Quantum dynamics with real wave packets, including application to three-dimensional (J = 0)D+H2→HD+H reactive scatteringStephen K. Gray and Gabriel G. Balint-Kurti Citation: J. Chem. Phys. 108, 950 (1998); doi: 10.1063/1.475495 View online: http://dx.doi.org/10.1063/1.475495 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v108/i3 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Quantum dynamics with real wave packets, including applicationto three-dimensional „J 50…D1H2˜HD1H reactive scattering

Stephen K. GrayTheoretical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439

Gabriel G. Balint-KurtiSchool of Chemistry, The University of Bristol, Bristol BS8 1TS, United Kingdom

~Received 25 July 1997; accepted 10 October 1997!

We show how to extractS matrix elements for reactive scattering from just the real part of anevolving wave packet. A three-term recursion scheme allows the real part of a wave packet to bepropagated without reference to its imaginary part, soS matrix elements can be calculatedefficiently. Our approach can be applied not only to the usual time-dependent Schro¨dinger equation,but to a modified form with the Hamiltonian operatorH replaced byf (H), wheref is chosen forconvenience. One particular choice forf , a cos21 mapping, yields the Chebyshev iteration that hasproved to be useful in several other recent studies. We show how reactive scattering can be studiedby following time-dependent wave packets generated by this mapping. These ideas are illustratedthrough calculation of collinear H1H2→H21H and three-dimensional (J50)D1H2→HD1Dreactive scattering probabilities on the Liu–Siegbahn–Truhlar–Horowitz~LSTH! potential energysurface. ©1998 American Institute of Physics.@S0021-9606~98!02803-7#

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I. INTRODUCTION

Accurate quantum dynamics methods have made anmense impact on the theory of molecular processes. Sigcant progress has occurred in developing and applying btime-independent and time-dependent quantum dynamThe use of time-dependent quantum dynamics,1,2 for ex-ample, has become progressively more popular and its usphotodissociation3–12 and reactive scattering13–20 has in-creased rapidly. Most time-dependent quantum dynamicsplications share the same general approach. An initial wpacket in some set of coordinatesx is set up, propagated intime by solving the time-dependent Schro¨dinger equation,

i\]c~x,t !

]t5Hc~x,t !, ~1!

and analyzed to extract relevant dynamical quantities. IfHamiltonian operatorH is independent of time, the wavpacket at timet1t is obtained from the wave packet at timt through

c~x,t1t!5exp~2 iH t/\!c~x,t !. ~2!

The wave packet approach above differs from traditiotime-independent scattering approaches in that a sipropagation generally does not yield information aboutpossible transitions, but rather about certain specific oSuch information is, however, obtained over a range ofergies in one calculation. This feature, which greatly simpfies matters, is one reason for the popularity of wave pacmethods. There are many similarities between wave pamethods and iterative methods for obtaining particular eigvalues and eigenvectors of large matrices. Iterative methsuch as the Lanczos method,21 involve some iteration pro-cess which is akin to a time propagation with discrete tistepst. The results of the iteration are analyzed in so

950 J. Chem. Phys. 108 (3), 15 January 1998 0021-9606/9

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fashion and the relevant ‘‘iteration-independent’’ informtion ~eigenvalues, eigenvectors! is inferred. The ‘‘spectro-scopic eigenvalue analysis’’ approach describedLanczos,22 for example, has all the features of the spectmethods used in wave packet calculations to infer boustates.23 In interesting recent work Chen and Guo24,25 havedeveloped approaches for obtaining bound and resonastates utilizing related ideas.

As the analogy to iterative methods above implies, italso possible to adopt the wave packet approach of targespecific dynamical information within a time-independeframework. A natural starting point is the time-independeGreen’s function.26–32 The approaches we develop in thpaper, while immersed in a time-dependent formalism, shcommon features with such time-independent methodsblurring of the distinction between time-independent atime-dependent approaches can occur when common stgies and numerical methods are employed.

The i 5A21 in Eq. ~1! forcesc to be, in general, com-plex. Typical wave packet calculations thus propagate bthe real and imaginary parts ofc. For large problems, thiscan be taxing both from computational and compumemory points of view. In this paper we show how it is onnecessary to evolve just the real part of the complete wpacket, and that this provides sufficient information to copute observables associated with the dynamics, includinSmatrices for reactive scattering processes. A three-termcursion allows the real part of the wave packet to be progated without reference to the imaginary part,8 and so ourideas allow one to compute dynamical information efciently.

We go on to show that the observable dynamical infmation~S matrix elements! may also be computed by solvina modified time-dependent Schro¨dinger equation in whichthe Hamiltonian is replaced by a function of the Ham

8/108(3)/950/13/$15.00 © 1998 American Institute of Physics

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951S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

tonian. The correct choice of functional mapping can dmatically simplify the propagation equations, permitting tdynamical information to be calculated in an efficient maner. In particular we choose the cos21 functional mappingdiscussed by Chen and Guo,24,25which leads to a Chebysheiteration. Chen and Guo24,25 used it to calculate bound staand resonance energies. This aspect of our work is aextension of the cos21 approach to more general scatteriprocesses.

Chebyshev iteration is the basis of Tal-Ezer and Ksloff’s original expansion of the time-dependepropagator.33 Recently, however, very important and usedevelopments have arisen through the use of Chebyshevrelated expansions in time-independent scattering theKouri and co-workers27,28 and Mandelshtam and Taylor29,30

have addressed how to properly include non-HermitHamiltonians or absorption at grid edges with variationsthe Chebyshev iteration and have focused on approximatime-independent Green’s functions. Our work also hmany features in common with this ‘‘time-independenwork and, in fact, can be viewed as an alternative, ‘‘timdependent’’ representation that leads to some of their imptant results. It is also important to note that Kroes, Nehauser, and co-workers have been working along sevrelated lines to the present work.32,34–36 For example, theyhave exploited the formalism of Kouri and co-workers27,28

and Mandelshtam and Taylor29,30 to use only real algebra in‘‘time-independent wave packet’’ propagation, and analyzthe performance of such approaches. This work also continsightful remarks on the similarities between timindependent and time-dependent approaches. They haveutilized the associated real algebra simplifications that ocwhen Mandelshtam and Taylor’s damped Chebyshev ittion scheme is used in constructing an actual time-depenwave packet based on a real initial condition.34–36 ~See alsothe related work in Ref. 8.! Furthermore, they have describean alternative method to that discussed later for extendthe effective propagation time when scattering resonanare present.35,36

Section II outlines the theory. Section III illustrates omethods with the computation of reaction probabilitifor the collinear H1H2→H21H and three-dimensiona(J50)D1H2→HD1H exchange reactions on the LiuSiegbahn–Truhlar–Horowitz ~LSTH! potential energysurface.37 A summary is given in Sec. IV. Appendices adevoted to more technical aspects.

II. THEORY

A. Quantum time evolution using only the real part ofthe wave packet

The starting point for our work is

c~x,t1t!52c~x,t2t!12 cosS Ht

\Dc~x,t !, ~3!

which is easily derived by adding Eq.~2! and its backwards(t→2t) counterpart.8 Unlike Eq. ~2!, Eq. ~3! does not in-volve i , which means that the real and imaginary parts oc

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can be propagatedindependentlyof one another given thaone has, in addition to an initial condition att50, the resultof the first time stept5t. Introducing the notation38,39

q~x,t !5Re@c~x,t!#,~4!

p~x,t !5Im@c~x,t!#,

whereq andp are always real-valued functions, one obtainfrom Eq. ~3!,

q~x,t1t!52q~x,t2t!12 cosS Ht

\D q~x,t !, ~5!

and a similar equation involving onlyp.In order to evolve just the real part,q, given some initial

conditionc(t50)5q(0)1 ip(0), onefirst obtainsq(t) viaEq. ~2!, which leads to

q~x,t!5cosS Ht

\D q~x,t50!1sinS Ht

\D p~x,t50!, ~6!

and then Eq.~5! may be repeatedly applied to obtainq(t) indiscrete time stepst.

Equation~3! must be modified for scattering applicationthat use absorption techniques. Absorption is necessary wa finite grid is used to represent relevant scattering coonates and wave packet amplitude must be prevented fbeing reflected back into the interaction region as it aproaches the grid edges.~Absorption also allows one to focus on just one scattering channel.40! Reference 8 discussedmodified Eq.~5!,

q~x,t1t!5AF2Aq~x,t2t!12 cosS Ht

\D q~x,t !G , ~7!

which leads to a particularly stable iteration procedure.representation ofA, which damps wave packet amplitudnear grid boundaries, is given in Sec. III.

B. Functional mapping of the Hamiltonian—the cos 21

representaion

When using time-dependent quantum dynamics to calate time-independent observables the role of the timemainly in the fact that the problem becomes an initial vaone, and a single calculation provides information overange of energies. Because time does not enter in any fumental way into the observables which are to be calculait is permissible to substitute a modified time evolution eqution for the time-dependent Schro¨dinger equation. ReplacingH with f (H) in Eq. ~1!, we obtain

i\]c~x,t !

]t5 f ~H !c~x,t !. ~8!

@Note: the ‘‘c’’ that satisfies Eq.~8! is not the same functionthat satisfies Eq.~1!, although we will see that similar information is present.# As long asf (E) is a one-to-one mappingeach eigenvalueE of H will be mapped onto one uniquvalue f (E). Furthermore,H and f (H) have the same station

No. 3, 15 January 1998

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952 S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

ary state eigenvectors. The particular function chosen isone which will simplify the iterative form of the Schro¨dingerequation, Eq.~3!, namely,

f ~H !52\

tcos21~Hs!, ~9!

with

Hs5asH1bs , ~10!

whereHs is a scaled and shifted Hamiltonian operator suthat its minimum and maximum eigenvalues lie between21and 1. If Emin and Emax are lower and upper bounds to thspectrum ofH, then as52/DE and bs5212asEmin withDE5Emax2Emin . Hs is the same scaled Hamiltonian as thused in the Chebyshev expansion of the propagator.33 ~Inmost wave packet applications,H is represented with discrete grids and basis functions, which leads to a finite valfor Emin andEmax.! The cos21 operator in Eq.~9! is assumedto be consistent with the principal values of the cos21(x)function, which, forx in the range (21,1) can be defined bya Taylor series about the pointx50. If cE is a stationarystate ofH with eigenvalueE, then it will also be a stationarystate of the abovef (H) with eigenvalue

f ~E!52\

tcos21~Es!, ~11!

with Es5asE1bs . We place a minus sign in front of thcos21 so that f (E) is a monotonically increasing functiowith E, ranging from f (Emin)52p \/t to f (Emax)50. Thisconvention leads to ‘‘incoming’’ and ‘‘outgoing’’ wavesalong a scattering coordinateR being associated withexp(2iukuR) and exp(1iukuR) plane wave terms, respectivel

With this choice forf (H), Eq. ~3! may be rewritten inthe simplified form

c~x,t1t!52c~x,t2t!12Hsc~x,t !. ~12!

Repeated application of this three-term recursion propagthe wave packet~or its real part only! forward in time stepst. Actually, the discrete dynamics isindependentof the valueof t, as is clear from Eq.~12!. Thus t is not absolutelynecessary and, for example, one could simply sett51 in allequations. However, we retaint because it is useful inchecking the correctness of our equations~e.g., observablesshould not depend ont! and because it is a reminder of thtime-dependent Schro¨dinger equation origins of the theoryEquation ~12! is also the Chebyshev iteration which hproved useful in recent scattering and resonance work24,25,29

and forms the basis of Tal-Ezer and Kosloff’s33 expansion ofthe propagator.

Thus the recursion relation for the propagation of treal part of the wave packet@see Eq.~5!# is

q~x,t1t!52q~x,t2t!12Hsq~x,t !. ~13!

If one is applying absorption, the equation, which is the mequation used in our calculations, is the analog of Eq.~7!,

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q~x,t1t!5A~2Aq~x,t2t!12Hsq~x,t !!. ~14!

Equation 14, with an appropriate choice for the represention of A, is the same damped Chebyshev recursion asone developed by Mandelshtam and Taylor in a differcontext.29 The initial step in the iterative process is

q~x,t!5Hsq~x,t50!2A12Hs2p~x,t50!, ~15!

which is the analog of Eq.~6!. The negative square rooarises because sin@2cos21(x)#52A12x2, where cos21(x) isthe principal value. If it is needed, it can be evaluated wan appropriate series representation.

C. Extraction of reactive S matrix elements

One approach to inferring observable information frowave packets is outlined in Ref. 5. In this section and Apendices A and B, we show how such ideas apply to justreal part of a wave packet evolving according to the usSchrodinger equation, and then outline the necessary mofications when a functional mapping,f (H), in particular thatgiven by Eq.~9!, generates the dynamics.

We specialize toA1BC reactive scattering, although thideas can be extended to other situations. The reactant cnel, corresponding to A1BC, is denoted by ‘‘a’’ and theproduct channel, corresponding to, say, AB1C, is denotedby ‘‘ c. ’’ Internal reactant~BC! states are labeledI and in-ternal product~AB! states are labeledF. The reactant Jacobcoordinates areRa5uRau, the distance between A and thcenter of mass of BC, andra , denoting the internal BC coordinates. The product Jacobi coordinates areR5uRu, thedistance between C and the center of mass of AB, andr , theinternal AB coordinates. Thera and r coordinates includethe relevant angles betweenRa andra , andR andr . Figure1 shows a potential energy contour map for collineH1H2→H21H in product Jacobi coordinates R,r. A possibanalysis lineR5R` is indicated in the product channel.

Consider an initial wave packetc(t50;I ) in the reac-tant channel a with BC in some particular interna~vibrational–rotational! stateI . First consider the case thathis wave packet is evolved according to the usual Sch¨-dinger equation, Eq.~1!. Wave packet amplitude along thanalysis line is denoted byc(R5R` ,r ,t;I ). Suppose the full~complex! wave packet along this cut is projected onto tfinal product states,fF(r ), to produce a set of timedependent coefficientsCF,I(t),

CF,I~ t !5E drfF* ~r !c~R5R` ,r ,t;I !. ~16!

These coefficients may be half-Fourier transformed otime to give energy-dependent coefficientsAF,I(E),

AF,I~E!51

2p E0

`

dt exp~ iEt/\!CF,I~ t !. ~17!

Appendix A outlines the technical analysis that leads toreactiveS matrix elements being given by the simple relati

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953S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

SF,Ic,a~E!52S \2kF

c kIa

mcma D 1/2

exp~2 ikFc R`!

AF,I~E!

g~2kIa!

, ~18!

where kIa and kF

c are wave vector components associawith the reactant and product channels and a total energma, and mc are ~A,BC! and ~AB,C! reduced masses, ang (2kI

a) is proportionate to the overlap of the initial wavpacket with incoming waves with wave vector2kI

a , Eq.~A7!.

Suppose now that we continue to consider ‘‘ordinarwave packet dynamics consistent with Eq.~1!, but use justthe real part,q(t), for analysis. The analog of Eq.~17! is

AF,Iq ~E!5

1

2p E0

`

dt exp~ iEt/\!CF,Iq ~ t !, ~19!

where

CF,Iq ~ t !5E drfF~r !q~R5R` ,r ,t !. ~20!

Appendix A shows that the reactiveS matrix elements arethen given by

SF,Ic,a~E!52S \2kF

c kIa

mcma D 1/2

exp~2 ikFc R`!

2AF,Iq ~E!

g~2kIa!

. ~21!

Comparing with Eq.~18!, the only difference when just threal part is used is an extra factor of 2.

Finally, consider the case of the real part of a wapacket generated via the modified Schro¨dinger equation ofSec. II B, i.e., quantum dynamics withH replaced by thefunctional mappingf (H). The corresponding analysissimilar to that above and in Appendix A, with the on

FIG. 1. Potential energy contour map for the collinear H1H2 reaction inJacobi coordinates. The LSTH potential surface is employed. The figuredepicts where an analysis lineR5R` might be drawn in the product regionand superimposes contours of the real part of an initial wave packet inreactant channel corresponding to a typical initial condition used incalculations.

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dE,

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change being thatE andH are everywhere replaced byf (E)of Eq. ~11! and f (H) of Eq. ~9!. Similarly, the continuumscattering functions entering into this detailed analysis mbe delta function normalized41 on the ‘‘f ’’ scale, as opposedto the E scale. The relation betweenE and f normalizedcontinuum wave functions isc(E)5ud f /dEu1/2c( f ), andsince theS matrix is an overlap of two continuum functionone has, assuming also that the dynamics has been geneby the functional mapping Eq.~9!,

SF,Ic,a~E!5ud f /dEuSF,I

c ~ f !

52\2as

t~12Es2!1/2S kF

c kIa

mcmaD 1/2

3exp~2 ikFc R`!

2AF,Iq ~ f !

g~2kIa!

, ~22!

where

AF,Iq ~ f !5

1

2p E0

`

dt exp~ i f t /\!CF,Iq ~ t !, ~23!

with CF,Iq now corresponding to the projection of the re

wave packetq at R5R` onto AB internal stateF.In practice, it is often the case that a time integral su

as that in Eq.~23! is evaluated within a trapezoidal rule~ordiscrete Fourier transform! approximation. One approacwould be to write

AF,Iq ~ f !←

t

2p (k50

N

~12dk,0/2!exp~ i f kt/\!CF,Iq ~kt!,

~24!

whereN is the number of iterations taken. In ordinary wapacket propagation, it is sometimes possible to evaluaterelevant variables to be transformed, such as correlafunctions, at intermediate times between time steps andgo beyond trapezoidal rule accuracy.1 In the present contextthe replacement Eq.~24! is, however, much more accurathan it appears. In fact, assumingN is sufficiently large andall the underlying assumptions used in the product analare satisfied,no approximation is introduced by this procedure. It turns out that one can formulate a purely discrbased definition ofAF,I

q ( f ) @the right-hand side of Eq.~24!#and carry out all the formal manipulations of Appendixwith the time integrals replaced by the corresponding dcrete sums. The existence of a particular discrete Fourieridentity involving delta functions, which is the analog of thkey relation used in Appendix A to formally evaluate thcontinuous time integrals, then makes the discrete theoryact. Appendix B sketches out these technicalities.

Notice that when Eq.~24! is inserted into Eq.~22! the tfactors cancel, leaving only thet factor in CF,I

q (kt). How-ever, as noted in Sec. II B, the discrete dynamics is invarto t and one could simply sett51. This result is physicallyreasonable since the energy resolved properties shoulddepend on the discrete time stept. Moreover, our resultingSmatrix expression is very similar in appearance to the reof employing Mandelshtam and Taylor’s damped Chebys

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954 S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

expansion of the time-independent Green’s function in Koand co-workers’ Kohn variational principle form forS ma-trix elements.29,30,28See also Kroes and Neuhauser,32 who inessence use a similar type of asymptotic product analyswe do along with these time-independent ideas. The mdifference with respect to all this work is that we have dveloped such expressions from a very different premnamely, the time-dependence of an effective Hamiltongiven by 2(\/t)cos21(Hs), coupled with established methods of analyzing time-dependent wave packets. Ourproach also, in principle, allows for the real part of a coplex initial condition to be propagated, although our specapplications~Sec. III! do not make use of this feature.

It is also possible toextrapolatethe calculated final stateamplitudesCF,I(kt) to k.N if N is sufficiently large suchthat, on average, the coefficients are in the process of daing out. A procedure for this is outlined in the Appendixbased on using Prony’s method.8,42 Such approaches, whilrequiring some care in their application, are very effectiveextracting the greatest possible amount of information ourelatively short iteration sequences.

Despite the derivation of the equations being somewinvolved, we note that the overall procedure is straightfward. One devises some initial condition in either the usmanner of time-dependent quantum mechanics, e.g., ancoming wave packet or~as in Sec. III! a superposition ofincoming and outgoing terms in the entrance channel. Othen propagates the associated real part with Eq.~14!. Spe-cific final state amplitudes in the asymptotic product chanare calculated at each iteration and saved for later analThe subsequent analysis consists of evaluating Eq.~24! atthe specificf (E) values of interest, which leads to theSmatrix, Eq.~22!. The following two examples serve to illustrate these points.

III. ILLUSTRATIVE CALCULATIONS

A. The collinear H 1H2˜H21H reaction

The collinear H1H2 reaction is an important test casfor new reactive scattering methods~see, e.g., Ref. 43!, andis perhaps not as trivial as it may seem because of the pence of sharp resonance features. We are interested intaining state resolved reaction probabilities for the H1H2

reaction,Pv I ,vF

R (E)5uSvF ,v I

c,a (E)u2, wherev I and vF denote

reactant and product vibrational states. Product Jacobi cdinatesR and r are adopted for theentire calculation.~Thisis convenient for obtaining the state-to-state reaction prabilities. For larger problems, other strategies might bequired.! Evenly spaced grids,Rk5kDR, k51,. . . ,NR , andr k5nDr , n51,...,Nr , are employed. The Hamiltonian operator becomes a simpleNRNr3NRNr matrix with elements

Hk,n,k8,n85dn,n8Tk,k8R

1dk,k8Tn,n8r

1dk,k8dn,n8V~Rk ,r n!,~25!

whereTR andTr are appropiate kinetic energy matrices cosistent with the grids.44–46Alternatively, and especially ifNR

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or Nr need to be large, Fourier transform techniques1,23couldbe used.V(R,r ) is taken to be the LSTH potential energsurface.37

The dynamics of the real part of the wave packet,q(t),is calculated assuming it arises from some particular iniand, in general, complex-valued wave packet,c~0!. While itis natural to employ, say, an incoming Gaussian wave pain the entrance channel forc~0! it is computationally expe-dient to employ areal-valued initial c~0!. Thus, q(0)5c(0). With this initial condition for c~0!, then q(t),which is needed to start the three-term recursion Eq.~14!, issimply q(t)5Hsq(0), whereHs is the scaled Hamiltonianmatrix representation of Eq.~10!. Kroes and Neuhauser,32 intheir time-independent wave packet work, employed esstially the same initial condition.@In our work, as shown inSec. II B, it is clear how to employ a complex-valuedc~0!,and then consider propagation and analysis of just thepart. The evaluation of the square root operator in Eq.~15!via an appropriate series representation would involve soadditional actions ofHs on a vector.# We thus take the initialwave packet in the reactant channel is taken to be real fution,

c~Ra ,r a ;t50;v I !5g~Ra ;k0 ,R0 ,a!fv I~r a!, ~26!

whereg is

g~Ra ;R0 ,a!5p21/4a21/2 expS 2~Ra2R0!2

2a2 D cos~k0Ra!.

~27!

Equation ~27! contains equal parts of a Gaussian wapacket moving inward inRa with wave vector2k0 and out-ward in Ra with wave vector1k0 . The initial condition iscentered atRa5R0 and the full width at half-maximum of itssquare is 2aAln 2.

The propagation is carried out in product Jacobi coornates, so that we must transform fromq(Ra ,r a ,t50;v I) toq(R,r ,t50;v I). Since](Ra ,r a)/](R,r )51 for the collinearcase, this simply means thatq(R,r ,t50;v I)5q(Ra ,r a ,t50;v I), whereRa5R/213r /4 andr a5R-r /2, using massesappropriate to H1H2. Thus for each grid point inR,r , onecalculates the correspondingRa and r a values and thenevaluates Eq.~26! to obtain the initial vectorq(t50). Wethen determineq(t)5Hsq(0), and repeatedly iterate withEq. ~14! in the obvious matrix-vector form consistent witthe above grid representation. Absorption is carried outboth the reactant and product channels, which correspondabsorbing in both the largeR andlarger limits. We employan absorption of the form

An,k,n8,k85dn,n8dk,k8AR~Rk!Ar~r n!, ~28!

with

AR~Rk!5 H exp@2CabsR ~Rk2Rabs!

2#, Rk.Rabs

1, Rk<Rabs~29!

and a similar equation forAr(r n). At each time step weevaluate the projections of the product vibrational stafvF

(r ) ontoq at the pointR5R` , which is just a set of rea

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955S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

numbersCvF

q corresponding to each vibrational statevF .

These numbers are saved for later analysis. After a proption has been carried out for a sufficient numberN of timesteps~or iterations!, in a separate calculation, we scan ovthe desired total energiesE, constructingf (E) via Eq. 11,and theS matrix elements with Eqs.~24! and ~22!. Noticethat with the initial condition chosen,g (k) is analytical andgiven by

g~k!51

2A a

2p3/2 Fexp@2 i ~k2k0!R0#

3expS 2a2~k2k0!2

2 D1exp@2 i ~k1k0!R0#

3expS 2a2~k1k0!2

2 D G , ~30!

which reflects the fact that the initial condition has bothcoming and outgoing components.~One of the terms willgenerally be very small.!

A variety of grid sizes and other details were examinExcellent results~see the next paragraph! were obtained withthe values listed in Table I. The initial wave packet wdesigned to have significant amplitude in thekv I

a range of

relevance to the desired energy interval to be examined.v I50, the range of energiesE of relevance is from abou0.45 to 1.8 eV~including the zero-point energy of H2, 0.270eV!. The Gaussian@Eq. ~27!# parameters listed in Table I leto an initial mean total energy of about 1 eV, with the wapacket having non-negligible amplitude in the energeticgion of interest.

Table II presents reaction probabilities at three differenergies and illustrates how convergence occurs withN. Ac-curate time-independent results28 based on the Kohn variational principle46–48are also given. ByN51500 the reactionprobabilities are converged to within60.005, and byN53000 the probabilities are converged to within60.001.The agreement of our results with the time-independentsults is excellent, with differences of60.002.~Of course, theconvergence is not strictly uniform over all energies owito sensitive resonance features.! We have also compared ouresults with the reaction probabilities calculated via tim

TABLE I. Parameters used in the collinear H1H2 and three-dimensionaD1H2 applications discussed in the text.

NR5Nr580, DR5Dr 50.15 a.u.Rabs5r abs58.5 a.u.,Cabs

R 5Cabsr 50.015 a.u.

Nj550 associated Legendre polynomials~three dimensional!Vcut56 eVa

(Emin /eV,Emax/eV)5(0.24,19)~collinear!, ~0.22,27! ~threedimensional!R056.5 a.u.,a50.5 a.u.uk0u57.94 a.u.~collinear!, 9.72 ~three dimensional!R`56.5 a.u.

aThe total potential is set equal toVcut if it exceedsVcut . The energy zero isseparated H1H2 with H2 at its equilibrium internuclear distance. In ththree-dimensional calculations, the same energetic cutoff is also appliethe diagonal centrifugal terms in Eq.~31!.

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a-

r

-

.

or

-

t

e-

-

independent techniques by Bondi and Connor49 and obtainedvery good agreement. For example, a comparison ofP0,0

R (E) points ranging from 0.43 to 2 eV had a root mesquare error of60.003 and a maximum error of 0.02, stijust a 3% difference, at the sharp resonance feature neaeV. It is also interesting to note that if one extrapolates oN5500 results via the technique discussed in the AppenC, results with the general quality of anN51500 propaga-tion are obtained. Figure 2 displays our calculatedP0,0

R (E)reaction probabilities over a wider range of energy. WhFig. 2 is consistent with our longest (N53000) propagation,the corresponding results forN51500 and alsoN5500 ifthe extrapolation trick is used, are essentially the same onscale of the graph.

It should be pointed out that our most accurate rescorresponded to relatively large grids with bothR and rranging between 0 and 12 a.u. with the total number of gpoints being 8038056400. ~Similar remarks here also apply to the three-dimensional calculations below.! Obviously,we could have omitted grid points46 at very smallR and rvalues, or where the potential is large. The maximumR andr values employed were also generous. If one uses onlypoints in each of theR and r grids, from 0 to 10 a.u., andcorrespondingly reduces other relevant parameters (R0 andR` set to 6 a.u.,Rabs set to 7 a.u.! reaction probabilities thagenerally agree to within6 0.003 with respect to the largegrid results in Table II are obtained and, e.g., the correspoing P0,0

R (E) curve is almost indistinguishable from Fig. 2.is therefore likely that smaller grids could also be employwe have not made a systematic study to obtain the smapossible grid. If only product coordinates~as here! are em-

to

TABLE II. Selected reaction probabilitiesPv I ,vFfor the collinear

H1H2(v I50)→H2(vF)1H(vF) reaction on the LSTH potential energsurface.N indicates the number of iterations used.

E/eV P0,0 P0,1 P0,2

N55000.4 0.0841 ••• •••1.1 0.312 0.389 •••1.4 0.0772 0.225 0.139

N5500, with extrapolation~see Appendix C!0.4 0.0843 ••• •••1.1 0.294 0.379 •••1.4 0.0597 0.224 0.103

N515000.4 0.0844 ••• •••1.1 0.294 0.380 •••1.4 0.0598 0.223 0.102

N530000.4 0.0844 ••• •••1.1 0.294 0.380 •••1.4 0.0598 0.223 0.108

S-matrix Kohn variational principlea

0.4 0.0830 ••• •••1.1 0.296 0.380 •••1.4 0.0596 0.224 0.107

aKohn variational principle results given in Ref. 28.

No. 3, 15 January 1998

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thhedad

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956 S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

ployed, the grids must be large enough to at least includeinitial wave packet. However, there is no reason why ot‘‘tricks’’ for avoiding large grids could not be employewithin the present context. For example, one could propagin the reactant coordinates initially, and then switch to prouct coordinates at an appropriate time.17 More sophisticatedconsiderations, such as those in the time-independentproaches in Refs. 28 and 29, can involve initial wave packthat require smaller grids.

Finally we note that, despite the use of the simple pruct coordinate representation and making no effort to redthe grid sizes, the collinear calculations are computationvery fast. AnN51500 calculation~with NR5Nr580! takes19 s of computer time on one processor of a Silicon GraphOrigin 200 computer with 180 MHz RS 10000 processor

B. Three-dimensional „J 50…D1H2˜HD1H

As a more challenging illustration of our approach, wchose to examine D1H2→HD1H in three dimensions withtotal angular momentumJ50. We now have a Hamiltonianoperator that is a function of the product Jacobi coordinaR, r , andg, whereg is the angle betweenR and r . The gdegree of freedom could be treated with a basis set ofmalized Legendre polynomials,P j (cosg), j 50,1,2,...,Nj

21. This has the disadvantage in that one must store rtively large potential matrices and, for some problems, escially if J.0, this can be a serious bottleneck if thereinsufficient computer memory available. An alternative isuse a grid of cosg values based on a Gauss–Legenquadrature rule for the potential matrix,50 which leads to thepotential energy being diagonal in all the coordinate degr

FIG. 2. The energy-dependent reaction probability for the collinH1H2(v I50)→2(vF50) exchange reaction on the LSTH potential surfa~Ref. 37!, calculated with the real wave packet method discussed in the

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isr

te-

p-ts

-e

ly

s

s

r-

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es

of freedom. We implemented both types of approach,outline only the basis set approach since the equationssimpler. The Hamiltonian matrix is

Hn,k, j ,n8,k8, j 85dn,n8d j , j 8Tk,k8R

1dk,k8d j , j 8Tn,n8r

1dn,n8dk,k8d j , j 8

j ~ j 11!

2 S 1

mRk2 1

1

mrn2D

1dk,k8dn,n8Vj , j 8~Rk ,r n!, ~31!

whereVj , j 8 represents the matrix elements between normized Legendre polynomialsP j and P j 8 .

We have carried out calculations to determine reactprobabilities from reactants withv I5 j I50. The initial con-dition, again consistent with a real initialc~0!, is similar tothe collinear initial condition above~since thej I50 normal-ized Legendre polynomial is a constant! is similar to thecollinear initial condition :

c~Ra ,r a ,ga ;t50;v I50,j I50!

5q~Ra ,r a ,cosga ;t50;v I50,j I50!

5g~R;k0 ,R0 ,a! P j I50~cosga!fv I50~r a!

5g~R;k0 ,R0 ,a!fv I 50~r a!

1

&

. ~32!

@The correspondingg (k), required for the final state analysis, is the same as before, Eq.~30!.# It is slightly more com-plicated to transform between product and reactant coonates in this case. First, note that](Ra ,r a ,cosga)/](R,r,cosg)5R2r2/(Ra

2ra2) leads to the relation

c~R,r ,cosg;t50!5Rr

Rar ac~Ra ,r a ,cosga ;t50!.

~33!

Thus we scan through theR, r , and cosg grid points, withthe latter chosen to be Gauss–Legendre quadrature poevaluate the correspondingRa ,r a ,cosga points via the cor-responding geometric relations and then evaluate the inwave function in the product coordinates via Eqs.~32! and~33!. If a normalized Legendre function basis set is beiemployed, we then expandc in terms ofNj such functions.The remaining details—propagation and analysis—arestraightforward extension of the collinear case above,need not be further outlined except to note that it is necesto multiply the calculated reaction probabilities by a factor2 in order to account for the fact that there are two possHD product channels, and the calculation is based onone channel.

Table I lists the relevant grid and other details, manywhich were the same as in the collinear calculations. TaIII presents reaction probabilities~from v I5 j I50! into finalHD vibrational statesvF50 and 1, summed over all finalj F ,and compares them to time-independent scattering resulKress et al.51 and Zhang and Miller.52 The agreement isclearly very good. Figure 3 shows the three-dimensional

r

t.

No. 3, 15 January 1998

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957S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

action probabilities on a finer scale, revealing the oscillatstructures seen and discussed in the work of ZhangMiller.47,52

Regarding convergence with respect toN, the three-dimensional dynamics converges much faster than the coear dynamics, presumably due to the fact that sharp~long-lived! resonance features are less important. Thus,N51000

TABLE III. Three-dimensional (J50) reaction probabilities for D1H2(v I

50,j I50)→H1HD(vF). ~Results are summed over all finalj F values.!

E/eV This work Kresset al.a Zang and Millerb

vF500.6 0.083 0.0808 0.0830.7 0.394 0.392 0.3940.8 0.421 0.420 0.4210.9 0.483 0.483 0.4841.0 0.470 0.469 0.4701.1 0.473 0.473 0.4781.2 0.485 0.483 0.4851.3 0.476 0.474 0.4881.4 0.495 0.494 0.5091.5 0.475 0.476 0.472

vF510.9 0.036 0.0349 0.0361.0 0.085 0.0851 0.0841.1 0.109 0.108 0.1081.2 0.133 0.134 0.1341.3 0.155 0.158 0.1551.4 0.157 0.161 0.1611.5 0.186 0.186 0.178

aReference 51.bReference 52.

FIG. 3. The energy-dependent final vibrational state reaction probabilfor the three-dimensional (J50)D1H2(v I50,j I50)→HD(vF5v)1H re-action on the LSTH potential energy surface, calculated with the real wpacket method discussed in the text.~We have summed over the variousj F

final state probabilities.!

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ynd

n-

iterations serves to converge the reaction probabilities.of the extrapolation trick~Appendix C! also works, and al-lows one to obtain comparable results with only 600 itetions. N51000 iterations of the three-dimensional problerequired 30 min to complete on the same Silicon Graph~RS10000 processor! machine used in the collinear calculations. As with the collinear calculations, we should emphsize that smaller grids could be used, with a correspondreduction in computational effort.

IV. SUMMARY

We discussed how one can generate scattering dynainformation from just the real part of an evolving wavpacket. We showed that it is possible to compute reactivSmatrix elements, and thus reaction probabilities and crsections, by analyzing of just the real part of the wapacket. Thus, depending on the information desired, it shosuffice in many wave packet calculations to simply propgate the real part. This can be done efficiently in a varietyways, e.g., via a three-term recursion or perhaps by diexpansion of the relevant cosine operator.8

It may seem surprising that so much can be inferfrom just the real part of a wave packet since the full wapacket is a complex-valued function. In the case of bouand even resonance states it is less surprising, since it shbe evident that Fourier transformation of just the real pwill contain the bound or quasibound state information.the case of scattering, which is more complicated, weable to extractS matrices or transition amplitudes from threal part because of causality,53 which builds in a relationbetween the real and imaginary parts. That is, we setproblem up such that there is initially no amplitude in somchannel of interest so that, for later times one can assuoutgoing waves are present in the channel. It is this simprinciple that made our analysis possible.~Causality is alsoused to prove various dispersion relations, i.e., relationstween real and imaginary parts of causal functions.53!

We went further, however, by discussing such real wapacket analysis in relation to Schro¨dinger equations withHreplaced byf (H). We specialized tof given by the interest-ing cos21 mapping previously introduced by Chen anGuo,24,25 and showed how to obtain reactiveS matrix ele-ments. This mapping is particularly convenient since it leato the Chebyshev iteration that has also been so effectiused in certain time-independent scattering theories,29,30 andthe Chebyshev expansion of the time-dependpropagator.33 One view of our theory is that it is extractininformation from the same iteration equation as in the timindependent theory of Mandelshtam and Taylor,29,30 exceptthat our intial conditions are different, and motivatedtime-dependent ideas and also our final state analysis isferent, and similarly motivated. Our approach also represean alternative, ‘‘time-dependent’’ way of obtaining equatiosimilar to those developed and applied by Kouri aco-workers,27,28Mandelshtam and Taylor,29,30and Kroes andNeuhauser.32

s

e

No. 3, 15 January 1998

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958 S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

We believe our approach allows us to retain the attrtive simplicity, particularly regarding initial and final statanalysis, of a time-dependent theory using wave pacwhile also ensuring good numerical efficiency. This simplity, of course, does have an associated disadvantage inthe grids employed in our approach have to be large enoto include a wave packet in the entrance channel. Tproved to be no difficulty for our test applications, includina nontrivial three-dimensional case.~In fact the simplicity ofour basic approach led to the three-dimensional case beivery straightforward extension of the collinear one.! How-ever, there is no reason why other strategies developeordinary wave packet propagation methods for treatingdifficulty could not be adopted in the present context.

We illustrated our approach with two reactive scatterproblems, the ‘‘standard’’ collinear H1H2 exchange reactionand the more challenging three-dimensional (J50)D1H2→HD1H reaction. In both cases our reaction probabties for individual transitions agreed well with availabtime-independent calculations on the same systems.

The efficiency and favorable computer memory requiments, at least in comparison with the most straightforwtime-dependent quantum approaches, suggest that ouproach should be of use for a variety of other problems.example, we plan to study the reactions of O1H2 with ourapproach, and also to extend it to total angular momeJ.0.

ACKNOWLEDGMENTS

S.K.G. thanks the EPSRC for a visiting fellowshwhich enabled this research to be undertaken. S.K.G.also supported by the Office of Basic Energy Sciences,vision of Chemical Science, U.S. Department of Energy,der Contract No. W-31-109-ENG-38. The authors thankEPSRC for computer time on the Columbus computer atRutherford Laboratory, provided under a grant to the Coputational Chemistry group. We are also grateful to J.NConnor, A. Ja¨ckle, and H-D. Meyer for providing us withcollinear reaction probabilities for comparison with our rsults.

APPENDIX A

This Appendix outlines how analysis of a complex wapacket evolving according to the usual Schro¨dinger equation,Eq. ~1!, can yieldS matrix elements, and how this analysismodified to treat just the real part of the wave packetrelies on ideas developed for photodissocation problem5

and is a more explicit, general version of an earlier discsion of the reactive scattering case.16 The coordinates andnotation introduced in Sec. II C are employed. Throughoasymptotic scattering functions describingRa and R aretreated as one-dimensional, Cartesian-like functions withsociated normalizations, which makes the analysis a lsimpler. ~The full radial functions are our wave functiondivided byRr or Rar a .! The use of such functions withoureference to, e.g., spherical harmonic expansions impliesour results are most directly applicable to the collinear a

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-

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J50 cases. Additional phase considerations and a moreplicitly radial treatment may be required in the associateJ.0 case.

The central object is the half Fourier transform of prouct state amplitudes in the asymptotic product region,~17!, which may also be expressed as

AF,I~E!51

2p E0

`

dtE0

`

dR exp~ iEt/\!d~R2R`!

3S E drfF* ~r !c~R,r ,t;I ! D . ~A1!

Writing c(t) as the initial wave packet acted on by the timevolution operator yields

AF,I~E!51

2p E0

`

dtE0

`

dR exp~ iEt/\!d~R2R`!

3E drfF* ~r !exp~2 iH t/\!c~Ra ,ra ,t50;I !.

~A2!

We now use the identity operator53

I 5E2`

`

dE8(F

ucF2~E8!&^cF

2~E8!u, ~A3!

with scattering functionscF2(R,r ,E8)5^R,r ucF

2(E8)&, nor-malized on the the energy scale,41 having asymptoticform54,55

cF2~R,r ,E!

;R→`

S mc

2p\2kFc D 1/2Fexp~ ikF

c R!fF~r !2(F8

~SF,F8c,c

!*

3S kFc

kF8c D 1/2

exp~2 ikF8c R!fF8~r !G

;Ra→`

2S ma

2p\2D 1/2

(I 8

~SF,I 8c,a

!*

~kI 8a

!1/2 exp~2 ikI 8a Ra!f I 8~r a!.

~A4!

In the above,ma and mc are the A,BC and AB,C reducemasses, kI

a5@2ma(E2e Ia)/\2#1/2 and kF

c 5@2mc(E2eF

c )/\2#1/2, with e Ia and eF

c denoting internal energies oBC and AB. Inserting Eq.~A3! into Eq. ~A2! yields

AF,I~E!51

2p E0

`

dtE2`

`

dE8(F8

E0

`

dR

3exp~ iEt/\!d~R2R`!E drfF* ~r !exp~2 iH t/\!

3cF82

~R,r ,E8!^cF82

~R,r ,E8!uc~Ra ,ra ,t50;I !&

51

2p E0

`

dtE2`

`

dE8(F8

E0

`

dR

No. 3, 15 January 1998

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959S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

3exp@ i ~E2E8!t/\#d~R2R`!E drfF* ~r !

3cF82

~R,r ,E8!^cF82

~R,r ,E8!uc~Ra ,ra ,t50!&,

~A5!

where the angular brackets (^•••u•••&) indicate integrationover all spatial coordinates. The initial wave packet is takto be

c~Ra ,r a,t50;I !5g~Ra!f I~r a!

5S E2`

1`

dk g~k!exp~ ikRa! Df I~r a!,

~A6!

where

g~k!51

2p E0

`

dRa exp@2 ikRa#g~Ra!. ~A7!

The integral in Eq.~A5! involving the initial wave packetbecomes, with the aid of Eqs.~A4!, ~A6!, and~A7!,

^cF82

~R,r ,E8!uc~Ra ,ra ,t50;I !&

52S 2pma

\2kI8a D 1/2

SF8,Ic,a g~2kI8

a!. ~A8!

Inserting Eq.~A8! into Eq. ~A5! and performing the integration overR the expression forAF,I(E) @Eq. ~A5!# simplifiesto

AF,I~E!52S 2pma

\2 D 1/2 1

2p E0

`

dtE2`

`

dE8

3(F8

exp@ i ~E2E8!t/\#SF8,Ic,a g~2kI8

a!

~kI8a!1/2

3E drfF* ~r !cF82

~R` ,r ,E8!. ~A9!

The integral

1

2p E0

`

dt exp@ i ~E2E8!t/\#

5\

2d~E2E8!1

i\

2pP

1

~E2E8!~A10!

occurs in Eq.~A9!, and was discussed in detail in Ref.where it was shown that its effect is to annihilate all incoing waves and to extract only the outgoing wave componof the integrand.@Equation~A10! is also an energy representation of iG1(E)/(2p), where G15(E2H1 i e)21.53#More specifically, the principal value term in Eq. A10 is5,56

i\

2pP

1

~E2E8!5

1

2

limt→`S i

p

~12cos@~E2E8!t/\#!

~E2E8!/\ D~A11!

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n

-nt

and arguments very similar to those in the Appendix of R5 can be used to show that, in the asymptotic limit of larR` ,

P E2`

`

dE8i\

2p~E2E8!exp~6 ik8R`!s~E8!

56\

2exp~6 ikR`!s~E!, ~A12!

wherek5k(E) andk85k(E8), ands(E8) is a function thatvaries on the real energy axis more slowly nearE85E thanthe highly singular and oscillatory exp(ik8R )/(E2E8) argu-ment. @This is a slightly more general conclusion than threached in Ref. 5. Actually, contour integration, couplwith the fact that asR` tends to` allows one to replace thefunction s(E8) with the constants(E) is the most directroute to Eq.~A12!.# We see that when the principal valupart operates on an outgoing wave term and is integraover energy, it results in an outgoing wave term at energyE,multiplied by \/2. Combined with the dirac delta functioterm in Eq.~A10!, this means that the outgoing wave partc2 in Eq. ~A9! yields an outgoing wave, multiplied by\. Onthe other hand, Eq.~A12! shows that when the principlevalue part acts on an incoming wave, and we then integover energy, a change of sign occurs and we get backincoming wave multiplied by2\/2. This term then cancelscompletely with the term arising from the delta function terin Eq. ~A10!, and the incoming wave part arising fromc2 inEq. ~A9! is annihilated. Inserting the asymptotic form foc2, Eq. ~A4! into Eq.~A9!, and using Eq.~A12!, we obtain,after integrating overr and summing overF8:

AF,I~E!52S mamc

\2kIakF

c D 1/2

SF,Ic,a exp~ ikF

c R`! g~2kIa!.

~A13!

Rearrangement of the above then leads to Eq.~18! of themain text. @An alternative approach is to useAFI(E)5 i ^R` ,FuG1(E)uc(t50)&/(2p), coupled with an appro-priate R→` form for ^R,FuG1uc(0)&. For discussions ofsuch asymptotic forms see, e.g., Ref. 57, p. 847, or Ref.p. 127, and note that in our case the prefactors will beferent owing to the nonspherical wave nature of our scating functions.#

We now turn to how just a few simple modificationsthe basic analysis above apply to the case of analyzingreal part of a wave packet generated by Eq.~1!. Equation 20may be rewritten as

CF,Iq ~ t !5

1

2 E drfF~r !@c~R5R` ,r ,t;I !

1c* ~R5R` ,r ,t;I !#, ~A14!

where the complex conjugate onfF has beeen omitted as thproduct wave functions may always be chosen to be rThe corresponding energy-dependent coefficientsAF,I

q (E)@see Eq.~A2!# are

No. 3, 15 January 1998

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960 S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

AF,Iq ~E!5

1

4p E0

`

dtE0

`

dR exp~ iEt/\!d~R2R`!

3E drfF~r !~exp~2 iH t/\!c~Ra ,ra ,t

50;I !1exp~ iH t/\!c* ~Ra ,ra ,t50;I !!

51

4p E0

`

dtE0

`

dRd~R2R`!E drfF~r !

3$exp@ i ~E2E8!t/\#c~Ra ,ra ,t50;I !

1exp@ i ~E1E8!t/\#c* ~Ra ,ra ,t50;I !%.

~A15!

Inserting Eq.~A3! to the left ofc andc* and following theanalysis of Eqs.~A6!–~A8! we obtain

AF,Iq ~E!52S 2pma

\2 D 1/2 1

4p E0

`

dtE2`

`

dE8(F8

SF8,Ic,a

~E8!

3Fexp@ i ~E2E8!t/\#g~2kI8

a!

~kI8a!1/2 1exp@ i ~E1E8!t/\#

3@ g~kI8

a!#*

~kI8a!1/2 E drfF~r !cF

2~R` ,r ,E8!. ~A16!

The second term in the square brackets gives, instead o~A10!, a term of the form

1

2p E0

`

dt exp@ i ~E1E8!t/\#

5\

2d~E1E8!1

i\

2pP

1

~E1E8!, ~A17!

which, following our previous arguments, leads to nonzcontributions from energiesE852E. Suppose, as is common in many formal developments, the zero of energychosen to lie at the onset of the continuum. Then the enerE of interest are positive and the energies for which E~A17! contributes are negative, corresponding to boustates. Since the initial wave packet is assumed to contaibound states, Eq.~A17! gives a zero contribution.

In actual propagations the energy scale is shiftedscaled@either Hs or f (H)#, so the actual~shifted! energiescontained in the initial wave packet have negative valuesthis case the term analogous to that in Eq.~A17! gives non-zero contributions only for very positive energy valuwhich are well separated from those contained withininitial wave packet. Theg (kI8

a) factor is zero for such contributions.@Note also that the second term in the curly bracets of Eq.~A16! contains the termg (kI8

a). This is the am-plitude of the initial wave packet which is going away frothe interaction region. This term is nearly always smallactual applications and may be made to be zero.#

Since the second term in the curly brackets of Eq.~A16!can be neglected, we see that Eq.~A16! is simply half the

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q.

o

ses.dno

d

In

e

-

corresponding quantity based on the full complex wapacket, Eq.~A9!. This leads to the result in the text, Eq.~21!.

APPENDIX B

In the main text and Appendix A we assume a continous time representation because it connects most direwith our previous work and because often~e.g., with propa-gation under an ordinary Hamiltonian operatorH! a continu-ous time representation is the most natural one. Howediscrete analogs to the arguments in Appendix A and SII C exist in relation to thef (H)52(\/t)cos21 Hs mapping.Consider simplyredefining AF,I

q ( f ) to be the discrete sum oEq. ~24! instead of the continuous time integral. Similarlreplaceall time integrals that enter into the detailed technicarguments with the corresponding discrete sums. Leu5 f t/\52cos21 Es. The E8 integrals in Appendix A mustthen be replaced by integrals overu8. The usual Fourierseriesrepresentation ofd(u2u8) can then be used to showthat

1

2p (k50

`

~12dk,0/2!exp@ ik~u2u8!#

5d~u2u8!

21

i

2pk~u2u8!, ~B1!

wherek(x5u2u8), a generalized function,58 is defined in anumber of equivalent ways, including

k~x!5 (k51

`

sin~kx!5 limN→`

cosx

22cosS N1

1

2D x

2 sinx

2

5P1

2cotS x

2D . ~B2!

~It is easy to verify the forms fork above have the samdefinite integrals.! Equation~B2! is the discrete analog of thimportant half-Fourier transform relation Eq.~A10! of Ap-pendix A. While the principal value term in Eq.~B2! appearsto be a little different from that in Eq.~A10!, it has the sameeffect under the analogous integrals overu8 that Eq.~A10!has in the corresponding integrals overE8. This is because12 cot(x/2)→1/x as x→0, and it is only this region of theintegral that contributes under the asymptotic assumption

Going through the analysis of Sec. II C and Appendixin detail with the above observations, one finds that the alog of Eq. ~21!, with E replaced byf , results.

APPENDIX C

In this Appendix we discuss how to effectively extraplate the time-dependent coefficientsCF,I(t5kt), k50,1, . . . ,N that are required to constructS matrix elements.Ideally one should have as long a time sequence as posto be inserted into Eq.~24!. If, for example, the number oiterationsN is not large enough, false oscillations of varyin

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sroo

aymsin-arpasola

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961S. K. Gray and G. G. Balint-Kurti: Quantum dynamics with real wave packets

severity, can be present in the resulting energy profilesreaction probabilities. However, it might be the case thatthe relevantCF,I(t) have already passed their maximum vues and are, on average, declining. If this is the case~whichcan obviously be checked in practice!, then the followingprocedure has been found to be very effective.

Assuming that afterN0,N iterations the coefficientsare, on average, declining, then the approximation

CF,I~kt!'(s51

I

bs exp@2 i f s~k2N0!t#

3expS 2gs

2~k2N0!t D , ~C1!

should be a very reasonable model fork.N0 . Equation~C1!is just a sum of damped complex exponentials. The 4I realparametersRe(bs), Im(bs), f s , and gs can be determinedsuch that one has a very good fit to the availableCF,I(kt)data fromk5N011, N012, . . . ,N. Assuming the resultangs come out positive~so that, on average, damping is occuring!, then the above formula can be used toextrapolatethatdata far beyondN. We use the Prony method8,42 for thispurpose. Alternative approaches, such as fidiagonalization59,60 could also be employed, probably witequal effectiveness. In fact filter diagonalization has alrebeen used to avoid long time propagations in some scattecalculations.35,36

Thus we use the Prony algorithm to fit the 4I real pa-rameters to the lastN2N0 points of CF,I(kt). In our par-ticular case theCF,I are real, which implies either one cousimplify the basic Prony method or apply it in its genecomplex form but expect to find that for each positive frquencyf s with amplitude and decay widthbs ,gs that therewill be another frequency~among theI obtained! with value2 f s , and amplitude and widthbs* ,g. ~For example, for thereal case there are really 2I independent parameters.!

Typically, we take the lastN2N05100 points of aniteration sequence and useI 520. We check that the resultyield positive gs values and that the sum of squares erbetween actual data points and the model is small. If nthen that is an indication thatN is not yet large enough toconfidently apply the extrapolation. With the results, one cevaluate Eq.~24! far beyondN, e.g., to convergence, bsimply using Eq.~A1!. As an example, the collinear problein the text withN5500 points was analyzed, using the la100 points. Without the extrapolation technique, i.e., usEq. ~24! with just theN5500 points yields a very oscillatory, poor approximation to Fig. 2. The root mean squerror of the associated probabilities is 0.05, and someticular oscillations can lead to individual errors near renance peaks, for example, of up to 0.3. However, extrapoing the sequence as discussed leads to results thaindistinguishable from Fig. 2 on the scale of the figure, wa root mean square error of 0.003. We also applied the tnique to the three-dimensional D1H2 case and found thainformation equivalent to a 1000 iterations could be obtainfrom just N5600 iterations. The three-dimensional case

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ofll

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hibited faster dynamics and presumably the lack of imptance of long-lived resonances contributed to the relativmodest~in comparison with the collinear case! gain achievedwith the technique. While obviously more involved, the tecnqiue outlined here, if applied carefully, can allow oneextract as much as possible out of a relatively short iterasequence.

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