Ž .Chemical Physics 246 1999 315–322www.elsevier.nlrlocaterchemphys

Nonadiabatic processes in solution: molecular dynamics andsurface hopping

Paola Cattaneo, Maurizio Persico ), Alessandro TaniDipartimento di Chimica e Chimica Industriale, UniÕersita di Pisa, Õ. Risorgimento 35, I-56126 Pisa, Italy`

Received 25 January 1999

Abstract

In this paper we present a technique to simulate nonadiabatic dynamics in solution, combining classical MolecularDynamics with the Surface Hopping algorithm. In this way we can investigate the mechanism of the photochemical reactionsin solution and the role played by the solvent. The evolution of the electronic wavefunction in a diabatic basis is introducedand a first example about the azomethane photochemistry and the possibility of future applications are briefly presented.q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

The final products and the mechanism of a photo-chemical reaction may be very different in solutionand gas phase. For instance, the photodissociationprocesses are often inhibited by the solvent, with theso-called cage effect. For describing the role playedby the solvent molecules, we can distinguish be-

w xtween static and dynamic effects 1,2 . In the firstcase, the most important action of the solvent is the

Ž .modification of potential energy surfaces PES , dis-placing minima and transition points, increasing theheights of the barriers and changing in this way thereaction paths. Another consequence of the PESvariations is the difference in intensity and positionof the absorption spectral bands in gas and solutionphase. To study this kind of solvent effects, whichmay be highly specific, we can use ab initio methods

) Corresponding author. Fax: q39-050-918260; E-mail:mau@hermes.dcci.unipi.it

Žapplied to a supermolecule solute plus a few solvent.molecules or to the solute embedded in a continuum

w xrepresenting the solvent 3,4 . On the other hand,several photochemical reactions in solution showthat the cage effect is largely independent on thesolvent characteristics. This may occur because thesolvent interacts dynamically with the excitedmolecule: it can subtract the excess vibrational en-

wergy before the dissociation vibrational relaxationŽ . xVR mechanism or favour the recombination of

Ž .product fragments structural mechanism , besidesmodifying the PES statically. In such cases, a dy-namical simulation is mandatory.

The exact quantum dynamics calculation is possi-ble only for very small systems, with few internalcoordinates. For describing larger molecules, withmore than three atoms, in the last decades severalsemi-classical methods have been developed, in or-der to treat the electronic degrees of freedom quan-tum-mechanically and the nuclear motion classically.Among these approaches, we may quote the mean-

0301-0104r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0301-0104 99 00191-3

( )P. Cattaneo et al.rChemical Physics 246 1999 315–322316

w x w xfield 5,6 and surface hopping 7,8 ones, whichdiffer in the way of coupling the quantum andclassical degrees of freedom. Recently, a new method

w xhas been proposed 9 to combine the two algo-rithms.

Ž .The dynamic VR or structural and static effectsof the solvent can be all taken into account bysimulation of nonadiabatic dynamics in solution withthe Molecular Dynamics algorithm. The first and

w xsimplest example 10,11 was the simulation of ex-Žcited excess electrons in simple fluids helium in this

.case , using the surface hopping algorithm. In asimilar way, Rossky and co-workers have simulatedw x12–14 the photoexcitation of an hydrated electron;they applied the algorithm developed by Websterand co-workers by combining the surface hoppingand the nonadiabatic scattering formalism of

w xPechukas 15,16 . During the same years, other smallŽ w xsystems proton transfer in solution 17,18 and H 2

w x.transport in water 19 have been studied by treatingthe hydrogen atoms quantum-mechanically and thedegrees of freedom of the solvent classically, inorder to include some important non-classical effectsŽzero-point motion, quantum transitions, tunneling,

. w xetc. . A photoexcited aqueous halide 20 has beendescribed by treating the halide and the watermolecules classically and the electrons quantum-mechanically. The cage effect has been investigated

w x w xfor I in liquid xenon 2 and argon 21 . ICN in2w xargon 22 provides the most complex example of

simulated photochemistry in condensed phase, in-cluding dissociation, isomerization and electronic re-laxation.

Several recent works make use of the empiricalŽ .valence-bond EVB model, originally put forward

˚w x Žby Warshel and Weiss 23 see also Aqvist andw x .Warshel 24 and references therein . The EVB model

is a powerful tool to fit non-trivial potential energysurfaces, describing for instance breakingrformation

Žof several bonds andror charge transfers see forw x.instance 18,25 . As far as each valence state can be

identified with a quasi-diabatic electronic wavefunc-tion, the model lends itself to represent also nonadia-batic effects: a recent application is the semiclassicalstudy of the photodissociation of NaI with one inter-

w xacting water molecule 26 .The aim of the present work is to describe a

strategy to simulate nonadiabatic dynamics for small

Žor medium-sized solutes with several non-trivial i.e..large-amplitude non-harmonic internal motions,

Ž .coupled with a non-atomic solvent like water . Themethod is based on a surface hopping algorithm inwhich the time evolution of the electronic wavefunc-tion is computed in a quasi-diabatic basis. The lattercan be determined by any suitable ab initio orsemiempirical technique, coupled with a general ‘di-abatization’ procedure put forward by one of usw x27,28 and recently extended to cope with several

w xreaction coordinates 29 . Other techniques to con-struct quasi-diabatic states have been proposed in the

Ž w x.past see for instance 30,31 and may be alsoapplied in this context. Within this strategy, thequasi-diabatic states are not necessarily related toVB states: for instance, in polyatomic systems withconical intersections, they may rather be constructedon the basis of symmetry considerations. In thisrespect, the procedure we propose is more generalthat the EVB approach, and also more rigorous, inthat the dynamical couplings can be more accuratelyminimized.

In Section 2 we describe the method, brieflyreviewing the determination of quasi-diabatic statesand focusing on the semiclassical treatment of thedynamics. In Section 3 we outline shortly a recentapplication to the photochemistry of azomethane:details about the results will be published elsewhere.

2. Method

2.1. Time eÕolution in a diabatic basis

We treat the nuclear motion classically and theelectronic wavefunction of the solute is evolved by

Ž .the time-dependent Schrodinger equation TDSE .¨All the information concerning the electronic statesand PES of the solute is contained in the matrix H ,dia

which represents the electronic hamiltonian in a� < : < :4quasi-diabatic basis h . . . h :1 N

ˆ² < < :H s h HH h 1Ž . Ž .i jdia i e l j

w x < :The quasi-diabatic states 30,31 h are definedi

so as to cancel, or at least minimize, the nonadiabatic² < < :or dynamical couplings h ErER h , where R isi a j a

an internal coordinate: this implies that their physical

( )P. Cattaneo et al.rChemical Physics 246 1999 315–322 317

character is, at most, only moderately dependent onthe molecular geometry. The adiabatic states, incontrast, may undergo abrupt changes when goingthrough avoided crossings or conical intersections: insuch cases, resorting to a quasi-diabatic descriptionis practically mandatory, in order to avoid sharppeaks or divergences in the nonadiabatic couplingsand cusps in the PES.

We construct the diabatic basis starting from the� < : < :4adiabatic one, c . . . c with a unitary transfor-1 N

mation C:

< : < : < : < :h . . . h Cs c . . . c 2� 4 � 4 Ž .1 N 1 N

The transformation matrix is chosen so as toobtain the maximum overlap with an appropriate set

< :of templates R .i

N2<² < : <V C s h R smax 3Ž . Ž .Ý i i

i

The templates, in turn, have an a priori diabaticcharacter, being constructed as antisymmetrizedproducts of atomic or fragment wavefunctions, or by

w xresorting to symmetry considerations 28,29 . Thisprocedure is currently implemented as part of the

w xCIPSI package 32,33 for multireference perturba-tion CI, but could be easily adapted to other ab initioor semiempirical methods.

We run trajectories over the adiabatic PES andintroduce the nonadiabatic transitions through the

w x‘fewest switches’ surface hopping algorithm 7,8 ;however, we calculate the time evolution of theelectronic wavefunction in the diabatic basis, as de-scribed later on. The adiabatic PES for the isolatedmolecule are the eigenvalues of H . For the simula-dia

tion of condensed phase dynamics we need to add asolvent–solvent and a solute–solvent potential. Bothcan be taken from standard Molecular Mechanicsparametrizations. The latter, however, may be conve-niently adapted, if necessary, by resorting to ab initiocalculations on the solute plus solvent super-molecule, to take into account the state-specificity ofthe solute–solvent interactions. We define thereforea new diabatic hamiltonian matrix including the so-lute–solvent interactions V :solv

H sol sH qV 4Ž .dia dia solv

The advantage of expressing V in the diabaticsolv

rather than in the adiabatic basis is that its depen-

dence on the solute internal coordinates will be muchmore regular. In fact, by definition the diabatic states

Žgo smoothly through funnel regions avoided cross-.ings or conical intersections , whereas the adiabatic

wavefunctions and charge distributions may changeabruptly. In most applications, it will suffice toconsider diagonal matrix elements of V , i.e. thesolv

interaction potentials between solute and solvent ineach diabatic state. In fact, the most important effectof solvent interactions on photochemical funnels isto displace them both in energy and in the internalcoordinate space of the solute: as a result, both theaccessibility of the funnel and the transition probabil-ities may be altered, the latter because of the geome-try dependence of H . By supermolecule calcula-dia

tions, one can also determine off-diagonal correc-Ž .tions V .solv i j

The solvent modified adiabatic PES are the eigen-values of H sol :dia

H solU sE U 5Ž .dia k k k

The forces acting on the solute nuclei are givenby the Hellmann–Feynman theorem:

E E E H solk dia†y sU U 6Ž .k kE R E Ra a

where R is a cartesian coordinate of one of thea

solute nuclei. For a solvent coordinate R , only theb

Žcoupling matrix derivatives are needed in addition.to the solvent–solvent forces :

E E E Vk sol†y sU U 7Ž .k kE R E Rb b

The classical dynamics of nuclei is carried outwith the well-known Verlet algorithm with periodicboundary conditions and Ewald summation to de-scribe the long-range Coulombic interactions.

We underline the importance of adding V be-solv

fore the diagonalization of H if the solute–solventdia

potential is assumed to be state-specific. Indeed,each diabatic state is endowed with specific featuresŽ .charge distribution, polarizability, etc. which deter-mine the intramolecular potentials. It is not physi-cally correct to attribute such features to the adia-batic states, at least in the proximity of surfacecrossings. The diagonalization of H qV allowsdia solv

us to take into account in a consistent way both the

( )P. Cattaneo et al.rChemical Physics 246 1999 315–322318

differences between electronic states with respect tosolvent interactions and, at least partially, the effectof the solvent reaction field on the adiabatic wave-functions. In particular, we take into account thesolute–solvent interaction during the calculation ofnonadiabatic couplings and transition probabilities.As a limiting case, let us consider a two-state systemin which the two diabatic surfaces cross in vacuo,

Ž .but not in solution or vice versa . Under suchcircumstances, it is important to take into account theV matrix in the construction of the adiabatic statessolv

and PES, in order to introduce or remove the avoidedcrossing according to the strength of the solute–solvent interaction. On the other hand, of course, ifthe solute–solvent potential does not depend on theelectronic state, V may be indifferently added tosolv

H before diagonalization, or to its adiabatic eigen-dia

values.The evolution of the electronic wavefunction is

based on the TDSE, expressed in the diabatic basis.For each trajectory, we write the electronic wave-

< Ž .:function C t at time t as:

< : < :C t s D h 8Ž . Ž .Ý k kk

w xIn our previous work 34,35 we used an adiabaticexpansion, which is completely equivalent, as thetwo basis sets span the same electronic subspace.The diabatic representation offers some advantages.First, we need the non-adiabatic couplings only whena surface hop takes place and we can avoid tocalculate them at each time step. Moreover, theequations and the computer code are simpler and therisk of divergences in the coupling matrix elementsis avoided. The electronic time evolution for a giventrajectory is theoretically independent on the elec-

Žtronic basis the equivalence of the adiabatic and.diabatic TDSE is shown in Appendix A . On the

other hand, the choice of the PES which govern thenuclear motion does make a difference. When theelectronic energy gaps are large and the Born–Op-penheimer approximation is valid, the adiabatic PESare obviously the best choice. For quasi-degeneratesituations, which are normally well localized in the

Ž .coordinate space crossing seams , the choice of theforces acting on the nuclei is connected with that ofthe criterium for nonadiabatic transitions. The sur-face hopping method is one of the simplest and most

easily adapted to MD calculations, either in thediabatic or in the adiabatic representation. Within the

Žprocedure here outlined nuclear trajectories com-puted on the adiabatic PES and electronic time evo-

.lution in a diabatic basis one may also take intoŽaccount small interactions such as spin–spin or

.spin–orbit couplings by adding them to the diabaticŽ . Ž .hamiltonian in the TDSE, Eqs. 9 and 10 below.

The TDSE for the electrons is:

< :E C tŽ .ˆ < :i" sHH C t 9Ž . Ž .elE t

Ž .By introducing expansion 8 we obtain the fol-lowing set of coupled differential equations for thecoefficients:

isolDsy H Q t D 10Ž . Ž .Ž .dia

"

The hamiltonian matrix elements depend on timethrough the nuclear coordinates Q. We solve thesystem with a five points finite differences schemeŽ .see Appendix B .

2.2. Surface hopping

The electronic wavefunction can also be expanded� < : < :4in the adiabatic basis c . . . c :1 N

< : < :C t s A c 11Ž . Ž .Ý k kk

Ž .The U matrix connects the adiabatic A andŽ .diabatic D coefficients:

DsUA 12Ž .To obtain the transition probabilities W betweenk l

the adiabatic states k and l we apply Tully’s ‘fewestw xswitches’ algorithm 8 . The transition probabilities

< < 2depend on the variation of r s A with respectk k k

to time:

d Ak)r s2 A 13Ž .˙k k k d t

d A d U†DŽ . kk ˙ ˙s s U D qU DÝ ž /lk l lk ld t d t t

i˙s U U A y A E 14Ž .Ý lk l i i k k

"l , i

To establish the last equation, one makes use ofthe relations: Ý H U sU E and Ý U U sd .j l j ji l i i l lk l i k i

( )P. Cattaneo et al.rChemical Physics 246 1999 315–322 319

The increment of r is:k k

Dr ,r Dts 2 Re A) A U DU˙ Ý Ýk k k k k l i l i kž /l i

s W r 15Ž .Ý k ™ l k kl

Xk lW s u yX 16Ž . Ž .k ™ l k l

rk k

˙Ž .where X s 2Re r Ý U DU , DU s U D t ,k l lk i i l i k l i l iŽ . Ž . Ž .U tqD t yU t and u X is the Heaviside func-l i l i

tion.The surface hopping takes place if W )W,k l

where W is a random number between 0 and 1. Aftera jump from state k to l, in order to conserve thetotal energy of the system we make up for the

Ž .variation of the potential energy E yE with anl kŽ .opposite variation in the kinetic one T yT .new old

When there is a hop, we need to calculate thenonadiabatic couplings

EŽa .G s c c 17Ž .k l k l¦ ;E Ra

because the momentum P , associated with the co-a

ordinate R , changes according to:a

PX sP qs GŽa . 18Ž .a a k l

The factor s is obtained by solving:2Ž . Ž .a aG P GŽ .k l a k l2s q2 sÝ Ýž /ž /m ma aaR

q2 E yE s0 19Ž . Ž .l k

where m is the nuclear mass associated with R .a a

One cannot expect semiclassical dynamics to re-produce quantitatively the quantum mechanical prob-

Ž w xabilities in all circumstances see Refs. 34–36 for.some accurate comparative studies , but there are

indications that the agreement improves when in-w xcreasing the dimensionality of the system 36 . In

particular, a typical drawback of the surface hoppingŽalgorithm is the failure of upward transitions from

.lower to higher PES when the kinetic energy is lessthan the potential energy gap: it is to be expectedthat such events will be less probable in condensedphase simulations, because the vibrational energyloss to the solvent takes the solute away from thefunnel region after a transition to the lower surface.

2.3. Initial conditions

A batch of trajectories is normally run, with initialconditions which obey a distribution of coordinatesand momenta, preliminarily defined on physicalgrounds. In the first applications of this simulationprocedure, we do not try to imitate a quantum me-chanical distribution and we rely on merely classicalconcepts. We start from pure solvent and we runsome picoseconds of MD simulation at constant

w xtemperature 37 . Some solvent molecules at thecenter of the cell are then replaced with the soluteand the simulation is run again for a few picosecondsat constant temperature, in order to equilibrate thesolute–solvent system. During this first step the so-lute remains in its lowest adiabatic PES. We samplethen the initial conditions for each trajectory at regu-

Ž .lar time intervals of the order of some tens of fs ,long enough to allow for a thorough exploration ofthe accessible phase space of the solute within about100 steps. At the end of each interval, a new trajec-tory starts with a classical Franck–Condon excita-tion: the electronic state switches from the ground tothe k-th excited without changing the nuclear coordi-nates and momenta. The excitation is done in theadiabatic representation, and the initial diabatic coef-ficients D are given by the appropriate eigenvectorU .k

Since the trajectory starts in the excited state, onemay stick to the constant temperature MolecularDynamics, or switch to the constant energy mode.The former choice corresponds to a very rapid equi-libration of the single cell, containing one excitedsolute molecule and some 100 or 200 molecules ofsolvent, with the thermal bath supplied by the sol-vent bulk. The latter choice is equivalent to theassumption that the cell does not exchange energy

Žwith the bulk during the time of the simulation a.few ps, for a typical photochemical event ; quite

clearly, this is correct in the limit of a very large cell.For finite cell sizes, the realistic behaviour will lie inbetween these two limiting cases.

3. Photochemistry of azomethane

w xIt is well known 38 that the photochemistry ofacyclic azoalkanes, after the n™p ) excitation, is

( )P. Cattaneo et al.rChemical Physics 246 1999 315–322320

different in solution and in gas phase. In gas phase atlow pressure, the only reaction which takes place isdissociation, which proceeds with high quantumyields. The photodissociation is inhibited in a largevariety of solvents and the most important reactionbecomes the trans–cis isomerization. On the otherhand, there are no definite experimental evidencesabout the mechanisms of the two reactions, the statesinvolved and the role played by the solvent. The

Ž .simplest azoalkane, azomethane CH NNCH ,3 3

shows the same behaviour. Since the inhibition ofdissociation is substantially independent of the sol-vent characteristics, the static deformation of thePES, due to specific solute–solvent interactions, isprobably of secondary importance: therefore, weconcentrate on the dynamical effects.

We have carried out ab initio calculations and ananalytic fit in order to obtain H for the two lowestdia

singlet states of azomethane in vacuo, as describedw xelsewhere 29,39 . Simulations of the isolated

w xmolecule photochemistry 40,41 in vacuo show thatthere is a very efficient IC from S to S at twisted1 0

geometries. In fact, the torsion pathway does notshow any barrier in S , and a conical intersection1

connects the two PES at twisting angles of about 908.This feature would make a treatment based on adia-batic electronic states quite unpractical: resortingmore or less explicitly to a diabatic representation ishere mandatory. Because of the very short lifetime ofS , ISC to the triplet manifold is probably not an1

important process. The dissociation takes place in theground state PES, in times of the order of about 100ps.

For the simulation of azomethane in solution, wechose water as solvent. We adopted a flexible,SPCE-derived, model for the water molecule internal

w xpotential and the water–water interactions 42,43 .The water–azomethane potential was built using the

w xtabulated data of BOSS 44 , without introducing anydifference between the two electronic states of thesolute. We have used a cell with 120 molecules ofwater and one of azomethane, with periodic bound-ary conditions. The Ewald treatment of Coulombicinteractions included 257 vectors in the reciprocalspace, a real space cutoff at half box side and theconvergence parameter as6.4. We carried out 200trajectories, with a twofold stop condition: either

Ždissociation occurs one of the NC bonds becoming

.longer than 8 bohr ; or, the internal energy ofazomethane becomes lower than 5 kcalrmol with

Žrespect to 2CH qN about 36 kcalrmol above the3 2.trans-azomethane minimum .

A detailed analysis of the simulation results willbe presented elsewhere: here we just want to showthat the method can be successfully applied and thatinteresting information can be drawn from the simu-lations. In agreement with experimental findings, inthe simulation the photodissociation is strongly sup-

Žpressed only 6% of the trajectories yield CH NNP3.qCH P , while trans–cis isomerization takes place3

Ž .40% . The internal conversion to ground state isalmost as fast as in vacuo. The inhibition of dissocia-tion is clearly due to energy transfer to the solvent,rather than to caging and recombination of the twofragments. Examples of the latter mechanism arefound in other reactions, such as the photodissocia-

w xtion of nitrosamines 45 .

4. Conclusions

This paper describes a comprehensive computa-tional model for condensed phase photochemistry.The method combines the traditional MD techniquefor treating the solvent and the ‘fewest switches’surface hopping algorithm for the nonadiabatic dy-namics of the solute, with a diabatic development ofthe electronic wavefunction. In the first application,briefly outlined in the previous section, we adoptedthe easiest choice for the solute–solvent interaction:a standard atom–atom potential, not specific of thesolute electronic state. In this way we were able todescribe successfully the vibrational relaxation of thesolute with energy transfer to the solvent; othereffects, related to momentum transfer and changes in

Ž .solvent structure caging, cavitation are also takeninto account.

In future applications, already feasible with thepresent software, we shall introduce intermolecularpotentials which depend on the solute electronicstate, as described in Section 2.1. The determinationof the interaction potentials, or at least of some oftheir most important parameters which are character-

Žistic of each electronic diabatic state for instance,w x.atomic charges 46 , can be done in the ab initio

framework. Many-body effects can be effectively

( )P. Cattaneo et al.rChemical Physics 246 1999 315–322 321

incorporated into pairwise potentials with the aid ofw xthe polarizable continuum model of the solvent 47 ,

which is currently applied also to excited state calcu-w xlations 4 . These improvements will be important in

describing electron-transfer and proton-transfer pho-toprocesses, solvatochromic shifts and solvent reor-ganization following absorption or emission, solva-tokinetic effects, and in general to obtain a betterthan semi-quantitative description of photochemicalevents.

Acknowledgements

This work was partially supported by the C.N.R.through the grant ‘Dynamics, reactivity and structureof molecular solutes’. We are grateful to E. Guardia`for providing us with the MD code we developedinto the present MD-surface-hopping program.

Appendix A. Equivalence of the adiabatic anddiabatic representations

The time-dependent evolution of the adiabaticcoefficients A is governed by the Schrodinger¨k

equation:

iŽa .˙ ˙Asy EAy R G A 20Ž .Ý a

"a

where E is the diagonal matrix of the E eigenvaluesk

and GŽa . is the matrix of the nonadiabatic couplingsassociated with the coordinate R , Ga . We want toa k l

show that this equation is completely equivalent toŽ .Eq. 10 .

The nonadiabatic couplings involve the deriva-tives of the eigenvectors U :k

E UŽa . †G sU 21Ž .

E Ra

Ž .The derivation with respect to the time of Eq. 12leads to:

E U˙ ˙ ˙ ˙ ˙DsUAqUAs R AqUA 22Ž .Ý a E Raa

Ž .On the other hand, by substituting Eq. 12 intoŽ .Eq. 10 we obtain:

i iDsy HUAsy UEA 23Ž .

" "

Ž . Ž .From Eqs. 22 and 23 we find the relation:i E U

˙ ˙UAsy UEAy R A 24Ž .Ý a" E Raa

Finally, by multipling on the left with the matrix† Ž .U we obtain the equivalent of Eq. 20 :

i E U†˙ ˙Asy EAy R U A 25Ž .Ý a

" E Raa

Appendix B. Numerical integration of the TDSE

We introduce suitable phase factors in the expan-Ž .sion 8 , so that the coefficients become:

D t sd t eyig k t 26Ž . Ž . Ž .k k

with

t X Xy1g t s" H Q t dt 27Ž . Ž . Ž .Ž .Hk k k0

Ž .The set of differential equations 10 is replacedby:

iyi Žg yg .k ld sy H d e 28Ž .Ýk k l l

" Ž .l /k

where H is an element of the H sol .k j dia

We want to propagate the g exponents and thek

d coefficients from time t to time tqD t. The gk k

derivatives are known at any desired time step, fromŽ .tqD t backwards g sH r" ; so g is obtained˙ k k k k

by a straightforward 5-points finite differences for-mula:

251g tqDt sg t qDt g tqDtŽ . Ž . Ž .˙k k k720

323 11q g t y g tyDtŽ . Ž .˙ ˙k k360 30

53q g ty2 DtŽ .˙ k360

19y g ty3Dt 29Ž . Ž .˙ k720

˙The d derivatives are not a priori known atk

time tqD t, so we obtain them together with the

( )P. Cattaneo et al.rChemical Physics 246 1999 315–322322

Ž .d tqD t coefficients, by solving a sistem of 2nkŽlinear equations n being the number of electronic

. Žstates . We have two sets of n equations two equa-. Ž .tions for each k . From Eq. 28 :

H tqDt eyi wg kŽ tqDt.yg lŽ tqDt.xDdŽ .Ý k l lŽ .l /k

˙y id tqDt sy H tqDtŽ . Ž .Ýk k lŽ .l /k

=eyi wg kŽ tqDt.yg lŽ tqDt.x d tŽ .l

30Ž .Ž . Ž .where Dd sd tqD t yd t . The second set ofl l l

Ž .equations is analogous to the 29 :

251 323˙ ˙Dd y d tqDt sDt d tŽ . Ž .k k k720 360

11˙y d tyDtŽ .k30

53˙q d ty2 DtŽ .k360

19˙y d ty3DtŽ .k720

31Ž .˙ Ž .The unknowns are the Dd and the d tqD t , ask k

all other quantities have been determined in theprevious time steps.

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