A semiclassical self-consistent-field approach to dissipative … · 2013-04-04 · A semiclassical...

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A semiclassical self-consistent-field approach to dissipative dynamics:The spin–boson problem

Gerhard StockInstitute of Physical and Theoretical Chemistry, Technical University of Munich,D-85748 Garching, Germany

~Received 5 October 1994; accepted 18 April 1995!

A semiclassical time-dependent self-consistent-field approach for the description of dissipativequantum phenomena is proposed. The total density operator is approximated by a semiclassicalansatz, which couples the system degrees of freedom to the bath degrees of freedom in aself-consistent manner, and is thus in the spirit of a classical-path description. The capability of theapproach is demonstrated by comparing semiclassical calculations for a spin–boson model with anOhmic bath to exact path-integral calculations. It is shown that the semiclassical model nicelyreproduces the complex dissipative behavior of the spin–boson model for a large range of modelparameters. The validity and accuracy of the semiclassical approach is discussed in some detail. Itis shown that the method is essentially based on the assumption of complete randomization ofnuclear phases. In particular, the assumption of phase randomization allows one to perform the traceover the bath variables through quasiclassical sampling of the nuclear initial conditions withoutinvoking any further approximation. ©1995 American Institute of Physics.

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I. INTRODUCTION

Dissipative quantum dynamics has been found to plakey role in many areas of theoretical physics.1 In the field ofchemical physics, condensed phase reactions such as eleand proton transfer processes have been investigated in tof a dissipative two-level system, often referred to as spboson problem.2Although there has been significant progrein the development of exact quantum-mechanical propation techniques3,4 straightforward basis-set methods evdently cannot account for multidimensional condensed phdynamics, since the computational effort increases expontially with the number of nonseparable degrees of freedoFeynman’s path integral formulation appears to be the oexact alternative that would avoid this problem,5–13 but onetrades the exponential increase of basis functions for amidable sum over paths, the number of which grows exnentially with propagation time. Even for moderate timegimes and despite considerable advances in the numeimplementation, the evaluation of this sum is rather cumbsome and often only feasible within some approximationthe problem under consideration allows for a perturbattreatment, ‘‘golden rule’’-type formulations14,15 ~for smallelectronic coupling!and reduced density-matrix theory16–21

~for small system–bath coupling! are widely used formula-tions that have proven to be quite valuable. In the case whthe coupling between the individual nuclear degrees of frdom may be assumed to be weak, time-dependentconsistent-field~TDSCF! methods, which approximate thtotal density operator by a product of operators dependonly on a single nuclear coordinate, have been successapplied to a variety of problems.22–31The validity of simpleTDSCF schemes, however, is not easy to predict, andhave also been shown to fail badly, e.g., in the descriptionnonadiabatic relaxation processes.26,27

Most path-integral and reduced density-matrix formutions are restricted to harmonic ‘‘bath’’ degrees of freedo

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Only very recently, formulations have been developedgeneralize the two-level system to a one-dimensional retion coordinate,9,18,20,21or to introduce general basis functions for the evaluation of the path integral.12 If one is inter-ested in investigating more realistic models for dissipatidynamics~including, e.g., multidimensional anharmonic reaction coordinates and kinetic couplings!, it therefore stillseems unavoidable that one resorts in one way or anotheclassical mechanics. For the description of nonadiabaticlaxation processes that involve intersections or avoidcrossings of the potential-energy surfaces, one has to invmixed quantum-classical schemes, such as surfahopping32–38 and classical-path theories.39–46 Both formula-tions share the idea that the ‘‘system’’ degrees of freedomdescribed quantum mechanically while the bath degreesfreedom are described classically, but differ in the way tsystem is coupled to the bath.

In the surface-hopping approach, classical trajectorare propagated on a single adiabatic potential-energy suruntil, according to some ‘‘hopping criterion,’’ a transitionprobability P2←1 to another potential-energy surface is caculated and, depending on the comparison ofP2←1 with arandom number, the trajectory ‘‘hops’’ to the other adiabasurface. The many existing variants of the method diffmainly in choice and degree of sophistication of the hoppicriteria. They all share the problem that, owing to the hoping processes, the coherence of the total wave functiondestroyed after a relatively short time. Classical-path meods, on the other hand, avoid this problem, by propagatthe classical degrees of freedom self-consistently in an avaged potential, the value of which is determined by thestantaneous populations of the different quantum states.well known from applications in reactive scattering thclassical-path methods may lead to unphysical results ifsystem is asymptotically in a single electronic state.47 Themean-field approximation, however, seems to be not tcritical for the description of bound-state relaxation dynam

156161/13/$6.00 © 1995 American Institute of Physicst¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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1562 Gerhard Stock: SCF approach to dissipative dynamics©

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ics. Using the so-called classical electron analog modeMeyer and Miller,42 it has been shown recently that thsimple classical-path approach describes surprisingly arately the ultrafast electronic and vibrational relaxation dnamics occurring in internal conversion48–50and simple pho-toisomerization processes.51

In this paper we propose a semiclassical TDSCFproach to describe dissipative quantum phenomena sucthe electronic relaxation dynamics of the spin–boson moContrary to common quantum-mechanical TDSCF schemwhich assume that the total density operator factorizessingle-mode operators, this ansatz couples the systemgrees of freedom to the bath degrees of freedom in a sconsistent manner, and is thus in the spirit of a classical-pdescription. We apply the ansatz to the spin–boson mowhich appears to be an ideal test problem for the approas there exists a wealth of approximate theories,2,52–57 andnumerical reference calculations8,10,13 to compare with.

We present computational results for a spin–bosmodel with an Ohmic bath, which are compared to expath-integral calculations. It is demonstrated that the seclassical model is able to reproduce the complex dissipabehavior of the spin–boson model rather accurately and flarge range of model parameters. Examples include the etron transfer dynamics in the low- and high-temperatureadiabatic and nonadiabatic regimes, and the transition fcoherent to incoherent relaxation dynamics.

The comprehensive discussion of the paper is devoteinvestigate and analyze the validity and accuracy ofsemiclassical TDSCF approximation. To this end, we dera generalized master equation for the electronic [email protected]., for the expectation value ofsz(t)#, which allows com-parison with analytical formulations of the spin–bosmodel.2,52–57Within the decoupling approximation this eqution is shown to be equivalent to the classical limit ofquantum-mechanical expression recently given by Vitali aGrigolini using a polaronic-transformation approach andglecting the nonclassical reaction force.56 Introducing fur-thermore second-order perturbation theory in the systebath coupling, we are able to make contact to the so-ca‘‘noninteracting-blip approximation’’ ~NIBA! of Leggettet al.2 The NIBA represents the best-known approximatiin the field of quantum dissipation, and has proven toquite valuable for the qualitative description of many dispative phenomena. We also compare the semiclassical mto related self-consistent-field formulations such as quantmechanical multiconfiguration TDSCF schemes.29,30 Wedemonstrate that the semiclassical theory can be obtafrom the quantum formulation by neglecting quantumechanical phase terms, which is equivalent to the assution of randomization of nuclear phases. The randomizaof nuclear phases is found to be the crucial condition forvalidity of the semiclassical approach. The conditionshown to be fulfilled, e.g., in the case of a continuous Ohmbath, and the implications on the applicability of the methis discussed in some detail.

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II. A SEMICLASSICAL SELF-CONSISTENT-FIELDANSATZ

A. Model system

We consider a two-level systemHS which is linearlycoupled to a harmonic bathHB through the system bath coupling HS–B , yielding the standard spin–boson Hamiltonian2

H5HS1HB1HS–B

5\Dsx1(j

~ 12Pj

2/mj112mjv j

2Xj2!11sz(

jK jXj ,

~2.1!

wheres i( i5x,y,z) are the Pauli matrices and1 representsthe unit matrix. As usual, the two-level system is charactized by the electronic couplingD, andmj , vj , andKj denotethe mass, the vibrational frequency, and the linear couplconstant of thej th oscillator. To be specific, we will refer tothe spin and the boson variables as electronic and nucdegrees of freedom, respectively.

For notational convenience we will henceforth set\51,and change to dimensionless coordinates and momenta

xj5~mjv j !1/2Xj ,

~2.2!pj5~mjv j !

21/2Pj ,

and define analogouslyCj5(mjv j )21/2Kj . Representing

furthermore the Pauli matrices in terms of diabatic electronbasis statesuw1&,uw2&, the spin–boson Hamiltonian takes thform

H5 (k51,2

uwk&hk^wku1$uw1&D^w2u1h.c.%,

hk5h06(jCjxj , ~2.3!

h05(j

12v j~pj

21xj2!.

The time-dependent dynamics of the spin–boson prolem ~2.3! is described by the Liouville–von Neumann eqution

id

dtr~ t !5@H,r~ t !#, ~2.4!

where r(t) represents the total density operator of systeand bath. We assume that att50 the electronic system is inthe uw1& electronic state and the bath is in thermal equilibriu

r~0!5uw1&^w1ue2bhB/Tr e2bhB, ~2.5!

where, as usual,b denotes the inverse temperature 1/kBT.Depending on the physical situation to be described, the bis initially in the equilibrium geometry of theuw1& electronicstate~i.e., hB5h1 , ‘‘tunneling case’’!, or in the equilibriumgeometry of the uncoupled bath HamiltonianHB ~i.e.,hB5h0 , ‘‘spectroscopic case’’!, where the latter case referto a photoexcitation of the electron from a neutral electronstate~e.g., uw0&! to the uw1& state.

3, No. 4, 22 July 1995t¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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B. Self-consistent-field ansatz

The main approximation that we are going to make isassume that for all times the total density operatorr(t) canbe written as a sum of products of system density operrS(t) and bath density operatorrB(t)

r~ t !5(a

waraS~ t !ra

B~ t !, ~2.6a!

raS~ t !5 (

k,k851,2

xa~k!~ t !uwk&^wk8uxa

~k8!* ~ t !, ~2.6b!

raB~ t !5uFa~ t !&^Fa~ t !u. ~2.6c!

Here, the electronic density operator is represented bycomplex numbersx~1! andx~2!, the squared moduli of whichrepresent the population probability in the electronic stuw1& and uw2&, respectively. The nuclear density operatorrepresented by the multidimensional state vectoruF&, whichwe will assume to be normalized, i.e.,^FuF&51. The indexalabels the different ‘‘configurations’’ of the TDSCF ansa~cf. Sec. IV C!, wa being the statistical weight factosampled, e.g., from a Boltzmann distribution. To give a cocrete example, consider the initial condition~2.5!, which re-quires for the system density operator thatuxa

~1!~0!u251 anduxa

~2!~0!u250, and for the bath density operator that

wa5e2bea Y(a

e2bea, ~2.7a!

raB~0!5ua&^au, ~2.7b!

where theua& are eigenstates ofhB such thathBua&5eaua&.Note that by virtue of ansatz~2.6! the electronic~system!

degrees of freedom are coupled to the nuclear~bath!degreesof freedom in a self-consistent manner. This is in contrascommon TDSCF formulations, which assume that the todensity operator factorizes in single-mode densoperators,22–31thus introducing a self-consistent coupling btween the individual nuclear degrees of freedom. In otwords, the ansatz~2.6! still leaves us with the problem tosolve the equations of motion for many nuclear degreesfreedom on asingleelectronic potential-energy surface.

Let us consider the equations of motion for the systand bath density operator, respectively, that result fromansatz~2.6!. It has been pointed out above that the summtion overa arises from the thermal initial distribution of thbath modes. This is to say that, without any further appromation to Eqs.~2.6!, we may solve the Liouville–von Neumann equation for the total density operatorr(t) by solvingthe Schro¨dinger equation for each individual configuratioraS(t)ra

B(t), and adding up these solutions according to E~2.6a!. Therefore, the equations of motion forra

S(t) andraB(t) @and thus forxa

(k)(t) anduFa(t)&# can directly be takenfrom the well-known wave-function TDSCF formalism,58

yielding

i xa~1!~ t !5^Fa~ t !uh1uFa~ t !&xa

~1!~ t !1Dxa~2!~ t !, ~2.8a!

i xa~2!~ t !5^Fa~ t !uh2uFa~ t !&xa

~2!~ t !1Dxa~1!~ t !, ~2.8b!


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i Fa~x,t !5~ uxa~1!~ t !u2h11uxa

~2!~ t !u2h2

1ReDxa~1!~ t !xa

~2!* ~ t !!Fa~x,t !. ~2.9!

It is seen that within the TDSCF approximation the eletronic ~system!equations of motion reduce to a coupled twolevel system with explicitly time-dependent coefficien^Fa(t)uhkuFa(t)&. The nuclear~bath!dynamics is describedby a single multidimensional wave functionFa(x,t) propa-gating in an averaged potential, weighted by the electropopulation probabilitiesuxa

(k)(t)u2 ~k51,2!.

C. Semiclassical approximation

For the sake of notational convenience, we will in thfollowing refer to a single nuclear coordinatex ~instead of(j xj !. To motivate an ansatz for the vibrational wave funtion Fa(x,t) we rewrite Eq.~2.9! as

i Fa~x,t !5~ 12v~p21x2!1~e1Cx!Pa~ t !

12ReDxa~1!~ t !xa

~2!* ~ t !!Fa~x,t !, ~2.10!

which is recognized as the time-dependent Schro¨dingerequation for a driven harmonic oscillator. It is well knowthat the solutionFa(x,t) of Eq. ~2.10! can be written interms of Gaussian wave packets as59,60

Fa~x,t !5 (x0 ,p0

ca~x0 ,p0!fx0 ,p0~a! ~x,t !, ~2.11!

where the expansion coefficientsca(x0 ,p0) are determinedby the initial conditions~2.7!, andx0 andp0 denote the initialvalue for the classical position and momentum, respectiveThe Gaussian wave packets

fx0 ,p0~a! ~x,t !5p21/4 exp$2 1

2@x2xa~ t !#2

1 ipa~ t !@x2xa~ t !#1 iga~ t !% ~2.12!

are completely described by the trajectoryxa(t), the corre-sponding momentumpa(t)5mxa(t), and the action integralga(t).

60 The trajectoryxa(t) is given by

xa~ t !5x0 cosvt1p0 sin vt2CE0

t

dt8 sin v~ t2t8!Pa~ t8!

[xa~0!~ t !1xa

~1!~ t !, ~2.13!

wherexa(0)(t) andxa

(1)(t) denote the unperturbed motion anthe motion due to the reaction forceCPa(t), respectively.The dynamics of the nuclear degrees of freedom thus enthe electronic equations of motion~2.8! through the expres-sion

^Fa~ t !uhkuFa~ t !&5 (x08 ,p08

(x0 ,p0

ca* ~x08 ,p08!ca~x0 ,p0!

3^fx08 ,p08

~a!~ t !uhkufx0 ,p0

~a! ~ t !&. ~2.14!

To obtain an expression that is computationally managable in the case of many nuclear degrees of freedom we hto make an additional approximation to Eqs.~2.6!. From a


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classical point of view, an obvious approximation to simplithe evaluation of the correlation function~2.14! is to neglectthe nondiagonal contributions, i.e.,

^Fa~ t !uhkuFa~ t !&' (x0 ,p0

uca~x0 ,p0!u2

3^fx0 ,p0~a! ~ t !uhkufx0 ,p0

~a! ~ t !&

5 (x0 ,p0

uca~x0 ,p0!u2hk@xa~ t !#, ~2.15!

wherehk[xa(t)] denotes the classical Hamitonian functioEquation~2.15! represents a classical approximation in thsense that the correlation function~2.14!depends no longeron the phases of the Gaussian wave packetsfx0 ,p0

(a) (t), but

are completely determined by the classical trajectoryxa(t).Equation~2.15!amounts to approximate the bath density oerator as

raB~x,x8,t !5Fa~x,t !Fa* ~x8,t !

' (x0 ,p0

uca~x0 ,p0!u2fx0 ,p0~a! ~x,t !fx0 ,p0

~a!* ~x8,t !.

~2.16!

Note that all summation indicesa and x0 ,p0 refer toinitial conditions of the Gaussian wave packetsfx0 ,p0

(a) (t).

Owing to the ansatz~2.6!, the system density operator depends only ona but not onx0 ,p0 . Using the semiclassicaapproximation for the bath density operator~2.16!, we maygeneralize the ansatz~2.6! such that the system density operator also depends ona and on x0 ,p0 . This way the sumover a becomes redundant, and we may rewrite Eqs.~2.6!with Eq. ~2.16! into a semiclassicalTDSCF ansatz for thetotal density operator

r~ t !5E dx0E dp0 w~x0 ,p0!rx0 ,p0S ~ t !rx0 ,p0

B ~ t !,

~2.17a!

rx0 ,p0S ~ t !5 (

k,k851,2

xx0 ,p0~k! ~ t !uwk&^wk8uxx0 ,p0

~k8!* ~ t !, ~2.17b!

rx0 ,p0B ~ t !5ufx0 ,p0

~ t !&^fx0 ,p0~ t !u. ~2.17c!

Formally, Eqs.~2.17!are rather similar to the quantummechanical TDSCF ansatz~2.6!, because thea summationweighted bywa has been converted to an integration overx0and p0, weighted byw(x0 ,p0). Note, however, that theBoltzmann factorwa refers to the initial vibrational stateua&,whereas the functionw(x0 ,p0) refers to the classical trajectory xx0 ,p0(t) with initial conditionsx0 andp0. The functionw(x0 ,p0) thus represents the initial distribution of thnuclear degrees of freedom, which may be sampled, efrom the thermal Wigner distribution61


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-

-

g.,

w~x0 ,p0!51

ptanhS 12 bv DexpH 2tanhS 12 bv D ~p0

2

1~x02d!2!J , ~2.18!

where, according to Eq.~2.5!, the coordinate shiftd is d50for hB5h0 andd52C/v for hB5h1 . The semiclassical TD-SCF approximation for the density operator~2.17! is a majorresult of this work. The TDSCF ansatz~2.17! represents to-gether with the equations of motion~2.8!,~2.9!and the initialconditions ~2.18! the working equations for the numericalresults presented below.

It should be noted that in the present application~i.e.,calculation of the electronic population probabilities for thespin–boson problem! the semiclassical TDSCF ansatz~2.17!reduces to a classical-path formulation.39–46 The theory de-veloped above can therefore be considered as a derivationthe classical-path formulation, which will turn out to be useful for the discussion of the approximations involved~seeSec. IV!. However, the semiclassical TDSCF ansatz~2.17! isonly equivalent to a classical-path scheme for diabatic potetials that are at most quadratic inx, and for the evaluation ofquantities associated with thediagonalmatrix elements ofthe system density operator~e.g., electronic populations,mean values of the bath degrees of freedom!. The calculationof electronic coherences, on the other hand, implies thevaluation of expression of the formF(t) uF~0!&, where thenuclear phases of the individual wave packets do contributthe ansatz~2.17!is therefore able to roughly account for pureelectronic dephasing effects.62

III. COMPUTATIONAL RESULTS FOR AN OHMIC BATH

We want to demonstrate the capability of the semiclassical TDSCF approach by presenting some computational rsults for the spin–boson model with an Ohmic bath. Quitrecently a number of groups have reported~numerically!ex-act path-integral calculations,8,10,13 which we will use tocheck the reliability and accuracy of the TDSCF method.

Within the spin–boson model~2.1! all properties of thebath enter through a single function called the spectral desity

J~v!5p

2 (jCj2d~v2v j !, ~3.1!

which in the case of an Ohmic bath has the form

J~v!5p

2ave2v/vc. ~3.2!

The strength of the system–bath coupling is characterized bthe dimensionless Kondo parametera52/ph, whereh is theusual Ohmic viscosity. The special density~3.2! exhibits awide variety of dissipative phenomena and has been studiextensively by a number of groups. It has a characteristlow-frequency behaviorJ(w)5hv, and is peaked at the cut-off frequencyvc , which defines the time-scale distributionof the bath dynamics.

The quantity of interest is the time-dependent electronipopulation

, No. 4, 22 July 1995t¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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P~ t ![^sz~ t !&

5E dx0E dp0 w~x0 ,p0!~ uxx0 ,p0~1! ~ t !u22uxx0 ,p0

~2! ~ t !u2!,

~3.3!

which is evaluated by solving numerically the electronequations of motion~2.8!, choosing according to Eq.~2.5!the electronic initial conditions asxx0 ,p0

(1) (0) 5 1 and

xx0 ,p0(2) (0) 5 0. The initial conditions for the nuclear vari

ablesxj (0),pj (0) can either be sampled from the Wignedistribution ~2.18!, or be obtained via the classical actioangle variables

xj~0!5A2nj11 sinqj1d j ,~3.4!

pj~0!5A2nj11 cosqj ,

where the phasesqj are picked randomly from the interva@0,2p#, and the quantum numbersnj are sampled accordingto the Boltzmann distributionP(nj )5exp~2bnjv j !. The co-ordinate shiftdj in Eqs. ~2.18! and ~3.4! reflects the initialmean position of the bath modes@d50 for hB5h0 and d52C/v for hB5h1 , cf. Eq. ~2.5!#, and has been chosen i

FIG. 1. The electronic populationP(t) as a function of the system–bathcouplinga50.13 ~a!, a50.25 ~b!, anda50.64 ~c! and the system param-etersD[1, kBT52.5D, vc52.5D. The TDSCF results~full line! are com-pared to exact path-integral~dotted line!and NIBA ~dashed line!calcula-tions taken from Ref. 8.


c

r-

accordance with the reference calculations we compare~i.e., for Figs. 1–3hB5h1 , for Figs. 4–6hB5h0!. Interest-ingly, it has been found that both distributions~2.18! and~3.4! give almost identical results. For the calculations reported here we have used action-angle initial condition~3.4!, which in the average seemed to do slightly better.

The bath has been simulated byNmod5400 harmonicmodes, the frequencies of which are equally distributed btweenvmin50.01vc andvmax54vc . This way the times cor-responding to the frequencyvmin and the line spacingDv5~vmax2vmin!/~Nmod21! are safely beyond the observedtime evolution of the system. Integration of the spectral desity ~3.1! over v yields the coupling parametersCj5[2/pDvJ(v j )]

1/2. To obtain converged results we haverun 2500 trajectories, which take about 1 h CPU time on aIBM RS 6000 work station.

To learn about the strong and weak points of the methowe wish to check the model in a large range of model parameters, such as the electronic couplingD, the system–bathcouplinga, and the temperatureT. As the transition to clas-sical mechanics can be characterized by

\bv→0, ~3.5!

the classical method should do well for ‘‘small\,’’ high tem-peratures, and for short times 2p/v, wherev correspond to atypical frequency of the problem.

FIG. 2. Same as Fig. 1, but forkBT50.625Danda50.32 ~a!,a50.51 ~b!,anda50.64 ~c!.


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Performing extensive quantum Monte Carlo computions, Mak and Chandler have studied in some detailtransition between coherent and incoherent relaxation behior of the spin–boson model.8 As a representative examplwe consider a case whereD[1, kBT52.5D, vc52.5D, andinvestigate the time evolution ofP(t) as a function of thesystem–bath coupling measured by the Kondo parametea.Figure 1 shows the comparison of exact~dotted line!, TD-SCF ~full line!, and NIBA ~dashed line!calculations, evalu-ated fora50.13 ~a!, a50.25 ~b!, anda50.64 ~c!, respec-tively. The computations illustrate nicely the transition froweakly coherent~a! to completely incoherent~c! relaxationdynamics. The TDSCF data are seen to reproduce the eresults almost quantitatively, and are thus a clear improment to the NIBA data of Ref. 8. As a second example, F2 shows the transition from coherent to incoherent relaxatin the case of relatively low temperaturekBT50.625D,where the system–bath coupling has been chosen asa50.32~a!, a50.51 ~b!, and a50.64 ~c!, respectively. Again thesemiclassical approximation is seen to reproduce the reence results fairly well.

To provide a challenge for the TDSCF approach, wnext turn to a case with rather strong system–bath coup~a52!, as has been recently studied by Egger and Mak eploying exact quantum Monte Carlo methods.13 Figure 3

FIG. 3. Comparison of the electronic populations as obtained by exact qutum Monte Carlo simulations~Ref. 13! ~dotted line!and TDSCF calcula-tions ~full line! calculations for the case of strong system–bath couplia52 and the parametersvc[1, kBT54vc . Panel~a! shows the case ofmedium electronic couplingD/vc50.8, panel~b! the case of strong elec-tronic couplingD/vc51.2.


-ev-

acte-.n

er-

eg-

shows the exact~dotted line!and TDSCF~full line! calcula-tions of the electronic populationP(t) for the parametersvc[1, kBT54vc . In the case of moderate electronic copling D/vc50.8 ~a!, the simulations exhibit a monoexponetial decay on the golden rule time scale. For stronger etronic couplingD/vc51.2 ~b!, the electronic population isseen to exhibit a rapid initial decay, which is followed byslower relaxation dynamics. The TDSCF results qualitativmatch the reference data, but predict a somewhat slowelaxation, particularly in the case~b! of both strong system–bath and strong electronic coupling. The method therefseems to do better in the nonadiabatic regime~D/vc!1! thanin the adiabatic regime~D/vc*1!, where the mean-potentiaapproximation may become problematic~cf. Sec. IV!.

Another parameter range of the model to check isbehavior at low temperatures. Dealing with an essentiaclassical model, the medium~Figs. 1 and 2! and high~Fig. 3!temperature cases studied above should certainly be easreproduce than the caseT→0. Using an influence–functionaapproach, Makarov and Makri recently reported exact paintegral calculations for the spin–boson model in the lotemperature regime.10 As an example of relatively strongsystem–bath couplinga50.5625, Fig. 4 shows the electronpopulation dynamics for the parametersD[1, kBT50.1D,vc52.5D. The electronic population is seen to undergoweakly coherent relaxation, which is somewhat exaggeraby the TDSCF calculation.

As a final test we wish to compare to long-time lowtemperature calculations, which have become availableimproved path-integral methods only very recently. Accoing to the criteria~3.5!, this parameter regime corresponjust to the quantum-mechanical limit. The time evolution ocoherently relaxing system~D[1, kBT50.2D, vc52.5D,a50.09! is presented in Fig. 5 for the first 30 time units. Uto aboutt'20, we are able to compare to the path-integdata of Makarov and Makri.10 Although the overall damping

n-

g

FIG. 4. The electronic populationP(t) in the low-temperature regime of thespin–boson model~D[1, kBT50.1D, vc52.5D, a50.5625!. The full linerepresents the TDSCF calculation, the dotted line represents an exactintegral calculations~Ref. 10!.

, No. 4, 22 July 1995t¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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is slightly exaggerated, the model is seen to nicely reprodthe coherent relaxation dynamics of the spin–boson modethe low-temperature limit.

IV. DISCUSSION

In light of the fact that the simple semiclassical modecapable of describing the complex relaxation dynamicsthe spin–boson problem, the obvious question arisesgeneral and how reliably the semiclassical TDSCF ans~2.17! is. This is particularly of interest because it has bereported by several workers that TDSCF methods candramatically in curve-crossing problems.25–27,46The remain-der of the paper is therefore devoted to discuss and anathe the validity and applicability of the semiclassical TDSCapproximation. To this end, we will~A! derive a generalizedmaster equation for the electronic populationP(t), whichallows comparison with analytical formulations of the spinboson model,2,52–57~B! discuss the nature of the semiclascal approximation by comparing to exact quantumechanical multiconfiguration TDSCF formulations,~C!discuss in some detail the condition of randomizationnuclear phases, which turns out to be the crucial condifor the validity of the semiclassical approach.

A. Generalized master equation description

It is well known that the theoretical description of disspative dynamics simplifies considerably if one is only inteested in the time evolution of the diagonal elements ofelectronic density matrix. In the case of the spin–bosmodel, e.g., one usually considers the time-dependent etronic population@cf. Eq. ~3.3!#

P~ t !5TrB@r11~ t !2r22~ t !#, ~4.1!

whererkk85^wkuruwk8& are the electronic matrix elemenof the total density operatorr(t) and TrB denotes the traceover the bath variables. If we restrict ourself to the calcution of P(t), we may change from the Liouville–von Neumann equation@describing the time evolution of the totadensity operatorr(t)# to a generalized master equation@de-scribing the time evolution of the electronic populatio

FIG. 5. The electronic populationP(t) for the parametersD[1, kBT50.2D,vc52.5D, a50.09. Compared to the exact path-integral results~Ref. 10!~dotted line!, the TDSCF calculation~full line! slightly exaggerates the overall damping of the beating.

J. Chem. Phys., Vol. 1Downloaded¬22¬Dec¬2000¬¬to¬132.230.78.190.¬¬Redistribution¬subje

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P(t)#, which will prove to be advantageous for the discussions of the approximations introduced below.

To derive an integrodifferential equation for the diagonelementsrkk , we insert the formal solution of Eq.~2.4! forthe off-diagonal elementsrkk8 into the equation of motion forthe diagonal elementsrkk , thus yielding

17

P~ t !52~2D!2 Re E0

t

dt8 TrB$e2 ih2~ t2t8!eih1~ t2t8!

3@r11~ t8!2r22~ t8!#%. ~4.2!

Equation~4.2! is still exact. In general, it is an operator equation with respect to the multidimensional boson field, whiccan only be solved within some approximation.

Intending to solve the integrodifferential Eq.~4.2! in aquantum-mechanical manner, one usually ends up employan approximation of the decoupling type,17,58

^e2 ih2~ t2t8!eih1~ t2t8!@r11~ t8!2r22~ t8!#&B5K~ t2t8!P~ t8!,~4.3a!

where

K~ t2t8!5^e2 ih2~ t2t8!eih1~ t2t8!&B ~4.3b!

is the nonlocal memory kernel and•••&B5TrB$•••%. UsingEqs.~4.3!, we obtain the generalized master equation for telectronic populationP(t)

P~ t !52~2D!2 Re E0

t

dt8 K~ t2t8!P~ t8!. ~4.4!

To elucidated the basic physics described by Eq.~4.4!, itis instructive to consider two simple limiting cases. In thlimiting case of vanishing system–bath coupling~i.e., nofriction!, Eq. ~4.4! reduces to the master equation for a twolevel system, which has the familiar Rabi solutionP(t)5cos 2Dt. In the limiting case that the electronic populatioP(t) does not change much within the time the memokernelK(t2t8) deviates from zero, we may setP(t8)'P(t)and the solution of Eq.~4.4! reduces to the standard goldenrule expression with ratek5~2D!2 Re*0

`K(t)dt. In general,the generalized master Eq.~4.4! therefore describes the com-petition between coherent quantum tunneling and localiztion of the system in a electronic state due to the interactiwith the bath.

Let us now introduce the semiclassical formulation oSec. II into the integrodifferential Eqs.~4.2! and ~4.4!.Within the TDSCF approximation, it is possible to solve ththe electronic equations of motion by integrating out the badegrees of freedom.25 This is so, because the trajectoryx(t)of Eq. ~2.13! represents a closed solution for the nucleadynamics. We may therefore replace the boson operatorshkin Eq. ~4.2! by their time-dependent mean values

03, No. 4, 22 July 1995ct¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html.

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e2 ih2~ t2t8!eih1~ t2t8!→expH 2 i Et8

t

dt^Fx0 ,p0~t!uh2

2h1uFx0 ,p0~t!&J ~4.5a!

5expH 2iCEt8

t

dt xx0 ,p0~t!J , ~4.5b!

where for clarity the dependency of the trajectoryxx0p0(t) onthe nuclear initial conditions has been noted explicitly. Instion of Eqs. ~2.17! and ~4.5! into the exact formula~4.2!yields

P~ t !52~2D!2 Re E0

t

dt8E dx0E dp0 w~x0 ,p0!

3Px0 ,p0~ t8!expH 2iCE

t8

t

dt xx0 ,p0~t!J . ~4.6!

Equation~4.6! is the desired integrodifferential equatiofor the electronic population in the semiclassical TDSCFproximation. Here,Px0 ,p0

(t8) denotes the electronic population evaluated for a single trajectoryxx0 ,p0(t) with initialconditions x0 ,p0 , whereasP(t) represents the averageelectronic population defined in Eq.~3.3!. Note that the traceover bath variables in Eq.~4.2! has been converted intoclassical average over thermally weighted initial states insemiclassical expression. Furthermore it should be notedthe semiclassical TDSCF ansatz~2.17! has been the onlyapproximation involved in the derivation of the integrodiffeential equation~4.6!, i.e., the semiclassical evaluation of E~4.2! does not require a decoupling approximation. In ordto make contact with other formulations of the spin–bosproblem, however, it is nevertheless instructive to consithe limiting cases of the decoupling approximation asystem–bath perturbation theory.

Let us first introduce the decoupling approximati~4.3!, which assumes that thePx0p0

(t8) in Eq. ~4.6! are in-dependent of the nuclear initial conditionsx0 ,p0 , and en-ables us to perform the integration over initial conditionsEq. ~4.6! analytically. Note that only the unperturbed soltion x(0)(t) ~2.13! is subject of the averaging process. Asuming that initiallyhB5h1 @cf. Eq. ~2.5!#, the integrationover the Wigner distribution~2.18!yields

K expH 2iCEt8

t

dt xx0 ,p0~0! ~t !J L

Cl

[E dx0E dp0 w~x0 ,p0!expH 2iCEt8

t

dt xx0 ,p0~0! ~t !J

5exp$22C2/v2@coth~ 12bv!~12cosv~ t2t8!!

2 i sin vt1 i sin vt8#%, ~4.7!

where the subscript Cl denotes a classical average. InseEqs. ~2.13! and ~4.7! in Eq. ~4.6! and going back to multi-mode notation (x→xj ), we finally obtain


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ting

P~ t !52~2D!2 Re E0

t

dt8 K ~0!~ t2t8!K ~1!~ t,t8!P~ t8!,

~4.8!

where

K ~0!~ t !5exp$ i e r t2Q2~ t !%, ~4.9!

K ~1!~ t,t8!5expH 2i(jCj2E

t8

t

dt2E0

t2dt1 sin v j~ t22t1!

3@12P~ t1!#J , ~4.10!

and

e r52(jCj2/v j

2, ~4.11!

Q2~ t !52(j

Cj2

v j2 coth~

12bv j !~12cosv j t !. ~4.12!

Hereer is the reorganization energy, andQ2(t) is recognizedas ubiquitous function from standard spin–boson theory2

Equation~4.8! is the master equation for the electronic population P(t) in the semiclassical TDSCF and decoupling approximations. Within these two approximations,K (0)(t2t8)represents the stationary~‘‘zero-order’’! memory kernel,while K (1)(t,t8) is a nonstationary memory function, takinginto account the classical reaction force to all orders. Together with the semiclassical TDSCF ansatz~2.17! the inte-grodifferential Eqs.~4.6!and~4.8! represent the central theo-retical results of this paper. Both equations are valid foarbitrary system–bath coupling, and may be therefore epected to describe dissipative quantum dynamics beyondusually considered low-coupling regime.

Let us compare the master Eq.~4.8! to recent work byGrigolini and co-workers56,57 who have obtained similar re-sults, but used completely different theoretical approachePerforming a coordinate transformation of the bath mode@sometimes referred to as ‘‘~pseudo!polaronic transforma-tion’’ #, and neglecting a nonclassical driving term of thenuclear motion, Vitali and Grigolini56 derived a master equa-tion for the electronic population@Eq. ~2.17! in Ref. 56#,which is rather similar to our Eq.~4.8!. The difference be-tween the two formulations arises, because Vitali anGrigolini performed a quantum-mechanical average over thboson fieldx


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K expTH 2i(jCjE

t8

t

dt x j~0!~t !J L

Qm

5K expH 2i(jCjE

t8

t

dt x j~0!~t !

22(jCj2E

t8

t

dt2Et8

t2dt1@ x j

~0!~ t2!, xj~0!~ t1!#J L

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~4.13a!

5K expH 2i(jCjE

t8

t

dt xj~0!~t !J L

Cl

3exp$2 iQi~ t2t8!1 i e r~ t2t8!%. ~4.13b!

Here expT denotes the time-ordered exponential, the Magnformula for the harmonic oscillator64 has been used in Eq~4.13a!, and the function

Q1~ t !52(j

Cj2

v j2 sin v j t ~4.14!

is well-known from standard spin–boson theory.2 It is seenthat the classically averaged result~4.7! is recovered by ne-glecting the commutator [x j

(0)(t2), xj(0)(t1)], which gives rise

to the additional phase terms in Eq.~4.13!. The quantum-mechanical approach of Vitali and Grigolini therefore leato the identical nonlinear memory functionK (1)(t,t8) as inEq. ~4.10!, but replaces the classical expressionK (0)(t) ofEq. ~4.9! by the quantum-mechanical memory kernel

KNIBA~ t !5exp$2Q2~ t !1 iQ1~ t !%, ~4.15!

which is recognized as the result of the NIBA of Leggeet al.2 The NIBA represents the best-known approximatioin the field of quantum dissipation. It has been derived vianumber of different theoretical approaches,2,53–56always re-quiring at one point or another the decoupling and perturtive approximations.

To investigate the consequences of the classical apprmationK (0)(t) to KNIBA(t), let us assume a continuous batin which caseQ1(t) turns out to be a smoothed step funtion. In the case of an Ohmic bath, for examplQ1(t)52a arctanvct, wherea denotes the Kondo parameteandvc is the cutoff frequency@cf. Eq.~3.2!#. Fort.0,Q1(t)may be therefore approximated by a constant,Q1(t)→pa,which is equivalent to a assumption of randomizationphases~see Sec. IV C!. Introducing this condition into Eq~4.15!and usinga51

2e rvc , the NIBA memory kernel reads

KNIBA~ t !5expH i p

2e r /vc2Q2~ t !J , ~4.16!

which differs from the semiclassical result~4.9! in the waythe electronic coupling is affected by the bath. In thquantum-mechanical expression the bath dynamics resula renormalization of the electronic coupling consta(D→DAcospa),2 in the semiclassical expression~4.9! weobtain a shift of the electronic energy gap by the reorgani


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tion energyer . In the assumed case of low system–bacoupling, however, both corrections should be of minor importance.

To conclude this section, it has been shown that tsemiclassical ansatz leads to a general master equationthe electronic population which is up to a quantummechanical phase factor equivalent to the polaronic transmation approach of Ref. 56 and to the NIBA of Ref. 2. Futhermore it should be stressed that the TDSCF ansatz~2.17!yields the same nonlinear memory functionK (1)(t,t8) as inRef. 56, indicating that the semiclassical approach takproperly into account the classical reaction field. Employithe Wigner–Weyl formalism, Bonciet al.57 showed that thetotal Liouvillian of the spin–boson system can be split uinto a semiclassical termLcl and a quantum correctionLQGD,where the equations of motion corresponding toLcl are iden-tical to our TDSCF Eqs.~2.8!and~2.9!. Both theories there-fore seem to indicate that the semiclassical TDSCF ans~2.17! can be understood as the neglect of the nonclassreaction field.

B. Relation to quantum TDSCF formulations

Another approach to investigate the nature of the aproximation~2.17! is to compare the semiclassical modelquantum-mechanical TDSCF formulations. Let us first cosider the most general ansatz for a TDSCF density operawhich in the position representation can be written as30

r~x,x8,t !5(rwr (

k,k851

2

uwk&^wk8u (i,i851

n

cr ,i~k!~ t !

3cr ,i8~k8!* ~ t ! )

j51

Nmod

f i j~k!~xj ,t !f i

j8~k8!* ~xj8 ,t !.

~4.17!

Heref i j(k)(xj ,t) denotes thei j th time-dependent expan

sion function of the coordinatexj pertaining to the diabaticelectronic stateuwk&, cr ,i

(k)(t) represents the correspondintime-dependent expansion coefficients labeled by the vecindex i 5 $ i 1 ,...,i Nmod%, andwr is a statistical weight factordescribing the thermal occupation probability of this cofiguration. In the parlance of electronic structure theory, wspeak of a ‘‘single configuration’’ ansatz if we consider ona single expansion function~i.e., n 5 $n1 ,...,nNmod% 5 1!,whereas we obtain a ‘‘full configuration interaction~CI!’’ansatz if we consider the complete expansion~4.17!, whichin the case of complete basis sets and converged expansums of course represents the exact density operator.

Intending to take into account many~say, hundreds!nuclear degrees of freedom, it seems to be clear that oneto drop the exact full CI scheme, which turns out to be rathcumbersome forNmod*10 vibrational modes.29–31 Single-configuration or simple multiconfiguration schemes, on tother hand, have often been found to yield reasonable resonly for rather short times~say, for one or two vibrationalperiods!.26–28,46This approach thus seems to be well suitefor the description of ultrafast decay processes~e.g., directphotodissociation!and for the modeling of structureless ab


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sorption and resonance-Raman spectra,28 but is not sufficientfor the description of bound-state relaxation processwhich typically last over many vibrational periods of the famodes.

Let us rewrite the semiclassical TDSCF ansatz~2.17!, inthe following form:

r~x,x8,t !5 (x0 ,p0

w~x0 ,p0! (k,k851

2

xx0 ,p0~k! ~ t !uwk&^wk8u

3xx0 ,p0~k8!* ~ t ! )

j51

Nmod

fx0 ,p0~xj ,t !fx0 ,p0

* ~xj8 ,t !.

~4.18!

Compared to the exact full CI ansatz~4.17!, we have madetwo major approximations in Eq.~4.18!:

~i! We use identical vibrational wave functions foboth diabatic electronic states, i.e.,

f j~1!~xj ,t !5f j

~2!~xj ,t !5f j~xj ,t !. ~4.19!

~ii! The sum over statistical weightswr and the doublesum over expansion functions with variationally detemined coefficients in Eq.~4.17!has been replaced ba quasiclassical sampling procedure of nuclear iniconditions.

Approximation ~4.19! results in the often discusseproblem that, instead of using an extra trajectory for eaelectronic state, there is a single trajectory propagated inaveraged potential~2.9!. It is well known from applicationsin reactive scattering, that this approximation may leadunphysical results if the system is asymptotically in a sinelectronic state but not in a mixture of states.47 In the case ofbound-state relaxation dynamics including internconversion48–50 and simple isomerization processes,51 how-ever, approximation~4.19! has not turned out to be a vercritical assumption. Note, e.g., that in all spin–boson callations reported above we have obtained a long-time limiP(t→`)5P12P250, corresponding to an equal popultion probability P15P25

12 in each electronic state. In th

case of nonzero bias, e.g., in internal conversion and electransfer processes, it has been found that the system typilocalizes in the lower adiabatic state, but is diabatically sin a mixture of states,65 which is nicely reproduced by thclassical model.48–51

C. Randomization of nuclear phases

At a first glance, however, approximation~4.19! seemsto be unnecessary, because it is only a minor extra efforcalculate two trajectories~one for each electronic state!instead of one. Allowing for two different vibrationawave functionsF~1! andF~2!, the corresponding electroniequations of motion can readily be obtained froEq. ~2.8! by making the replacementsF(t)uhkuF(t)&→^Fk(t)uhkuFk(t)& and D→^Fk(t)uDuFk8(t)&. Whilewithin the semiclassical approximation~2.15! the diagonalterms^Fk(t)uhkuFk(t)& are again independent of the phaof the wave packets, the off-diagonal term^Fk(t)uDuFk8(t)&now explicitly depends on these phases. One of the pitfall


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using a superposition of Gaussian wave packets, howevethat although the sum of classical trajectories~2.13! oftengives a rather accurate description of the nuclear dynamthe corresponding sum of phase-dependent matrix elemdoes not.60,66,67Although this shortcoming can be in principle improved by introducing additional semiclassical phafactors the resulting methods are too cumbersome for hdling problems with many degrees of freedom.66,67

The approximation~4.19! therefore avoids the problemsassociated with nuclear phase factors which often leadsnonsensical results for the electronic dynamics after a ratshort time. On the other hand, the simple ansatz~4.18! is ofcourse not able to describe the molecular dynamics in cawhere this nuclear phase information is important. Restriing ourselves, however, to cases involving many weakcoupled degrees of freedom, one might expect to a laextent a randomization of the nuclear phases, which wojustify the neglect of this phase information. Dissipativelectronic relaxation as described by the spin–boson mowith an Ohmic bath thus appears to be a perfect candidatethe semiclassical TDSCF approach. By definition, a baconsists of many weakly coupled vibrational modes, thefore randomization of phases appears to be plausible.

The second approximation~ii! of quasiclassical samplingof the density operator also is only appropriate under tassumption of a complete randomization of nuclear phasAs has been shown above, the quasiclassical sampscheme can be considered as an average over a sumGaussian wave packets, where the off-diagonal terms, cosponding to an interference of the individual trajectoriehave been neglected~see also the discussion in Ref. 14!. Toillustrate this point, note that the electronic population~3.3!is evaluated in the same way for a pure initial state~i.e., forT50! as in the case of finite temperature. Regardlesswhether the initial state of the system is given as a cohersuperposition of basis states~pure initial state!or by an in-coherent superposition of basis states~mixed initial state!,we thus perform in both cases a completely incoherent sumation over trajectories.

To give an example of where the classical method hasbreak down, consider a spin–boson problem where the ovall system–bath couplinga has been put into a few vibra-tional degrees of freedom. Unlike the case of many weacoupled vibrational modes in a quasicontinuous bath, owould expect in the case of a few strongly coupled vibrtional modes the nuclear phases to be important. We hchosen a system similar to the model shown in Figs. 4 an~D[1, vc52.5D, T50!, and considered four vibrationamodes with frequenciesv50.01, 1.34, 2.67, 4.0. The quantum propagation of the density operatorr(t)5uC(t)&^C(t)uhas been performed by expandinguC(t)& in a direct-productbasis constructed from diabatic electronic states aharmonic-oscillator states, and solving the resulting systof coupled first-order differential equations using a RungKutta–Merson scheme with adaptive step size.68,69 Figure 6shows the classical~full lines! and quantum~dotted lines!calculations evaluated for a system–bath coupling of~a!a50.09 and~b! a50.5625, respectively. While in the exacquantum calculation the coherent electronic oscillati


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~;cosDt! is seen to be perturbed only very weakly by tinteraction with the four vibrational modes, the classicsimulation nevertheless predicts a slow decay of the etronic population in Fig. 6~a!. Even more dramatically is tfailure of the model for four strongly coupled modesshown in Fig. 6~b!. The quantum population exhibits irregular coherent beatings, whereas the classical populationbasically decayed after the first period and fluctuates arozero.

One may note, however, that most quantum TDSCFpath integral calculations have reported only results foror two electronic or vibrational periods, which are evenproduced here in the worst-case scenario of Fig. 6~b!. Fur-thermore it is interesting to note that the quantum resobtained for four modes@Figs. 6~a!and 6~b!#and four hun-dred modes~Figs. 5 and 4! show in both cases the saminitial time evolution of the electronic population. Withit&4, corresponding approximately to the period of the lespacing of the four bath modes, the system cannot disguish whether it is damped by a few or by a infinite numbof modes.

As expected, the assumption of a complete randomtion of nuclear phases breaks down for a spin–boson plem with a few strongly coupled modes. To study this somwhat vague randomization condition in more detail, let usback to the perturbative analysis of the spin–boson probas considered in Sec. IV B. It was shown above thatsemiclassical memory kernel~4.9! is ~up to a somewhat dif-ferent renormalization of the electronic coupling! equivalentto the NIBA memory kernel~4.15!, if

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v j2 sin v j t → const. ~4.20!

FIG. 6. Time evolution of the electronic populationP(t) for a four-modespin–boson model~D[1, vc52.5D, T50!. The TDSCF~full lines! andexact quantum~dotted lines!calculations are shown for a total system–bacoupling of ~a! a50.09 and~b! a50.5625.


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Note that, as the sumQ1(t) of oscillatory functions becomestime independent, all the oscillatory contributions havecancel each other, which is indistinguishable to the casethe phases of the individual contributions are random.

Figure 7 illustrates this consideration by showing thfunctionQ1(t) obtained for the four-mode model of Fig. 6~a!~full line! and for the quasicontinuous model of Fig. 5~dot-ted line!, respectively. While in the case of an Ohmic bacondition ~4.20! is fulfilled after a short initial transient, thefour-mode calculation is seen to exhibit an oscillatory riswhich is mainly determined by the lowest frequency of thmodel. It is instructive to perform a simple numerical expement and add to each vibrational frequency inQ1(t) a ran-domly chosen phase, i.e., sin(v j t)→sin(v j t12ph j ), wherehj are uniform deviates. In the case of four vibrationmodes, the resulting functionQ1(t) depends completely onthe phases 2phj , whereas the four-hundred mode calculatiofor Q1(t) is hardly affected by the modification, becausethat case the phases of the vibrational modes are alrerandomized.

It should be emphasized, however, that there are caseelectronically coupled models with few nuclear degreesfreedom that do show electronic [email protected].,P(t→`)→0# and phase randomization. Allowing, e.g., focoordinate-dependent electronic coupling~e.g., D→D xc!,one obtains a conical intersection of the potential-energy sfaces instead of an avoided crossing, which is known to rresent a very effective relaxation mechanism.68,69 It has beenshown for a number of systems including ethylene1,65,68,69

pyrazine,65,69 and benzene1,70 that these vibronic couplingmodels exhibit ultrafast electronic and vibrational relaxatiobehavior, if at least threestrongly coupled vibrational de-grees of freedom are involved. Interestingly, it has befound that this relaxation dynamics can be described wellclassical-path methods,48–51where it proved advantageous tuse dynamical corrections~‘‘Langer-like modifications’’! assuggested by Meyer and Miller.42 The crucial condition forthe validity of the classical TDSCF model therefore seemsbe astrong enough electronic relaxation dynamics that th

h

FIG. 7. Time evolution of the functionQ1(t) obtained for the four-modemodel of Fig. 5~a! ~full line! and for the quasicontinuous model of Fig. 4~dotted line!.


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nuclear phases get randomized rapidly enoughnot to be im-portant for the overall time evolution of the system.

V. CONCLUSIONS

We have outlined a semiclassical TDSCF approachdescribe dissipative quantum phenomena such as thetronic relaxation dynamics of the spin–boson model.have introduced a semiclassical approximation for the tdensity operator, which couples the system degrees of fdom to the bath degrees of freedom in a self-consistent mner and is therefore in the spirit of a classical-path formution. It has been shown that the semiclassical ansatz leaa general master equation for the electronic population whis up to a quantum-mechanical phase factor equivalent topolaronic-transformation approach of Vitali and Grigolini56

and to the NIBA of Leggetet al.2 Compared to an exacquantum-mechanical TDSCF representation of the total dsity operator, the semiclassical ansatz has been shownbased on two main approximations, namely~i! the use ofidentical vibrational density operators for both diabatic eltronic states and~ii! the assumption of complete randomiztion of nuclear phases.

The mean-potential approximation~4.19!has been intro-duced to avoid the problems associated with spurious phfactors, and has proven to represent an appropriate asstion for the description of the dissipative quantum dynamof the spin–boson model. Unlike the case of reactiscattering processes, the mean-potential approximationnot turned out to be a very critical assumption for the moeling of bound-state relaxation dynamics.48–51 The assump-tion of complete randomization of nuclear phases, onother hand, has been found to represent the central appmation of the semiclassical approach. Most important,randomization assumption makes it possible to performtrace over the bath variables through quasiclassical sampof the nuclear initial conditions without invoking any furtheapproximation. The crucial condition for the validity of thsemiclassical TDSCF approach therefore is a strong enoelectronic relaxation dynamics that the nuclear phasesrandomized rapidly enough to be unimportant for the ovetime evolution of the system.

To demonstrate the capability of the outlined approawe have compared semiclassical calculations for a spboson model with an Ohmic bath to exact path-integral cculations. It has been found that the semiclassical monicely reproduces the complex dissipative behavior ofspin–boson model for a large range of model parametincluding the transition from coherent to incoherent relaation dynamics and the electron-transfer dynamics innonadiabatic and adiabatic regime.

It is clear, however, that in the case of the spin–bosmodel the semiclassical approach cannot beat exact pintegral methods, which will soon be able to predict the timevolution of the system for considerably long timregimes.10–13The spin–boson model has mainly been choto be able to compare to analytical formulations and exreference calculations. The benefit of a classical methoogy is rather the possibility to easily extend the formulati


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to a quite general class of problems. In the following, wtherefore want to briefly point out some obvious generaliztions and extensions of the method.

Owing to the structure of the equations of motions~2.8!and ~2.9!, it is first of all clear that the method readily generalizes from the case of a two-state system to aN-statesystem, which also implies the possibility to represent a geeral reaction coordinate in a quantum-mechanical mannFurther it is straightforward within the semiclassical methodology to evaluate any time-dependent observable of the stem ~e.g., diabatic and adiabatic electronic population proabilities, electronic autocorrelation function! and the bath~e.g., mean values of the position, momentum, and enecontent of a particular vibrational mode!.49–51In particular, ageneral formulation of the nonlinear spectroscopy of eletronically coupled systems has been recently developwithin a classical-path framework.49 Applied to the descrip-tion of time- and frequency-resolved pump–probe spectrocopy, it has been found that the semiclassical approach pvides a promising tool for the modeling and interpretationmodern femtosecond pump–probe experiments.

Finally it should be stressed that the diabatic Hamiltonians hk and the electronic couplingD are by no meansrestricted to potential functions up to second order in thcoordinatex. Recent results for a simple model of nonadiabatic cis–trans isomerization have shown that the semiclasical model is able to describe the electronic and vibrationrelaxation dynamics on strongly anharmonic potentiaenergy surfaces.51 In principle, this opens the possibility of amore realistic description of electron-transfer processessolution by using a molecular-dynamics~MD! simulation toaccount for the bath dynamics. The combination of semiclasical methodology and MD simulations would provide ainteresting alternative to the more usual indirect approacwhere a MD simulation is used to generate a harmonic sptral density, for which in a second step the dynamical prolem can be solved with path-integral methods.71–73

ACKNOWLEDGMENTS

The author thanks Wolfgang Domcke for numeroustimulating and helpful discussions in all stages of this worThanks also go to William H. Miller for insightful and criti-cal comments on the manuscript and to Nancy Makri aReinhold Egger for sending preprints of Refs. 10 and 1This work has been supported by the Deutsche Forschungemeinschaft.

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