REVISTA MEXICANA DE FíSICA -17SlJJ>LEl\IENT() l. :~n-3(l

Simulation oCmolecular transitions using c1assical trajectories

A. Donoso and e.e. MartensDe¡}{lrtment of Chemistry. Ullil'ersity ofCalifomia

Irvi"e, California, 92697.2025 USA

Recibido el 29 de febrero de 2()(X);aceplado el 21 de junio de 2000

MARZ0200J

In the presenl worl.;. we describe the implemcntatíon of a scmiclassical method to study physical-chemical processes in molecular systemsv.'hcre cJcctronic state transitions and quantum coherences playa dominant role. The mcthod is hascd on c1assical trajectory propagation onIhc llndcrlying coupled c1cclronic surfaces and is dcrivcd from the scmiclassical I¡mit of Ihe quantum Liouville equation. Unlike previousclassicaltrajeclory-hascd ITlcthods, quantum c!cclronic coherences are trcatcd naturally within this appro<:H:has complex weighted trajectoryensl'mhles propagaling on the average electronic surfaces. Thc mcthod is tcsted on a model prohlem consistíng of one-dimensional motion ont\Vocro<;sing:ekctronic surfaces. Exccllent agrcement is ohtained when compared lo Ihe cxact results ohlained by wavepacket propagation.The method is applicd lo model quantum wavepacket inlerferometry. where two wavcpackels. differing only in a relative phase. collide inthe region where the IWO electronic surfaces cross. The dependence al' the resulting population Iransfer on the initial relative phase 01' thewavcpackets is perfectly captured by our c1assical trajectory melhod. COlllparison with an alternalivc method. surface hopping. shows thatour approach is appropriate for modelling quantum interference phenornena.

KI'."H'on/s: Elcrtronic coherence; molecular physics

En este trabajo se descrihc la implementación de un método semiclásico para estudiar procesos físico-químicos en sistemas molecularesdonde las transiciones entre eSlados electrónicos y las coherencias cuánticas juegan un papel predominante. El método se has.l en la propa.gación de trayectorias clásicas sobre las correspondientes superficies electrónicas acopladas y se deriva a partir del límite semiclásico de laecuaci6n cuántica de Liouville. A diferencia dc mélodos previos hasados en trayectorias clásicas. dentro de este esquema. las coherenciaselectrónicas cu5nticas son tratadas de manera natural como ensamhles de trajeclorias con pesos complejos. moviéndose en una superficieelectrónica promedio. El método es probado en un prohlema modelo. el cual consiste en el movimienlollnidimensional sohre dos superficieselcl.."tr6nicas acopladas que sc cruzan. Los resultados concuerdan muy hien con los cálculos exnctos ohtenidos mediante propagación depaquetes de onda. El método es aplicado para modelar inlerferometría de paquetes de onda. donde dos paquetes de onda. los cuales difierensólo en una fase relativa. chocan en la región donde las superficies electnlnicas se cruzan. Nuestro método. hasado en trayectorias clásicas.caplura perfectamenle la dependencia en la fase relativa de la transferencia de pohlación resultante. Comparación con un mélodo alternativo.muestra que nuestro método es apropiado para modelar fenómenos de interferencia cuántica.

/)('Jail'torc.c Coherencia electrónica; física molecular

I'ACS: 34.10.+x; 31.70.,: 31.15.-p

1. Introductinll

Until recently. Ihe details 01' elementary dynal1lical processesin chemical syslems could only be infcrred indirectly frommeasurements with time resolution many mders of magnitu-de slower than the intrinsic dynal1lical time scales. The rapi-dly advancing lleld nf l'emtochemistry now allows us lo watchchemical reactions unfnld in real time r 1]. Thc interprctationof the wealth 01' detail provided by ultrafast experiments po-ses fascinating hut challcnging opportunities fm themy" Thehighly noncquilinriulTl initial enscmblcs and subscquent non-linear nonseparanle dynamics induced by ultrafast laser exci-tation ¡cad to conditions that can violatc the Iimits ol' validity01"the traditional equilihrium statistical mechanical or line-ar rcsponse-hased thcories of condensed phase rcaction rales<tul! dynamics 12~-lI. Direct computer simulation of many-hody systems is an increasingly powerful tool for moJelingcondensed phase ultrafast dynamics 15]. Still, a fully quan-tUI1l mechunical treatmenl 01' the electronic anu nuclear de-grecs of frecdom for a many-body system is an intractablenumerical problcm, und will prohably remain so fm the fo-

rcsecanle futurc. It is thus neccssary 10 tum ro simplifyingapproximations and conceptual models in oruer for progress10 he made. A varicty of approximatc methods for simulatingmany-hody chemical dynamics have bren developed and ap-plied tu a range 01' physical prohlcms. Despite the inlrinsica-Ily quantum nature of molecular systems, classical trajcctoryintcgralion oflen works surprisingly well. For many prohlems01' physical interest, however, the nuclear and eleclronic de-grces 01' freedom cannot he lIncoupled, and elcctronic transi-tions unavoidably accompany the motion 01' the nuclei. Thisintroduces an intrinsically quantum mechanical feature intothe dynumics. NUlllerOllS attcmpts have heen maJe to dcvelopmcthods which allow the intuitive and compulational advan-lages 01' classical mcchanics to be combined with the possibi-lity 01' Iransitions hctwccn clcctronic statcs. Lanuau IGJ. Zc-ncr 171. and Stuckclhcrg [8\ develnpcd carly theories 01' clcc-tronic transitions induced by scattering cvcnts descrihed bysimple known c1assical trajcctories [9]. The c1assical motionof the nuclei was assumed ro he independent of the electronicdynamics in these early studics. which is a good approxima-tion in ¡he Iimit of high kinetic energy collisions. In gene-

SIMULATlO~ OF MOLECULAR TRANSITIONS USING CLASSICAL TRAJECTORIES 31

(1 )

ral, though. lhe coupled clcctronic-nuclcar dynarnics mus! hesol ved simultaneously anl! sclf-consistcnlly, ano several ap-pmaches lo this prohlcm have been proposcd 'lOl. includingtime-dependen! mean field, sclf-consistcnt cikonal [11] ap-proximulions. Ihe c1assical electron mode!, and lhe vcry po-pular surfacc hopping mcthod. introduccd hy Tully and Prcs-ton 112] ano slil1 under active dcveJopmcnl. In mosl 01' theseprevious attcmpts al cornbining c1assical and quanlUm clc-lIIcnls inlo ulle consistenl dynamical thcory, Ihe prohahilisticdcscription 01' lhe quantum subsystem in tcrms 01' its wavc-function or densily malrix is courJed lo a c1assical descriptionfonnulatcd in Icrms of ¡ndependent classical trajectorics, eit-her individually or as pan 01' a final average over an enselllhle.Recently, Martens el al. proposed an al!crnative approach toclassical dynamics on coupled electronic surf~lces, hased 011asemiclassical limit 01' Ihe l11ultistate quanlulIl Liouville equa-¡ion for the elcctronic.nuclcar dynamics !13]. The rcsllltingforrnalisl11 yiclds cqualions 01' motion for cOllpled phase spa-ce distrihution functions corresponding lo hOlh the clectro-ni<: slate-dcpendenl nuclear prohability distributions and thecleclronic coherence bctween states. Thcse equations 01" 1110-lion have the fonn of a generalized nnnequilihriurn stalisticalIIlcchanics, anJ the electronie coupling lIlanifests itself as ex-Ira terms in Ihe classical Liouville-Iike cquations governingthe distrihulions. These addilional tenns aet as "sources" and"sinks" of prohahility llow hetween slates, and also contro!the generation, evolutiol1, and decay nf electronic coheren-ces. In marked contrast lo nlher approaches which introdu-ce c1assical l11echanics in tcrms of individual trajectories orc1assical palhs, Ihe formalislll is developed from the olltselin terms of Ihe coupling helween the distrihution fllnclionsreprescnting poplIlations or coherences Ihel11selves. Tlle ahi-lity lo treal electronic cohercnces naturally within a classical-like contexl is a unique advanlage of this approach. In a pre-viotls puhlication [13J, Martens el al. descrihed in detail thegencral forlllalislll 01' our coupled Liollville-Iike represcnla-lion of classical molecular dynamics on coupled eleclronicslatcs. In addition, they provided analytical results in simplif-ying lilllils antl made comparisons wilh conventional dcscrip-l;olls of nonadiahalic dynamics, such as Landau-Zencr lhe-{lry 10.7.01. They also prescnted a dirccl numerical solulion01' Ihe gcneralil.ed Liouville panial differcntial equalions fora one-dimensional, I\vo-stale model syslem. The results we-re l'oll1pared wilh eoupled stale wavepackcl calculations, andc.'(ccl!ent agreemcnt was ohscrved, justifying a posteriori theaprroxirnalions !nade in the derivation of lhe semiclassicalcl)ualions 01' motion. Howc\.er, the grid solution of lhe pha-se space parlial differcntial eqllations presented requires evel1more nurnerical cflort lhan Ihe exact qllantull1 approach, and,as it sl<lI1ds, uoes not conslitule a practicalmclhod for Ircatingmany-hody couplcd-state dynamics. An efficient trajeclory-hased implementation is dcsirahle, preferahly in lhe form of;¡ rclalively minor modifkatiol1 of the sl¡mdard cIassical mo-lecular dynalllics compulcr simulation. The challenge of dc-veloping such a mcthod is lo use classical trajectories as anumerical tool to solve equations of motion lhat are, at a fun-

damcntal level, morc general than Hamilton's equations fororhits in phase space. In this paper, we prescnt a trajectoryimpiemenlalion of lhe coupled surface semiclassical Liouvi-111.'approach to nonadiahatic dynamics. Such an implcmen-talion is straighlforward for dynamics on a single electronicsurface: In Ihal case, evolution of the phase space distrihu-lion function solving Liouville's equation is achieved simplyhy propagaling <Inenscrnhle of trajectories sampled fmm theinitial phase spacc densily hy the corresponding Harnilton'sequations. A gcneralization ofthis Hamillonian trajectory ap-proach is required to incorporale lhe cffeel of lhe nonclassi-cal coupling lcrms in the Liouville-Iike equalions of mOlinn.In Seco 2, wc presenl such a generalization and compare lheresults 01" its applicalion with numerically exacl quantum re-Slllts for a simple hut nontrivial (lile-dimensional two-stalemodel prohlcm in lhe diahatic representation. In Seco 3, theclassical lrajeclory-melhod is applied to model quanturn wa-vcpacket inlerferometry. where two wavepackels, differingonly in a relative phase, collide in tlle region where lhe Iwoelcclmnic surfaces cross. A discusion is presented in Sec. 4.

2. Simple two-stale system

Our developl11cnt hegins with lhe cxact quantum rnechanicalLiouvllle equallon for Ihe densily operalor. given hy [15-lí]

. 01' '.," 01 = [H.p).

where f¡ is Ihe Hamlllonlan of the syslem and [f¡,¡jJ deno-les lhe COllll11utator of Ji and p. In lhe case of vihrationaldynamics on a single elcclronic surface, the c1assical ana-loguc of (1) is the corrcsponding c1assieal Liouvillc equa-lion of nonequilihrium stallsllcalmechanies Op/81 = {H. p}where p = p( 11.1'. 1.) and l/ = l/ ('1,1', 1) are now funelions01" the classical phase space variables. To lowcst order in h,the cornrnUlalor of two opcralors is related to Ihe c1assicalPoisson hrackcl 01' the corresponding phase space funclionsvia [A,E] -+ ¡"{A. B} + 0(""). In praclice, a numeriealsolulion 01" the c1assical Liouville equalion can be aCCOIll-plished (to within Slalistica! uncertainty) hy inlegrating Ha-millon's equa!lons 'j = Ol//Ol' and Ti = -oH/Oq usinga Jinile ensemhle of Initlal eondilions {(li)(O).Pj(O))}(j =1,2, .... LV) sampled fmm Ihe prohahility distribulion givenhy Ihe Inilia1 phase space densily p(q. 1', I = O). Thc den-slly ofe"olvlng trajeclOrles {(ll](t).I'J(t))} in phase spacethen approxilllatcs p(q, /l, t), Ihc exact solution of the Liou-ville cquafioll. The classical limit in lhe more general case01" a Hami!tonian that is a marrix of operalors rcpresenlingvihralional dynamics on Illultiplc elcclronie states is not.asslraightforward. The coupling hetween lhe highly quanturnIllechanical elcctronic slatcs inlroduces an intrinsically non-c1assi<:al cOlllponcnt lo Ihe cvolulion orille full densily {J. No-nellleless. a consistenl sellliclassicallirnit can be derived !13J.Here, we hrietJy review Ihe approach l'or lhe case ol' one di-mensional vihrational mOliotl on two coupled eleclronic sta-

Re" Mex. I.ú. 47 SI (2001) 30-36

.12 A. DONOSO AND c.e. t\1AHTENS

iF+ r; (PII - (22)' (7)

tes; lhe rnclhod can he casily gcneralized 10 lreal N couplcdsurfaces ami" vihrational degrces 01'[rccdom. Thc Hamilto-nian ami dcnsity opcrators for lhe two-statc problem are givcnin the eleetronie {II) .12)} hasis hy

(9)_ (p - po)']o .,_17;;

1>(1' - ['0) =' 1>('1 - 'lo, J' - 1'0)

('J - 'Jo)'

where cach t5-funclion in lhe SUlll is lwo-dimensional. amiis cenlered on a l1lel11berof lhc enscmhle at rYl\t) ==["y') (1), [lj") (1)l. The variahles {" j"1 (I)} eorrespond to ¡herelalive weights 01"each Illcmher of lhe ensemhle, and proha-hilit)' l10w hetween electronic states manifests itself in the ti-me variations 01'these weights. Por lhe off-diagonal coheren-ce (111, these coeflicienls are complex numbers. and incorpo-rale lhc nonclassical phase rcsulting fmm the term involvingthe tiifferenee potential [see Eq, (7)]. A smooth representa-,ion 01' p,,(r. 1) is ohtaineti hy eonvoluting Eq. (8) with a testfune'ionq,(r): 1',,(r.l) = IId'r'q,(r - r')¡i,,(r'.I), \Veuse a two4dimcnsional phase spacc Gaussian lest function inlhe nutllericalcalclllations helow, 01"lhe form

whe1"ef1r¡ and f11' are the widlhs ofthe test function in q anJ JI,

respecli\'ely: Ihesc widlhs are dClcrmined by numerical con-silkralions in praclicc. The c\'olution 01"Pij arises through thcnow nf the corresponding trajeclories of the ilh ensemhle.{r~iJ)(f) }, ami lhc coeffkients of the trajectories, {a ~I'J)(r) }.The lime e\"olulion ofthe {r~I'}l(t)} are governeJ hy con\'en-tional Hamilton's equations, gcnerated by Ihe ith Hamilto-nian with ini,ial eontiitions {r!:il(O)} anti {a,:il(O)}, Thetime dependence of the weighls {a~:j) (t)} is induced by lhenone\asical elcctronic coupling. In practice. lhe nlllllerical in-tegration nf the equations 01"motion for lhe full sel of dyn<I-mical variabks {r;l'.il (t), (l~:j) (t)} is accomplished by COI1l-hining. within a given time step, lhe conventional c1assicalmolecular dynamics propagalion of the rhase space variables\vith;¡ lincar algebraie calculation ofthe updated coefficicnts.The delails 01"Ihese calculations can be found in Ref. 22.

We tcslcd our trajcctory implcllIenlation hy lrealing a mo-del olle-dimensional system eOllsisling of ll1otiOIlof a parti-ele \vilh mass IJI 011two coupled exciled cleclronic surfaces.The syslcm is cxciled at f = O hy a sudden optical cxcita-tion frolll a harmonic ground elcctronic stale. Thc two ex-cited slales treated explieit\y in {he simulation coosist of arepulsi\'e exponelllial potential. U1(q) = Ae-1(q-q¡) - [J

which is initially popu\alcd oy lhe optica\ excitatioll. andwhich is crossed hy a hOlllld Morsc pOlential, ú"Aq) =D (e-1(t/-I/:.!) _ 2(.-(fI-IJ:.d] .. Thc potential curves are shownin f-ig. 1..Thl:sC lw() diabatic cur\'es are couplcd by an off-diagonal lcon 1111 = \ '(11), which is taken to he a Gaus4sian funclion ccnten:d at the crossing poinl (/r. \"((1) =\ (1I,-.I(If-I/ ...):!.

The initial l)lIantutll state 01"Ihe system is a 10cali7.ed(iaussian wavcpackct on Ihe replllsive surfaee, ccntered at thegrollnd slalc eí]llilihrium hond Icngth (J¡.

(6)

(5)

(4 )

(2 )

(~11

P21ti =

. 21'= CIII'II + {1'.RepI2} - --,¡Imp12,

. 2\/= C221'22 + {V.Rep12} + --,¡Ill1PI2.

U/Jll (' ) 1--- Ln-i ..••' (lll+ry_{F,Pll+P11}JI -

D/JII

JID(ll'l

ni

Here. we have defined a sel 01"Liouville oper::nors (id =.{H".fl(i U) "nti Co! {~(Hll + llnJ.J):atHl Ihe frequency term ...•.' in Eq. (7) is given by Lv'(q) ==(JIII - 11,,) Ir},

The phase space densily (l¡¡(I', t) is approxim,Hcd by anensemhle 01"trajeetmies

ror i ;lIld j = 1.2. For simplicity, \Ve assut1lc in what followslha! lhe off-diagonal clclllcnls of lhe Hami1tonian are indivi-duall\' HCl"mitian: ¡[lO) = iL¡ == fl: sec Rcf. 13 for a dc\'c-, --lopmcllt 01"lhe general case. \Vc can no\\' derive scrniclassicalcqualions of lllo1ion fOl"lhe phasc space functions correspon-ding lo Ihe density malrix element opcralors Pij. Each 01'Iheterms of Eqs. (...1-)is wrilten cxplicitly in terms of comllluta-lors alld anticol1llllutators 01"Hcrmitian operators, which have\l,.'ell-dclilled scmic\assiea\ Iimits. A systematic expansion inpowers 01"I1can he developcd using the Wigner-t\'loyal ap4proach Pi-lO!. Por nur present purposes, we keep only Iheleading le[lns. atHl Il1us considcr the semiclassical limiL Poranticollllllulalors. (he transcription lo the semiclassical limitis given hy ¡..l. ¡J)+ --t 2/\('1. [I)D(r¡, [1) + O(h'). To lowestorde1"in/,. Ihc n:sulling classica\4likc equations 01"motion co~rresponding to Eqs. (4) are

respee,ively. In this seetion, the eicetronie states {jI), 12)}are choscll lo he diahatic oasis functions, lile adiahatic rc-prcscnlalion has heen lIscd in Re!'. 14. Suhstituting Eqs. (2)and (3) intn Eq. (1) givcs lhe quantum mcchanical LiOllVillccqualioll

and

/\'1'

1",(1',1) = 2:>\"\1)6(1' - 1')")(1))j:::1

(X)(JI/u.!) I ¡.¡ ['1IJ:..1 .)]

'/'('1, f = 11)= -=- ex!, -T, ('1 - 'Ie)-" " - ~

(10)

Rel'. Ale.\". Fú. 47 SI (2001 l :m-3ú

SIMULATlON OF MOLECULAR TRANSITIONS USING CLASSICAL TRAJECTORIES 33

cxact quantumsemiclassical

¡500 2000 2500 3000time ¡au

1000

P12

500

0.8

~o.~ 0.6;;¡"3o.oo. 0.4

0.2

OO

1211

U¡(q)

107654

-0.02

8 9q lau

FI(;URE l. Potential curves used in numerical calcul.ttions.

0.02

~.!" 0.01»~o

U,(q)oo

O~

""o -0.01o.

FIGURE 2. Comp;¡rison of semicl;¡ssical trajcctory and cxacl quan-tum populalíons of statcs 1 and 2 as functions of time.

T¡\BI.E 1. NUlllcrical valucs 01' paramelcrs. in alomic uníls

..1 2.2782 x 10-2 jlo 1.2 x ll)-:l

11 2.2782 x 10-' fjc G.l:J315

'1' 5.5 In 10 x lO'IJ 1.8225 X 10-2 W 4.0 X 10-.1

1/2 5.8 'le 5.7

rhe lllllllcricnl valucs nI' lhe paramClcrs are givcn in Tahle 1.In our model, lhe Jaser pulse couplcs Ihe ground slalc cxclu-sivcly lo lhe rcpulsivc statc 1, and thus lhe ¡n¡tial populationun Ihe hound statc 2 is zcro. As lhe syslem cvolvcs. Ihe cou-pling ICrln \1('1) will induce population transfcr hClwccn (heslalcs.

Ex;¡ct quantul11 I11cchanical calculations were pcrformcdlIsing lhe standard FFT-hascd grid mcthod of KoslnlT [201.gCllcraJizcd lo trcal couplcd wavcpacket dynamics on multi-pie clcclronic surfaces.

In Fig. 2, \Ve sho\V lhe populalions of the statcs I anJ 2as <l fUllclion 01' lime for hOlh the exact and scmiclassicalmolccular dynamics. In OUt"trajeclory impletllcnlalion. Ihepop1l1ation 01' lhe 11th clcclronic slale is gi\'cn hy p¡!(t) ==Trfl,,(r,r) = 2:) "j',) (1) (/, = 11,22). The resullsol'ourIrajcclory-bascd implclllcnlalion are compared \Villl (he co-rn..'sponding llull1crically exacl qllanlum mcchanical quallli-tics. The c1assical trajeclory resll1ts are in ncarly quantilativcagrcclllenl wilh thc quanllll11 wavcpackcl dynalllics.

3. Wa\'cpackct Interfcromctry

In Ihis scclioll, \VC considcr in more dClail lhe [rcalmenl ofeleclronic cohcrcllcc in lhe conlext of our sCl11iclassical I.iou-vilh: dpproilL"h. \Ve fOL"u:-; Ol! ¡\11 iucalizcu ultrafa:-;t cxpcrimcllt.\\'hich \Ve call nonaJiahatic wílvcpacket inlerferolllelry. in\\'hich lhe cohercnt rhase relalion hel\Vecn 1\\'0 inilially ropu-bled elcctronic stales plays a dominant role in Ihe suhsequent

nonadiabalic dynalllics. This manifestly coherenl case is animportanl and challcnging lesl for arproximale mcthods formodcling nonadiahalic dynamics. An experimental realiza-lion 01' this general scenario is heing pursued in ongoingwork 1231. \Ve consider an idealized pump-probe experimentconsisting of a cohcrent ultrafasl excitalion from the lowcst-Iying ground elcclronic stJtc lo a cohcrcnt superposilion oftWD excited eleclronic slales 11) am' 12) using a pair 01' laserpulses wilh a well-defincd ami cOlllrollahle phase rclation <p.A diahatic rerrescntation 121,241 is aJopled for lhe dcscrip-lion of lhe clcclronic slalcs. For simplicity, we assume in ouranalysis Ihat Ihe Jaser pulses crcating the initial eoherent su-pcrpnsilion are very 5horl, amllhal caeh eouples the groundslalc lo a single upper surfaee. Eaeh of the pulses Ihen pre~pares a copy 01' lhe inilia! vihralional statc IV) of lhe groundelcclronic slalc on the appropriale excited surfacc. As lheyevolvc subseqllenlly, lhe lwo components overlap in lhe rc-gion nI' crossing pOlenliai curves, and lhe dynamics is domi-nated hy quantlll11 interferencc of lhe possiblc pathways avai-lahle to lhe Illoving wavcpackets. Thc resuIting erfeet is largeir bOlh initia1 componenls are comparable in magnilude, andir lhe relative phase of lhe 1\1/0 cOl1lponents is well-delincd.Furlher, lhe sign and magnilude 01' lhe effecl can he conlro-lIed cxlernalIy by manipulaling Ihe phase (jJ.

Thc IWe) cxcilcd slales trealeJ explicitly in Ihe simulationanJ lhe coupling tefln are chosen to hc lhe same as those usedin Ihe prcvious seclion. \Ve simulate the couplcd state nuclcardynamies of Ihis system using our Irajeclory implementation01' lhe semiclassical Liouville method [22]. and compare lheresults Wilh Ihose 01' nUlllerically exacl eoupled-slate wave-packel propagatiol1. Following Ihe developmenl of lhe prc-ViOllS seclion. \Ve SCIcCI inilial wa\'cpackels orlhe form

(1 I )

ReJ'. Akr. Fú. 47 SI (2001) JO-JfJ

A. DONOSO AND Ce. MARTENS

0.8exact quantum

0.7semiclassical

P22(7)

.~~0.6

/ ..I ..~ /' -.e

8<;; 0.5 " ,-;;Q, \o , \ /Q, ,

0.4 " ," /

~--: ./0.3 Pl1(7)

0.2O 2 3 4 5 6

Ó Irad

$ =0.3

$=0.0

exact quantumsemiclassical

200 400 600 800 1000 1200 1400

time lau

.140.7

0.65

0.6

0.55

o:.: 0.5

0.45

0.4

0.35

0.3O

FH,lfRE 3. Popubtion of slalCs 1 and 21's. time. Composilc rcsullscomp<lring the cxact quantum anu scmiclassical rcsults. for the ca.se with PI (O) = Pl(O) = 0.5. antl several valucs of (he relativcphasc o. Thc populalion transfcr strongly dependent on the phasc.

FIGURE 4. Final populalions \'S. (he relalive phase angle o. TheFigure compares Ihe quamum and scmiclassical statc populations¡¡Ilhe tinal time ol'T = 1100 au.

where v(q - '!r) is a rcal.valucd mínimum unccrtainty con.llguration spacc Gaussian givcn hy Eq. (10) ccnlcrcd al ([,,=.t.S (¿lIomic units), PI (O) is lhe ¡n¡tial statc I population. nor-m¡¡lizel! sueh Ihal P, (O) + P2(0) = 1, anl! 6 is lhe relativephasc oí"lhe ¡nitial wavcpackels. Thc samc valllc 01'<J" is eho-sen for hOlh w~lVcpackcts. lo Illodci a suddcn vertical (mn-sitioll from a lowcr-Iying harmonic ground stale wilh cquili-hrium distance <j ;;:;:; <JI" The ¡nitial expcctation value of tht.::momcntuln is zero for cach component of w(q, O).

Exact quantum mcchaoical results are ohtained using lhestandard numcrical Illcthods devcloped for wavcpackct pro~pagation hy Kosloff 120]. The semiclassical tr<tjcctory calcu-lalinos are performed lIsing the oUlllcrical algorithms descri-hed (he prcvious sectioo ami, in more dctail, in Re!'. 22.

In Fig. 3. we prescnt composite results for several cal-culations performed with equal ioitial populations 00 stalcst ami 2 (P, (O) = P2(0) = 0.5), ¡¡nl! with a nUlTlher01 va-lues of the initial relative phase (jJ. We note thallhc rcsults arestrongly dependent on lb, the relativc phase 01'the initial wa-vepackets. In Pig. 4, we show the lb dependence of the finalst¡¡le I anl! state 2 populalions, PI (T) ¡¡nl! P2(T), respecli-\'ely. whcre T ;;:;:;] ]00 au corrcsponds to the 1I0al time 01'theIlumerical simulations.

It is found that the mcthod yiclds cxccllent agrcement forholh the timc-depcndent populations and the detailed evol-ving states. This dcmonstrates that our semiclassical appma-ch is capahlc 01'capturing the intrinsically quantum mcchani-l.."al phcnomcnon of clcelronic cohcrcnl'e, even whcn lhe eo-herence plays a cenlral and dominant role in the dynamics 01'c1el'lronic Iransition.

\Ve compared our semiclassical method with Ihe widelylIscd .wlface hoppillg algorithm. which has proven lo he aoaccllrale and efficient 1001for modeling coupled electronic-nuclear dynamics in a wide rangc of physical settings. \Vhile

rOl" !he case 01' incohercnl dynamics hoth methods producedsimilar results in ahout the same CPU time (roughly, 1/2 hourin a 533 MH/. 164 LX Alpha processor), in the case uf w¡¡-vepackcI intcrfel"Omctry. surfacc hopping failed to properlycapture Ihe inlerfcrometric quantum transition dyoamics. ForIhese olle Jimensional syslems. the CPU time for both theexacl and scmiclassical calclllations are quite modest. No-nelheless, wc ShOllld note Ihal the semiclassical ca1culation(and the surface hopping vcrsion) are actually slower lhan theexac! wavcpacket calculations rOl"the results presenlcd here.'fhe promise 01' the Illethod is that il will allow rnany-hodysysteJ1ls that can not he tackled wilh exact qanlurn rnethodsto be trealed. Wc have indeed very reccntly applied the fll-11illlplcmenlalioll nI' the Illelhod successflllly to modcl elec-tmnic rclaxalion 01' a model diatomic in a liquid, a severalhundrcd atom system, in ahollt 5 hours on a 533 MHz Alphaprocessor. For coherent pmccsses such as the une descrihedhere, approxirnate hut qualitatively accurate results can, infact, he ohlained with ahollt the same effort as a conventionalclassical moleculnr dynamics simulatioo. A puhlication 00this suhject is in preparation. The origin of the shortcomingsnI' Ihe fewest.switches surfacc hopping rnelhod is the inde-penden! nature of Ihe Irajectory ensemhles undcrlyiog surfacehopping. Each trajcctory delermines ils own fate using a sto-chastic algorithm and perhaps its individual representation ofIhe e,"olving elcclronic degrces 01'freeJom. Interfereoce het-ween distincI trajeclories is not a part of the theory. and thedccision nf one IrajcC{ory 10 undcrgo a lransition, or not, isunaffected hy Ihe resl of thc ensemhle. This is often a go-0<.1approximation for the commoniy encouotered problem ofelcctrooic population relaxatioo. Hcre, the electronic cohe-rencc [112 is ahsent initially, and must he crcated by the evol-ving populations. \Vhen crealed. PI'!. is localized around theregions of crossing (or avoided crossing in the adiabatic re-

Rel'. Mi'X. Fú. 47 S 1 (200 1) 30....J6

SIMULATION OF MOLECULAR TRANSITIONS USING CLASSICAL TRAJECTORIES 3S

prcsentation) ol' the potential surfaces. This region beco mesinfinilesimal!y smal! as Ii -+ O. Thc probability of clcctro-nic transilion can be incorporated accurately ¡nlo the mcthodin lhis c1assical ¡¡mil, as done in surfacc hopping, w¡thau!rcsorting lo a fully cohcrcnt theory. If the clcctronic coheren-ce is ¡arge initially, 5uch as is the case in the nonadiabaticwavcpackct interfcrornctry proccss, the approximation thalindividual trajcctorics undcrgo localizcd, ¡ndependen! tran-silions brcaks down. lnterferencc bctwcen various paths andproccsscs can play an important role, and thus interdcpen-denee al' trajectories w¡thin and betwcen cnscmblcs mus! beincludcd. Thc scmiclassical Liouville method cmployed hereallows this quantum cffect to be incorporated into a classicalmolecular dynamics simulation.

4. Conclusions

In Ihis rcview, we have descrihed a trajectory-hascd imple-lllentalion 01' our rccently proposed semiclassical-limit ap-proach to molecular dynamics on coupled electronic surfa-ces. Building on the convcntional method of modeling sin-gle surface dynamical processes in the classical Iimit usingIrajectory integration and ensemble averaging, we presentedan augmented dynamical description of multi-state classicaldynamics, hased on adding additional time-dependent cocffi-eicnls lo each membcr of lhe couplcd ensembles. Physically,these ncw dynamical variahles represent the evolving weights(ami phases ror the electronic coherence) of Ihe lrajeclories.AH 01'the nonclassical effccts 01'electronic coupling and co-hercll<:e are descrihed hy the equations of 1l10tion for Ihese<:ocflkienls. whilc the phase space trajecturies thell1selvesohey conventional Hamilton's equations,just as in Ihe classi-cal single state case. Despite the use of classical trajecloriesin our numcrical implernentation, we stress that our approa-eh is explicitly a theory of lhe evolulion uf the distrihulionfunctions Pij (f, t) themselves not 01'individual, independentIrajectories. This is an important poinl, and at the heart 01'lhedescription of nonclassieal proccsses. Electronic coherencein our theory is nol a property 01' any individual trajectory,hut results from lhe "interference" of a whole ensemble 01'

1. "Chcmical Rcactions and their Control on the FemtosecondTi-me Scale". Admnces in Chemical Physics, cditcd by P.Gaspardand 1.Burghardt, (Wiley.New York, 1997)Vol. 101.

2. J.T. Hyncs.Annu. Rev. Phys. Chem. 36 (1985) 573.3. T. Fonesca. J.A.N.F. Gomcs, P.Grigolini. and F. Marchcsoni.

Ad" Che",. Phys. 62 (1985) 389.

4. D. Chandler. J. Stat. Phys. 42 (1986) 49.

5. M.P.Allen and D.Tildesley,J. Computer Simulatioll oI Liquids.(CJarcndon.Oxford. 1987).

G. LO. Landau. Phys. Z. Sow. 1 (1932) 46.7. C. Zencr. Proc. Roy. Soco A 137 (1932) 696.

trajeclories modeling an evolvinglunction. This function, inturn. both conlrols and is iolluenccd by Ihe evolution of olherfUllctions (the probahility dislrihulions on the two surfaces)and lhus influences lhe generalized dynamics of many othertrajectories. When uscd as a numerical method for solving thecoupled slate partial differcntial equalions for the Pij (r, ti.the members 01'all the enscrnbles hecome entwined in a webof mutual interactions lhat is quite different fmm their rigo-rous indepemlence in a conventional single surface moleculardynamics simulalion. We conjecture that many of the fun-damental problems associated with previous approaches tomodeling clectronic transitions wilh c1assical trajectories, aredue to their focus on single independent trajectories coupledto quanturn degrees of freedom. In the exact quantum mecha-nica! dcscription of multi-state dynamics, is the evolving wa-vepa<:kets on each state that couple lo each other. This roug-hly translales in the classicallimit to interactions between theentire evolving ensembles. The work presented here constitu-tes a first successful attempt al huilding a c1assical trajectory-hased irnplementation of the general semiclassical approachto nonadiabatic dynamics. Wc have shown that an augmenta-lion of a standard molecular dynamics program can be madewhich allows certain intrinsically quantum mcchanica! pro-perties. such as electronic population transfer and the evo-lution 01"clectronic coherence, to be rnodeled with excelJentaccuracy, at least in principie. More work needs to he done,howcver, bcfore lhis approach hecomes an cfficient nurneri-cal Illethod ror modeling many-hody systems. Refinementscurrently under developrnent inelude methods fm incorpora-ting birth. death. and '"retiremenC' of trajectories to allow thetotal numher of interacting mcmhers ol' the ensemhles to heminimized while keeping the ensembles ovcrlapping in pha-se space regions whcrc coupling is operative; the use of moreclahoratc higher-order represenlations 01'the short-time pro-pagator. al10wing larger time steps to he employed; and lheincorporation 01"dynamical approximations ¡nto the trajec-tory equations 01"motion. permitting an explicit and dctailedcalculation 01'the full electronic coherence to be avoided forapplications where it is not required.

8. E.CG. Slllekelherg./Ic1v. Phys. Acta 5 (1932) 369.

9. E.E. Nikitin. T/U'ory oI Elementary Atomic amI Molecular Pro-ces.w:s i" Cm.es. (Oxford Univcrsity Press, Oxford. 1974).

10. K.S. Lam and T.F. Gcorgc, "Semiclassical Theory of Electro-nically Nonadiabatic Transitions in Molecular Collision Pro-cesses", in Scmiclassical Ml'thods in Molecular Scattering andSpcctroscopy. cditcd by M.S. Child. (D. Reidel, Reidel Dordre-eht. 1980).

11. DA Micha. J. Chcm. Phys. 7K (1983) 7138.

12. J.C. Tully and R.K. Preston. 1. Che",. Phys. 55 (1971) 562.

13. CC Martens and J.-Y. Fang,J. Che",. Phys. 106 (1997) 4918.

Re\'. Mex. PÚ. 47 SI (2001) 30-36

36 A. DONOSO ANO e.e. MARTENS

14. A. Donoso and Ce. Martcns. 1.Chem. Phys. 112 (2000) 2980.

15. J.V Neumann. Mathematicaf FOU1ulatio/lS oJQuwltwn Mecha.flics. (Princeton University Prcss, Princcton. 1955).

16. U. Fano. Rel'. Mol!. Phys. 29 (1957) 74.

17. S. Mukamcl. PrincipIes o/ NOfllinear aplical SpectroJCoPY.(OxfonJ University Prcss. New York. 1995).

18. ~1. Hillery. R.F. ü'Connell. M.O. Scully, and E.P. Wigner.Phys. Rl'por/s 106 (1984) 12.

10. K. Takahashi. Prog. T/U'OI: Phys. Su""l. 98 (1989) 109.

:20. D. Kosioff and R. Kosloff. 1. COl1llnlt. Phys. 52 (1983) 35.

21. J.e. Tully. in Moderll MetllOdsfor Multidimensimwl DynamicsComplI/arioll in Cf¡c/IIütry. editcd hy D.L. Thompson (WorldScicnlific. Singaporc. 1998).

22. A. Donoso and ce Martens. I "hys. Chem. 102 (1998) 4291.

23. D. Kohcn. R. Zadoyan. V.A. Apkarian. and e.e. Martcns.(work in progrcss).

24. K.S. Lam and T.E Gcorgc. in Semiclassical MetllOds ;/1 Mo-leClllar Scatterillg {l/ul SI'('ctrosCOI'Y. ediled by M.S. Child (D.Rcidcl. Rcidel Dordrccht. 1980).

Rel'. Me.\". Fú'. 47 SI (2001) 30-36