Timedependent discrete variable representations for quantum wave packet propagationEunji Sim and Nancy Makri Citation: The Journal of Chemical Physics 102, 5616 (1995); doi: 10.1063/1.469293 View online: http://dx.doi.org/10.1063/1.469293 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/102/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multidimensional time-dependent discrete variable representations in multiconfiguration Hartree calculations J. Chem. Phys. 123, 064106 (2005); 10.1063/1.1995692 Time-dependent discrete variable representation method in a tunneling problem J. Chem. Phys. 118, 5302 (2003); 10.1063/1.1553977 A time-dependent discrete variable representation method J. Chem. Phys. 113, 1409 (2000); 10.1063/1.481959 A timedependent discrete variable representation for (multiconfiguration) Hartree methods J. Chem. Phys. 105, 6989 (1996); 10.1063/1.471847 Exact wave packet propagation using timedependent basis sets J. Chem. Phys. 90, 5566 (1989); 10.1063/1.456410

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Time-dependent discrete variable representations for quantum wave packetpropagation

Eunji Sim and Nancy MakriSchool of Chemical Sciences, University of Illinois, Urbana, Illinois 61801

~Received 7 October 1994; accepted 4 January 1995!

We present an efficient method for exact wave function propagation with several degrees of freedombased on time-dependent discrete variable representations~TD-DVR! of the evolution operator. Thekey idea is to use basis sets that evolve in time according to appropriate reference Hamiltonians toconstruct TD-DVR grids. The initial finite basis representation is chosen to include the initialwavefunction and thus the evolution under the bare zeroth order Hamiltonian is described at eachtime by a single DVR point. For this reason TD-DVR grids offer optimal representations intime-dependent calculations, allowing significant reduction of grid size and large time steps whilerequiring numerical effort that~for systems with several degrees of freedom! scales almost linearlywith the total grid size. The method is readily applicable to systems described by time-dependentHamiltonians. TD-DVR grids based on the time-dependent self-consistent field approximation areshown to be very useful in the study of intramolecular or collision dynamics. ©1995 AmericanInstitute of Physics.

I. INTRODUCTION

Recent advances in experimental detection techniqueshave stimulated considerable interest in developing quantummechanical schemes for studying the dynamics of poly-atomic molecular systems. In order to simulate femtosecondexperiments or to calculate low-resolution spectra, solvingthe Schro¨dinger equation in the time domain is usually thepreferred approach. Many successful numerical time-evolution schemes have been developed during the last de-cade, which integrate the time-dependent Schro¨dinger equa-tion on a grid or in a basis set.1 With present day computers,exact schemes are feasible in practice for systems of a fewatoms, while truly polyatomic systems are usually dealt withusing approximate techniques. An important exception in-volves problems where a one-~or two-! dimensional systemis coupled to a bath of~harmonic or anharmonic! degrees offreedom. Recently developed path integral methods2 havemanaged to deal very efficiently with such situations and thedynamics of system–bath Hamiltonians can now be calcu-lated exactlywith any number of degrees of freedom fortimes sufficiently long to observe interesting chemical ef-fects. Furthermore, exploiting the damping introduced bythose condensed phase environments characterized by dissi-pative baths of continuous spectral densities allows the dy-namics to be obtainediteratively for arbitrarily long times.3

The present paper reports progress in the development ofefficient numerical schemes for dealing with anharmonicHamiltonians of several degrees of freedom that cannot beexpressed in system–bath form. Often such problems requirethe use of prohibitively large grids or basis sets and thereforeare not amenable to study by numerically exact quantummechanical schemes. An additional difficulty arises becausemost conventional approaches require use of small time stepsfor convergence, although the size of the time step is not themost serious concern since the numerical effort for iterativetime evolution scales only linearly with the number of timeincrements.

Assuming that the eigenstates of the Hamiltonian are notavailable, perhaps the simplest generic iterative procedurefor calculating the evolution of a wave function involvesmultiplication of a propagator matrix times a wave functionvector and therefore scales generally asM tot

2 , whereM tot isthe dimension of the wave function in the chosen discreterepresentation. Use of position representations4 requiressome care because the conventional real time propagator is ahighly oscillatory function of coordinates and thus not suit-able for numerical integration. This problem is easily over-come by employing momentum5 or energy projectionoperators6 to filter out the rapidly oscillatory components ofthe propagator, or by fitting the latter to smooth functions.7

However, even after the appropriate smoothing of the propa-gator, uniform grid representations of wave functions requirehigh densities of points. Energy eigenstate basis sets often~for bound problems! provide more compact representationsand therefore are preferable in matrix multiplicationschemes. Another grid-based approach utilizes the so-calledpseudospectral methods.8,9 These schemes make use of twodistinct representations to evaluate the action of the shorttime evolution operator on the wave function. High effi-ciency is achieved by using fast Fourier transforms10 ~FFT!to perform the kinetic energy operation, resulting in ascheme that scales asM tot log2M tot . The favorable scalingof the FFT, which is almost linear with the total number ofgrid points, becomes increasingly important ifM tot is large,as is the case when a few coupled degrees of freedom areinvolved. However, FFT algorithms require high-density uni-form grids; in practice, on the order of 102 grid points perdegree of freedom are necessary for convergence and thislimits the practical applicability of the method to systemswith three at most degrees of freedom.

A particularly useful idea for treating unbound coordi-nates is to perform the propagation in an interactionrepresentation11,12 in order to prevent wave packet spreading.Because the interaction wave function is free of the ballisticcomponent characteristic of the Schro¨dinger wave function,

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much smaller grids are possible. The efficiency of the inter-action representation depends critically on the choice of ze-roth order Hamiltonian. The simplest choice consists in iden-tifying the reference Hamiltonian with the kinetic energy,leading to uniform grid representations well suited for FFT-type or even analytic treatment.11 More powerful schemesresult from more elaborate choices of the reference Hamil-tonian. If the latter includes potential terms, energy represen-tations have been shown to be more economical.12 However,the use of energy representations leads to fullM tot

2 scalingwith basis set size.

For a large class of systems, use of discrete variablerepresentations13,17 ~DVR! offers maximal efficiency. Direct-product DVR methods offer the advantage of compactnesscharacteristic of energy basis sets, while requiring numericaleffort that ~in systems with several degrees of freedom!grows almost linearly with the total numberM tot of gridpoints. In particular, potential-optimized discrete variablerepresentations15 ~PO-DVR!, which incorporate eigenstateinformation corresponding to physically meaningful zerothorder Hamiltonians, have been shown to allow convergencewith rather sparse grids. Yet, even in the potential-optimizedDVR, the required grid often becomes larger than practical,even in systems of only two or three degrees of freedom; thisis often the case when dissociative coordinates are involved,or simply when the number of states accessible to the wavepacket is high—regardless of the magnitude of the couplingterms in the Hamiltonian.

Clearly, combination of PO-DVR grids with interactionrepresentations should offer significant advantages in suchcases. The approach proposed in this paper is similar in spiritto an interaction DVR, although we adopt a propagatormethod in the Schro¨dinger representation rather than inte-grating the interaction differential equation. One reason forthis choice is that propagator-based schemes automaticallyguarantee unitarity; although time evolution in the interac-tion representation can be cast in the exponential propagatorform,12~b! the required procedure is quite cumbersome due tothe time dependence of the interaction Hamiltonian. Mostimportantly, a long-term goal of our work is to incorporatepowerful procedures in path integral schemes in order to dealefficiently with many-body dynamics; for this purpose it isnecessary to formulate methods in the exponential propaga-tor form. The time-dependent interaction Hamiltonian is notwell-suited for direct construction of propagators, and forthis reason we use the Schro¨dinger Hamiltonian; instead, weintroduce time dependence into the DVR grid.

Thus, in order to alleviate the large grid problem weintroduce atime-dependentDVR ~TD-DVR! that evolveswith the moving wave function according to some suitable~possibly time-dependent! zeroth order Hamiltonian. Thefirst step is to identify a physically relevant but separablereference Hamiltonian.~Note that the exact dynamics ofseparable degrees of freedom can be obtained easily in spiteof the presence of dissociative coordinates and/or spread-outwave functions.! Next, the time evolution operator is splitsymmetrically and the wavefunction of the fully coupledHamiltonian is represented on a time-dependent DVR gridwhich moves according to the dynamics of the reference

problem. Thus, the size of the TD-DVR grid is determinedmerely by the potential coupling terms; therefore this grid isoften much more compact than that required for direct solu-tion of the full Schro¨dinger equation. In the limit where thecoupling terms vanish, the TD-DVR grid reduces to a singlepoint.

To motivate the idea outlined above, consider a photo-dissociation problem where the wave packet on the excitedsurface evolves out of the interaction region and begins tospread toward an exit channel. The best potential-optimizedDVR treatment would use the eigenstates of the separablepart of the Hamiltonian~discretized inside a box! to con-struct a DVR grid on which the full Schro¨dinger equationwould be integrated. In order to prevent spurious reflectionof the outgoing wave packet the box must be made suffi-ciently large, resulting in fairly large density of energeticallyaccessible basis functions and eventually in a dense and spa-tially extended grid. In fact, the use of a DVR may not offera significant advantage over simple uniform grids in thiscase. When the grids of all degrees of freedom are combinedthe memory requirements may be impractical. Alternatively,consider the basis consisting of the initial wave packet aug-mented by a few orthogonal functions~e.g., vibrationally ex-cited states of the ground potential surface! and suppose onefollows their evolution in the zeroth order Hamiltonian com-prised of the upper-surface single-particle potentials. Ofcourse each of the necessary calculations would require alarge grid, but obtaining one-dimensional dynamics is alwaysa relatively easy task. In order to calculate the time evolutionunder the full Hamiltonian we propose using the basis com-prised of the propagated basis functions to construct apotential-optimizedTD-DVR grid. This grid, which will re-main small throughout the course of time evolution~eventhough the actual TD-DVR states may be quite spread out!,will in essence travel with the moving wave packet allowingaccurate but economical representation of the multidimen-sional wave function. If similar TD-DVR grids are con-structed in all degrees of freedom the savings in computa-tional effort for a multidimensional problem can be verysubstantial.

Section II describes the procedure for a general time-dependent reference Hamiltonian. In Sec. III we discuss theuse of the time-dependent self-consistent field ansatz to gen-erate a mean-field-type TD-DVR grid. Two numerical ex-amples are presented in Sec. IV; the first example, whichinvolves a collision problem, shows that TD-DVR grids canallow dramatic savings in numerical effort. Significant sav-ings are also observed in a model of anharmonic intermo-lecular dynamics, which is treated as a second application.Concluding remarks and a discussion of future directionsappear in Sec. V.

II. TIME-DEPENDENT DISCRETE VARIABLEREPRESENTATIONS

Consider a system ofN degrees of freedomxk describedby a HamiltonianH. The Hamiltonian may include time-dependent terms commonly arising from classical radiationfields. We begin with the common assumption that the initialstate has a product form, i.e.,

5617E. Sim and N. Makri: Time-dependent discrete variable representations

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C0~x1 ,x2 ,...,xN!5)k51

N

Fk~xk ;0!. ~2.1!

Although this assumption facilitates presentation, it is notessential for the scheme that we develop. The goal is tocalculate the time evolution of the initial wave function.

A. Split propagator with time-dependent reference

We begin by identifying a separable reference Hamil-tonian

HREF~ t !5 (k51

N

HkREF~ t !.

The reference Hamiltonian will be used to generate zerothorder solutions of the time-dependent Schro¨dinger equationwhich will define a basis set. Once the zeroth order wavefunction is known, the entire numerical effort will concen-trate on correcting that wave function by including multidi-mensional correlation. Clearly, a good choice of the zerothorder solution should lead to converged results with modestnumerical effort. If the description of the system of interestinvolves explicit time-varying terms the reference Hamil-tonian may be time-dependent. In addition, time-dependentmean field potentials often lead to adequate approximations,and thus the reference Hamiltonian may be time-dependenteven if the full Hamiltonian is independent of time. We alsodefine the~possibly time-dependent! correlation potential

HCORR~ t !5H2HREF~ t ! ~2.2!

and split the time-evolution operator for propagation fromtime t to t1Dt as follows:

U~ t1Dt,t !'UCORR~ t1Dt,t1Dt/2!UREF~ t1Dt,t !

3UCORR~ t1Dt/2,t !. ~2.3!

HereUREF is the time-evolution operator for propagation un-der the action of the separable reference Hamiltonian andUCORR is the time-evolution operator that corresponds to thecorrelation potential. Because it is assumed that the latterdoes not depend on momentum operators, the correspondingtime-evolution operator involves a simple time integralwhere no time-ordering is necessary,

UCORR~ t2 ,t1!5expF2i

\ Et1

t2HCORR~ t !dtG . ~2.4!

The error arising from the splitting of the time evolutionoperator in Eq.~2.3! is proportional toDt3, implying that thetime step must be kept small. Although such a conditionapplies to all split propagator schemes, it is important toemphasize that the actual size of time step allowed dependsvery strongly on the choice of reference. This is so becausethe terms omitted from Eq.~2.3! are also proportional to the~nested! commutator between the reference and the remain-ing part of the Hamiltonian. While the traditional splittinginto kinetic and potential energy parts8,9 requires indeedsmall time steps for convergence, physically motivated ref-erence Hamiltonians have been shown2 to allow large timeincrements. This point is also illustrated by the numericalexamples presented in Sec. IV.

B. Time-dependent basis set

Consider a basis set ofMk functions $fk,nk(0),

nk51,...,Mk% for each degree of freedomk, consisting ofthe initial wave functionFk(0) plus a set of mutually or-thogonal basis functions for that degree of freedom. In thecommon case where the initial multidimensional wave func-tion is an eigenstate of the separable part of the Hamiltonian,the above basis will simply consist ofMk eigenstates ofHk .If the initial state is a displaced Gaussian in one or morecoordinates, one can choose appropriately displaced har-monic oscillator basis functions. We now propagate allMk

basis functions for each degree of freedomk using the ref-erence Hamiltonian for that coordinate to obtain

fk,nk~ t !5Uk

REF~ t,0!fk,nk~0!. ~2.5!

The reference Hamiltonian is assumed Hermitian and thecorresponding time evolution operator is unitary. Thereforepropagation underHREF(t) preserves the orthonormality ofthe basis functions,

^fk,nk~ t !ufk,n

k8~ t !&5dnk ,nk8. ~2.6!

Thus, the above scheme leads to adynamicalorthogonalbasis of functions that include the zeroth order solution. Sucha basis set is expected to offer a very compact representationof the multidimensional wave function provided that thechoice of zeroth order Hamiltonian is good. The advantagesof a time-dependent basis set are easily appreciated if onerealizes that in the separable limit asingledynamical basisfunction would be sufficient; this is not true of any staticbasis for noneigenstate initial preparations. If the zeroth or-der wave function is reasonable, though not exact, thisscheme should require only a few basis functions per degreeof freedom and thus the total size of the calculation shouldremain manageable for several coupled coordinates.

C. Time-dependent discrete variable representation

The final and most crucial consideration concerns theactual representation of the full wave function evolving un-der the action of the total Hamiltonian. If the propagation iscarried out directly in the time-dependent basis definedabove, operation withUREF will require calculation of thefull M tot3M tot propagator matrix, whereM tot5Pk51

N Mk isthe dimension of the wave function vector, as the potentialcoupling is not local in this basis set. The optimal grid,which combines the advantages of the dynamical basis and atthe same time yields adiagonalcorrelation potential propa-gator, is offered by a time-dependent discrete variable repre-sentation. DVR quadratures have proved very successful invibrational eigenvalue problems. Although they have alsobeen employed in time-dependent calculations,16,17it appearsthat the advantages offered by discrete variable representa-tions have not been fully exploited in wave function propa-gation. Apart from allowing sparse grids, DVR methods areknown to scale asM tot(k51

N Mk because the most time con-suming step in calculating the matrix representation of theHamiltonian~or the propagator! is now the representation ofHREFwhich is separable, thus reducing toN one-dimensionaloperations.

5618 E. Sim and N. Makri: Time-dependent discrete variable representations

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Making use of the above ideas, we perform at each timestep a unitary transformation on the dynamical basis$fk,nk

(t)% to define the TD-DVR states$uk,nk(t)% such that

^uk,nk~ t !uxkuuk,nk8~ t !&5xk,nkDVR~ t !dnknk8, ~2.7!

wherexk,nkDVR(t) are the DVR eigenvalues for thekth degree of

freedom at timet. Note that the matrix elements ofxk willgenerally be complex attÞ0, but the Hermitian character ofthe coordinate operator guarantees real TD-DVR eigenvaluesand orthogonal states. If the initial wave function is an eigen-

state of the separable part of the Hamiltonian, the TD-DVRwill coincide at t50 with the PO-DVR. However, the TD-DVR grid may differ dramatically from the PO-DVR at latertimes, providing improved representation of the evolvingwave function.

We use this TD-DVR to evaluate the action of the splittime-evolution operator, Eq.~2.3!, on the wave function.Taking advantage of the DVR relations, Eq.~2.7!, the wavefunction is propagated by one time stepDt in the followingway:

^u1,n1~ t1Dt !•••uN,nN~ t1Dt !uC~ t1Dt !&

5expF2i

\ Et1Dt/2

t1Dt

HCORR~x1,n1DVR ,...,xN,nN

DVR ;t8!dt8G (m151

M1

••• (mN51

MN

^u1,n1~ t1Dt !•••uN,nN~ t1Dt !uUREF~ t

1Dt,t !uu1,m1~ t !•••uN,mN

~ t !&expF2i

\ Et

t1Dt/2

HCORR~x1,m1

DVR ,...,xN,mN

DVR ;t8!dt8G ^u1,m1~ t !•••uN,mN

~ t !uC~ t !&. ~2.8!

Since the reference Hamiltonian is separable, the corresponding propagator matrix can be factored into a product of one-dimensional propagator matrices,

^u1,n1~ t1Dt !•••uN,nN~ t1Dt !uUREF~ t1Dt,t !uu1,m1~ t !•••uN,mN

~ t !&

5^u1,n1~ t1Dt !uU1REF~ t1Dt,t !uu1,m1

~ t !&•••^uN,nN~ t1Dt !uUNREF~ t1Dt,t !uuN,mN

~ t !& ~2.9a!

and the summations in Eq.~2.8! can be performed sequentially. Each one-dimensional TD-DVR propagator matrix can becalculated using completeness along with the DVR transformation relations,

^uk,nk~ t1Dt !uUkREF~ t1Dt,t !uuk,n

k8~ t !&5 (

mk51

Mk

^uk,nk~ t1Dt !ufk,mk~ t1Dt !&^fk,mk

~ t !uuk,nk8~ t !&. ~2.9b!

D. Summary of procedure

In summary, the procedure we propose is the following:One begins by identifying an appropriate separable zerothorder Hamiltonian. Att50 an orthogonal basisfk,nk

(0) is

constructed for each degree of freedomk. This basis is usedto calculate the initial DVR states and express the initialwave function in the DVR representation. At each timet onemust proceed according to the following steps to propagatethe wave function by one time incrementDt:

~1! Propagate each of the one-dimensional orthogonal basisfunctionsfk,nk

(t) for one time step byHkREF to obtain

fk,nk(t 1 Dt). ~This propagation can be performed us-

ing basis set or grid methods and will generally requiretime increments much smaller thanDt.!

~2! For each degree of freedomk diagonalize the coordinateoperatorxk in the basis$fk,nk

(t 1 Dt)% to determine the

DVR states$uk,nk(t 1 Dt)% and eigenvalues$xk,nkDVR(t

1Dt)%.~3! Multiply the wave function vector~in the DVR represen-

tation! by the half-step propagator vector that corre-sponds to the nonseparable potential interactions,

expF2 i

\ Et

t1Dt/2

HCORR~x1,n1DVR ,...,xN,nN

DVR ;t8!dt8G3^u1,n1~ t !•••uN,nN~ t !uC~ t !&

[^u1,n1~ t !•••uN,nN~ t !uc8~ t !&. ~2.10!

This step requiresM tot operations.~4! Calculate the reference propagator matrix in the mixed

TD-DVR representation, i.e., between DVR states attime t and t1Dt,

^uk,nk~t1Dt!uUkREF~ t1Dt,t !uuk,n

k8~ t !&,

for each degree of freedom according to Eq.~2.9b!.~5! Multiply the vectorc8(t) that was calculated in step~3!

by the reference propagators computed in step~4!,

(mk51

Mk

^uk,nk~t1Dt!uUkREF~ t1Dt,t !uuk,mk

~ t !&

3^u1,n1~ t !•••uk,mk~ t !•••uN,nN~ t !uc8~ t !&

[^u1,n1~ t !•••uk,nk~ t1Dt !•••uN,nN~ t !uck9~ t !&.

~2.11!

5619E. Sim and N. Makri: Time-dependent discrete variable representations

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This procedure is repeated fork51,...,N and involves atotal ofM tot(k51

N Mk operations.~6! Finally, multiply the vectorcN9 (t) from step~5! by the

propagator vector that corresponds to the potential cou-pling terms for the second half step to obtain the wavefunction at timet1Dt,

expF2i

\ Et1Dt/2

t1Dt

HCORR~x1,n1DVR ,...,xN,nN

DVR ;t8!dt8G3^u1,n1~ t1Dt !•••uN,nN~ t1Dt !ucN9 ~ t !&

5^u1,n1~ t1Dt !•••uN,nN~ t1Dt !uc~ t1Dt !&. ~2.12!

Exact results are obtained once convergence has beenachieved with respect to the number of DVR points as wellas the time step.

Clearly, the most time-consuming part of the calculationis the multiplication by the reference propagators, step~5!above. IfM basis functions are used on the average for eachdegree of freedom the total number of operations will beNMtotM 5 NMtot

111/N . Thus, forN@1, the method scales al-most linearly with the dimensionM tot of the wave functionvector. This advantage is due to the use of a system-specificDVR quadrature16,17and is to be contrasted to the usual qua-dratic scaling of propagation schemes if ordinary energy ba-sis sets are used.

In summary, the appealing feature of the time-dependentDVR grid that we have introduced is that it adjusts with timeto the motion under a closely related~but simple! reference.In fact, in the limit where the potential coupling terms re-sponsible for correlation effects are turned off, the exact timeevolution is described by asingle grid point. As the movingwavepacket consists of several energy states, each TD-DVRgrid point incorporates information about multiple eigen-functions of the zeroth order Hamiltonian with the particularcoefficients relevant to the problem under consideration. If areasonable choice of reference has been made, the TD-DVRgrid will travel and expand or contract appropriately as timeprogresses, providing a representation of the wave packetthat will generally be superior to those obtainable by staticdiscrete variable representations.

III. TD-DVR BASED ON THE TIME-DEPENDENT SELF-CONSISTENT FIELD APPROXIMATION

For many processes the separable part of the Hamil-tonian ~comprised of the kinetic energy and single-particlepotential terms! provides a simple, yet useful choice of ref-erence for the TD-DVR scheme. Better zeroth order systems,if available, may lead to even greater contraction of the re-quired grid. A natural choice is offered by the time-dependent self-consistent field~TDSCF! approximation.18

This idea, which dates back to Dirac, exploits the time-dependent variational principle to obtain the ‘‘best’’ productwave function; i.e., the underlying Hamiltonian is separable~though time-dependent!. This procedure leads to aSchrodinger-type equation of motion for the wave functionof each degree of freedom that involves a time-dependentmean field potential, i.e., the average of the many-body po-tential with respect to all other particles. Although theTDSCF scheme cannot describe correlation effects, it allows

energy transfer among the coupled coordinates. The TDSCFapproximation has been found to provide a fairly accuratepicture of the dynamics in several cases, including ro-vibrational energy transfer and photodissociationprocesses.19 For those problems for which TDSCF is at leastqualitatively correct, representing the wave function on thedynamical TDSCF-DVR grid should lead to accurate resultswith far less numerical effort compared to other basis set ortime-independent DVR methods.

The idea of using the TDSCF scheme to generate a set oftime-dependent basis functions was first proposed byCampos-Martinez and Coalson.20 These authors used a directexpansion of the wave function in an orthogonal TDSCF-generated basis and derived coupled differential equationsfor the motion of the time-dependent expansion coefficients.Because the TDSCF basis is often superior to static basissets, considerable reduction of the number of expansion co-efficients was observed. However, these equations involvethe usual matrix-vector multiplication and thus the schemescales like typical energy basis set algorithms, i.e., asM tot

2 . Itis precisely in this regard that the present scheme shouldoffer numerical advantages; converting the TDSCF basis setto a moving DVR grid leads to simple multiplicative poten-tial operations while allowing decomposition of the multidi-mensional reference DVR matrix into several one-dimensional matrices; these features result in overall effortthat scales essentially linearly with grid size. At the sametime, the TDSCF-referenced DVR propagator permits use oflarge time steps, resulting in further acceleration of numeri-cal calculations.

To this end, we employ TDSCF-generated basis func-tions to define a TDSCF-DVR grid. According to the TDSCFansatz, the evolution of the initial wave function is given bythe product function

CTDSCF~x1 ,x2 ,...,xN ;t !5eia~ t !)k51

N

FkTDSCF~xk ;t !. ~3.1!

HereFkTDSCF are the solutions to the TDSCF equations

i\]

]tFk

TDSCF~xk ;t !5HkTDSCF~xk ,pk ;t !Fk

TDSCF~xk ;t !,

~3.2!

whereHkTDSCF include time-dependent mean-field potentials,

HkTDSCF~ t !5 )

k8Þk

N

^Fk8TDSCF

~ t !uHuFk8TDSCF

~ t !&, ~3.3!

anda(t) is an unimportant phase.To proceed, we use the TDSCF Hamiltonian

HTDSCF~ t !5 (k51

N

HkTDSCF~ t ! ~3.4!

to propagate the initial basis functions. Notice, however, thatthe TDSCF potential depends on the initial wave function,and thus the basis functions orthogonal to the initial state arenot propagated self-consistently under the Hamiltonian of

5620 E. Sim and N. Makri: Time-dependent discrete variable representations

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Eqs. ~3.3!–~3.4!. Although it is possible to propagate eachmember of the basis self-consistently by performing indi-vidual TDSCF calculations, such a procedure would notmaintain the orthogonality of the basis functions at latertimes, complicating the calculation. Propagation of all basisfunctions by the same Hamiltonian guarantees orthogonalityat any time.20

Before concluding this section, we wish to emphasizethat the TDSCF approximation is not always successful andtherefore may not provide a useful reference. It has beenshown21 that the simple TDSCF approach fails to describecorrectly wave packet splitting and therefore one should notuse the TDSCF reference propagator in tunneling problems;the adiabatic potential offers a reference by far superior toTDSCF in such cases and leads to optimal splitting of thepropagator.2

In summary, if the TDSCF description of the dynamicsis semiquantitative for a given system, the TDSCF-DVR gridcan offer a systematic way of correcting the TDSCF result,converging rapidly to the exact dynamics.

IV. NUMERICAL EXAMPLES

In this section we present two examples that illustratethe advantages of TD-DVR grids for treating the dynamicsof molecular processes. We describe a simple collision prob-lem as well as a model of intramolecular nonlinear dynam-ics.

Both examples involve only two degrees of freedom inorder to allow comparison with standard grid or basis setmethods without extraordinary computational effort. TheTDSCF ansatz is expected to provide a reasonable startingpoint in both cases and is therefore used as the reference.

A. Collinear He–H 2 scattering

We treat a simple model20,22 for the collision dynamicsof a He atom with a hydrogen molecule in the collinear ar-rangement. The Hamiltonian has the form

H5px2

2mx1

py2

2my1V~x,y!, ~4.1!

wherex and y are coordinates arising from linear transfor-mation of the Jacobi coordinates for this system. Specifically,x is the vibrational coordinate of H2 andy is the distance ofthe He atom from the H2 equilibrium center of mass whenthe outer H atom is considered as fixed.22 These coordinatesare scaled such thatmx51 andmy52/3. Thepotential con-sists of a Morse term for the H2 vibration and an exponentialrepulsive term for the He–H2 interaction,

V~x,y!5D~12e2bx!21e2a~y2x!, ~4.2!

whereb50.158,D520.0, anda50.3. ~All energies are inunits of the vibrational quantum of H2.! The initial state ischosen as a product of the ground vibrational state of H2 anda Gaussian wave packet of negative momentum for He cen-tered sufficiently far from the interaction region. Specifically,the incoming He wave packet has the form

fy~y;0!5a1/4

p1/4 e2~a/2!~y2y0!21 ip0~y2y0! /\, ~4.3!

wherey0524,a50.125, andp0523.56.Since the initialstate is not an eigenstate of the zeroth order Hamiltonian inthe y direction, the initial basis set for they coordinate wasconstructed by multiplyingfy(0) by y

k, k50,1,...; sincethis procedure does not yield orthogonal basis functions,these functions were orthogonalized by the Gram–Schmidtprocedure. Apart from minor differences in initial conditions,this model is essentially identical to the one considered byCampos-Martinez and Coalson.20 These authors demon-strated the rapid convergence achieved by using TDSCF-generated basis functions. Our goal here is to evaluate theadditional gain in efficiency offered by converting theTDSCF basis set into a TDSCF-DVR grid, such that thewell-known advantages of DVR methods can be fully ex-ploited.

Figure 1 shows the absolute square of the overlap be-tween the exact wave function and that obtained with thesimple TDSCF approximation, with a modest size TDSCF-DVR grid, and with time-independent PO-DVR grids. It isseen that the single-configuration TDSCF wave function de-viates significantly from the exact one as soon as the inter-action region is reached. However, as Campos-Martinez andCoalson have pointed out, the TDSCF evolution is qualita-tively correct and can lead to a useful basis set, such thatessentially exact results are obtained with a set of 5313TDSCF-generated basis functions or DVR points. This num-ber of grid points is by far not adequate if not allowed tomove during the collision process, and the results of a 5313static DVR are extremely poor. The superiority of theTD-DVR is apparent from Fig. 2, which shows the static andTDSCF-based grids at different times. The grid correspond-ing to modest size time-independent PO-DVR’s cannot ad-equately span all regions of space visited by the wave packet

FIG. 1. The absolute square of the overlap between the exact wave functionand that obtained with the various propagation schemes discussed in Secs. IIand III for collinear He–H2 scattering. The time units are given in femto-seconds and also in terms of the vibrational frequencyv0 of the H2 mol-ecule. Solid line, 5313 TDSCF-DVR. Triangles, single-configurationTDSCF approximation. Circles, 5313 time-independent PO-DVR. Dashedline, 203130 time-independent PO-DVR.

5621E. Sim and N. Makri: Time-dependent discrete variable representations

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during the collision. By contrast, the TD-DVR grid is con-structed to follow the motion of the evolving wave function,allowing very economical representation. A grid of at least203130 PO-DVR points is necessary in this case for semi-quantitative accuracy~5% error in the wave function over-lap! and only a larger PO-DVR grid can match the accuracyof the 5313 TDSCF-DVR. With such dense grids, DVR’susually offer no special advantages compared to uniform gridmethods. Thus, it is seen that allowing the DVR grid to movewith the wavepacket leads to reduction of the grid size by atleast a factor of 4 in the~bound! x coordinate and a factor of10 in the ~dissociative! y coordinate. Contour plots of thewave function obtained by the simple TDSCF approxima-tion, the 5313 TDSCF-DVR grid and the converged resultsof the split propagator method with a 1503250 uniform gridare shown in Fig. 3.

In order to compare the relative efficiency of the variousschemes discussed above we estimate the leading number ofoperations each scheme requires for accuracy. The static PO-DVR requires a minimum of 2600 points; thus, according toSec. II, the most time consuming step requires 43105 opera-

tions. Use of the TDSCF basis manages to shrink the basisset to 65, but since energy basis sets scale as full matrixmultiplication the total number of operations is about 43103

per time step. The TDSCF based DVR, on the other hand,combines compactness with favorable scaling and requiresonly 1170 operations. For completeness we note that the splitpropagator FFT method requires a grid of 1503250 and thusabout 43105 operations. Finally, the TDSCF-DVR scheme~as well as the TDSCF basis20! permits time steps that areconsiderably larger than those possible in the PO-DVR or theFFT-based split propagator algorithm.

In conclusion, this simple two-dimensional exampleshows that the use of TD-DVR grids, particularly a TDSCF-DVR in this case, can lead to significant savings in time-dependent studies of collision~or dissociation! dynamics.These savings should be even more pronounced in systemswith more degrees of freedom.

B. Modified Henon–Heiles potential

As a second test of the TD-DVR methodology we con-sider here the dynamics in a two-degree-of-freedom bound

FIG. 2. Static PO-DVR~open circles! vs dynamical TDSCF-DVR grid~solid circles! at various times.~a! v0t50; ~b! v0t53; ~c! v0t56; and~d! v0t59,wherev0 is the vibrational frequency of the H2 molecule.

5622 E. Sim and N. Makri: Time-dependent discrete variable representations

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potential with strongly anharmonic mode coupling. Follow-ing Meyer et al.,23~a! our model is described by a modified~bound! Henon–Heiles Hamiltonian of the form

H5 12 ~px

21py2!1 1

2 ~x21y2!1l~xy22 13 x

3!

1 116 l2~x21y2!2 ~4.4!

with l50.2. The Henon–Heiles Hamiltonian24 has provideda useful model for numerous classical or quantum studies ofnonlinear dynamics25 and should present a challenge to ourmethod.

The initial state is a displaced Gaussian wave packet ofthe form

C0~x,y!5a1/4

p1/4 e2~a/2!~x2x0!2

b1/4

p1/4 e2~b/2!y21 ip0y/\

~4.5!

with x051.8, p051.2, a50.89, andb52.08. Theinitialbasis set was constructed as described in Sec. IV A.

Figure 4 shows the survival probability,

P~ t !5u^C~ t !uC0&u2 ~4.6!

as obtained with the various schemes discussed above. Notethat one period of the uncoupled problem is equal to 2p. It isseen that the simple TDSCF approximation yields poor re-sults for times longer than one period of vibration. Neverthe-less, TDSCF provides a reasonable basis, and nearly quanti-tative results are obtained with a 9315 TDSCF-DVR grid. Inthis case, use of the TDSCF-generated basis~in the energyrepresentation! results in 23104 operations per time step,while conversion to a TDSCF-DVR requires only 3.43103

operations per time step. By contrast, if a static PO-DVR isused, a minimum of 14330 grid points are necessary for

FIG. 3. Contour plots of the evolving wave function obtained by~a! simpleTDSCF; ~b! 5313 TDSCF-DVR grid; and~c! exact results obtained withthe split propagator algorithm at selected times. The wave functions arelabeled by the value ofv0t, wherev0 is the vibrational frequency of the H2molecule.

FIG. 4. Time evolution of the survival probability for a displaced Gaussianwave packet on the modified Henon–Heiles potential~see Sec. IV!. TheTDSCF results are plotted as a dotted line. The dashed line indicates resultsobtained with a 14330 static PO-DVR grid. The results of a 9315 TDSCF-DVR are shown as markers. The solid line corresponds to exact quantumresults obtained with the split propagator algorithm.

5623E. Sim and N. Makri: Time-dependent discrete variable representations

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similar accuracy, requiring 1.83104 operations. Thereforeadopting the TD-DVR approach results again in significantreduction of numerical effort.

V. CONCLUDING REMARKS

The TD-DVR grid representations introduced in this pa-per offer significant advantages in wave packet propagation.The construction of basis sets that follow the motion of thewave packet allows accurate description of the dynamicswith minimal basis size. Transformation of such time-dependent basis functions into TD-DVR grids leads to maxi-mum efficiency, as the numerical effort at each time step ofpropagation becomes almost linear with grid size. At thesame time, the underlying use of suitable basis functions,which follow the exact dynamics of appropriate zeroth orderHamiltonians, allows large time steps, accelerating calcula-tions even further.

As with many other methods in quantum mechanics, thesuccess of the TD-DVR scheme depends critically on thechoice of zeroth order Hamiltonian. For TD-DVR to offeradvantages over time-independent grid methods one must beable to find a solvable reference Hamiltonian that leads to areasonable zeroth order description of the dynamics. Al-though TD-DVR’s are extremely efficient for dealing withwave packet spreading in position space, they will not facili-tate calculations in cases of strongly chaotic motion wherethe wave function spreads beyond control in the spacespanned byanyproduct-type basis.

In many cases the TDSCF ansatz leads to a semiquanti-tative picture of the dynamics. In such cases the TDSCFevolution of the initial state can be used to define a time-dependent reference Hamiltonian. The examples presented inSec. IV demonstrated the substantial reduction of grid sizeachievable by employing a TDSCF-DVR grid, compared tothe grid required if a static PO-DVR is used. Within theTDSCF-DVR scheme the simple TDSCF limit is recoveredwith a single grid point. Thus, the TDSCF-DVR approachprovides a natural and very efficient way of correcting theTDSCF approximation while retaining the almost linear scal-ing with grid size typical of DVR method. While multicon-figuration schemes23~a! ~MC-TDSCF! should generally con-verge with smaller basis sets, the TDSCF-DVR method mayoutperform them due to the more efficient scaling of DVRschemes. In addition, the methodology developed in this pa-per requires only one-dimensional unitary transformationsand basis propagation, leading to simple computer code.

We expect that TD-DVR methods will allow fully quan-tum calculations on larger molecular systems than currentlypossible by other approaches. Among many interesting ap-plications, the TD-DVR grid should prove very useful forstudying the dynamics of weakly bound polyatomic clusters.Such systems have already been studied with the simpleTDSCF approximation and its perturbative or semiclassicalcorrection.26 The dynamics of these clusters is characterizedby a strong classical component, such that wave packets re-main sufficiently localized during dissociation, and TDSCFis believed to be at least qualitatively correct. The TD-DVRmethod with TDSCF reference should provide an efficient

and systematic way of correcting the TDSCF zeroth orderapproximation, leading to exact quantum results.

Finally, we anticipate the ideas developed in this paperto prove useful in real time path integral calculations. A pow-erful path integral methodology for studying the dynamics ofa system coupled to a bath has been developed recently2

which relies on multidimensional DVR quadratures forevaluating the reduced dimension path integral over systemvariables. TD-DVR grids should offer optimal representa-tions for molecule–surface scattering path integral calcula-tions, and we are presently pursuing such problems.

ACKNOWLEDGMENTS

This work has been supported primarily by the NationalScience Foundation through a Young Investigator Award. Wewish to thank Professor Rob D. Coalson for useful discus-sions. N. M. also acknowledges generous funding by theArnold and Mabel Beckman Foundation, the David and Lu-cile Packard Foundation, the Alfred P. Sloan Foundation, andthe Research Corporation, and is grateful for a Beckman Fel-lowship at the Center for Advanced Study of the Universityof Illinois during the spring of 1994.

1See several articles in the special thematic issue on Time-DependentMethods for Quantum Dynamics, Comp. Phys. Commun.63 ~1991!.

2~a! N. Makri, Chem. Phys. Lett.193, 435 ~1992!; in Time-DependentQuantum Molecular Dynamics, edited by J. Broeckhove and L. Lathouw-ers ~Plenum, New York, 1992!, p. 209;~b! M. Topaler and N. Makri, J.Chem. Phys.97, 9001 ~1992!; Chem. Phys. Lett.210, 285 ~1993!; 210,448 ~1993!; J. Chem. Phys.101, 7500~1994!; ~c! D. E. Makarov and N.Makri, Phys. Rev. A48, 3626~1993!; ~d! G. Ilk and N. Makri, J. Chem.Phys.101, 6708~1994!.

3D. E. Makarov and N. Makri, Chem. Phys. Lett.221, 482 ~1994!; N.Makri and D. E. Makarov, J. Chem. Phys.~in press!.

4D. Thirumalai, E. J. Bruskin, and B. J. Berne, J. Chem. Phys.79, 5063~1983!.

5N. Makri, Chem. Phys. Lett.159, 489 ~1989!.6N. Makri, J. Chem. Phys.97, 2417~1993!.7O. A. Sharafeddin, D. J. Kouri, N. Nayar, and D. K. Hoffman, J. Chem.Phys.95, 3224~1991!; D. K. Hoffman, N. Nayar, O. A. Sharafeddin, andD. J. Kouri, ibid. 95, 8299~1991!; D. K. Hoffman and D. J. Kouri,ibid.96, 1179~1992!.

8M. D. Feit and J. A. Fleck, Jr., and A. Steiger, J. Comput. Phys.47, 412~1982!; M. D. Feit and J. A. Fleck, Jr., J. Chem. Phys.78, 301 ~1983!.

9D. Kosloff and R. Kosloff, J. Comput. Phys.52, 35 ~1983!.10E. O. Brigham,The Fast Fourier Transform~Prentice–Hall, EnglewoodCliffs, NJ, 1974!.

11J. Z. H. Zhang, J. Chem. Phys.92, 324~1990!; Chem. Phys. Lett.160, 417~1989!.

12~a! S. Das and D. J. Tannor, J. Chem. Phys.92, 3403 ~1990!; ~b! C. J.Williams, J. Qian, and D. J. Tannor,ibid. 95, 1721~1991!.

13D. O. Harris, G. G. Engerholm, and W. D. Gwinn, J. Chem. Phys.43,1515 ~1965!; A. S. Dickinson and P. R. Certain,ibid. 49, 4209~1968!.

14~a! J. V. Lill, G. A. Parker, and J. C. Light, Chem. Phys. Lett.89, 483~1982!; J. Chem. Phys.85, 900~1986!; ~b! J. C. Light, I. P. Hamilton, andJ. V. Lill, ibid. 82, 1400~1985!; ~c! S. E. Choi and J. C. Light,ibid. 92,2129~1990!; ~d! Z. Bacic and J. C. Light,ibid. 85, 4594~1986!; 86, 3065~1987!; Annu. Rev. Phys. Chem.40, 469 ~1989!.

15J. Echave and D. C. Clary, Chem. Phys. Lett.190, 225 ~1992!.16J. C. Light, R. M. Whitnell, T. J. Park, and S. E. Choi, inSupercomputerAlgorithms for Reactivity Dynamics and Kinetics of Small Molecules,NATO ASI Series C, edited by A. Lagana~Kluever, Dortrecht, 1989!, Vol.277, p. 187.

17U. Manthe and H. Koppel, J. Chem. Phys.93, 345 ~1991!.

5624 E. Sim and N. Makri: Time-dependent discrete variable representations

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18A. D. McLachlan, Mol. Phys.8, 39 ~1964!.19For a review of applications of TDSCF in chemical physics see the reviewby R. B. Gerber and M. A. Ratner, Adv. Chem. Phys.97, 4781~1992!, andreferences therein.

20~a! J. Campos-Martinez and R. D. Coalson, J. Chem. Phys.93, 4740~1990!; ~b! 99, 9629~1993!.

21N. Makri and W. H. Miller, J. Chem. Phys.87, 5781~1987!.22D. Secrest and B. R. Johnson, J. Chem. Phys.45, 4556~1966!.

23~a! H.-D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett.165,73 ~1990!; ~b! U. Manthe, H.-D. Meyer, and L. S. Cederbaum, J. Chem.Phys.97, 3199, 9062~1992!.

24M. Henon and C. Heiles, Astron. J.69, 73 ~1964!.25See, for example,~a! A. J. Lichtenberg and M. A. Lieberman,Regular andChaotic Dynamics~Springer, New York, 1991!; ~b! M. C. Gutzwiller,Chaos in Classical and Quantum Mechanics~Springer, New York, 1990!.

26Z. M. Li and R. B. Gerber, J. Chem. Phys.99, 8637~1993!.

5625E. Sim and N. Makri: Time-dependent discrete variable representations

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