Classical/quantal method for multistate dynamics: A computational studyTodd J. Martinez, M. BenNun, and Guy Ashkenazi Citation: The Journal of Chemical Physics 104, 2847 (1996); doi: 10.1063/1.471108 View online: http://dx.doi.org/10.1063/1.471108 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/104/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonadiabatic wave packet dynamics: Predissociation of IBr J. Chem. Phys. 105, 5647 (1996); 10.1063/1.472374 Pulse length control of Na+ 2 photodissociation by intense femtosecond lasers J. Chem. Phys. 105, 971 (1996); 10.1063/1.471939 Simulation of ultrafast dynamics and pump–probe spectroscopy using classical trajectories J. Chem. Phys. 104, 6919 (1996); 10.1063/1.471407 Toward preresonant impulsive Raman preparation of large amplitude vibrational motion J. Chem. Phys. 104, 1272 (1996); 10.1063/1.470786 Ultrashort pulse Chirp parmeter determination by interferometric methods AIP Conf. Proc. 160, 232 (1987); 10.1063/1.36863

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Classical/quantal method for multistate dynamics: A computational studyTodd J. Martinez,a) M. Ben-Nun, and Guy AshkenaziThe Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904 Israel

~Received 11 July 1995; accepted 6 November 1995!

We discuss a classically-motivated method for modeling ultrashort laser pulse optical excitation.The very same method can be used to treat the breakdown of the Born–Oppenheimerapproximation. The results are compared to numerically-exact quantum mechanics for a modelproblem representing excitation from theX ~ground! state to theB ~excited! state of moleculariodine. Expectation values and finalB state populations are predicted quantitatively. The methodprovides a new way to simulate pump–probe experiments in particular and multistate dynamics ingeneral. The method appears extendible to multidimensional problems. We argue that the increaseof effort with dimensionality will be similar to that encountered in classical mechanical simulationsas opposed to the exponential scaling of numerically-exact quantum mechanical propagationtechniques. ©1996 American Institute of Physics.@S0021-9606~96!00107-2#

I. INTRODUCTION

Recent experimental advances have made possible thecreation of ultrashort~,100 fs! laser pulses. Clever use ofthis technology has led to many fascinating experiments.Typically, a ‘‘pump’’ pulse is used to prepare a localizedinitial state. The ensuing dynamics are interrogated using‘‘probe’’ pulses. High-resolution control of the time delaybetween pump and probe allows direct experimental visual-ization of the time evolution of the initial wave function.Examples of the experiments we speak of include directprobing of the bond-breaking process,1 study of the influenceof solvation and curve-crossing on vibrational dephasing,2

and elucidation of the dynamics of the hydrated electron.3

These pulses have also been used to demonstrate the feasi-bility of controlling reaction pathways.4 The rapid onslaughtof experiments involving ultrashort laser pulses demands thedevelopment of theoretical methods which can model andexplain them.

Theoretical modeling of pump–probe experiments withultrashort pulses is conceptually straightforward. Any of thesingle-surface wave function propagation methods, such asthe Fourier method of Kosloff and Kosloff,5 can be modifiedto include several surfaces and their couplings. The increasein effort is at most quadratic with the number of surfaces,which is not at all unreasonable. More sobering is the factthat the numerically-exact wave function propagation meth-ods are already untenable on modern computers for single-surface calculations with more than four degrees of freedom.It remains to be seen whether the recently developed multi-configuration time-dependent self-consistent-field methods6,7

will alter this situation dramatically, although preliminary in-dications are encouraging.

For large molecules and/or multipicosecond time scales,an approximate method is necessary. It is well-known thatthe ultrashort laser pulse creates a highly localized wave-packet on the excited state surface.8 This localization impliesthat the wave packet approximates a coherent state, which is

the most classical-like object in quantum mechanics.9 Hence,we expect that an approach based on classical mechanicalconsiderations should be capable of capturing the essence ofthe phenomena. However, the immediate difficulty is how tointerpret the multistate nature of the problem within theframework of classical mechanics, which is only well-defined for particles moving on a single potential energy sur-face.

A similar problem was faced long ago when theoreticalchemists attempted to describe photodissociation absorptionspectra. Clearly, at least two surfaces are involved, yet work-ers desired a method which only required propagation on asingle surface. In the special case of ad-function pulse, thewave function created on the excited state surface is given bythe initial wave function on the ground state multiplied bythe transition dipole function. Heller10 showed that absorp-tion spectra in the continuous wave~cw! limit could be ob-tained by Fourier transformation of the survival probability^c(0)uc(t)&, wherec~0! is the wave function which wouldbe created on the excited state by ad-function pulse andc(t)is obtained by single-surface propagation on the excitedstate. There is thus no need to treat both surfaces or thelaser-matter interaction explicitly if the pulse is ad-functionin either of the complementary Fourier domains of time orenergy ~cw-limit!. In both cases, quasiclassical samplingtechniques can be used, leading to a computational prescrip-tion which involves nothing more complicated than classicaltrajectories.11,12,13

Because these approaches are by now quite standard, itis worthwhile to ask why they are inappropriate for the prob-lems we wish to study. First, the treatment is incorrect whenthe laser pulse is of finite temporal duration.14 Secondly, it isnot obvious how to generalize it to sequences of pulses, e.g.,the ‘‘pump–probe’’ techniques or sequences designed to con-trol photofragmentation pathways. Finally, no information isavailable on the amount of population transferred from onesurface to the other. Such branching ratios are of primaryinterest in the study of nonadiabatic effects~the breakdownof the Born–Oppenheimer approximation! and also in con-trol experiments.

a!Also at Department of Chemistry and Biochemistry, University of Califor-nia, Los Angeles, California 90024-1569.

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There is a strong analogy between the intersurface cou-pling induced by nonadiabaticity and the coupling inducedby a laser pulse. This is most clearly seen by considering the‘‘dressed-state’’ picture of optical excitation,15 where the in-teracting surfaces are the excited state and the ground stateshifted upward~‘‘dressed’’! by the energy of the photon. Theprimary differences are that the nonadiabatic coupling istime-independent and coordinate-dependent, while couplingdue to a laser pulse is time-dependent and often approxi-mated as coordinate-independent. We hope to develop amethod which works equally well for both types of prob-lems. However, in this paper we concentrate on assessing theaccuracy of our approximate method for the specific case ofoptical excitation. Modifications necessary to treat nonadia-batic problems successfully will be presented in a future pub-lication. The approximate method we describe includes thelaser pulse explicitly and is thus applicable to pump–probetype experiments, yet it is little more complicated than clas-sical mechanics once the intersurface coupling is removed. Avariant of this method has been previously applied to modelpump–probe experiments in solution.16

II. THEORY

We want to develop an approximate method which iswell-founded from a quantum mechanical standpoint yet re-tains a strong resemblance to classical mechanics. The re-semblance to classical mechanics is desired because themethod must be applicable to molecules with many degreesof freedom. The method should quantitatively reproduce thefinal population in the excited state, since we plan on adapt-ing it to address curve-crossing problems in the future. Fi-nally, we do not impose stringent requirements on the agree-ment between our method’s approximate wave functions andthe exact ones. Instead we want our wave functions to pro-vide the appropriate weighting for classical trajectories in thequasiclassical sense. This viewpoint is similar to that ex-pressed by the surface-hopping approaches to multistatedynamics.17,18,19

It is well-known from the work of Heller andcolleagues10,20–24that Gaussian basis functions are useful inelucidating links between classical and quantum mechanics.These functions have the form

x~R;R,P,g,a!5S 2a

p D 1/4 exp@2a~R2R!2

1 i P~R2R!1 ig#, ~1!

where the parametersR and P are expectation values ofposition and momentum,a ensures at least the uncertainty-principle mandated width in coordinate and momentumspace andg is a purely quantum-mechanical phase. Atomicunits are used throughout the manuscript. Given an arbitrarybasis set, the time-dependent variational principles25,26,27

~TDVP! provide a means of propagating a wave function intime. Although much of Heller’s work used the local har-monic approximation to derive equations of motion for theparameters of the Gaussian functions, he also proposed the

use of a variational principle to generate the equations ofmotion.28 Such an approach was later adopted by Heatherand Metiu.29 Skodje and Truhlar30 and Metiu andco-workers31,32 also suggested the use of more than oneGaussian basis function per degree of freedom—an approachwhich improved the results greatly in anharmonic potentials.Subsequently, Kay showed how to avoid the linear depen-dence problems which can arise when variational principlesare used to propagate Gaussian basis functions.33,34 Typicalapplications of Gaussian wave packet propagation tech-niques have allowed for the time-evolution of the width pa-rametera, but ‘‘frozen’’ Gaussian basis functions have alsobeen used.35 Almost all work using the TDVP with a Gauss-ian basis representation of wave functions has been directedat propagation on a single potential energy surface. Notableexceptions are the work of Sawada and Metiu,36,37 andHerman.38 Coalson has pursued the synthesis of perturbationtheory and Gaussian basis functions for multisurfaceproblems.39–42 As expected, low-order perturbation theoryencounters severe problems when the intersurface couplingis not weak.43 Within the framework of the local harmonicapproximation, Neria and Nitzan44 have developed a methodfor extracting nonadiabatic rates and Dehareng45,46,47 hasstudied the effects of nonadiabaticity on electronic spectra.

Of all these previous attempts, our method bears themost resemblance to work in the Metiu group.29,31,32,36,37

However, there are significant differences. The focus of thatwork was on raising the Gaussian wave packet propagationmethods to quantitative accuracy. In contrast, we force a di-rect connection to classical mechanics. Thus, for example,we do not allow arbitrary variation in the parameters of theGaussian functions. In particular, the average position andmomentum are constrained to evolve classically and thewidth of the Gaussian basis functions is time-independent.This allows us to interpret results in the familiar quasiclassi-cal context, averaging over an ensemble of trajectories tocompensate for the sparse basis used in each trajectory. Ul-timately, our use of a Gaussian basis set is dictated by a needto give meaning to classical mechanics on multiple elec-tronic states rather than a desire to obtain the detailed infor-mation available from numerical solution of the Schro¨dingerequation. This motivation is quite different from previousworkers who have used Gaussian basis functions.

A. General considerations

We will present two variants of a method which is basedon a Born–Oppenheimer expansion with time-dependentnuclear wave functions. Both include distinct nuclear wavefunctions on each electronic state, allowing access to dy-namical information on the individual states. In the multiple-spawning version~Sec. I B!, a basis set is constructed whichsamples the Franck–Condon region throughout the time ofthe intersurface coupling. In the interstate-correlated version~Sec. I C!, the nuclear wave functions evolve in a coupledmanner while the intersurface coupling is significant.

2848 Martinez, Ben-Nun, and Ashkenazi: Classical/quantal method for multistate dynamics

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B. Multiple-spawning time-displaced basis functions

The general multistate Hamiltonian operator can be writ-ten as

H5(I

uI &HII ^I u1(IÞJ

uI &HIJ^Ju, ~2!

where the electronic states~which we will take to be ortho-normal! are denoted in bra-ket notation and theHII andHIJ

operators act only on nuclear wave functions. Our methoduses the following multiconfigurational frozen Gaussiannuclear wave function ansatz for a problem with any numberof electronic states,

c~R,t !5(I

(k

NI

ckI ~ t !xk

I ~R!uI &, ~3a!

xkI ~R![x@R;Rk

I ~ t !,PkI ~ t !,gk

I ~ t !,a I #, ~3b!

where the indexk labels theNI basis functions on surfaceI ,and all time dependencies of the basis functions are explic-itly denoted. The phase factorsg(t) are actually redundantwith the complex coefficientsck

I (t), but we leave them ex-plicit for later convenience. Since each electronic state has itsown nuclear wave function, one has direct access to dynami-cal quantities on each individual state. This stands in contrastto mean-field approaches which use only one nuclear wavefunction for all states.48–51 In order to make a clear connec-tion to classical mechanics we choose the time evolution ofthe parametersRk

I and PkI in each Gaussian according to

Hamilton’s equations, which is practically equivalent to us-ing the local harmonic approximation of Heller. The equa-tions of motion for the phases can be chosen arbitrarily andwe use those obtained by Heller in the local harmonic ap-proximation. Explicitly, the propagation equations are

RG kI 5 Pk

I /m, ~4a!

PG kI 52F]VI~R!

]R GRkI, ~4b!

gkI 5@~ Pk

I !222a I #/2m2VI~RkI !, ~4c!

wherem is the appropriate reduced mass,VI is the potentialenergy for stateI , and the dot denotes the time derivative.Given the orthonormality of the electronic states, the remain-ing equations of motion are given by the TDVP@or equiva-lently by substituting the wave function ansatz of Eqs.~3!into the time-dependent Schro¨dinger equation# as

cJ52 i ~SJ!21H @HJJ2 iSJ#cJ1(IÞJ

HJIcIJ , ~5!

where SJ is the ~time-dependent! overlap matrix of theGaussian basis functions on electronic surfaceJ, HJI is thesub-block of the Hamiltonian matrix describing the interac-tion between basis functions on electronic statesI andJ, andSJ is the matrix representation of the right-acting time-derivative operator, i.e.,

~cJ!k5ckJ , ~6a!

~SJ!kl5^xkJux l

J&, ~6b!

~HJI !kl5^xkJuHJIux l

I&, ~6c!

~SJ!kl5 K xkJU ]

]tx lJL . ~6d!

The matrix inversion is performed after regularization of theoverlap matrix using a singular value decomposition.52 Thethreshold for retained singular values is typically set at131024, and the results are not very sensitive to the precisevalue.

Having detailed the equations of motion, we mustspecify how the basis function parameters and initial condi-tions should be chosen. The width parameter for each sur-face,a, is taken as real and time-independent, i.e., we workin a basis of ‘‘frozen’’ Gaussians with no position-momentum correlation. Since it is hard to find compellingarguments as to what the precise value of the width shouldbe, it is particularly gratifying to find the results are ratherinsensitive to it. For example, we find that changing its valueby a factor of 10 only affects the final excited-state popula-tion by less than 1%. In the model problem presented herewhich involves a bound→bound transition, we choose theground state width as the harmonic frequency of theX~ground! potential and use the same width for the basis func-tions on theB ~excited! state.

The initial conditions for the basis functions on the ex-cited state remain to be determined. Since we know thatpopulation transfer occurs preferentially at the crossing ofthe dressed state potential curves~in the Franck–Condon re-gion!, we must have excited state basis functions there dur-ing the pulse. Thus, we determine the initial conditions byplacing all excited state basis functions at the crossing pointand integrating each one backward in time byt i1k(td/Nb),wherek is the index of the basis function,td is the durationof the pulse,Nb is the total number of basis functions for theexcited state, andt i is the time when the pulse begins. Withthis prescription, the basis functions cover the crossing pointas best they can during the important time when the pulse isoperative. Note that the back-propagation is completelyclassical—the only parameters which need to be propagatedareRk

I andPkI . Since each of the basis functions differs in the

time it arrives at the crossing point, and hence the time whenit begins to be populated~or ‘‘spawned’’!, we call this themultiple-spawning time-displaced basis. In order to deter-mine the magnitude ofPk

I for each basis function, we need tospecify the classical energies of the basis functions. If wewere dealing with cw excitation, we would certainly wantthe basis functions to be centered on the classical orbit of therelevant energy.53 However, the pulses of interest are ul-trashort, implying an associated energy uncertainty in thepulse. Therefore, we choose the classical energy for eachbasis function from a Gaussian distribution corresponding tothe Fourier transform of the time-domain pulse envelope.The direction ofPk

I at the crossing point is chosen randomly.Since the phases,g, are arbitrary, we simply set the initialvalues for these parameters to zero.

2849Martinez, Ben-Nun, and Ashkenazi: Classical/quantal method for multistate dynamics

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The choice of a width in the classical energies of thebasis functions may seem rather peculiar since the Gaussianbasis functions already possess an uncertainty in position andmomentum. However, recall that we force the centers of thebasis functions to propagate classically and the width to betime-independent. Hence, the use of a small basis set willonly be possible if the classical centers mimic the evolutionof the quantum wave function. Such will not be the case ifthey all have the same energy. As a concrete example, weoffer the case of a Gaussian initial state evolving on a con-stant potential energy surface. The wave packet shouldspread with time, but an ensemble of classical trajectorieswith the same energy and initial positions chosen appropriateto the initial state will not reproduce this behavior. On theother hand, if the initial positions and momenta of the clas-sical trajectories are chosen from the appropriate distribu-tions given by the initial state, the trajectories will have anenergy width which leads directly to an accurate prediction9

of the extent of spreading in the ‘‘wave packet.’’If the physical problem of interest involves a stationary

initial state, one should diagonalize the Hamiltonian for theinitial state in a basis of fixed Gaussians~R and P are con-sidered time-independent! centered on the relevant classicalorbit. If one is interested in the evolution of the dynamicalhole created during the pulse,54 the same procedure as de-noted above for the excited state can be used for the groundstate. The moving basis functions so generated should obvi-ously be employed in addition to the fixed basis functionswhich describe the initial stationary state.

The description of the method is thus complete. Ratherthan ‘‘sewing quantum mechanical flesh onto classicalbones’’ as Berry and Mount have so eloquently describedsemiclassical methods,55 we have attempted to sew classicalflesh onto quantum mechanical bones. The appeal to quan-tum mechanics is necessary to give concrete meaning to themultistate dynamics. However, once this is accomplished, wefeel no compulsion to use sufficient basis functions to pro-vide a faithful representation of quantum mechanical wavepacket propagation. In this respect we differ substantiallyfrom the aforementioned extensions of Heller’s work whichwere aimed at improving the quality of the Gaussian wavepacket ansatz, for example in anharmonic potentials. Instead,the Gaussians which clothe the classical trajectories will beused only to give meaning to the nuclear wavefunction over-laps which modulate population transfer. The finite width ofthe Gaussians gives rise to a Franck–Condon region~insteadof a point!, and therefore to a continuous population transferover a finite time instead of an instantaneous ‘‘hopping.’’

Precisely because in each of our calculations there areinsufficient basis functions to reproduce quantum mechanicalwave packet propagation, an ensemble of such calculationsmust be run in the fashion of classical molecular dynamics.Because the particular case we use to benchmark the methodhere is one-dimensional, the only difference between thevarious ensemble members will be the classical energy andmomentum of the unpopulated Gaussian basis functions.However, in the generaln-dimensional case, the crossing‘‘point’’ is an n21 dimensional seam and the initial place-

ment of the unpopulated basis functions~prior to the back-propagation! should be determined by a Monte Carlo sam-pling of the Franck–Condon region. Such a Monte Carlosampling is consistent with our desire to interpret the methodin light of classical mechanics. Each member of the en-semble will have a basis set capable of describing correctlythe fraction of population transferred, but is not required toreproduce the overall dynamics correctly. In particular, theresults obtained from a single member of the ensemble maybe inordinately sensitive to the placement of the basis func-tions. By averaging over the initial conditions of the createdpopulation, we destroy this sensitivity. Since the Schro¨dingerequation will determine the relative weights of the basisfunctions, the details of the Monte Carlo sampling are unim-portant subject to adequate coverage of the Franck–Condonregion during the population transfer.

Extension of the method to cases involving more thanone laser pulse is possible. The prescription given above fordetermining initial conditions of the unpopulated basis func-tions is repeated for each pulse. Note that each pulse willhave its own t i , td , energy distribution, and crossingpoint~s!.

The scheme we have described conserves normalizationrigorously. This is of paramount importance in the treatmentof multisurface problems, since without norm conservationbranching ratios cannot be determined. Indeed, a lack ofnorm conservation provided an obstacle to Dehareng’s pre-vious attempts45,46,47to use frozen Gaussian basis functionsto elucidate the effects of nonadiabaticity on electronic spec-tra. The quantum mechanical energy is also conserved. Bothof these quantities are conserved as a consequence of usingthe TDVP, as has been pointed out by other authors.56,57

Since the width parameters are time-independent and thesame for all basis functions, the classical energy, defined as

EClassical5(I

(k

~ckI !* ck

I @~ PkI !2/2m1VI~Rk

I !# ~7!

would be conserved if all the basis functions ran on classicalorbits of the same energy.31 Because of the energy dispersionwe use in modeling the pulse, the classical energy is onlyconserved when there is no intersurface coupling. Thisshould not be construed as a disadvantage—the energy un-certainty of an ultrashort laser pulse is an important part ofthe physical description as has been discussed by Gruebeleand Zewail.58

The computational cost of the method is formally cubicin the number of basis functions used per surface and qua-dratic in the number of surfaces involved. In practice, theeffort required for the matrix inversion can be reduced byusing an iterative scheme—one only requires the product ofthe inverse and a known vector. Such methods scale qua-dratically with matrix dimension.59 However, as long as thepulses considered are very short and the effective couplingbetween the surfaces is highly localized in coordinate space,we expect that it should often be sufficient to include ten orfewer basis functions per surface per crossing event regard-less of dimensionality. These requirements~short coupling

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time and localized coupling region! are often satisfied innonadiabatic problems as well. It is clear that our methodwould require many basis functions to model cw excitationaccurately, because of the long-lived~in principle infinite!intersurface coupling.

Finally, the method has been set up such that it is trans-parent to move between the classical and quantum mechani-cal basis set pictures. Once the surfaces cease to be con-nected by the pulse, one can revert to traditional frozenGaussian propagation35 @only Eqs.~4! are solved# until theintersurface coupling again becomes numerically significant.The phase information from the basis function coefficientsshould be incorporated into theg factors and weights,a,assigned to each basis function and surface. Thus,

akI 5A~ck

I !* ckI , ~8a!

aI5~cI !†SIcI, ~8b!

gkI→gk

I 1atnFRe~ckI !Im~ckI !G . ~8c!

Traditional frozen Gaussian propagation does not conservenormalization,60 but we know that the total population oneach surface should be constant in the absence of intersur-face coupling. Therefore, we must renormalize the basisfunction coefficients when the coupling again becomes nu-merically significant, prior to restarting the solution of Eq.~5!. This is accomplished by defining the new basis functioncoefficients as

ckI 5ak

IAaI /~cI !†SIcI ; ~9!

While this procedure is clearly a further approximation, itprovides a consistent means of applying the quasiclassicalideas~that a swarm of classical trajectories with the properinitial conditions will approximate quantum wave packetevolution! to multisurface dynamics. This approximation toour method is reminiscent of surface-hopping methods. Theimportant difference is the retention of phase informationand thus the ability to describe both intersurface~electronic!and intrasurface~vibrational! interferences correctly. This iscrucial in the study of molecular control as in the experi-ments of Scherer and co-workers.61We leave detailed discus-sion of this issue to a future publication. As our model prob-lem involves only one pulse, it should be clear that we havenot used this approximation here.

C. Interstate-correlated nuclear wave functions

Equations~3! express the nuclear wave functions foreach electronic state as a sum over Gaussians. The time de-pendent parameters in these Gaussians are chosen to ensurethat the Franck–Condon overlap between the coupled elec-tronic states is finite whenever the interstate coupling is on.From a computational standpoint, it is clearly desirable touse as few Gaussians as possible. In fact, an alternative formusing a single nuclear basis function per electronic state ispossible,16 whereupon Eq.~3! takes the form

c~R,t !5(JcJ~ t !xJ~R!uJ&. ~10!

There are two differences with the earlier procedure. One isthat the time dependent nuclear wave functions are not pre-scribed beforehand but are chosen by the condition that thewave function of Eq.~10! satisfies the time dependent Schro¨-dinger equation. In other words, both the linear parameters~thec’s! and the nuclear wave functions in Eq.~10! are to bechosen. The other difference is that for every multiple-spawning run, a set of trajectories is to be computed. Eachmember of this ensemble of trajectories has its own wave-function associated with it. The reason is that in this point ofview the wave function serves only to transform from theHeisenberg to the Schro¨dinger picture,16 i.e., it needs to sat-isfy the time dependent Schro¨dinger equation.

When a single nuclear wave function~per electronicstate! is used in the total wave function, Eq.~5! has thesimpler form

cJ52 i H SHJJ2 i K xJU ]

]txJL D cJ1(

IHJIcIJ . ~11!

There is no need to introduce an overlap matrixSJ and all thematrices and vectors are reduced to numbers. Here, thesquare of the complex coefficients [cI(t)] * cI(t)# is thepopulation of theI th electronic state and the cross terms[cI(t)] * cJ(t)# serve as a measure for the electronic coher-ence between statesI andJ.

It is however necessary to specify the nonstationarynuclear states which appear in Eq.~10!. The condition thatthe total wave function satisfies the Schro¨dinger equationimplies that

i ^Juc&5 i ~ cJxJ1cJxJ!5^JuHuc&5cJHJJxJ

1(IÞJ

cIHJIx I . ~12!

The solution of these exact coupled partial differential equa-tions can be approximated by choosing all nuclear wavefunctions to be Gaussians. Unlike the basis functions of Eq.~3!, these Gaussian wave functions are meant to mimic thecoupled wave functions and hence need not be the same asthose of Eq.~3!. As discussed below, the present Gaussianswill describe a correlated nuclear motion on the differentelectronic states.

A priori one can not be sure that the use of a single basisfunction ~per electronic state! is sufficient to reproduce theexact behavior of Eq.~12!. The use of nonstationary Gauss-ian wave functions implies that for a laser-induced coupling,all the interstate coupling terms in Eq.~11! are a product ofthe laser pulse, the electronic transition dipole and a timedependent Franck–Condon overlap integral between theground and the excited wave functions.~This is in the Con-don approximation in which the electronic transition dipoleis considered to be independent of the inter nuclear separa-tion. In the more general case the transition dipole does de-pend on the nuclear coordinates and it cannot be taken out ofthe overlap integral.! Hence, the time scale for the effective

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coupling between the states is determined both by the exter-nal short laser pulse and by the time interval over which theFranck–Condon overlap integral is significant. Physically, itis obvious that for a short pulse we want the former to be animportant component. The multiple-spawning time-displacedbasis ensures this by making sure that throughout the excita-tion stage there is at least one~and possibly even more thanone!, ground and excited, overlap integral that significantlydiffers from zero. In the present, single nuclear wave func-tion framework, correlation between the two nuclear func-tions is established via Eq.~12!.

For an optical excitation, the Franck–Condon principleimplies that the excited state is formed in a narrow region,close to the inner turning point. This results in a Franck–Condon overlap integral which is very sensitive to smallchanges in momentum and/or position. For the model prob-lem we discuss in the next section, this effect is further en-hanced by the high vibrational energy content of the excitedmolecule which is more than half of the well depth at thewavelengths of interest. Hence, if a single Gaussian param-etrized as in Eqs.~3! is used, one finds that the decline in thevalue of the overlap integral~between the ground and theexcited state! is faster than the duration of the pulse. Sincethis is true for each trajectory in the ensemble and for theensemble as a whole it implies an averaged coupling be-tween the electronic states that is too weak. Therefore, for asingle Gaussian, one cannot use uncorrelated values of theposition and momentum on the different surfaces to computethe overlap integral. This is sad but it is only to be expectedin view of the correlation in the nuclear motion on the elec-tronic states, as imposed by Eq.~12!.

The simplest way to overcome this problem is to use abrute force correlation, namely a clamped nuclei approxima-tion. In this approximation during the time that the shortpulse is operating the nuclei~on all surfaces! are kept fixedin their initial positions. This simple solution was testedagainst the exact quantum mechanical computation. For thefield parameters of interest~duration'50 fs! it turns out thatthe clamped nuclei approximation overestimates the popula-tion transfer to the excited state.~For very short pulses itdoes better.!

In order to successfully use only one Gaussian basisfunction per electronic state, the method must be modified totake into account the correlation established during the exci-tation stage. We first consider ad-like excitation. As has beenemphasized by Heller20 and discussed in the introduction, aninstantaneous optical excitation is equivalent to launchingthe ground state wave function times the dipole operator,mIJux

I&, on the upper electronic state. For a pulse of finiteduration the correlation is more complicated in that there is acontinuous buildup of the population on the upper electronicstate with the instantaneous increment in the wave functionbeing proportional to the wave function on the ground state.This is quite evident from Eq.~12!. The required correlationcan be simply imposed in the following manner. In each ofour runs we use the ground state Hamiltonian to propagateand hence determine the parametersRI and PI that param-etrize the ground state wave function,x I , whereas the param-

eters,RJ and PJ, of the excited state wave function,xJ, aredetermined using a ‘‘mean-field’’ potential which is given bya weighted sum of the ground and excited state Hamiltonianswith the weights given by the population of the state. Oncewe determine these interstate-correlated nuclear wave func-tions we can, analytically, compute the interstate coupling@HIJ in Eq. ~11!# and propagate the electronic equations.Thus, in each of our runs we simultaneously propagate thecoupled nuclear and electronic equations and an ensemble ofsuch runs is to be computed. Note, that the use of a corre-lated nuclear wave function does not imply that the Franck–Condon integral between the two states equals unity as thetrajectories are not confined to be at the crossing point of the‘‘dressed’’ ground and excited state. As demonstrated in thenext section this procedure compares very favorably with theexact quantal results.

III. MODEL AND RESULTS

The problem we use to benchmark the approximatemethod represents the laser-induced excitation from theXstate to the bound region of theB state of molecular iodine.Both electronic state potential energy curves are modeledusing the Morse functional form with appropriate parameterslisted in Table I. The potential energy curves are depicted inFig. 1, along with the ‘‘dressed’’X state for a carrier fre-quency of 507 nm. A laser pulse with Gaussian envelope of50 fs at full-width, half-maximum is used and the carrierfrequency is varied between 474 nm and 543 nm. The tran-sition dipole is taken independent of coordinate and set at0.461 D.61 We have chosen two laser intensities for thisstudy—2.631011 W/cm2 ~0.001 93 atomic field units! and1.031012W/cm2 ~0.003 86 afu!. The initial wave function istaken as the ground state harmonic oscillator wave functionwith the appropriate frequency.

The numerically-exact method with which we gauge theperformance of our method uses the first Magnusapproximation62 to treat the time-dependent Hamiltonian

c~ t2!'e2 i*t1

t2H~ t8!dt8c~ t1!5e2 iOc~ t1!. ~13!

The Newton polynomial interpolation scheme63 is used toexpress the propagator as a power series inO. The Fourier

TABLE I. Simulation parameters in atomic units.

Morse parameters Vi(R)5Di@12e2b i (R2Ri )#21Ti

D/hartree b/bohr21 R/bohr T/hartreeuX& 0.056 65 0.987 22 5.038 6 0.0uB& 0.023 58 0.897 47 5.715 9 0.071 822

Pulse parametersa e(t)5Am exp@22 ln 2(t2t0)2/s2#

t051 240.233 atub s52 480.466 atub m50.181 aduc

AWeak50.001 93 afud AStrong50.003 86 afud

aThese parameters describe the slowly-varying envelope of the pulse as inEq. ~14!.b1 atu5atomic time unit50.024 fs.c1 adu5atomic dipole unit52.542 D.d1 afu5atomic field unit55.1431011 V/m2.

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method5 is used to generate the result ofOc. The rotatingwave approximation~RWA! is employed, whereupon theHamiltonian operator takes the form

H5F HX1v0 e~ t !

e* ~ t ! HBG , ~14!

wheree(t) is the slowly varying envelope of the pulse,v0 isthe carrier frequency, andHX and HB are the single-surfaceHamiltonian operators for theX ~ground! and B ~excited!states, respectively. Because the problem chosen is one-dimensional, we are able to easily follow the numerically-exact quantum mechanical simulations for up to a picosec-ond. Such simulation times are necessary if one is to judgethe quality of the excited state dynamics predicted by theapproximate method.

In the multiple-spawning method, a single fixed Gauss-ian basis function is used on the ground state. Fromnumerically-exact simulations, this wave function is station-ary to within graphical resolution for the time scale of thelower intensity pulse, and therefore this approximation isquite good. Ten basis functions are used for the excited state,which is sufficient to converge the branching ratio to foursignificant figures.

For the interstate-correlated method, the nuclear propa-gation consists of two trajectories, one on the ground stateand one on the excited state. The reported results are over anensemble of 100 such runs. The initial conditions in theground state are chosen from the proper ground vibrationalstate stationary distribution of iodine whereas on the excitedstate we sample them uniformly from the classically allowedFranck–Condon region. The excited state sampling ensuresthat each of the excited state trajectories in our ensemblespends a somewhat different duration in the Franck–Condonregion and hence our initial population is not too localized.As for the multiple-spawning propagation, the excited statepopulation has a width in energy which is determined by theuncertainty in energy which is implied by the limited shortduration of the pump pulse. Hence, we first determine theenergy of the excited state trajectory by sampling it from aGaussian distribution. Once we determine the energy we ran-

domly choose the position to be in the classically allowedFranck–Condon region~this is a quite narrow region in be-tween the inner turning point on the excited state, at thechosen energy, and the outer turning point of the ‘‘dressed’’ground state! and then determine the initial momentum~itsdirection is chosen randomly!. The excited state Gaussiandoes not propagate until its population is numerically signifi-cant ~.1310210 in these computations!.

The first results we show concern theB state populationas a function of time. Experimentally, one measures laser-induced fluorescence with a probe laser and this is taken tobe proportional to the population in the probed state. In manysituations, it may not be crucial to reproduce numericallyexact populations. For example where one pumps populationand then probes dynamics on the excited state with a secondpulse, it is sufficient to provide a reasonable description ofthe initial state prepared by the pump laser. However, it iscrucial to model these populations correctly when branchingratios are directly measured as when one attempts to controlthe fragmentation path of a molecule.4 Figure 2 shows theBstate population for both the weak~lower panel! and strong~upper panel! fields with a carrier frequency of 507 nm. Thisfrequency places the crossing point in the dressed state pic-ture at the equilibrium distance of the ground state potentialand hence gives rise to maximum absorbance. Notice that theapproximate method~in both the multiple-spawning andinterstate-correlated forms! is essentially quantitative in itsprediction of the total population pumped into theB state.Some overpopulation is visible in the multiple-spawningtreatment of the strong field, which is a consequence of thestationarity assumption for the ground state. As discussed

FIG. 1. The ground (X) and excited (B) electronic state potentials of theiodine molecule used in the computation. The dressed potential is the groundstate potential shifted up by the external field energy~507 nm in this ex-ample!. The arrow illustrates the electronic excitation.

FIG. 2. The excited state population as a function of time~in fs! for low andhigh excitation. Full line, exact quantum calculation. Dashed line, Multiple-spawning. Dotted line, interstate-correlated wave functions, averaged over100 trajectories. Lower panel, laser intensity 2.631011W/cm2. Upper panel,laser intensity 1.031012 W/cm2.

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previously, this can be remedied by adding nonstationary ba-sis functions to the ground state. In Fig. 3, we show theresults of the fixed nuclei approximation, which clearly over-populates the excited state.

In Fig. 4 we show the finalB state population as a func-tion of the carrier frequency of the pulse for the strong field.In the dressed state picture, the limits chosen go from acrossing point at the inner turning point of theX state to oneat the outer turning point. It is in these limits that one wouldexpect any classically-based method to have the most severedifficulties. In fact, we find quite good and consistent agree-ment for all the frequencies, with a maximum deviation ofabout 3% in the final population.

The expectation value of position on theB state isshown in Fig. 5. For the interstate-correlated wave functions,the expectation value~at each point in time! is computed asa weighted ensemble average, where the weight of each ex-cited state trajectory is given by its population. Again, theagreement with numerically-exact quantum mechanics is vir-tually quantitative.

Finally, in Figs. 6 and 7 we present direct comparison of

exact and multiple-spawning wave functions at the time ofthe pulse maximum and after the pulse is numerically insig-nificant. The multiple-spawning wave functions are ratherclose to the true wave functions. Our criterion that theyshould provide the correct weighting for the various trajec-tories is quite clearly satisfied.

IV. DISCUSSION

The method we have presented is aimed at exploiting theclassical analogy to quantum mechanics which is made allthe more accurate when ultrashort pulses are used. Therefore,it is perfectly positioned for simulating and elucidating thephenomena involved in pump–probe experiments. Further-more, it can be easily modified to treat nonadiabatic effectsin the form of the breakdown of the Born–Oppenheimer ap-

FIG. 3. As in the lower panel of Fig. 2 but comparing the exact quantumcalculation~full line! to the fixed nuclei approximation~dashed line!. In thisapproximation the nuclei are frozen in their initial positions during the ex-citation. Notice the extreme over population of the excited state.

FIG. 4. FinalB state population for strong field strength as a function ofcarrier frequency.

FIG. 5. Expectation value of position on the excitedB state predicted by thedifferent methods.~Full line, exact quantum calculation. Dashed line,multiple-spawning. Dotted line, interstate-correlated wave functions, aver-aged over 100 trajectories, each one weighted by its population at each pointin time.! Due to the anharmonicity of the potential and to the energy uncer-tainty of the pulse, the initially localized population spreads. On a longertime scale it will relocalize.

FIG. 6. Exact~dashed line! and multiple-spawning~full line! B state nuclearwave function for strong field att53000 atu, approximately 10 fs after themaximum of the pulse.

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proximation. We have shown that the method is virtuallyquantitative for a model one-dimensional problem.

The success of the method revolves around the assump-tions that the intersurface coupling is simultaneously local-ized in time and space. The formal scaling behavior withdimensionality is at worst cubic if multiple-spawning is usedand linear for the interstate-correlated wave functions. Thusour method is computationally much less demanding thannumerically-exact quantum mechanics. In fact, the interstate-correlated wave functions have already been used to studythe excited state dynamics of the iodine molecule solvated byrare gas atoms.16

Recent work by several groups64–67 has focused on theuse of Gaussian basis sets as a means to accurately propagatequantum mechanical wave packets. Although superficiallysimilar to our method in the use of Gaussian basis functions,we would like to stress that these methods are philosophi-cally quite different from ours. We are content with the pre-diction of averaged quantities and hence there is no reason torequire accurate wave functions. Instead we exploit the clas-sical analogy to quantum mechanics as fully as possible.Similar in spirit, but strikingly different in details, is recentwork by Kinugawa68 which exploits simplifications inherentin the direct Monte Carlo evaluation of operator matrix ele-ments as opposed to the computation of the full wave func-tion.

Because of the encouraging agreement we obtain here,future work will concentrate on adapting the multiple-spawning method to treat nonadiabatic effects and multidi-mensional problems.

ACKNOWLEDGMENTS

The authors are grateful to Professor R. D. Levine formany helpful comments and suggestions regarding the re-search and the manuscript. T.J.M. thanks the United States–Israel Educational Foundation and the Institute of Interna-

tional Education for a Fulbright Junior PostdoctoralResearcher Award and the University of California Office ofthe President for a Postdoctoral Fellowship. M.B.N. is aClore Foundation scholar. The Fritz Haber Research Centeris supported by the MINERVA Gesellschaft fu¨r die Fors-chung, mbH, Munich, Germany. This work was supported bythe Stiftung Volkswagenwerk.

1R. Zadoyan, Z. Li, P. Ashjian, C. C. Martens, and V. A. Apkarian, Chem.Phys. Lett.218, 504 ~1994!.

2N. F. Scherer, D. M. Jonas, and G. R. Fleming, J. Chem. Phys.99, 153~1993!.

3J. C. Alfano, P. K. Walhout, Y. Kimura, and P. F. Barbara, J. Chem. Phys.98, 5996~1993!.

4J. L. Herek, A. Materny, and A. H. Zewail, Chem. Phys. Lett.228, 15~1994!.

5D. Kosloff and R. Kosloff, J. Comp. Phys.52, 35 ~1983!.6H. D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett.165, 73~1990!.

7A. D. Hammerich, U. Manthe, R. Kosloff, H. D. Meyer, and L. S. Ceder-baum, J. Chem. Phys.101, 5623~1994!.

8S. O. Williams and D. G. Imre, J. Phys. Chem.92, 6636~1988!.9C. Cohen-Tannoudji, B. Diu, and F. Laloe¨, Quantum Mechanics~Wiley,New York, 1977!, Vol. 1, pp. 559ff.

10E. J. Heller, Acc. Chem. Res.14, 368 ~1981!.11S. Goursand, M. Sizun, and F. Fiquet-Fayard, J. Chem. Phys.65, 5453

~1976!.12E. J. Heller, J. Chem. Phys.68, 2066~1978!.13R. Schinke,Photodissociation Dynamics~Cambridge University, Cam-bridge, 1993!, pp. 93–108.

14M. V. Rama Krishna and R. D. Coalson, Chem. Phys.120, 327 ~1988!.15T. F. George, I. H. Zimmerman, J. Yuan, J. R. Laing, and P. L. DeVries,Acc. Chem. Res.10, 449 ~1977!.

16M. Ben-Nun and R. D. Levine, Chem. Phys.201, 163 ~1995!.17J. C. Tully and R. K. Preston, J. Chem. Phys.55, 562 ~1971!.18J. C. Tully, J. Chem. Phys.93, 1061~1990!.19P. J. Kuntz, J. Chem. Phys.95, 141 ~1991!.20E. J. Heller, J. Chem. Phys.62, 1544~1975!.21D. J. Tannor and E. J. Heller, J. Chem. Phys.77, 202 ~1982!.22K. C. Kulander and E. J. Heller, J. Chem. Phys.69, 2439~1978!.23G. Drolshagen and E. J. Heller, J. Chem. Phys.82, 226 ~1985!.24N. E. Henriksen and E. J. Heller, J. Chem. Phys.91, 4700~1989!.25J. Frenkel,Wave Mechanics~Clarendon, Oxford, 1934!.26P. A. M. Dirac, Proc. Cambridge Philos. Soc.26, 376 ~1930!.27A. D. McLachlan, Mol. Phys.7, 39 ~1964!.28E. J. Heller, J. Chem. Phys.64, 63 ~1976!.29R. Heather and H. Metiu, Chem. Phys. Lett.118, 558 ~1985!.30R. T. Skodje and D. G. Truhlar, J. Chem. Phys.80, 3123~1984!.31S. Sawada, R. Heather, B. Jackson, and H. Metiu, J. Chem. Phys.83, 3009

~1985!.32R. Heather and H. Metiu, J. Chem. Phys.84, 3250~1986!.33K. G. Kay, J. Chem. Phys.91, 170 ~1989!.34K. G. Kay, Chem. Phys.137, 165 ~1989!.35E. J. Heller, J. Chem. Phys.75, 2923~1981!.36S. Sawada and H. Metiu, J. Chem. Phys.84, 227 ~1986!.37S. Sawada and H. Metiu, J. Chem. Phys.84, 6293~1986!.38M. F. Herman, J. Chem. Phys.82, 3666~1985!.39R. D. Coalson, J. Chem. Phys.86, 6823~1987!.40R. D. Coalson and J. L. Kinsey, J. Chem. Phys.85, 4322~1986!.41R. D. Coalson, J. Chem. Phys.86, 995 ~1987!.42R. D. Coalson, Chem. Phys. Lett.147, 208 ~1988!.43M. V. Rama Krishna, J. Chem. Phys.93, 3258~1990!.44E. Neria and A. Nitzan, J. Chem. Phys.99, 1109~1993!.45D. Dehareng, Chem. Phys.84, 393 ~1984!.46D. Dehareng, Chem. Phys.110, 375 ~1986!.47D. Dehareng, Chem. Phys.120, 261 ~1988!.48D. J. Diestler, J. Chem. Phys.78, 2240~1983!.49M. Jacon, P. Guerin, O. Atabek, and H. Le Rouzo, J. Chem. Phys.97,2490 ~1992!.

FIG. 7. Exact~dashed line! and multiple-spawning~full line! B state nuclearwave function for strong field att55000 atu, after the influence of the pulseis completely negligible.

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50H. D. Meyer and W. H. Miller, J. Chem. Phys.70, 3214~1979!.51G. Stock and W. H. Miller, J. Chem. Phys.99, 1545~1993!.52W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,Nu-merical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.~Cambridge University, Cambridge, 1992!, pp. 51–63.

53M. J. Davis and E. J. Heller, J. Chem. Phys.71, 3383~1979!.54U. Banin, A. Bartana, S. Ruhman, and R. Kosloff, J. Chem. Phys.101,8461 ~1994!.

55M. V. Berry and K. E. Mount, Rep. Prog. Phys.35, 315 ~1972!.56F. Arickx, J. Broeckhove, E. Kesteloot, L. Lathouwers, and P. Van Leuven,Chem. Phys. Lett.128, 310 ~1986!.

57R. D. Coalson and M. Karplus, J. Chem. Phys.93, 3919~1990!.58M. Gruebele and A. H. Zewail, J. Chem. Phys.98, 883 ~1993!.

59L. A. Hageman and D. M. Young,Applied Iterative Methods~Academic,San Diego, 1981!.

60M. F. Herman, J. Chem. Phys.85, 2069~1986!.61N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys.95,1487 ~1991!.

62W. Magnus, Comm. Pure Appl. Math7, 649 ~1954!.63R. Kosloff, Annu. Rev. Phys. Chem.45, 145 ~1994!.64M. A. Sepulveda and E. J. Heller, J. Chem. Phys.101, 8004~1994!.65M. F. Herman and E. Kluk, Chem. Phys.91, 27 ~1984!.66E. Kluk, M. F. Herman, and H. L. Davis, J. Chem. Phys.84, 326 ~1986!.67K. G. Kay, J. Chem. Phys.100, 4432~1994!.68T. Kinugawa, Chem. Phys. Lett.235, 395 ~1995!.

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