simple two-state thrmode model of theS1(np* ) and S2(pp* ) states ofpyrazine,43 which has been adopted by several authors astandard example of electronic relaxation.36,44–46Taking intoaccount a single coupling mode (n10a) and two totally sym-metric modes (n1 ,n6a), Domcke and co-workers have identified a low-lying conical intersection of the two lowest ecited singlet states of pyrazine, which has been showntrigger IC and a dephasing of the vibrational motion onfemtosecond time scale.43,44A variety of theoretical investi-gations, including quantum wave-packet studies44,45 and ab-sorption and resonance Raman74 as well as time-resolvedpump-probe spectra70 have been reported for this system.the present work we adopt this model to study the appl


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bility and limitations of the mean-path approximation and tdiscuss the dynamical effects ofg modifications introducedabove.

To ensure convergence, we have run 2500 trajectoriesall simulations, although less than a hundred trajectories asufficient to recognize the essential structures. The initiconditions for the nuclear variablesxj (0),pj (0) can either besampled from the Wigner distribution,68 or be obtained viaclassical action-angle variables, which was found to give amost identical results.34We have chosen the electronic initialconditions asx r

(1)(0)51 andx2(2)(0)50, which corresponds

to the electronic action-angle variables~2.33! n(0)51, qarbitrary. Interestingly, it did not make a difference whethewe setq50 for each trajectory, or whether we performed aMonte-Carlo integration over the electronic angle variableqas suggested by Meyer and Miller.28 In the case ofg50, onefurthermore needs to assumen(0)512h, where h is asmall number~say, 1026), to ensure that one does not start ina singularity of the differential equation~2.30!. The quantumpropagation of the wave functionuC(t)& has been performedby expandinguC(t)& in a direct-product basis constructedfrom diabatic electronic states and harmonic-oscillator stateand solving the resulting system of coupled first-order diffeential equations using a Runge-Kutta-Merson scheme wadaptive step size.44,70

The comparison of the semiclassical simulations~fulllines! and exact quantum calculations~dotted lines!is com-prised in Fig. 1, which shows the diabatic electronic population probabilityP2(t) @panels~a!#, the mean positionx(t) ofthe vibrational moden6a @panels~b!#, and the state-specificenergiesHk(t) andVc(t) @panels~c!#. Each semiclassicallyevaluated observable is plotted for three different valuesthe dynamical correctiong, namely forg50 @panels~1!#,g51

4 @panels~2!#, andg512 @panels~3!#.

To get an impression of the IC dynamics of the modelet us first look at the quantum results~dotted lines!of theelectronic population probabilityP2(t) of the initially pre-paredS2 state. The electronic population exhibits a slightlyoscillatory initial decay on a time scale of about 50 fs, whicis followed by pronounced quasiperiodical recurrences of thpopulation. For times larger than 500 fs~not shown in Fig.1!, P2(t) fluctuates around its long-time limit of' 0.33. It isinteresting to note how the electronic decay exhibited bP2(t) is reflected in the vibrational relaxation of the systemPanels~b! shows the time evolution of the mean positionx(t) of the totally symmetric moden6a , which, in the ab-sence of vibronic coupling, would exhibit coherent harmonioscillations. Owing to the electronic relaxation process, thvibrational motion is strongly damped and undergoes vibrtional dephasing. Note that, analogously to the electronic rlaxation, the vibrational dephasing process is completewithin ' 500 fs.

The simulation results obtained by the semiclassicTDSCF model with different values forg are seen to be inrather good agreement with the exact results forP2(t) andx(t). The model nicely reproduces the initial decay, recurences and long-time limits of these observables. Note, hoever, that the overall damping ofP2(t) as well as ofx(t) istoo weak forg50 @panels~1!, corresponding to a standard



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FIG. 1. Comparison of the results obtained by semiclassical simulations~full lines! and exact quantum calculations~dotted lines!for the two-state three-modesmodel of pyrazine. Panels~a! show the diabatic population probabilityP2(t), panels~b! the mean positionx(t) of the vibrational moden6a , and panels~c!the state-specific energiesHk(t) andVc(t). Each semiclassically evaluated observable is plotted forg50 @panels~1!#, g5

14 @panels~2!#, andg5

12 @panels~3!#.

-

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hu

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n

o

an-ath-

ap-f

hen-al-essp-

tic

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icalofofita-el

classical path model#, and too strong forg512 @panels~3!,

corresponding to the classical electron analog model#. Thevalue ofg51

4 @panels~2!# appears to be the appropriate dynamical correction for this model system. In particular, fog51

4 the semiclassical results of bothP2(t) andx(t) are inphase with the quantum calculation, indicating that the oveall coupling is correct.

To understand the effects of the dynamical correctioand to investigate the validity of the mean-path approximtion, let us now consider the quantum results for the meenergiesHk(t) and Vc(t), which are shown in panels~c!.Recall that initially the wave packet corresponding to thvibrational ground state in theS0 has been impulsively ex-cited into theS2 . Owing to this preparation of the system athe time t50, there is initially no vibrational energy con-tained in theS1 state@H1(0)50#, while the vibrational en-ergy contained in theS2 state is given by the stabilizationenergy@H2(0)5e2#. The initial decay ofP2(t), correspond-ing to an initial increase of the population in theS1 state, isreflected by a rapid increase of the vibrational energy in tS1 state to' 1.5 eV. This is because the excess energy dto the initial S2 excitation leads through theS2→S1 transi-tion to highly excited vibrational levels in theS1 state. Thevibrational energy of theS2 state, on the other hand, is seeto decrease slightly to' 0.3 eV. Furthermore, the relaxationprocess is accompanied by the mean diabatic coupliVc(t) oscillating around its long-time value of'20.5 eV.

It is quite interesting to compare the quantum results fHk(t) andVc(t) to the corresponding semiclassical data obtained for different values of the dynamical correctiong. As


r

r-

s-n

e

t

ee

g

r-

has been anticipated in the theoretical analysis of the mepath approximation in Sec. II C, the standard classical pmethod~corresponding tog50! underestimates both the diabatic energy gapDE(t) and the mean couplingVc(t). Notethat in the present model for pyrazine the classical pathproximation predicts a long-time limit for the energy gap oDE(t)'0.4 eV, whereas the quantum value is' 1.2 eV. Themain flaw of the mean-path approximation thus is that tS1 and theS2 get about the same amount of vibrational eergy of the system, which is in contrast to the quantum cculation, showing that the vibrational energy mainly gointo theS1 state, i.e., the vibrational relaxation mainly takeplace in theS1 state. This inadequacy of the mean-path aproximation cannot be completely remedied by theg modi-fications, but, as has been shown in Sec. II C, increasinggalso increases the energy gapDE(t). More important for therelaxation process is the value of the overall nonadiabacoupling Vc(t), which is correctly reproduced forg5 1

4

@panel~c2!#. A value ofg5 12 corresponding to the classica

electron analog model@panel~c3!#, is seen to exaggerate thdiabatic couplingVc(t), which in turn causes a too fast relaxation of the electronic population probabilityP2(t) andthe mean positionx(t), respectively.

To summarize, it has been shown that the semiclassTDSCF model augmented with a dynamical correctiong51

4 reproduces the electronic and vibrational relaxationthe three-mode two-state model of pyrazine almost quanttively. Similar studies on the relaxation dynamics of modsystems for C2H4

1 ,72 ozone,75 and C6H6149 have reconfirmed

this finding.


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C. The C˜B˜X IC process in the benzene cation

Recently Koppel, Domcke, and Cederbaum hapresented interesting quantum calculations of the ultraIC process in the benzene cation~Bz1).48,49 In his latestwork,49 Koppel has included five vibrational modes aexplicitly accounted for the vibronic interactions betwethe X 2E1g , B

2E2g , and C2A2u electronic states of Bz1.

Adopting this five-mode three-state model for Bz1, he hasperformed time-dependent quantum wave-packet proptions and showed that the~initially excited! C state decaysirreversibly into theX state within' 250 fs.

In a preceding letter,37 we have employed the classicelectron analog model28 to reproduce the quantum resultsRef. 49. In the present work, we discuss theC→B→X IC

f

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ga-

lf

process Bz1 in some greater detail, addressing, in particulathe interplay between electronic and vibrational relaxationthis system. Furthermore it is shown that in the case ofelectronic multistate model a large value for the dynamiccorrectiong ~e.g.,g5 1

2 as employed in the classical electronanalog model!may lead to dynamical inconsistencies in thsemiclassical simulation.

In the case of Bz1, the model Hamiltonian~2.1!as givenby Koppel amounts to an electronic five-state system labeby k5Xa ,Xb ,Ba ,Bb ,C, i.e., the second excited electronicstate (C2A2u) is represented byuw C&, while uw Xa&,uw Xb& anduw Ba&, uw Bb& denote the degenerate electronic groun(X 2E1g) and first excited (B 2E2g) electronic state, respec-tively. Adopting matrix notation, the electronic potential matrix reads:49

V5V0111EX1(

i51

2

k i~X!Qi (

i56

9

l i~X!r ie

2 iu i 2(i54

5

k iQi 0 0

(i56

9

l i~X!r ie

iu i EX1(i51

2

k i~X!Qi 0 (

i54

5

k iQi 0

2(i54

5

k iQi 0 EB1(i51

2

k i~B!Qi (

i56

9

l i~B!r ie

2 iu i (i516

17

l i r ieif i

0 (i54

5

k iQi (i56

9

l i~B!r ie

iu i EB1(i51

2

k i~B!Qi (

i516

17

l i r ie2 if i

0 0 (i516

17

l i r ie2 if i (

i516

17

l i r ieif i EC1(

i51

2

k i~C!Qi

2 . ~3.18!

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The potential matrix~3.18! features three different kinds ovibronic interactions: the Jahn-Teller coupling owing to thdegeneracy of theX and B states through the moden62n9 ~Wilson numbering!, the coupling of theC and Bstates by the pseudo-Jahn–Teller active modesn16 andn17,and the coupling of theB andX states by the modesn4 andn5 . As has been discussed in detail elsewhere,

47–49,52each ofthe vibronic interactions leads to a multidimensional conic

,

l

intersection of the corresponding PESs.To be able to perform quantum-mechanical wave-pack

propagations, the Hamiltonian is simplified by neglecting thpresumably weaker coupled modesn2 ,n5 ,n7 ,n9 , andn16.Ignoring furthermore the degeneracy of the remaining vibrational modesn6 ,n8 ,n17 and theX and B electronic states,the model system~3.18! reduces to the three-state five-modeHamiltonian:49

H5 ~T1V0!11S EX1k1~X!Q12 (

i56,8l i~X!Qi k4Q4 0

k4Q4 EB1k1~B!Q11 (

i56,8l i~B!Qi l17Q17

0 l17Q17 EC1k1~C!Q1

D . ~3.19!


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Using the model Hamiltonian~3.19!, Koppel performedlarge-scale dynamical quantum calculations by solving nmerically exactly the time-dependent Schro¨dinger equation.As an initial state it has been assumed that the system ithe vibrational ground state of neutral benzene shiftedvertically to theC PES, corresponding to ionization from thelectronic and vibrational ground state of neutral benzeAll parameters of the model Hamiltonians~3.18!and ~3.19!are taken from Ref. 49. For a further discussion of tHamiltonian~3.18!see Refs. 48 and 49.

Figure 2 compares the quantum~a! and semiclassicalTDSCF~b! calculations of the diabatic population probabiltiesPk(t) for the five-mode three-state model@Eq. ~3.19!#ofBz1. The quantum calculation exhibits an oscillatory poplation decay of the initially excitedC electronic state, whichis reflected in a rapid rise and similar beatings of theB statepopulation. After a somewhat delayed onset, the populatof the X electronic state exhibits a non-oscillatory rise andominates over both theC and B electronic states after' 120 fs. The TDSCF calculation, which has been evaluafor g5 1

4, is seen to reproduce the features of the quantreference, e.g., transient beatings and long-time limits, fawell. Except for the amplitude of the quantum beats andclassically somewhat to slowB→X decay, the agreement oquantum and semiclassical calculations is almost quanttive.

Let us now consider the electronic relaxation dynamof the more realistic model system@Eq. ~3.18!# for Bz1,where the remaining vibrational modesn2 ,n5 ,n7 ,n9 , andn16 are included and the degeneracy of the~pseudo!Jahn-

FIG. 2. Time-dependent population probabilitiesPk(t) of the C ~full line!,B ~dotted line!, andX ~dashed line!diabatic electronic states of the benzencation. Panel~a! shows the quantum calculation49, panel~b! the semiclas-sical calculation for the five-mode three-state model. The semiclassicalculation of the diabatic populations for the sixteen-mode five-state modethe benzene cation is shown in panel~c!.


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Teller modes and theX and B electronic states is explicitlytaken into account. Figure 2c shows the diabatic populatioprobabilities of the resulting sixteen-mode five-state modof Bz1. The comparison of the five-mode three-state mod~b! and the sixteen-mode five-state model~c! reveals that theinclusion of additional electronic and vibrational degrees ofreedom results in~i! a large suppression of the coherenquantum beating and~ii! a faster and more complete IC pro-cess, i.e., within' 300 fs the initially excitedC electronicstate is almost completely decayed into the electronic groustateX of the system.

It is interesting to compare the results of Fig. 2 with thecorresponding data obtained with the classical electron anlog model,37 where the dynamical correction isg5 1

2. Whilethe electronic population decay of the five-mode three-stamodel is forg51

2 only slightly faster than in Fig. 2~b!, theclassical electron analog simulation predicts for the sixteemode five-state model of Bz1 an about two times larger ratefor the B→X transition.37Although it is clear from Sec. II Dthat a larger dynamical correction results in stronger vibroncoupling and thus in a faster electronic relaxation, the effecof theg modifications should certainly not dominate the dynamics of the system.

To check the consistency of the TDSCF calculation witdynamical corrections, we may compare the time-dependeenergiesEsum(t) ~3.17a! and Etot(t) ~3.17b!. As has beenshown in Sec. III A, in an exact calculation the total vibrational energyEsum(t) is equal to the loss of total electronicenergyEtot(t), whereas simulations employing theg modi-fications may lead toEsum(t) Þ Etot(t). Figure 3 demon-strates this effect by showing the total vibrational energEsum(t) ~full lines! and the loss of total electronic energyEtot(t) ~dotted lines!for g5 1

4 ~a! and forg512 ~b!. While the

mean energies obtained by the TDSCF model withg514 ap-

proximately fulfill the consistency criteriaEsum(t)5Etot(t),the two energies are seen to diverge considerably in the caof g51

2, leading to a spurious additional increase of the tot

al-of

FIG. 3. Comparison of the total vibrational energyEsum(t) ~full lines! andthe loss of total electronic energyEtot(t) ~dotted lines!for g5

14 ~a! and for

g512 ~b!. The energies have been evaluated semiclassically for the sixte

mode five-state model of the benzene cation.


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vibrational energyEsum(t) by almost 3 eV. Discarding thedynamical corrections completely, on the other hand, is nsatisfying solution either, because forg50 the TDSCF cal-culation considerably underestimates the subsequentB→Xrate. Compared to the two-state case discussed abovevalidity of the mean-path approximation and the choicethe dynamical corrections are more critical for multistate stems, and require careful checks of the simulations.

Let us finally employ the semiclassical model to invetigate theC→B→X IC process of the sixteen-mode fivestate model of Bz1 in more detail. As has been pointed oabove, in the semiclassical formulation it is quite easycalculate observables pertaining to theadiabatic representa-tion, while the actual propagation is performed in thediaba-tic representation. As an example, Fig. 4~a! shows theadia-batic population probabilitiesPk

ad(t) defined in Eq.~3.8!. Itis seen that the adiabatic excited-state populationsP2

ad(t),P1ad(t) decay on the same time scale as the correspon

diabatic populations shown in Fig. 2~c!, although the initdecay of P2

ad(t) is even faster than in the diabatic caswithin only ' 15 fs, the population of theC state has de-cayed to 95% into the lower-lyingB state. Furthermore thecoherent transients exhibited by the diabatic populationsalmost completely suppressed in the adiabatic representa

To study the microscopic origin of this ultrafast eletronic relaxation, we particularly want to address the qu

FIG. 4. Adiabatic population probabilitiesPkad(t) @panel ~a!# and time-

dependent energy content of various vibrational modes@panels~b! and~c!#,obtained by a semiclassical calculation for the sixteen-mode five-state mof the benzene cation. Panel~b! shows the time-dependent energy contentthe totally symmetric modesn1 and n2 ~dashed line!, the nontotally symmetric coupling modesn4 andn5 ~full line!, the pseudo-Jahn–Teller activmodesn16 andn17 ~dotted line!, and the Jahn–Teller active modesn62n9

~dashed-dotted line!. Panel~c! shows the time-dependent energy contentthe Jahn–Teller active moden6 ~full line!, n7 ~dashed-dotted line!,n8

~dashed line!, andn9 ~dotted line!, respectively.


a

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-

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tion of intramolecular energy redistribution, i.e., how the intial photoexcitation energy of almost 5 eV is distributeamong the vibrational modes of the molecule. To this enFigs. 4~b!and 4~c!show the time-dependent energy contenof some of the vibrational modes included in the model. Amentioned above, the model contains four different kindsvibrational modes, the total vibrational energies of which aplotted in Fig. 4~b!: The totally symmetric modesn1 andn2 ~dashed line!, the nontotally symmetric coupling moden4 and n5 ~full line!, the pseudo-Jahn–Teller active moden16 andn17 ~dotted line!, and the Jahn–Teller active moden62n9 ~dashed-dotted line!. Figure 4~b!reveals that most ofthe electronic excitation energy is assumed by the degeneJahn–Teller modesn62n9 , whereas the other vibrationaldegrees of freedom participate only scarcely in the intramlecular energy redistribution of the molecule. As a furtheillustration, Fig. 4~c! shows how the vibrational energy isdivided up among the Jahn–Teller modesn62n9: the mostimportant mode isn8 ~dashed line, it also holds the largesvibronic coupling constants, see Ref. 49!, while the remaiing modesn6 ~full line!, n7 ~dashed-dotted line!, andn9~dotted line!accept about the same vibrational energy. Nothowever, that the low-frequency moden6 with frequencyv650.075 eV is considerably higher vibrationally excitethan the high-frequency moden7 with frequencyv750.393eV.

Figure 4 nicely elucidates the complex interplay of utrafast electronic and vibrational relaxation in a polyatommolecule. Furthermore it represents an interesting examof the highly efficient intramolecular energy redistribution omultimode Jahn–Teller systems. As has been discusseddetail in Ref. 52 for the example of a two-mode Jahn–Tellmodel of BF3

1 , the mode-mode interaction results in a paticularly strong distortion of the adiabatic PESs, which iturn gives rise to the ultrafast IC process observed in thesystems.

IV. SUMMARY AND CONCLUSIONS

We have outlined a semiclassical TDSCF approach fthe description of ultrafast IC processes of polyatomic moecules. The formulation represents a generalization of famiar classical-path methods and can be considered as a cbination of three well known techniques:

~i! A TDSCF ansatz for the total density operator~2.37!which couples the electronic degrees of freedomthe nuclear degrees of freedom in a self-consistemanner, and describes the vibrational density operain terms of Gaussian wave packets.

~ii! Approximation of thesemiclassicalexpansion coeffi-cients of the density operator~2.37! through quasi-classical sampling of the nuclear initial conditions~2.39!. This approximation requires to employsingle vibrational density operator for all electronistates~i.e., the mean-path approximation!, and to asume rapid randomization of the nuclear phases.

~iii! In the case of coordinate-dependent diabatic coupliVkk8(x), dynamical corrections are employed to thoff-diagonal elements of the electronic density oper

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tor, which correct for the inadequacy of the mean-papproximation in these cases.

By virtue of point ~i!, both electronic and nuclear degrees of freedom are represented by a density operwhich allows to employ standard density-matrix rules, efor the evaluation of time-dependent mean values and colation functions. Furthermore it is straightforward within thTDSCF formalism to make contact to well-known appromations of dissipative quantum dynamics34 such as the socalled noninteracting-blip approximation of Leggett aco-workers.39 Also note that in the calculation of electroncorrelation functions, the ansatz~2.39! takes into consider-ation the nuclear phases of the individual wave packets. Ctrary to standard classical-path methods, the semiclasTDSCF formulation therefore allows to roughly account fpure electronic dephasing effects.76

Point ~ii! ensures that the method is practical and is trapplicable for complex systems including many degreesfreedom. We therefore do not attempt to evaluate the seclassical Greens function, thus avoiding complex valuedjectories and complicated equations of motion which havebe evaluated iteratively or have to satisfy ‘‘double-endeboundary conditions. The approximation~2.39! to the ‘‘rig-orous’’ semiclassical expansion~2.37! relies on the assumption of randomization of nuclear phases, which makes it psible to perform the trace over the nuclear variables throquasiclassical sampling of the nuclear initial [email protected]. ~2.45!#.

We have discussed in some detail the validity of tmean-path approximation which is the second assumpimplied in the ansatz~2.39!. Comparing the equations omotion pertaining to a multiconfiguration~MC! ansatz~2.4!to the corresponding equations of a single-configuration~SC!ansatz~2.12!, we have shown that for constant diabatic cpling elementsVkk8 the mean-path approximation shouldappropriate in many cases, which is in accordance withfindings of paper I. In this case the equations of motionthe nuclear degrees of freedom are identical in the SCMC formulations, and the tuning of the electronic energy gDE(t) due to linearly coupled modes is described correcby the SC ansatz. In the case of coordinate-dependent dtic coupling Vkk8(x), however, it has been found that thmean-path approximation becomes problematic, whichgeneral results in a underestimation of the vibronic coupland thus of the overall relaxation rate. As a remedy for tinadequacy of the mean-path approximation, we haveployed dynamical corrections to the off-diagonal elementsthe electronic density operator, as has been suggesteMeyer and Miller.28 The effects, merits as well as pitfalls, othese modifications have been investigated in detail. Inticular, we have given a consistency criterion~3.17!, whichallows to determine an upper limit for the dynamical corretions, and have suggested a value ofg5 1

4. ~In accordancewith the well known modifications due to Langer,62 Meyerand Miller have used a value ofg5 1

2.!Employing ~i! a two-state three-mode model of th

S12S2 conical intersection in pyrazine,43 and ~ii! a three-state five-mode and a five-state sixteen-mode model of


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the

C→B→X IC process in the benzene cation,49 we have un-dertaken detailed computational studies to demonstratecapacity of the semiclassical TDSCF approach for the dscription of ultrafast IC processes. The comparison to exaquantum-mechanical basis-set calculations reveals thatsimple semiclassical method reproduces almost quantitively electronic population probabilities and mean values fothe energy content and the position of individual vibrationamodes. It should be noted that the complex electronic anvibrational relaxation dynamics occuring on intersectinPESs is not reproduced by other approximative propagatitechniques. Simple quantum-mechanical TDSCF methodfor example, neglect the correlations between the separanuclear degrees of freedom, and therefore are able to accoonly for the very short-time dynamics of the system.10,46

Even for in principle exact methods, such as path-integra5

and multiconfiguration TDSCF12 propagations, the calcula-tion of electronic population probabilities turns out to bedemanding task because of these mode-mode correlatioThe semiclassical TDSCF model explicitly accounts fothese correlations through the coupled equations of motioEqs. ~2.16! and ~2.17!. It therefore represents a suitablemethod for the modeling of ultrafast electronic relaxatioprocesses in polyatomic molecules, where typically sever~3-10! strongly coupled vibrational modes are involved inthe IC process.52,54

On the other hand it is clear that the semiclassicTDSCF propagation is no ‘‘black-box’’ method, which canbe applied blindly to any problem. Particularly in cases wheseveral energetically well separated coupled electronic staare involved, careful checks of the simulations are advisabincluding, e.g., sensitivity tests with respect to the choice othe dynamical corrections, initial conditions, and excitatioenergies. Another stringent test for the underlying approxmation of the semiclassical model is the description of reative processes. First results for simplecis-transisomerizationmodels have shown that the semiclassical approach seembe capable to model the relaxation dynamics of this elemetary chemical reaction.38 The extension of the approach to-wards nonadiabatic photoisomerization processes isprogress and will be the subject of a further publication.

ACKNOWLEDGMENTS

The author thanks Wolfgang Domcke for numeroustimulating and helpful discussions. This work has been suported by the Deutsche Forschungsgemeinschaft.

1See papers inUltrafast Phenomena IX, edited by P. F. Barbara, W. HKnox, G. A. Mourou, and A. H. Zewail~Springer, Heidelberg, 1994!.

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5M. Winterstetter and W. Domcke, Phys. Rev. A47, 2838~1993!;48, 4272~1993!; S. Krempl, M. Winterstetter, H. Plo¨hn, and W. Domcke, J. Chem.Phys.100, 926~1994!;

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s

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. 44

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3, No. 8, 22 August 1995ct¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

a semiclassical self-consistent-field approach to ... · and realistic solvent potentials!,...

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