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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
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
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
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�
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
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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,
À;Á i<Â�à H@ ÄÆÅÈLj W É t6 ÊL˺̈ 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
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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
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�
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
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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
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>à 0.Ä Å
31ª Æ
The solution of Eq. Ç 31ª È
can³ be found in the followingway:| we first write Ɉ We as� a powerseriesin Ê$Ë orF in themassratio� (mÌ /
éMÍ
)� 1/2 if
²scaledvariablesareusedÎ
ψ We Ð ÑnÒ Ó 0ÔÕ×Ö
nÒ Øˆ We
ÙnÒ Ú . Û 32
ª ÜSubstitutingÝ
this expressionin Eq. Þ 31ª ß
and� grouping bypowersà of á ,� we obtainthe following recursionrelations:fornâ ã 0:
Äi� ä
HW ,� åˆ We
æ0Ô çéèëê
0Ä ì
33ª í
5808 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Nielsen, Kapral, and Ciccotti
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and� for nâ î 0,Ä
i� ï
H� ˆ
W ,� ðˆ We
ñnÒ ò 1 óõô¹ö 1
2÷ ø H� ˆ W ,� ùˆ We
únÒ ûõüþý 1
2÷ ÿ��ˆ We
�nÒ � ,� H� ˆ W
�. � 34
ª �An�
analogousset of recursionrelationsmay be writtenfor�
the partial Wigner transform of the full quantumme-chanical³ canonicalequilibrium densitymatrix. Theserecur-sion� relationsaregivenin AppendixB whereit is shownthat�ˆ We
Q (ßR,� P)
�and ˆ We(
ßR,� P)
�are identical to (
ß �)�. We shall
exploit´ this featurebelow wherethe solutionsof the mixedquantum-classical¶ recursionrelationsarepresented.
It is convenientto analyzethe structureof the recursionrelationsandobtaintheir solutionsin anadiabaticbasis.Theadiabatic� statesaredefinedto be theeigenstatesof thequan-tumã
isa no ddfunction of P. This, along with the fact that F(
ßHW® ) i
�s an
even¯ function of P$
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
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 ^ (
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
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
e wish to take the mixed quantum-classicallimit ofthis_
expressionbut this cannotbe doneby a simple expan-
5810 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Nielsen, Kapral, and Ciccotti
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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 ሠWe
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
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.
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,
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
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  eU i à tI AÄ ˆ W ÅVƈ We
Ç Tr È dRÿ
dPAW É eU Ê i Ë tÌ Íˆ We ÎÏ Tr� Ð
dRÿ
dPAW ш 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 /
5812 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Nielsen, Kapral, and Ciccotti
Downloaded 25 Sep 2001 to 142.150.225.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
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
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
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.
1J.ë
C. Tully, in Modernì
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ited by B. J. Berne,G. Ciccotti, and D. F. Coker ÷ Wø orld Scientific,Sin-gapore,1998ù .
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