statistical mechanics of quantum-classicalsystems · infm and dipartimento di fisica,...

Statistical�

mechanics of quantum-classical systemsSteve�

Nielsen and Raymond KapralChemical Physics Theory Group, Department of Chemistry, University of Toronto,Toronto, ON M5S 3H6, Canada

Giovanni�

CiccottiINFM�

and Dipartimento di Fisica, Universita ‘‘La Sapienza,’’ Piazzale Aldo Moro, 2, 00185 Roma, Italy�Received�

29 May 2001;accepted13 July 2001�The statisticalmechanicsof systemswhoseevolution is governedby mixed quantum-classicaldynamics�

is investigated.The algebraicpropertiesof the quantum-classicaltime evolution ofoperators� andof thedensitymatrix areexaminedandcomparedto thoseof full quantummechanics.Theequilibriumdensitymatrix thatappearsin this formulationis stationaryunderthedynamicsanda methodfor its calculationis presented.Theresponseof a quantum-classicalsystemto anexternalforce

which is appliedfrom the distantpastwhenthe systemis in equilibrium is determined.Thestructure� of the resulting equilibrium time correlation function is examined and thequantum-classical� limits of equivalentquantumtime correlationfunctionsarederived.The resultsprovide a framework for the computationof equilibrium time correlation functions for mixedquantum-classical� systems. © 2001

�American Institute of Physics. � DOI:

�10.1063/1.1400129�

I.�

INTRODUCTION

Often�

situationsarise where it is appropriateto studycomposite� dynamicalsystemswith interactingquantumme-chanical� andclassicaldegreesof freedom.In condensedmat-ter�

physicssuchsituationsoccur when one is interestedinthe�

dynamicsof a light quantumparticle or set of quantumdegrees�

of freedom interacting with more massiveparticles. 1–3 Specific

�examplesincludeproton transfer,4

�sol-�

vation� dynamicsof an excesselectron,5�

and nonradiativere-laxation�

processesof moleculesin a liquid-state environ-ment.� 6

�In�

thesecircumstancesit is not feasibleto attemptafull quantumsolution of the Schrodinger

�equationfor the

entire� system.Consequentlyone is led to considerthe dy-namics� of a quantumsubsystemcoupledto a bathwheretheenvironmental� degreesof freedom are treated classically.Such�

mixed quantum-classicalsystemsarise in other con-texts�

aswell.7

Dif�

ferent formulationsof the dynamicsof such mixedquantum-classical� systemshaveappearedin the literature.Inthese�

reduceddescriptionsof the quantumdynamicsthe en-vironmental� degreesof freedom are accountedfor by theinclusion!

of dissipativeand decoherencetermsin the equa-tions�

of motion,8–10 through�

multistate Fokker–Planckdynamics� 11 or� representationsby quantum stochasticprocesses. 12 Other

�approachesutilize a more detailedtreat-

ment� of the classical environment.These include simpleadiabatic dynamicswherethe classicalsystemevolveson apotential energy surfacedeterminedfrom a single adiabaticeigenstate,� or Ehrenfestmeanfield modelswherethe classi-cal� evolution is governedby a meanforce determinedfromthe�

instantaneousvalue of the quantumwave function.13,14

These"

descriptionsproducea definiteclassicalevolutionbutdo�

not alwaysgive physicallycorrectresults.If we relax thecontinuity� of thetrajectoryof theclassicalvariables,15 we# areled to considersurface-hoppingalgorithms.16–19 Evolution

equations� for thequantum-classicaldensitymatrix wheretheevolution� operator is expressedin terms of a quantum-classical� brackethavealsobeenstudied2

$0–26 and their solu-

tions�

have been formulated in terms of surface-hoppingtrajectories.� 26–28

In�

this paper we develop the statistical mechanicsofmixedquantum-classicalsystems.We takeasa startingpointthe�

evolutionequationfor the mixed quantum-classicalden-sity� matrix.21,23–26,29This

"equationmaybederivedby partial

W%

igner transformingthe quantumLiouville equationoverthe�

bath degreesof freedomcorrespondingto massivepar-ticles�

andexpandingthe resultingevolutionoperatorto lin-ear� orderin themassratio (m& /

'M(

)) 1/2,* wherem& and M

(are the

characteristic� massesof the quantumsubsystemand bathparticles, respectively.26

$Theresultingevolutionequationcan

be+

recastasan integralequationin which classicaltrajectorysegments� are interspersedwith environment-inducedquan-tum�

transitions and corresponding bath momentumchanges.� 27

$The"

canonicalequilibrium densitymatrix for quantum-classical� systemsis constructedto be stationaryunder thequantum-classical� evolution. Its form is derived and com-pared with its full quantumanalog.A linear responsederiva-tion�

is carriedout to determinetheresponsefunctionandtheformsof theequilibriumtime correlationfunctionsappearingin quantum-classicalsystems.Mixed quantum-classicaldy-namics� and the associatedcorrelationfunctionspresentdif-ferences

from their full quantumanalogs:Identitiesamongquantum� correlation functions hold only approximatelyinthe�

quantum-classicallimit andpropertiessuchastime trans-lation�

invariancearealsoonly approximatelyvalid.The"

paperis organizedasfollows: SectionII presentstheformal structureof mixed quantum-classicaldynamicsandcontrasts� it with that of full quantummechanics.In Sec.IIIwe# discussthe canonicalequilibrium densitymatrix and inSec.�

IV a linear responsederivationof the responsefunction

JOURNAL,

OF CHEMICAL PHYSICS VOLUME 115, NUMBER 13 1 OCTOBER2001

58050021-9606/2001/115(13)/5805/11/$18.00 © 2001 American Institute of Physics

Downloaded 25 Sep 2001 to 142.150.225.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

is carriedout and a generaldiscussionof the propertiesofmixedquantum-classicalcorrelationfunctionsis given.Con-cluding� remarksare madein Sec.V andAppendixesA–Ccontain� additionaldetails.

II.-

QUANTUM-CLASSICAL DYNAMICS

Consider.

a quantumsystemwhich may be partitionedinto!

two interactingsubsystems,a quantumsubsystemwithparticles of massm& ,/ and a quantumbath with particlesofmassM, (/ M 0 m& )

). In order to describethe distinctive fea-

tures�

of quantum-classicaldynamics,we begin with a briefoverview� of the algebraicstructureof quantumandclassicaldynamics�

in the Wigner representationof the quantumbath.

A. Quantum and classical dynamics

The"

von Neumannevolution equationfor the quantummechanicaldensitymatrix 1ˆ is

243ˆ 5 t6 78

t6 9;: i<=?> H@ ˆ ,/ Aˆ B t6 CED ,/ F 1G

where# H@ ˆ is!

theHamiltonianof thesystem.Its formal solutionis

Hˆ I t6 JLK eM N iLtO Pˆ Q 0R SLT eM U iHtO /V WYXˆ Z 0R [ eM iHtO /V \ ,/ ] 2with# iL

< ˆ _ (`i</' a

)) b

H,/ c the�

quantumLiouville operator.And

alternativeform of the evolution equationmay beobtained� by takingthepartialWignertransformoverthebathdegrees�

of freedom.This partial Wigner transform of thedensity�

matrix is definedby

eˆ W f R,/ P gLhji 2 kmlonLp 3q

Nr

dzes iP t zu /V v R w zx

2 yˆ R z zx2

,/ {3| }

while# the correspondingtransformof an operatorA is givenby+

AW ~ R,/ P �L� dzs

e � iP � zu /V � R � zx2

A R � zx2

. � 4�Both�

the densitymatrix and quantumoperatorsretain theirabstract operatorcharacterin thequantumsubsystemdegreesof� freedombut arenow functionsin the � R,

�P� phase spaceof

the�

quantumbath.To proceedwe usethefact thattheWignertransform�

of a productof operatorsis given by30q

�AB � W � R,/ P �L� AW � R,/ P � eM �� /2

ViBW � R,/ P � ,/ � 5� �

where# the left andright actingdifferentialoperator� is!

thenegativeof the Poissonbracketoperator,

�� P R ¡ ¢ R £ P ,/ ¤ 6¥ ¦

and the arrowsindicatethe directionsin which the gradientoperators� act.Using this resultandtaking thepartialWignertransform�

of Eq. § 1 we# find

©4ªˆ W « R,/ P,/ t6 ¬

t6 ®;¯ i<°�± HWeM ²´³ /2

Vi µˆ W ¶ t6 ·L¸º¹ˆ W » t6 ¼ eM ½´¾ /2

ViHW ¿

À;Á i<Â�Ã H@ ÄÆÅÈÇˆ W É t6 ÊLËºÌˆ W Í t6 Î H@ ÏÆÐÒÑ

Ó;Ô iL< ˆ

W Õˆ W Ö t6 ×Ø;ÙjÚ HW ,/ Ûˆ W Ü t6 ÝEÝ Q . Þ 7ß àIn�

theseequationswe havedefinedthe right (H@ áãâ

))

and left(`H@ äãå

))

actingoperators,

Hæãçéè HW ê R,/ P ë eM ì´í /2V

i,/ î8ï ð

Hñãòéó eM ô´õ /2V

iHW ö R,/ P ÷where# the partially Wigner transformedHamiltonianHW is

H@ ˆ

W ø R� ,/ Pù úLû Pù 2$

2ü

M( ý pþ ˆ 2

$2ü

m& ÿ VW � q�ˆ ,/ R� � . � 9� �

Here pþ ˆ and q�ˆ are the momentumand position operators,respectively� , of the quantumsubsystemand VW(

`q�ˆ ,/ R� )

)is the

partially Wigner transformedtotal potentialenergy operator,which# is the sumof the quantumsubsystem,quantumbath,and subsystem-bathpotentialenergies.

The"

secondequalityin Eq. � 7ß � defines�

thequantumLiou-ville� operatoriL

< ˆW in the partial Wigner representationand

the�

associatedLie bracket(H@

W , )/ Q . More generallythe Liebracket+

of two partially Wigner transformedoperatorsis de-fined as

A ˆ

W ,/ B� ˆ W � Q i<�� A �� B

� ˆW � B

� ˆWA ��

� i<�� AWeM �� /2

ViBW � BWeM �! /2

ViAW " . # 10$

Here%

the A &�'

and A (�)

operators� aredefinedasin Eq. * 8ï + with#the�

replacementH@ ˆ

W , A ˆ

W .The"

formal solutionof Eq. - 7ß . is!

/ˆ W 0 R� ,/ Pù ,/ t6 132 eM 4 iH576 tO /V 839ˆ W : R� ,/ Pù ,0/ ; eM iH<7= tO /V >? eM @ iLW

A tO Bˆ W C R� ,/ Pù ,0/ D . E 11FAd

similar setof equationsmay be written for the evolu-tion�

of anyquantumoperatorA ˆ . In theWignerrepresentation

these�

equationsand their solutions, respectively, take theform

dAs ˆ

W G t6 Hdts I iL

< ˆWA ˆ

W J t6 K3LNM H@ ˆ W ,/ A ˆ W O t6 PQP Q R 12Sand

A ˆ

W T t6 U3V eM iLWA tO A ˆ W W eM iHXZY tO /V [ A

ˆWeM \ iH]Z^ tO /V _ . ` 13a

bW%

e shall drop the dependenceof quantitieslike A ˆ

W(`R�

,/ Pù ))

on� the bath phasespacecoordinateswhen confusionis un-likely to arise.c

The"

Wigner transformof a productof operatorssatisfiesthe�

associativeproductrule,

5806 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Nielsen, Kapral, and Ciccotti


dABC e W fNgQg AWeM h�i /2

ViBW j eM k�l /2

ViCW m

nNo AWeM prq /2V

i s BWeM tru /2V

iCW vQv ,/ w 14xwith# an obvious generalizationto productsof ny operators.�Consider.

a quantumoperatorC z AB which# is theproductoftwo�

operators.Sincethe time evolutionof C maybewrittenas C(

`t6 )) { A

ˆ (`t6 )) B� ˆ (

`t6 )) , its partial Wigner transformis

CW | t6 }3~ A ˆ

W � t6 � eM �� /2V

iB� ˆ

W � t6 � . � 15�W%

e may also obtain this result by consideringthe actionofthe�

Liouville operatoron the operatorproductin the partialW%

igner representation:

iL< ˆ

WCW � i<�� H@ ˆ WeM �r� /2

Vi � A ˆ WeM �r� /2

ViB� ˆ

W �Q�� i<��Q� AWeM �� /2

ViBW � eM �� /2

ViHW �

�N� iL< ˆWAW � eM �� /2

ViBW � AWeM �¡ /2

Vi ¢ iL< ˆ

WBW £ . ¤ 16¥In writing the last line of this equationwe have usedtheassociative property in Eq. ¦ 14§ . From this result we mayimmediatelycomputetherepeatedactionof iL

< ˆW on� CW and,

using¨ thedefinitionof theexponentialevolutionoperatorasapower series,find Eq. © 15ª by

+identificationof thetwo power

series� expansions.We shall makeuseof this lengthierrouteto�

Eq. « 15¬ in the sequelwherethe analogousdemonstrationfor

mixed quantum-classicalsystemsis not so straightfor-ward.#

The quantummechanicalLie bracket,eitherin its origi-nal� form as (i

</'

)) ®

A ˆ ,/ B� ˆ ¯ or� in its partially Wigner trans-

formed

form (A ˆ

W ,/ B� ˆ W))

Q ,/ satisfiesthe Jacobiidentity,°AW ,/ ± BW ,/ CW ² Q ³ Q ´Nµ CW ,/ ¶ AW ,/ BW · Q ¸ Q

¹NºBW ,/ » CW ,/ AW ¼ Q ½ Q ¾ 0.

R ¿17À

W%

e may remark on several limiting situationsof thisgeneralÁ formulation of quantumdynamics.If the quantumbath+

is absentand the systemcomprisesonly quantumsub-system� degreesof freedom,we simply havethe usualquan-tum�

dynamical description in terms of the von Neumannequation� Â 1Ã . If one considersthe quantumbath dynamicsalone without a quantumsubsystem,one has the ordinaryW%

igner representationof quantummechanicsand all par-tially�

Wigner transformedoperatorsbecomesimple phasespace� functions:AW(

`R,/ P)

) ÄAW(

`R,/ P)

). Theclassicallimit of

the�

quantumbathdynamics,which consistsin keepingonlyterms�

of order Å 0Æ

in!

the evolution operator, is obtainedbythe�

following truncationof thepowerseriesexpressionof theexponential� operator:exp(ÇÉÈ /2

'i<)) Ê

1 ËÍÌÏÎ /2'

i<. In this limit

the�

bracket(H@

W , )/ Q reduces� to the PoissonbracketÐ H@ W ,/ Ñand the Wigner representationof the quantum Liouvilleequation� becomesthe classicalLiouville equation, ÒÔÓ C /

' Õt6ÖØ×ÚÙ H@ W,/ Û C ÜÞÝØß iL

<C à C(

`t6 )) , whosesolution may be written

asá

C â R� ,/ Pù ,/ t6 ã3ä eM å iLCæ tO ç

C è R� ,/ Pù ,0/ éê eM ë 1/2H

ìWA í tO î

C ï R,/ P,0/ ð eM 1/2ñ Hì

WA tO . ò 18ó

The Poissonbracketis a Lie bracketandsatisfiesthe Jacobiidentity. Of course,productsof classicalphasespacefunc-tions�

satisfy the associativeproperty. Consequently, bothquantum� andclassicaldynamicshavea Lie algebraicstruc-ture�

and productsof quantumoperatorsor classicalphasespace� functionssatisfyan associativeproductrule.

B. Quantum-classical dynamics

The"

formulation of quantum dynamics in the partialWô

igner representationallows the limit of a mixed quantum-classical� systemto betakeneasily. In suchquantum-classicaldynamics,�

the full quantumdynamicsof the subsystemistaken�

into accountwhile the bath, in isolation, evolvesac-cording� to the classicalequationsof motion. This limit istaken�

by replacingH@ õ÷ö

and H@ øÞù

by+

their expansionsto firstorder� in ú :

H@ ûÞüþý ÿ � �� H

@ ˆW 1 �

��2i< ,/

�19

H�� 1 ��2i< HW .

The"

full systemevolution,which includestheinteractionbetween+

the quantum subsystemand classical bath, isthen�

given by the mixed quantum-classicalLiouvilleequation:� 2

�0–26

��ˆ W � R,/ P,/ t6 ��

t6 "! i<#%$ H@ ˆ W ,/ &ˆ W ' t6 (*),+ 1

2ü -/. H@ ˆ W ,/ 0ˆ W 1 t6 243

57698ˆ W : t6 ; ,/ H@ ˆ W <>=?"@BA H@ ˆ W ,/ Cˆ W D t6 EFEHG"I i

< J ˆ Kˆ W L t6 M . N 20ü O

The last equalitiesin Eq. P 20Q define�

the mixed quantum-classical� Liouville operatorR ˆ ,/ and the bracketwhich takesthe�

form23,24

SA ˆ

W ,/ B� ˆ W THU i<VXWZY/[H\ B

� ˆW ] B

� ˆW ^`_Hacb

d i<e A

ˆW 1 f

g�h2ü

i< B� ˆ

W i B� ˆ

W 1 jkml2ü

i< A ˆ

W

n i<oXp AW ,/ BW q,r 1

2s`t

AW ,/ BW u�v7w BW ,/ AW x>y ,/ z 21{

where| }/~H� is definedas �� in Eq. � 19� with| HW � AW . Thebracket�

reducesto thequantumLie bracket(i� �

)� � 1 � A� ˆ W ,� B� ˆ W �

when| the classicalbath is absentand to the classicalLiebracket� �

Poisson�

bracket�� AW ,� BW � when| the quantumsub-system� is absent.

Using�

this notation, the evolution equationsfor �ˆ W(�t� )�

and� an operatorA� ˆ

W are� given, respectively, by��

ˆ W � t� ��t� �¡ B¢ HW ,� £ˆ W ¤ t� ¥F¥ ¦ 22§

and�

5807J.¨

Chem. Phys., Vol. 115, No. 13, 1 October 2001 Statistical mechanics of quantum-classical systems


dA© ˆ

W ª t� «dt© ¬B HW ,� AW ® t� ¯F¯ ,� ° 23±

in²

apparentcompleteanalogywith purequantumandclassi-cal³ dynamics.However, a numberof importantdifferencesexist´ which we now examine.

Considerµ

the product of three operatorsin the mixedquantum-classical¶ limit. From Eq. · 14 we| have

¹ABC º W » AW 1 ¼

½m¾2i� BW 1 ¿

ÀmÁ2i� CW

Â A� ˆ

W 1 ÃÄ�Å2Æ

i� B

Ç ˆW 1 È

ÉmÊ2Æ

i� CW

ËÍÌ 2

4ÎÏÎFÎ

AW Ð BW ÑÓÒ CW ÔÕBÖ AW ×ÙØ BW Ú CW ÛÏÛFÛ ,� Ü 24Ý

where| the last term is Þ (ß à 2)

�. Thus,the associativeproduct

rule is no longerexactbut is valid only to terms á (ß â 2�)�. This

hasimportantimplicationsfor thedynamics.In particular, inthisã

limit, considertheactionof themixedquantum-classicalLiouville operator on the product of two operators,CWä A� ˆ

W(1ß åçæmè

/2é

i�)�BÇ ˆ

W :

i� ê ˆ CW ë i

�ì í�î�ï A

� ˆW 1 ð

ñmò2Æ

i� BÇ ˆ

W

ó i�ô A

� ˆW 1 õ

öm÷2Æ

i� BÇ ˆ

W ø�ùûúüBý i� þ ˆ AW ÿ 1 �

��2i� BW

�A� ˆ

W 1 ��2Æ

i� � i� ˆ B

Ç ˆW �� . � 25

Æ �Thus, while Eq. � 16� is exact, the correspondingquantum-classical³ equationis valid only to terms � (

ß �)�. From this

result� it follows that

CW � t� �� e� i ˆ t! CW "$# e� i % ˆ t! A� ˆ W & 1 '(*)2Æ

i� + e� i , ˆ t! BÇ ˆ W -�.�/�02143

5 AW 6 t� 7 1 89*:2i� BW ; t� <>=@?�A�B�C . D 26E

Consequentlyµ

, the evolution of a composite operator inquantum-classical¶ dynamicscannotbedeterminedexactlyintermsã

of the quantum-classicalevolution of its constituentoperators,F but only to terms G (

ß H)�, in contrastboth to quan-

tumã

andclassicaldynamics.This resulthasimplicationsfortheã

computationof time correlationfunctions discussedinSec.I

V.The Jacobi identity involving the quantum-classical

bracket�

is valid only to terms J (ß K

):�

LA� ˆ

W ,� M BÇ ˆ W ,� CW NON�P$Q CW ,� R A� ˆ W ,� BÇ ˆ W SOS�T$U BÇ ˆ W,� V CW ,� A� ˆ W WOWX�Y[Z�\�] ; ^ 27Æ _

thus,ã

the quantum-classicalbracket is not strictly a Liebracket.�

As a consequence,Poisson’s theorem,which statesthatã

thePoissonbracket ` orF thequantumLie bracketa ofF anytwoã

constantsof themotion is alsoa constantof themotion,fails. More generally, from this resultwe concludethatprod-uctsb or powersof theconstantsof themotionareconstantsoftheã

motion only to c (ß d

).�

Thee

formal solutionsof theequationsof motionthatpar-allel� thoseof full quantummechanicsalsorequiresomedis-cussion.³ For example,the formal solutionof Eq. f 23g can³ bewritten| as

A� ˆ

W h t� i�j e� i k ˆ t! AW lnmpo e� i q rts t! /u v AWe� w i x ytz t! /u {�| . } 28~Here�

the operator� specifies� the rule that must be usedtoevaluate´ the exponentialoperatorsin the secondequality inEq. � 28� . AppendixA givesthe detailsof the rulesthat mustbe�

followed in order for the formal solution involving theexponentials´ of the left �� and� right �2�� operatorsF to beequivalent´ to thatobtainedusingthequantum-classicalLiou-ville� operator.

Given�

this compact formulation of mixed quantum-classical³ dynamicsthat exploits the formal similarity withfull�

quantummechanics,we turn to a descriptionof the sta-tisticalã

mechanicsof such systems.In the next sectionweconsider³ the determinationof the equilibrium densitymatrixfor a mixed quantum-classicalsystemand then turn in thefollowing�

sectionto a moregeneralstudyof thepropertiesofequilibrium´ time correlationfunctions.

III. EQUILIBRIUM DENSITY MATRIX

Thee

quantummechanicalequilibrium canonicaldensitymatrix� hasthe familiar form

�ˆ e�Q � Z� � 1e� �� H,� � 29

Æ �where| Z

�is²

thepartitionfunction,Z� �

Te

r exp(�� H� ˆ ).�

Its partialW�

igner transformis given by

�ˆ WeQ � R,� P �¡ dz

©e iP ¢ z£ /u ¤ R ¥ z¦

2 §ˆ e�Q R ¨ z¦2

. © 30ª «

W�

e denotethecorrespondingcanonicalequilibriumden-sity� matrix in the quantum-classicallimit by ¬ˆ We(

ßR,� P)

�which| is definedto betheapproximationto ˆ We

Q (ßR®

,� P¯ )�

that isstationary� underquantum-classicaldynamics:

i� ° ˆ ±ˆ We ² i

�³µ´�¶2·¹¸»ºˆ We ¼¾½ˆ We ¿ À¹Á»Â>Ã 0.Ä Å

31ª Æ

The solution of Eq. Ç 31ª È

can³ be found in the followingway:| we first write Éˆ We as� a powerseriesin Ê$Ë orF in themassratio� (mÌ /

éMÍ

)� 1/2 if

²scaledvariablesareusedÎ

Ïˆ We Ð ÑnÒ Ó 0ÔÕ×Ö

nÒ Øˆ We

ÙnÒ Ú . Û 32

ª ÜSubstitutingÝ

this expressionin Eq. Þ 31ª ß

and� grouping bypowersà of á ,� we obtainthe following recursionrelations:fornâ ã 0:

Äi� ä

HW ,� åˆ We

æ0Ô çéèëê

0Ä ì

33ª í



and� for nâ î 0,Ä

i� ï

H� ˆ

W ,� ðˆ We

ñnÒ ò 1 óõô¹ö 1

2÷ ø H� ˆ W ,� ùˆ We

únÒ ûõüþý 1

2÷ ÿ��ˆ We

�nÒ � ,� H� ˆ W

�. � 34

ª �An�

analogousset of recursionrelationsmay be writtenfor�

the partial Wigner transform of the full quantumme-chanical³ canonicalequilibrium densitymatrix. Theserecur-sion� relationsaregivenin AppendixB whereit is shownthat�ˆ We

Q (ßR,� P)

�and ˆ We(

ßR,� P)

�are identical to (

ß �)�. We shall

exploit´ this featurebelow wherethe solutionsof the mixedquantum-classical¶ recursionrelationsarepresented.

It is convenientto analyzethe structureof the recursionrelationsandobtaintheir solutionsin anadiabaticbasis.Theadiabatic� statesaredefinedto be theeigenstatesof thequan-tumã

subsystemwith fixed classicalcoordinates,

h�ˆ

W R� �� ;R� ��

E� ��

R� ��

;R� �

,� � 35ª �

where|

h�ˆ

W R� !#" P$ ˆ 2%

2Æ

mÌ & VW ' q(ˆ ,� R� ) . * 36ª +

Teaking matrix elementsof Eqs. , 33

ª -and� . 34

ª /,� respec-

tivelyã

, we find,

iE� 0103254

We

60Ô 798:83;=<

0,Ä >

37ª ?

iE� @1@3A5B

We

CnÒ D 1E�F1F3GIHKJ iL

� L1LNMPOWe

QnÒ R9S:S3TIUWVXYX�Z

J[ \1\3]

, ^Y^�_5` We

anÒ b=cYc�d .e

38ª f

This setof equationsis equivalentto gihYh�j i� kml1l3n , oporq5s WetYt�u=v 0Ä

with| wWexYx�y written| asa powerseriesin z . In theseequations,

usingb the definitionsin Ref. 26, we havethe classicalLiou-ville� operator,

iL� {:{3|I} P

MÍ ~~ R

� � 1

2Æ � F� W�� F

�W�3�P� ��

P$ ,� � 39

ª �where| F

�W��W�� ;R

� ��VW(

ßq(ˆ ,� R� )/

� �R� ��

;R� �

is²

the Hellmann–Feynmanforce for state � ,� the energy differenceE �1�3� (ß R)

�� E � (ß R)� �

E N¡ (ß R)�, and the term responsiblefor nonadia-

batic�

transitions,

J[ ¢1¢3£

, ¤Y¤�¥I¦K§ P

MÍ d

© ¨ª©1 « 1

2Æ S¬ :® ¯¯ P

$ °²±3³µ´�¶· P

$M

d© ¸N¹µº�»* 1 ¼ 1

2S¬ ½3¾9¿rÀ* ÁÁ P Â²ÃªÄ . Å 40Æ

InÇ

this equationthe nonadiabaticcouplingmatrix elementis

d© È1ÈNÉ=ÊÌË�Í

;R ÎÐÏÏ R Ñ�ÒÔÓ ;R Õ ,� Ö 41×while| the quantityS

¬ Ø:Ùis²

definedas

S¬ Ú:ÛÝÜßÞ

FWà:áÝâ FWãåä ã:æ ç P

Md© è:é ê 1

ë E� ì:í

d© î:ï P

$M

d© ð:ñ ò 1

,� ó 42ô õ

and� specifiesthe momentumtransferto andfrom the classi-cal³ subsystemarisingfrom quantumtransitions.

Equation ö 37ª ÷

providesà no informationaboutthe diago-nal elementsø We

(0)ù ú1ú�ûýü

We(0)ù þ

but�

requiresthat the off-diagonal

elements´ ÿWe(0)ù ��

be�

zero.Similarly, Eq.�38ª �

for nâ � 0�

pro-vides� no information about the diagonal elements � We

(1)ù

� �We(1)ù �

but�

relatesthe off-diagonalelements� We(1)ù ��

toã

thediagonal�

elements� We(0)ù �

. For nâ � 1 Eq. � 38ª �

determines�

thediagonal�

elements� We(ùn� )� �

in termsof theoff-diagonalelements

ofF thesameorder, andtheoff-diagonalelements� We(ùn� � 1) �� in

termsã

of the diagonalandoff-diagonalelementsof ! We(ùn� )� "�"�#

.Thus,providedwe know thediagonalelements$ We

(0)% &

thereã

issufficient� information to determinethe completesolutionof'

We(( ) .Thee

methodusedto solveEq. * 38ª +

for�

the diagonalele-ments� ,

We(%n� )- .

requires� somediscussion.Writing this equationexplicitly´ for the diagonalelementswe have

iL� /10

We

2n� 3547698:<;>=@? 2ACB J[ DD , EFE@GIH We

Jn� KMLFL@NPO ,� Q 43R

where| we haveused S WeTT�UWV X WeY�Z\[ * and] J[ ^�^

, _`_baWc J[ dd

, e@f5g* and]theã

fact thatJ[ h�h

, i`i<j 0�

whena realadiabaticbasisis chosen.Theleft-handsideof Eq. k 43l consists³ of a linearself-adjointdifm

ferential operatoriL� n

acting] on the diagonalelementsoftheã

density matrix, iL� o1p

We(qn� )r sutwv

H�

Wx ,� y W(qn� )r z|{

and,] due to thePoisson}

bracket form of this operator, one can determine~W(qn� )r �

onlyF up to a constantof themotionundertheadiabaticHamiltonian�

H�

W� . To insurethat a solutioncanbe found wemust invoke the theoremof the Fredholmalternativewhichrequiresthat the right-handsideof Eq. � 43� be

�orthogonalto

theã

null-spaceof iL� �

.31�

Thise

null-spaceconsistsof the con-stants� of themotionundertheadiabaticHamiltonianH

�W� —at

least�

any function F�

(ßH�

W� (ßR�

,� P$ )�) of this Hamiltonian.Hence

we� mustverify that the following condition is satisfied:

dR©

dP ��<��b� 2�� J[ �� , �F�@�I� We

�n� �M�`�b�P� F � HW� �1¡ 0.

� ¢44£

W¤

e show in Appendix C that ¥ (ßJ[ ¦¦

, §`§b¨I© We(ªn� )« ¬`¬b

)�

isa no ddfunction of P. This, along with the fact that F(

ßHW® ) i

�s an

even¯ function of P$

since° it is a function of the Hamiltonian,guarantees± the validity of Eq. ² 44

ô ³.

Thus,we may write the formal solutionof Eq. ´ 43µ as¶·

We

¸n¹ ºM»7¼�½

iL� ¾1¿1À 1 ÁÂÄÃ>ÅbÆ 2

Æ Ç�ÈJ[ É�É

, ÊFÊ@ËIÌ We

Ín¹ ÎMÏ`ÏbÐPÑ ,� Ò 45

ô ÓandÔ the formal solutionof Eq. Õ 38

ª Öfor� ×ÙØÚ×ÜÛ

asÔÝ

We

Þnß à 1 á5ââ�ãWä i

�E åå�æ iL

� ç�ç�èIéWe

ênß ëMìì�íWîÚïð`ðbñ

J[ òò�ó

, ôFô@õIö We

÷nß øMù`ùbú .û

46ô ü

Using�

the formal resultsoutlined above,we may con-structý the form of the mixed quantum-classicalcanonicalequilibriumþ densitymatrix. In AppendixB we showthat

ÿWe

�0� ��

Z�

0� � 1e �� H

WA � ,� Z

�0� �� dR

©dPe �� H

�WA � ,� � 47

ô �andÔ that � We

(1)� ��

0.�

Then,taking n� � 0�

in Eq. 46! we" obtain,after# somealgebra,

5809J.¨



$We

%1 &(')'+*�,.- i

� P

Md© /0/2143

We

506 7(8 9

2 : 1 ; e< =?> E@ ACBEDGF

H 1 I e< J�K E@ LNMPO

E� Q0Q2R S 1 TVUXW0W+Y4Z . [ 48

ô \W¤

e may then usetheseresultsin the recursionrelationstoobtainF all higher order terms in the expansion.The higherorderF termsare more difficult to evaluateand, as we shallshow] in the following section,theexplicit solutionsto ^ (

ß _)�

will" sufficesinceall resultswe canobtainarereliableonly tothisã

order.

IV. TIME CORRELATION FUNCTIONS

Teo deriveexpressionsfor transportpropertiesin termsof

equilibrium` time correlationfunctions,onemayconsidertheresponseof a systemto an external force using linear re-sponse] theory or monitor the equilibrium fluctuationsin asystem.] In this sectionwe examinelinearresponsetheoryformixed quantum-classicalsystems.

A. Response function

Standarda

descriptionsof linear responsetheory32b

assume#thatã

a systemwhich is in thermalequilibrium in the distantpastà is subjectedto a time dependentexternal force. Theresponsec of thesystemto this externalforceis determinedbycomputing³ theaveragevalueof somepropertyat time t� usingbtheã

densitymatrix determinedto linearorderin theforce.Weadopt# a similar point of view hereexceptthatwe supposethesystem] follows quantum-classicaldynamicsinsteadof fullquantumd mechanics.

The mixed quantum-classicalsystemis subjectedto atimeã

dependentexternalforce F(ßt� )� which couplesto theob-

servable] AW† e †f is theadjointg and# is appliedfrom thedistant

past.à The partially Wigner transformedHamiltonian of thesystem] in the presenceof the externalforce is

HW h t� iNj HW k AW† F l t� m . n 49o

The evolution equationfor the density matrix is obtainedfrom�

Eq. p 22Æ q

by�

replacingrts)u and# vtw)x by�

Hy z|{

(ßt� )� andH

y }|~(ßt� ),�

respectively, to yield��ˆ W � t� ��

t� �� i� ��N� 1 � Hy �|�� t� ��ˆ W � t� �N��ˆ W � t� � Hy �|�� t� ��. �¡ i� ¢ ˆ £ i

� ¤ Â¥ F ¦ t� §¨§¨©ˆ W ª t� « ,� ¬ 50

®where" H|° (

ßt� )� ±³²t´)µ·¶³¸º¹N»

F(ßt� )� and i

� ¼ Â hasa form analogous

toã

i� ½ ˆ with¾ A

� ˆW† replacingc H

� ˆW ,� i

� ¿ Â¥ À (

ßA� ˆ

W† ,� ). The formal

solution] of this equationis foundby integratingfrom t� 06 toã

t� ,�Áˆ W Â t� ÃNÄ eÅ Æ i Ç ˆ È tÉ Ê tÉ 0Ë Ì(Íˆ W Î t� 06 Ï

ÐtÉ 0ËtÉdt© Ñ

eÅ Ò i Ó ˆ Ô tÉ Õ tÉ ÖØ× i� Ù ˆ A Úˆ W Û t� Ü�Ý F Þ t� ß�à . á 51 â

Choosingµ ã

ˆ W(ßt� 06 )� to be the equilibrium densitymatrix,äˆ We ,� definedto beinvariantunderquantum-classicaldynam-

ics, i� å ˆ æˆ We ç 0,

èthe first term on the right-handside of Eq.é

51 ê

coincidesë with ìˆ We andí is independentof t� 06 . We can

now assumethat thesystemwith HamiltonianHW is in ther-mal� equilibrium from t� î.ï³ð upb to t� 06 . With this boundarycondition,ë to first order in the externalforce Eq. ñ 51

òyieldsó

ôˆ W õ t� öN÷�øˆ We ù úXûtÉdt© ü

eÅ ý i þ ˆ ÿ tÉ � tÉ �� i� � ˆ A¥ �ˆ WeF � t� �� . 52

�The nonequilibriumaveragevalue of any operatorBW

over� the densitymatrix ˆ W(ßt� ),� B

�(ßt� )� � Tr

� ��dR©

dP BW �ˆ W(ßt� )�

maybecomputedto determinetheresponseof thesystemtothe�

externalforce.Using Eq. � 52 �

we¾ may write this as

�B�

W � t� �� ! tÉdt© "

Tr� #

dR©

dP BWeÅ $ i % ˆ & tÉ ' tÉ (*) i+ , ˆ A -ˆ WeF� .

t� /0132 4!5tÉ

dt© 687:9

AW† ,; BW < t� = t� >?@?BA F C t� DE

F G!HtIdt© JLK

BAM N t� O t� PQ F� R t� ST ,; U 53

VwhereW X BW(

ßt� )� Y BW(

ßt� )� Z\[ BW ] . In this equationwe made

use^ of integrationby partsandcyclic permutationsunderthetrace_

to movethe evolutionoperatoronto BW and` i+ a ˆ

A ontobB� ˆ

W(ßt� c t� d )� . Thenusingthedefinitionof i

+ e Â as` thequantum-

classicalf bracket and the definition of the average g fh ˆ

W ij Tr k�l dR©

dP f W mˆ We for any operatorfh ˆ

W ,; we arrive at theformn

in the secondline. The prime on the tracedenotesthefactn

that the traceis over the quantumsubsystemstates.Thelast line definestheresponsefunction o BA whichW is givenbyp

BA q tr s�t3u\v:w Ax ˆ W† ,y Bz ˆ W { tr |@|B} . ~ 54

�The�

responsefunction in Eq. � 54 �

wasW derivedby consider-ing a systemfollowing quantum-classicaldynamics,in equi-librium in the past, subject to a time dependentexternalforce.n

This quantum-classicalresponsefunction shouldnowbe�

comparedwith the reductionof the responsefunction fora` fully quantummechanicalsystemto its quantum-classicalform.

The�

quantumexpressionfor the responsefunction is32b

�BAMQ � tr ��3� i

+� Tr � A†,y �ˆ e�Q � B � tr �� i+� Tr � A†,y B � tr ��ˆ e�Q ,y � 55

�which,W taking the partial Wigner transform of the secondequality� in Eq. � 55

�,y may be written as

�BAMQ � tr �¡3¢ Tr £ dR

©dP

i+¤¦¥ AW

† e§ ¨ª© /2«

iBW ¬ tr ® BW ¯ tr ° e§ ±ª² /2

«iAW

† ³µ´ˆ WeQ

¶3· Tr� ¸

dR©

dPi+¹¦º Ax »½¼ B

z ˆW ¾ tr ¿BÀ B

z ˆW Á tr Â Ax Ã½ÄLÅµÆˆ We

Q

Ç3ÈÊÉÌË Ax ˆ W† ,y Bz ˆ W Í tr Î@Î Q Ï . Ð 56

ÑT�o obtain the first equality we have usedthe fact that the

partialÒ Wigner transformof Tr AB Ó Tr Ô�Õ dR©

dP AWBW .W¤

e wish to take the mixed quantum-classicallimit ofthis_

expressionbut this cannotbe doneby a simple expan-



sionÖ in × . Mixed quantum-classicaldynamicsis definedbythe_

evolution equation Ø 20Ù whereW the evolution operatorisobtainedb by an expansionto first order in ÚÜÛ orb the massratioÝ Þ ; however, the time-evolveddensitymatrix containsallordersb in ß sinceÖ it involvestheexponentialof theevolutionoperatorb . The sameargumentsapply to operatorsat time trand` the equilibrium densitymatrix.

As�

anansatzonemayreplacethe time dependentopera-tor_

Bz ˆ

W(ßtr )à and the quantumequilibrium densitymatrix áˆ We

Q

by�

their quantum-classicalforms. In addition, we may re-placeÒ thequantumLie bracket( , )Q by

�its quantumclassical

analog,` ( , )Q â (ß

, ) which is equivalentto replacingthe ex-ponentialÒ operatorexp(ã!ä /2

éi+)à

by its expansionto linear or-derå

in æ ,y exp(çéè /2é

i+)à ê

1 ëíìïî /2é

i+. After thesereplacements

the_

responsefunction takesthe form

ðBAM ñ tr ò�ó3ô Tr

� õdR©

dPi+ö¦÷ Ax ˆ W

† ø 1 ùíúüû /2é

i+ ý

Bz ˆ

W þ tr ÿ� BW

�tr �� 1 �� /2

éi+

AW† � �ˆ We

�� Ax ˆ W† ,y Bz ˆ W � tr �� ,y � 57

�whichW is exactly the result in Eq. � 54

�obtainedb by perform-

ing�

a linear responsederivation directly on the mixedquantum-classical� equationsof motion. This entitlesone touse^ the above intuitive correspondencerule to convert aquantum� mechanical correlation function to its mixedquantum-classical� analog.

InÇ

thelinearresponseapproachto dynamicalphenomenathere_

arevarioususefulequivalentwaysto representthe re-sponseÖ function. Moreover, since the responsefunction isessentially� an equilibrium correlationfunction, its computa-tion_

is generallysimplifiedby usingthesymmetrypropertiesofb the equilibrium system.As we shall seein the next sec-tions_

the situation is much less favorable in the quantum-classicalf casesincemanyof therigorousequivalencesestab-lished in quantum or classical responsetheory are onlyapproximately` true in the quantum-classicallimit. The firstand` most important equivalenceis obtained through theKubo�

identity; therefore,we begin our discussionwith thiscase.fB. Kubo transformed correlation functions

In quantummechanics,correlationfunctions often ap-pearÒ in Kubo transformedform. Making useof the quantummechanical� identity,

i+�! A†,y "ˆ e�Q #%$

06&

d© ')(

ˆ e�QA† *,+ i+ -.0/

,y 1 58 2

in the first equality of Eq. 3 55 4

weW may write the responsefunctionn

as

5BAMQ 6 tr 798;:

06<

d© =

Tr A† >,? i+ @A0B

B C tr D Eˆ e�QFHG

06I

d© J

Tr K dR©

dP L A†W M N i

+ OQPSRT

eU VXW /2Y

iBW Z tr [�[ \ˆ WeQ ,y ] 59

^

whereW we have again written the secondline in partiallyW¤

igner transformedform. Using the correspondencerule totake_

thequantum-classicallimit of this expression,we obtain

_BA ` tr a9b;c

06d

d© e

Tr f dR©

dP g AW† h i i

+ jk0lmon

1 p�qr /2é

i+ s

Bz ˆ

W t tr u�u vˆ We . w 60x y

W¤

e shall shownow thatEqs. z 57 {

and` | 60x }

are` not equaland` agreeonly to ~�� . The quantummechanicalidentity�Eq.� �

58 ��

used^ to obtain the Kubo transformedcorrelationfunctionusestheexplicit form for thethermaldensitymatrix

ofb �ˆ e�Q � Z� � 1eU �� H

� ˆ. In thequantum-classicalcasethestructure

ofb �ˆ We is complicated� seeÖ Sec.III � and` we do not haveaclosedf form expressionfor it. Consequently, it is necessarytoobtainb an expressionfor �ˆ We that

_is analogousto �ˆ e�Q soÖ that

manipulations� parallel to those in the quantumderivationmay� be performed.We shall seethat this is possibleonly toterms_ �

(ß � 2�).à

The unnormalizedequilibrium quantumdensitymatrix,� ˆe�Q ,y satisfiesthe Bloch equation,

�� ˆe�Q��¡ ;¢ H £ ˆ e�Q ¤;¥�¦ ˆ e�QH. § 61

x ¨

T�aking its partial Wigner transformwe may write it in the

form

©�ª ˆWeQ

«�¬®;¯ H°²±�³ ˆ WeQ ´;µ�¶ ˆ

WeQ H·¹¸ ,y º 62

x »

withW the boundarycondition ¼ ˆ WeQ (ß ½¿¾

0)À Á

1.The normalizedformal solution Â taking

_into accountthe

boundary�

conditionÃ is�

Äˆ WeQ Å Z

� Æ 1 Ç eU È�É HÊÌË 1 Í9Î Z� Ï 1 Ð 1eU ÑÓÒ HÔÖÕØ× . Ù 63

x ÚByÛ

the usual rule the quantum-classicallimit should beobtainedb by the replacementH

Ü ÝQÞàß)á. However, by direct

calculationf one may show that (exp(âäã�åçæ�è )1)à

and(ß1 exp(é¿ê�ëíì�î )

à) agreeonly to first orderin ï .33

bMoreover, to

first orderin ð they_

alsoagreewith ñˆ We whoseW explicit formto_

this orderis givenin Eqs. ò 47ó ô

and` õ 48ó ö

. Thus,to this orderthese_

expressionsmay be usedinterchangeably.The Kubo identity ÷ 58

øin the partialWigner representa-

tion_

is

ùûúˆ We

Q ,y AW† ü

Q ý06þ

dÿ ��

ˆ WeQ eU �� /2

YiAW

† �� i+ � �

,y � 64x �

and` its quantum-classicalanalog,which holdsonly approxi-mately� , is

��ˆ We ,y AW

† ��06�

dÿ ��

ˆ We 1 ��2i+ AW

† �� i+ �! "�#%$'&)( *

+-,/.%0'13254 . 6 65x 7

This�

resultis derivedasfollows: performingtheintegralover8weW obtain

5811J.9



i+ :�;/<>=@?

ˆ We 1 AB�C2i+ DFE eU GIH J�K AW

† eU LNMIO P�QSR

T@Uˆ We 1 V

W�X2Y

i+ Ax ˆ

W†

Z>[ Z\ ] 1 ^ 1eU _N`Ia b�cSd 1 e

f�g2Y

i+ hFi eU j H

k lnmAx ˆ

W† eU oNpIq r�sSt

u@vˆ We 1 w

x�y2i+ AW

† z%{'|)} 2 ~ ,y � 66x �

whereW we haveusedoneof the ‘‘equivalent’’ expressionsfor�ˆ We ,y andwe remindthereaderthat the interpretationof � ( )�

is�

given in Appendix A. Next, expandingthe exponentialoperatorb andusingthequantum-classicalassociativeproductrule � valid� to terms � (

� � 2�)à �

weW may showthat

�1eU �� S� 1 �

��2Y

i+ � eU �N�I� ��%��)� 2 ,y ¡ 67

¢ £

1 ¤¥�¦2i+ § eU ¨�©�ª «�¬ 1 �® eU ¯N°I± ²�³�´%µ�¶)· 2 ¸ ,y ¹ 68

¢ º

and` usetheseresultsto manipulateEq. » 66¢ ¼

. We have

i+ ½�¾/¿>À

Z Á 1eU ÂNÃIÄ Å�Æ�ÇFÈ eU ÉIÊ Ë�Ì AW† eU ÍNÎIÏ Ð�ÑSÒ

Ó@Ôˆ We 1 Õ

Ö�×2i+ AW

† Ø%Ù'Ú)Û 2Ü Ý

. Þ 69¢ ß

T�o further simplify the right-handside of this equationwe

use^ the relation

eU à�á�â ã�ä�åFæ eU çIè é�ê AW† eU ëNìIí î�ïSð�ñ AW

† eU òNóIô õ�ö�÷%ø�ù)ú 2Ü û

,y ü 70ý þ

whichW maybeprovedby directexpansionof theexponentialoperators.b ThenusingEq. ÿ 68

¢ �weW have

i+ ��

Z\ � 1A

x ˆW† 1

��2Y

i+ � eU �� 1 ��ˆ We 1 �

��2Y

i+ Ax ˆ

W†

�� "! 2 #

$�% AW† 1 &

')(2i+ *ˆ We +�,ˆ We 1 -

.�/2i+ AW

† 0�13254 2 6

7 i+ 8:9�;

ˆ We ,y Ax ˆ W† <�=�> ?"@ 2 A . B 71

ý C

StartingD

with Eq. E 54F G

and` usingEq. H 65¢ I

givesJK

BA L tr M�N�O Tr P dRÿ

dP Q AW† ,y BW R tr STSVUˆ We

WYX Tr� Z

dRÿ

dP []\ˆ We ,y Ax ˆ W† ^ Bz ˆ W _ tr `

aYb0cd

dÿ e

Tr f dRÿ

dP AW† gVh i

+ i�jlkm

1 no�p2i+ B

z ˆW q tr r sˆ We tYu v5wyx . z 72

ý {

The last equalityhasbeenobtainedby substitutingEq. | 65¢ }

and` thenusingcyclic permutationof thetraceandintegrationby�

parts.Following~

Kubo, we maydefinea newcorrelationfunc-tion,_

also in the mixed quantum-classicallimit, by

�Ax ˆ

W ;Bz ˆ

W � tr ��V� 1�0c�

dÿ �

Tr� �

dRÿ

dP Ax ˆ

W† �� i

+ �)�y��

1 ��)�2Y

i+ B

z ˆW � tr � �ˆ We . � 73

ý �

Using�

this notation, the responsefunction may be writtensimplyÖ as

�BAM � tr ��y� Ax ˆ W ;B

z ˆW tr ¡£¢�¤�¥ ¦"§l¨ . © 74

ª «Equation� ¬

74ª

may� be useful in practicalcalculations.How-ever� , since the equivalencebetweenthe two forms of theresponseÝ function ® Eqs.

� ¯57° ±

and` ² 74ª ³µ´

cannotf be establishedto_

all ordersin ¶ ,y themagnitudeof their numericaldifferencemustbe evaluatedin specificapplicationsin order to deter-mine the applicability of Eq. · 74

ª ¸.

C.¹

Time translation invariance

Forº

both quantumand classicalsystemsin equilibriumthe_

time evolution of an observablegeneratesa stationaryrandomprocess.In particular, ensembleaveragesof observ-ables` areindependentof time andtime correlationfunctionsdo»

not dependon the time origin but only on time differ-ences.� For quantum-classicaldynamicstheensembleaverageofb an observableis independentof time,

Tr� ¼

dRÿ

dP AW ½ tr ¾T¿ˆ We À Tr� Á

dRÿ

dP Â eU i Ã tI AÄ ˆ W ÅVÆˆ We

Ç Tr È dRÿ

dPAW É eU Ê i Ë tÌ Íˆ We ÎÏ Tr� Ð

dRÿ

dPAW Ñˆ We ,y Ò 75ª Ó

whereW we haveusedthe stationarityof Ôˆ We ,y while for thetime_

correlationfunction in Eq. Õ 57° Ö

weW only have×ÙØ

AW ,y BW Ú tr ÛTÛ£Ü�ÝßÞáà AW â�ã£ä ,y BW å tr æYç�èTè£é�ê�ë ì"íyî . ï 76ª ð

Indeed,ñ

for the exactequality to hold oneneeds

AW† ò�ó�ô 1 õ

ö)÷2Y

iø BW ù tr úüû£ý�þ AW

† 1 ÿ��2Y

iø BW

�tr � �� .

77ª �

A special caseof this is, for any operatorsCW and` DW ,yCW(

�tr )à � 1 (

� ��/2�

iø)à �

D� ˆ

W(�tr )à � (

�CW � 1 � (

� ��/2�

iø)à �

D� ˆ

W)(à

tr ).à Anecessary� andsufficientconditionfor this to be true is

iø � ˆ CW 1 �

��2iø DW

�! iø " ˆ CW # 1 $%�&2Y

iø D

� ˆW ' CW 1 (

)�*2Y

iø + iø , ˆ D

� ˆW - . . 78

ª /



However, we haveshownin Eq. 0 251 that2

this is trueonly to3547698. Thus,time translationinvarianceis valid only approxi-

mately: in quantum-classicaldynamics.

V. CONCLUSION

The;

resultsobtainedin this paperprovide the basisforthe2

computationof equilibrium time correlationfunctionsinmany-bodyquantum-classicalsystems.The numericalcom-putationÒ of time correlationfunctionsentailsboth the simu-lation<

of the evolution of dynamicalvariablesor operatorsthat2

enterin thecorrelationfunctionof interestandsamplingover= a convenientweight function (exp(>@? H

A ˆW))à

which ispartÒ of the quantum-classicalcanonicalequilibrium densitymatrix. Algorithms havebeendevelopedfor the simulationof= quantum-classicalevolutionandtheir furtherdevelopmentisB

a topic of continuingresearch.Samplingmethodsfor thecalculationC of quantum-classicaltime correlationsrequiread-ditional»

considerations.Typically, in simulationsof classicalequilibriumD time correlationfunctionsthe ensembleaverageisB

replacedby a time averageandthe informationneededtosampleE from the equilibrium distribution is obtainedfrom along moleculardynamicstrajectory. The validity of suchaprocedureÒ hingeson the assumedergodicity andstationarityof= the systemandreplacesdirect samplingfrom the equilib-rium distributionusingMonte Carlo methodsby a time av-erage.D This ensembleaveragemethodhasbeenusedocca-sionallyE but sinceit is not necessaryand is algorithmicallymore complex, requiring both Monte Carlo and moleculardynamics»

programs,its practicalimportancehasbeenminor.Forº

quantum-classicalsystemsneitherthe assumptionof er-godicityF nor stationaritycanbemadeandsamplingfrom theequilibriumD distribution or anothersuitableweight functionmustbe carriedout to evaluatethe correlationfunctions.

The;

analysispresentedin this papergivesall of the in-formationG

neededto computeequilibrium time correlationsin the quantum-classicallimit in a consistentfashion and,thus,2

providesa useful way to treat a large classof many-bodyH

systemswherethe environmentaldegreesof freedomhaveI

a classicalcharacter.

ACKNOWLEDGMENTSJ

This work was supportedin part by a grant from theNaturalK

Sciencesand Engineering ResearchCouncil ofCanada.L

Acknowledgmentis madeto the donorsof The Pe-troleum2

ResearchFund,administeredby theACS, for partialsupportE of this research.Oneof theauthorsM S.N.

N OwouldP like

to2

acknowledgea Walter C. SumnerMemorial Fellowship.

APPENDIX A

The;

meaningof the formal solutionin thesecondequal-ity in Eq. Q 28R willP beestablishedhereby comparingthetwoforms in this equation

AW S tr TVUXWZY eU i [ \^] t_ /Y ` AWeU a i b c^d t_ /Y ef ,y g A1handi

AW j tr kVl eU i m ˆ t_AW . n A2o

The first few terms p thought2

of asa powerseriesexpan-sionE in it

q/� r�s

are,i from Eq. t A2u ,yeU i v ˆ t_ AW w AW x it

qy{z}|�~�� AW � AW � ��V� tr 2

2� 2� �}��9�}�� AW

�X�� AÄ ˆ W � ��V�!�}�� AÄ ˆ

W ¡£¢ ¤�¥�¦ AÄ ˆ

W § ¨�©9ª�«�¬®

¯ itq 3°

6¢ ± 3

° ²�³�´�µ9¶�·�¸9¹�º�» AW ¼X½ ¾�¿9À�Á�Â�Ã AW Ä Å�Æ®Ç

ÈXÉ�Ê�Ë�ÌÍÌ�Î}Ï�Ð AW Ñ£Ò Ó�Ô�ÕÖØ×}Ù�Ú�Û AW Ü Ý�Þ9ß�à�á�âã!ä}å�æ�ç9è�é�ê AW ë£ì í�î�ï!ð}ñ�ò�óõô

AW ö ÷�ø®ùÍù£ú�û�üý!þ}ÿ�� AÄ ˆ

W��

AÄ ˆ

W � �� ,� � A3 !

and,i from Eq. " A1 #

,�$

e% i & '�( t_ /) * AÄ ˆ

We% + i , -/. t_ /) 0

1 AÄ ˆ

W 2 itq35476�8�9�: A

Ä ˆW ; AÄ ˆ

W < =�>�?�@ tA 2�2B C 2 DFE�G H�IJ�K�L A

Ä ˆW

M 2N�O�P AW Q R�S�T AW U V�WX�Y�Z\[�] itq 3°

6¢ ^ 3

° _7`�a�b�cd�e�fg�h�i AW

j 3k l�m�no�p�q

AÄ ˆ

W r s�t�u 3k v�w�x

AÄ ˆ

W y z�{|�}�~� AÄ ˆ

W � ��\�� . � A4 �

For theseexpressionsto agreewe must make assignmentslike<�7��

AÄ ˆ

W � �� 1

3� �� AW �¡ ¢�£¤�¥�¦�§©¨�ª�«�¬® AW ¯ °�±\²�²�³�´�µ¶¸·�¹�º®»

AÄ ˆ

W ¼ ½�¾¿�À�Á�Â�Â . Ã A5 Ä

In generalthe term Å ((� Æ�Ç�È

)É jÊAW(

� Ë�Ì�Í)Î kÏ)Î

is composedof ( jÐÑ

kÒ)!/Î

jÐ! kÒ! separatetermseachwith a prefactorof j

Ð! kÒ!/( jÐÓ

kÒ)Î! . Eachof theseseparatetermscorrespondsto a unique

associationi of terms indicating the order in which the ÔÖÕoperators= act.

APPENDIX B

This;

appendix concernsthe relationship betweenthequantum× Øˆ We

Q (�RÙ

,� PÚ )Î

and quantum-classicalÛˆ We(�RÙ

,� PÚ )Î

equi-libriumÜ

densitymatricesin thecanonicalensemble.We beginbyH

derivingtherecursionrelationsfor thequantummechani-calC equilibrium densitymatrix.

The thermaldensitymatrix Ýˆ WeQ is stationaryunderthe

quantum× dynamicsin Eq. Þ 7ª ß ,�0à á©â

Hã ˆ

W ,� äˆ WeQ å

Q

æ içèêé Hã ˆ Weë ìîí /2

ïi ðˆ We

Q ñ5òˆ WeQ eë óõô /2

ïiHã ˆ

W ö . ÷ B1ø ù

5813J.ú



If we substitutethe expansion,

ûˆ WeQ üþý

nÿ � 0�

� �ˆ WeQ � nÿ �� nÿ ,� B2

ø into Eq. � B1� andi groupby powersof weP obtain � using� thefactG

that the PÚ

dependence»

of Hã

isB

of the form PÚ 2/2

�M� �

0à ��

iç �� 1 � HW ,� �ˆ We

Q � 0� � �

,� ! B3"0à #%$ 0

� &iç '

Hã ˆ

W ,� (ˆ WeQ ) 1 * +-, 1

2. /ˆ We

Q 0 0� 1�2

Hã ˆ

W 3 12. Hã ˆ

W 465ˆ WeQ 7 0

� 8:9,� ; B4

ø <0à =%> 1 ? i

ç @Hã ˆ

W ,� Aˆ WeQ B 2

C D E-F 1

2G H We

Q I 1 J�K Hã ˆ

W L 1

2G Hã ˆ

W M%Nˆ WeQ O 1 P

Q 1

8R

iç S We

Q T 0� U�V

2CHã ˆ

W W 1

8R

iç Hã ˆ

W X 2C Y

ˆ WeQ Z 0

� [,� \ B5

ø ]

0à ^%_ 2 ` i

ç aHW ,� bˆ We

Q c 3d e f-g 1

2 hˆ WeQ i 2

C j�kHW l 1

2HW m%nˆ We

Q o 2C p

q 1

8R

iç r We

Q s 1 t�u 2CHã ˆ

W v 1

8R

iç Hã ˆ

W w 2C x

ˆ WeQ y 1 z|{ 1

48} ~ We

Q � 0� ��

3dHã ˆ

W

� 1

48HW � 3

d �ˆ We

Q � 0� �

,� � B6�andi soon. Theseresultsshouldbecomparedwith themixedquantum-classical× recursionrelationsgiven in Eqs. � 33

k �andi�

34k �

.NoticeK

that the first two recursionrelationsareidenticalinB

the quantum-classicalandfully quantummechanicalsys-tems2 �

seeE Eqs. � 33k �

andi � 34k �

for n� � 0,à �

B3� ,� and � B4�� .Hence,we cancalculatethe first two termsfor the quantummechanical� system� inB the partially Wigner transformedrep-resentation� � andi the results will also be valid for thequantum-classical× system.

In quantummechanicswe have for the unnormalizedcanonicalC equilibrium densitymatrix, � ˆ We

Q

� ˆWeQ �¡ eë ¢¤£ H

k ˆ ¥W

¦ 1 §6¨ Hã ˆ

W ©ª 2

C2G Hã ˆ

Weë «¬ /2ï

iHã ˆ

W ® . ¯ B7ø °

This powerseriesin ± mustbeconvertedinto a powerserieswithP respectto ² . Extractingthe first two termsin this newpower³ seriesgives

´ ˆWeQ µ eë ¶¸· H

k ˆW¹ º¼»

2iç

½Hã ˆ

W¾P

¿nÿ À 0

�Á ÂÄÃ6ÅÇÆ

nÿ È 2Én� Ê 2 Ë !

Ì¼ÍjÎ Ï

0�

nÿ Ðn� Ñ 2

GjÐ Ò

Hã ˆ

Wnÿ Ó j

Î Ô HWÕRÙ H

ã ˆWjÎ ÖØ×ÚÙÜÛ

2C Ý

,� Þ B8ø ß

afteri isolatingthezerothandfirst ordertermsandprovingbyinductionB

to first order in à that2

áHnÿ â 2 ã

W

ä1stå æ|çéè

2iç ê H

ã ˆWë

P

ìjí î

0�

nÿ ïn� ð 2 j

ñ òHW

nÿ ó jí ô Hã ˆ

WõR

HWjí

. ö B9÷The operatorsHW andi ø HW /

� ùR do»

not commuteandso theorder= mustbe preservedasshown.

It is possibleto show that in the adiabaticbasisrepre-sentationE ú properlyû normalizedü these

2are the resultsquoted

inB

Eqs. ý 47} þ

andi ÿ 48} �

. We haveestablishedthe equivalencebetweenH

the secondterm on the right-handsideof Eq. � B8ø �

andi Eq. � 48� byH

a direct calculationwhich showsthat thematrix representationof this term in Eq. � B8� is the sameasthe2

Taylor expansionof Eq. � 48} �

inB

powersof .

APPENDIX C

Here we establish that (�J� � �

, �� We(�nÿ )� ��

)�

isa no ddfunctionG

of PÚ

. Using the definitions in Sec. III, this isequivalent� to the condition that � ((

� �(�P/�M )�

d� ��! *" 1

2. E# $ $!%

d� & &!'* (

� (/� )

PÚ

))� *

We(�nÿ )� +�+!,

)�

be an odd function of PÚ

or= ,choosingC to work with a real adiabaticbasisso that d

� - -!.is

real,� this simplifies to the conditionsthat PÚ /

(� 0

We(�nÿ )� 1 132

)�

and4/� 5

PÚ 6

(� 7

We(�nÿ )� 8 8!9

)�

be odd functionsof PÚ

. The effect of multi-plicationû by P or= differentiation with respect to P is to

changeC the parity of the factor : (� ;

We(�nÿ )� < <3=

)�, if indeed this

factor has a definite parity; thus, we must show that>(� ?

We(�nÿ )� @ @!A

)�

is an evenfunctionof PÚ

. For technicalreasonsit

willP alsobeusefulto showthat B (� C

We(�nÿ )� D�D!E

)�

is an oddfunctionof= P.

WF

e usean inductive proof. The statementsare true for

nG H 1. From Eq. I 48J weP seethat K We(1)� L�LNM

0à

andthat O We(1)� P�P!Q

is pure imaginaryandodd in P for RTSURWV . AssumingXZY\[We

]nÿ ^`_�_!acb even� function of P dfe ,� gWh ,� nG ,� i C1

j klnm\o

We

pnÿ q`r r3sct odd= function of P

Ú ufv,� wWx ,� nG ,� y C2

j zweP show that theseconditionsare true for nG { 1. From Eq.|46} }

weP have

~��We

�nÿ � 1 �`� �3�c�� 1

E ��!� � PÚM

��R��

We

�nÿ �`� �!�c�

� 1

2G � F� �� F

� �!�c ¢¡¡ PÚ £�¤\¥ We

¦nÿ §`¨ ¨3©cª

«¬ ®�¯ d° ± ±�² 1

2E ³�³�´¶µµ P · P

M ¸�¹�º We

»nÿ ¼¾½�¿�À!ÁcÂ

Ãd° Ä3Å�ÆÈÇ 1

2G E# É!ÊÌË�Í¶ÎÎ P

Ú Ï PÚM� Ð�Ñ�Ò We

Ónÿ Ô¾Õ Õ�ÖØ× .

ÙC3Ú Û

Eachof the termson theright-handsideis evenwith respect

to2

P soE Ü (Ý Þ

We(ßnÿ à 1) á�á!â )ã is even in P. The analysis ofä

(Ý å

We(ßnÿ æ 1) ç ç!è )ã is similar andwill be omitted.Thus,the state-

ment is true for né ê 1 completingthe inductiveproof.

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5815J.�



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