Download Transport properties of quantum-classical systems

THE JOURNAL OF CHEMICAL PHYSICS 122, 214105 共2005兲
Transport properties of quantum-classical systems
Hyojoon Kima兲 and Raymond Kapralb兲
Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S
3H6, Canada
共Received 8 March 2005; accepted 7 April 2005; published online 2 June 2005兲
Correlation function expressions for calculating transport coefficients for quantum-classical systems
are derived. The results are obtained by starting with quantum transport coefficient expressions and
replacing the quantum time evolution with quantum-classical Liouville evolution, while retaining
the full quantum equilibrium structure through the spectral density function. The method provides
a variety of routes for simulating transport coefficients of mixed quantum-classical systems,
composed of a quantum subsystem and a classical bath, by selecting different but equivalent time
evolution schemes of any operator or the spectral density. The structure of the spectral density is
examined for a single harmonic oscillator where exact analytical results can be obtained. The utility
of the formulation is illustrated by considering the rate constant of an activated quantum transfer
process that can be described by a many-body bath reaction coordinate. © 2005 American Institute
of Physics. 关DOI: 10.1063/1.1925268兴
I. INTRODUCTION
Transport properties, such as diffusion and viscosity coefficients or rate constants, are some of the most basic quantities that are used to characterize the dynamical behavior
systems. For equilibrium systems statistical mechanics provides well-defined expressions for transport coefficients in
terms of time integrals of flux–flux correlation functions.1
The evaluation of these correlation function expressions entails sampling over an equilibrium ensemble of initial conditions and time evolution of dynamical variables or operators.
While such calculations are routinely carried out for classical
many-body systems, their evaluation for quantummechanical systems is a very challenging problem. Part of
the difficulty stems from the fact that no methods exist for
solving the time-dependent quantum equations of motion for
a large condensed phase system; thus, in contrast to classical
systems, direct calculations of transport properties by quantum molecular dynamics are rarely attempted.2
In some instances a full quantum-mechanical treatment
is unnecessary. In many applications the quantum character
of certain degrees of freedom 共termed the subsystem兲 must
be accounted for, while the remainder of the system 共bath兲
with which they interact may be approximated by classical
mechanics.3–6 For example, a decomposition of this type is
appropriate for a subsystem composed of light particles, like
electrons or protons, interacting with a solvent of heavy molecules. Thus, transport properties, such as quantum-particle
diffusion coefficients, rate coefficients of proton or electron
transfer processes and vibrational relaxation rate coefficients
in the condensed phase, may be computed in a mixed
quantum-classical framework.
Assuming the dynamics is described by the quantumclassical Liouville equation,7–12 the linear response theory
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
b兲
0021-9606/2005/122共21兲/214105/10/$22.50
yields expressions for transport coefficients6,13 the evaluation
of which entails carrying out quantum-classical evolution of
operators and sampling over the quantum-classical equilibrium density. More general expressions for time correlation
functions have been derived by taking the quantum-classical
limit of the quantum correlation function.14 The evaluation of
these correlation functions involves forward and backward
quantum-classical time evolution of the operators and sampling based on the spectral density that retains the full quantum equilibrium structure. The structure of the propagator in
this correlation function expression is reminiscent of that in
the forward-backward influence functional technique,15 and
it may be possible to employ similar approximations to the
propagator in a quantum-classical context. Linearization
methods have also been used to incorporate nonadiabatic effects in the evaluation of time correlation functions.16–19
In this article we construct general quantum-classical expressions for transport properties, starting from a full quantum treatment of the entire many-body system. The transport
coefficient formulas again retain the full quantum equilibrium structure of the system and entail carrying out quantumclassical Liouville evolution of operators but allow much
more flexibility in how the quantum-classical limit is taken.
The resulting expressions are flexible enough to be applicable to a variety of transport properties, including the calculation of the rate constants of activated nonadiabatic reactions.
In Sec. II we derive a number of general expressions for
quantum transport coefficients in terms of Wigner transforms. These expressions provide a convenient separation of
the quantum equilibrium structure in the spectral density
function from the time evolution of operators. While formally exact, the results in this section do not provide a computationally tractable route to the evaluation of transport
properties because of the instabilities inherent in the simulations of the Wigner-transformed expressions. The spectral
density function plays a central role in this formulation. To
122, 214105-1
© 2005 American Institute of Physics
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214105-2
J. Chem. Phys. 122, 214105 共2005兲
H. Kim and R. Kapral
examine its structure, in Sec. III we consider a single harmonic oscillator in thermal equilibrium for which this function can be analytically determined. In Sec. IV we employ
the results of Ref. 14 to take the quantum-classical limit of
the evolution equation for the spectral density and use this
expression to obtain computationally useful expressions for
the transport coefficients. The results are illustrated by deriving expressions that can be used to simulate the rates of
activated nonadiabatic chemical reactions. The conclusions
of the study are given in Sec. V.
II. QUANTUM TRANSPORT COEFFICIENTS
For a quantum-mechanical system in thermal equilibrium a transport coefficient ␭AB may be determined from the
time integral of a flux–flux correlation function,1
␭AB =
冕
⬁
dt具ĵA ; ĵB共t兲典 =
0
1
␤
冕
⬁
dt
0
冓
冔
i
关ĵB共t兲,Â兴 ,
ប
冕
冓
冔
1 i
关B̂共t兲,Â兴 ,
␭AB共t兲 = dt⬘具ĵA ; ĵB共t⬘兲典 = 具Â;B̂共t兲典 =
␤
ប
0
共2兲
where we assumed 关B̂ , Â兴 = 0.20 The transport coefficient may
then be obtained from the plateau value of ␭AB共t兲.21
A. General expressions for ␭AB„t…
=
1
␤ZQ
1
␤ZQ
冕
冕
␤
ˆ
˙
d␭ Tr共Â共− iប␭兲B̂共t兲e−␤H兲
˙
d␭ Tr共Â共t1 − iប␭兲
␤
d␭
dX1dX2共Ȧ兲W共X1,t1兲BW共X2,t2兲
0
1
共2␲ប兲2␯ZQ
冕
冓 冏
冓 冏
⫻ R1 +
⫻ R2 +
Z1
2
Z2
2
⫻ e共i/ប兲Ĥt⬘B̂共t2兲e−共i/ប兲Ht⬘e−␤H兲,
ˆ
共3兲
where t⬘ ⬅ t + t1 − t2. To insert the times t1 and t2, we used the
fact that the time evolution of an operator Ô is given by
ˆ
Ô共t兲 = e共i/ប兲ĤtÔe−共i/ប兲Ht.
We partition the entire quantum system into a subsystem
plus bath so that the Hamiltonian is the sum of the kinetic
energy operators of the subsystem and bath and the potential
energy of the entire system, Ĥ = P̂2 / 2M + p̂2 / 2m + V̂共q̂ , Q̂兲,
dZ1dZ2e−共i/ប兲共P·Z1+P2·Z2兲
冏
e共i/ប兲Ĥ共t⬘+iប␭兲 R2 −
冏
Z2
2
e−␤H−i/បH共t⬘+iប␭兲 R1 −
ˆ
ˆ
冔
Z1
2
冔
共4兲
.
In writing this equation we used the fact that the matrix
element of the operator Ô共t兲 can be expressed in terms of its
Wigner transform OW共X , t兲 as
冓 冏 冏 冔
R−
=
Z
2
Ô共t兲 R +
冕
1
共2␲ប兲␯
Z
2
dPe−共i/ប兲P·ZOW共X,t兲,
共5兲
where ␯ is the coordinate-space dimension and the Wigner
transform is defined by
冕
冓 冏 冏 冔
dZe共i/ប兲P·Z R −
Z
2
Ô共t兲 R +
Z
2
共6兲
.
We use the notation R = 共r , R兲, P = 共p , P兲 and X
= 共r , R , p , P兲 where again the lowercase symbols refer to the
subsystem and the uppercase symbols refer to the bath
We define the spectral density by
1
共2␲ប兲2␯ZQ
冕
dZ1dZ2e−i/ប共P·Z1+P2·Z2兲
冓 冏 冏 冔
冓 冏 冏 冔
⫻ R1 +
0
ˆ
⫻
冕 冕
W共X1,X2,t兲 =
0
␤
1
␤
OW共X,t兲 =
We first establish some general relations for the transport
coefficients that will prove useful in the subsequent reduction to the quantum-classical limit. Writing the second equality in Eq. 共2兲 in detail and inserting arbitrary time variables t1
and t2, we can write the transport coefficient ␭AB共t兲 as
␭AB共t兲 =
␭AB共t兲 =
共1兲
˙
where ĵA = Â = 共i / ប兲关Ĥ , Â兴 is the flux of Â, with an analogous
expression for ĵ B, 关·,·兴 is the commutator, and the angular
ˆ
brackets 具Â ; B̂典 = 共1 / ␤兲兰0␤d␭具e␭ĤÂe−␭HB̂典 denote a Kubotransformed correlation function, with ␤ = 共kBT兲−1. The equiˆ
−1
librium quantum canonical average is 具¯典 = ZQ
Tr¯ e−␤H
where ZQ is the partition function. In simulations it is often
convenient to consider the time-dependent transport coefficient defined as the finite time integral of the flux–flux correlation function,
t
where lowercase and uppercase symbols refer to the subsystem and bath, respectively. In Sec. IV we shall show how
the transport coefficients for a system partitioned in this way
can be evaluated in the quantum-classical limit. For the
present, however, it is convenient to first take a Wigner transform 关22兴 over all degrees of freedom, subsystem plus bath,
and later single out the subsystem and bath degrees of freedom for different treatments. Introducing a coordinate representation 兵Q其 = 兵q其兵Q其 of the operators in Eq. 共3兲 共calligraphic symbols are used to denote variables for the entire
system兲, making a change of variables, Q1 = R1 − Z1 / 2, Q2
= R1 + Z1 / 2, etc., and then expressing the matrix elements in
terms of the Wigner transforms of the operators, we have
⫻ R2 +
Z1
2
Z2
2
e共i/ប兲Ĥt R2 −
ˆ
ˆ
Z2
2
e−␤H−共i/ប兲Ht R1 −
Z1
2
, 共7兲
which satisfies the following relations:
W共X1,X2,t兲* = W共X2,X1,− t兲,
共8兲
W共X1,X2,t − i␤ប兲 = W共X1,X2,t兲* .
共9兲
The last equality may be written as
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214105-3
J. Chem. Phys. 122, 214105 共2005兲
Transport properties of quantum-classical systems
W̃共X1,X2,− ␻兲 = e−␤ប␻W̃共X1,X2, ␻兲,
where the Fourier transform
⬁
dtei␻t f共t兲. If we let
= 兰−⬁
1
W̄共X1,X2,t兲 =
␤
冕
is
共10兲
defined
␤
d␭W共X1,X2,t + iប␭兲,
as
f̃共␻兲
共11兲
0
冕
dX1dX2共Ȧ兲W共X1,t1兲BW共X2,t2兲
⫻W̄共X1,X2,t + t1 − t2兲.
共12兲
Equation 共12兲 involves the quantity 共Ȧ兲W共X1 , t1兲 which is
not easy to compute. Starting with the Heisenberg equation
˙
of motion, Â = 共i / ប兲关Ĥ , Â兴, and taking its Wigner transform
we have,
i
d
AW共X1,t1兲 = 共HW共X1兲e共ប⌳1/2i兲AW共X1,t1兲
dt
ប
冉 冊
2
ប⌳1
= HW共X1兲sin
AW共X1,t1兲
2
ប
Equivalent forms of the evolution equation can be derived as
follows: Taking complex conjugates on both side of Eq. 共17兲
gives
共13兲
共14兲
where the direction of an arrow indicates the direction in
which the operator acts. To obtain this equation we used the
relation 共ÂB̂兲W = ÂW共X兲e共ប⌳/2i兲B̂W共X兲 for the Wigner transform of a product of operators.22
Using the properties of the phase-space derivatives of
the Wigner-transformed Hamiltonian and integration by
parts, one may establish that
冕
⳵
W̄共X1,X2,t兲 = − iLW共X2兲W̄共X1,X2,t兲.
⳵t
冕
dX共AW共X,t兲兲iLW共X兲G共X兲,
共15兲
for any function G共X兲. Making use of this result, it follows
that the transport coefficient can be cast in the form
␭AB共t兲 = −
冕
⳵
W̄共X1,X2,t兲 = 21 共iLW共X1兲 − iLW共X2兲兲W̄共X1,X2,t兲. 共20兲
⳵t
␭AB共t兲 =
⫻
dX1dX2AW共X1,t1兲BW共X2,t2兲
共21兲
Comparing this result with Eq. 共16兲, we have
⳵
i
关W共X1,X2,t − iប␤兲 − W共X1,X2,t兲兴
W̄共X1,X2,t兲 = −
⳵t
ប␤
=−
2
ImW共X1,X2,t兲,
ប␤
共22兲
where Im stands for the imaginary part, and we used Eq. 共9兲.
Thus, we have
2
ប␤
冕
dX1dX2AW共X1,t1兲BW共X2,t2兲
⫻ImW共X1,X2,t + t1 − t2兲.
共23兲
We may choose the times t1 and t2 to yield various forms
for the correlation functions. For example, setting t1 = 0 and
t2 = t, Eq. 共16兲 reduces to
冕
dX1dX2AW共X1兲BW共X2,t兲
⫻iLW共X1兲W̄共X1,X2,0兲
=
dX1dX2AW共X1,t1兲BW共X2,t2兲
⳵
W̄共X1,X2,t + t1 − t2兲,
⳵t
冕
− W共X1,X2,t + t1 − t2兲兴.
␭AB共t兲 = −
dX1dX2AW共X1,t1兲BW共X2,t2兲
冕
i
ប␤
⫻关W共X1,X2,t − iប␤ + t1 − t2兲
⫻iLW共X1兲W̄共X1,X2,t + t1 − t2兲
⬅−
共19兲
An alternative proof of these expressions is given in Appendix A. From these results it also follows that23
␭AB共t兲 =
dX共iLW共X兲AW共X,t兲兲G共X兲
=−
共18兲
*
= iLW and Eq. 共6兲. If we then
where we used the relations iLW
exchange variables X1 ↔ X2 and t ↔ −t, we get
Here, the Wigner-transformed Hamiltonian is HW共X兲
= P2 / 2M + VW共R兲 and ⌳ is the negative of the Poissonbracket operator,
⌳ = ⵜឈ P · ⵜជ R − ⵜឈ R · ⵜជ P ,
共17兲
From the last equality in Eq. 共2兲, the transport coefficient
can be written in a form involving the commutator of  and
B̂共t兲. Performing a set of manipulations similar to those used
above, we may show that ␭AB共t兲 is also given by,
− AW共X1,t1兲e共ប⌳1/2i兲HW共X1兲兲
⬅ iLW共X1兲AW共X1,t1兲.
⳵
W̄共X1,X2,t兲 = iLW共X1兲W̄共X1,X2,t兲.
⳵t
⳵
W̄共X2,X1,− t兲 = iLW共X1兲W̄共X2,X1,− t兲,
⳵t
we can write the transport coefficient as
␭AB共t兲 =
where we have written the evolution equation for W̄ as
共16兲
冕
dX1dX2共iLW共X1兲AW共X1兲兲BW共X2,t兲
⫻W̄共X1,X2,0兲.
共24兲
Finally, we observe that the initial value of W,
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214105-4
J. Chem. Phys. 122, 214105 共2005兲
H. Kim and R. Kapral
W共X1,X2,0兲 = e−共2i/ប兲P2·共R2−R1兲
⫻
1
ZQ共冑2␲ប兲2␯
冓
冕
⫻ 2R2 − R1 −
dZe−共i/ប兲共P1−P2兲·Z
Z
2
冏 冏
ˆ
e −␤H R 1 −
Z
2
冔
, 共25兲
involves only a single propagator, while W̄ defined in Eq.
共11兲, which appears in the transport coefficient expression,
still involves two imaginary time propagators.
The Wigner representation results obtained in this section involved no approximations and are as difficult to solve
for a many-body quantum system as the original expressions
for the transport coefficients. However, as we shall see in
Sec. IV, they form a convenient starting point for approximations leading to quantum-classical limit expressions.
III. SPECTRAL DENSITY FOR A HARMONIC
OSCILLATOR
The spectral density W plays a central role in the expressions for the transport coefficients and its calculation is a
difficult task, even at t = 0. Consequently, it is instructive to
examine its structure for a single harmonic oscillator where a
complete analytical solution may be obtained.
For a single harmonic oscillator, the system Hamiltonian
is given by Ĥ = 共1 / 2兲m␻2共p̂⬘2 + q̂2兲, where m and ␻ denote
mass and frequency, respectively, and we have rescaled the
momentum operator as p̂⬘ ⬅ p̂ / 共m␻兲. The propagator for this
Hamiltonian is well known24 and is given by
ˆ
具q兩e−␤H兩q⬘典 =
冉
a
2␲ sinh ␤ប␻
冊 冋
1/2
a
exp − 兵共q2 + q⬘2兲
2
册
⫻coth ␤ប␻ − 2qq⬘ csch ␤ប␻其 ,
冉冊
a
␲
2
共26兲
exp关− a兵共r21 + r22 + p1⬘2 + p⬘22兲c1
+ 共r1r2 + p1⬘ p2⬘兲c2共t兲 + 共r1 p2⬘ − p1⬘r2兲c3共t兲其兴,
共27兲
where x = 共r , p兲 and
c1 = coth共␤ប␻/2兲,
c2共t兲 = − 2共c1 cos ␻t + i sin ␻t兲,
W共x1,x2,0兲 =
冉冊
a
␲
c3共t兲 = 2共c1 sin ␻t − i cos ␻t兲.
From this explicit expression for W共x1 , x2 , t兲 one can easily
see that the relations in Eqs. 共8兲 and 共9兲 are satisfied.
The initial value of W is
exp关− a兵共共r1 − r2兲2 + 共p1⬘ − p2兲2兲c1
⬘
共29兲
The phase factor couples the position and momentum variables in the two x1 and x2 phase spaces. If we define x12
= x1 − x2 and xc = 共x1 + x2兲 / 2, with a similar change of variables for the momenta, Eq. 共29兲 can be written as
冉冊
a
␲
2
2
⬘2兲c1
exp关− a兵共r12
+ p12
⬘ 兲其兴,
− 2i共pc⬘r12 − rc p12
共30兲
where now the phase factor couples the relative and center of
mass positions and momenta. In Fig. 1, we plot W共x1 , x2 , 0兲
versus r12 and p12
⬘ for ␤ប␻ = 1 and rc = p⬘c = 1.
The fact that W共x1 , x2 , 0兲 is symmetric with respect to
the plane p12
⬘ = −r12 for p⬘c = rc = 1 is evident in the figure. The
real part of W contains a Gaussian function that is sharply
peaked around r12 = p12
⬘ = 0 and the imaginary part of W is
small in the vicinity of the origin. When r12 and p12
⬘ are
small, we can represent the function in a multipole expansion
and keep only the first order term to get
W共x1,x2,0兲 ⬇
共28兲
2
− 2i共r1 p2⬘ − p⬘1r2兲其兴.
W共x1,x2,0兲 =
with a = m␻ / ប. Substituting Eq. 共26兲 into Eq. 共7兲 specialized
to the single harmonic oscillator and integrating over z1 and
z2, we obtain
W共x1,x2,t兲 =
FIG. 1. Real 共upper panel兲 and imaginary 共lower兲 parts of W共x1 , x2 , 0兲 as a
function of r12 and p⬘12 for rc = pc⬘ = 1 and ␤ប␻ = 1. Results are reported in
terms of the dimensionless units 共m␻ / ប兲1/2r and 共m␻ / ប兲1/2 p⬘.
冋
册
a
a
exp − 共pc⬘2 + r2c 兲 ␦共x12兲
␲c1
c1
⬅ ␳We共xc兲␦共x12兲.
共31兲
In this case W reduces to the Wigner-transformed equilibrium density matrix ␳We for the center of mass variables of
two phase spaces. This result is also expected to hold for
general potentials. For other values of pc⬘ and rc, we find that
the peak rotates around the origin, maintaining its shape.
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214105-5
Transport properties of quantum-classical systems
J. Chem. Phys. 122, 214105 共2005兲
FIG. 3. Real 共upper panel兲 and imaginary 共lower panel兲 parts of W共x1 , x2 , t兲
for ␻t = ␲ / 4.
FIG. 2. Plots of the real 共upper panel兲 and imaginary 共lower兲 parts of W
versus 共dimensionless兲 r12 along the line p12
⬘ = −r12 for ␤ប␻ = 0.1 共solid lines兲
and ␤ប␻ = 10 共dashed lines兲.
In order to illustrate the quantum effects more clearly,
W共x1 , x2 , 0兲 is plotted in Fig. 2 for ␤ប␻ = 0.1 and ␤ប␻ = 10,
corresponding to small and large quantum character, respectively.
As the value of ␤ប␻ becomes larger, negative values of
the real and imaginary parts of the W function become significant. For ␤ = 0, W is proportional to a delta function as in
Eq. 共31兲.
tively, we can rewrite Eq. 共27兲 as W共x1 , x2 , t兲
= W共x1共t兲 , x2 , 0兲. In more general situations it is not possible
to obtain such a simple form involving the phase-space coordinates at time t.
In Fig. 3, we plot W共x1 , x2 , t兲 at ␻t = ␲ / 4 with the other
parameters, the same as in Fig. 1. Comparing this figure with
Fig. 1, one can see that the peak moves along the line p12
⬘
= −r12 with little change in its shape. Figure 4 shows plots W
versus r12 with p12
⬘ = −r12 for three different time values. The
A. Equation of motion for W
The general form of the equation of motion for W̄共t兲, and
hence W共t兲, was given in Eqs. 共17兲, 共19兲, and 共20兲. For a
harmonic oscillator the operator sin共ប⌳ / 2兲 can be replaced
by the first term in its expansion, so that
⳵
W共x1,x2,t兲 = 兵W共x1,x2,t兲,HW共x1兲其,
⳵t
=兵HW共x2兲,W共x1,x2,t兲其,
共32兲
where 兵·,·其 denotes the Poisson bracket; thus, W共t兲 evolves
through the classical equations of motion. This is not surprising since the quantum and classical evolution of p and r are
the same for a single harmonic oscillator and quantum effects enter through the initial condition in the Wigner representation. We may verify that these equations of motion also
follow directly from the analytical solutions by differentiation with respect to time. In particular, noting that
⳵c2 / ⳵共␻t兲 = c3 and ⳵c3 / ⳵共␻t兲 = −c2, we can easily confirm the
above equations of motion.
Finally, we observe that using the explicit expressions
for the time evolution of the position and momentum, r共t兲
= r cos ␻t + p⬘ sin ␻t and p⬘共t兲 = p⬘ cos ␻t − r sin ␻t, respec-
FIG. 4. Plot of W共x1 , x2 , t兲 function vs r12 for ␻t = 0 共solid line兲, ␻t = 7␲ / 8
共long dashed line兲 and ␻t = 9␲ / 8 共short dashed line兲.
Downloaded 30 Jun 2005 to 142.150.225.29. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
214105-6
J. Chem. Phys. 122, 214105 共2005兲
H. Kim and R. Kapral
dynamics is periodic, as expected with strongly oscillatory
behavior for long times. It is of interest to note that when
␻t = ␲ the W function depends on the difference variables
only through the phase factor,
冉
冊冉冊
␲
a
W x1,x2,
=
␻
␲
2
兺
␭AB共t兲 = −
␣1,␣1⬘,␣2,␣2⬘
⫻
exp关− a兵共r2c + p⬘c 2兲4c1
共33兲
For the harmonic oscillator it is also possible to obtain
analytical expressions for general quantum correlation functions, and these results are given in Appendix B.
⳵ ␣⬘␣ ␣⬘␣
W̄ 1 1 2 2共X1,X2,t兲,
⳵t
冕兿
2
=
IV. QUANTUM-CLASSICAL SYSTEMS
In this section we show how to take the quantumclassical limit of the general expressions for the transport
coefficients given in Sec. II. As discussed earlier, the general
equations in the Wigner-transformed representation are intractable as they stand, except for very simple harmonic systems, since they are equivalent to a full quantum mechanical
treatment of the system plus bath. By taking the quantumclassical limit of these expressions we can obtain transport
coefficient expressions that are amenable to solution using
surface-hopping methods. The computation of the initial
value of W is still a challenging problem but far less formidable than the solution of the time-dependent Schrödinger or
von Neumann equation for the entire quantum system since
it involves only imaginary time propagators.
To make connection with surface-hopping representations of the quantum-classical Liouville equation,7 we first
observe that AW共X1兲 can be written as
dZie−共i/ប兲共P1·Z1+P2·Z2兲
冓 冏 冏 冔
冓 冏 冏 冔
冓 冏 冏 冔
dz1e共i/ប兲p1·z1 r1 −
z1
z1
ÂW共X1兲 r1 +
,
2
2
where ÂW共X1兲 is the partial Wigner transform of Â, defined
as in Eq. 共6兲, but with the transform taken only over the bath
degrees of freedom. The partial Wigner transform of the
Hamiltonian is ĤW = P2 / 2M + p̂2 / 2m + V̂W共q̂ , R兲 ⬅ P2 / 2M
+ ĥW共R兲, where ĥW共R兲 is the Hamiltonian for the subsystem
in the presence of fixed particles of the bath. The adiabatic
eigenstates are the solutions of the eigenvalue problem,
ĥW共R兲兩␣ ; R典 = E␣共R兲兩␣ ; R典. We may now express the subsystem operators in the adiabatic basis to obtain
兺
␣1␣1⬘
冕
冓
dz1e共i/ប兲p1·z1 r1 −
冓
␣1␣1⬘
⫻ AW
共X1兲 ␣1⬘ ;R1兩r1 +
z1
兩␣1 ;R1
2
冔
z1
,
2
冔
⫻具␣⬘1 ;R1兩 R1 +
Z1 共i/ប兲Ĥt
Z2
e
R2 −
兩␣2 ;R2典
2
2
⫻具␣⬘2 ;R2兩 R2 +
Z2 −共i/ប兲Hˆt⬙
Z1
e
R1 −
兩␣1 ;R1典
2
2
⫻
1
1
,
ZQ 共2␲ប兲2␯h
共37兲
with t⬙ = t − i␤ប. From the definition in Eq. 共37兲 one may
show that these matrix elements satisfy the symmetry properties,
W␣⬘1␣1␣2⬘␣2共X1,X2,t兲* = W␣2␣2⬘␣1␣⬘1共X2,X1,− t兲,
W␣1⬘␣1␣2⬘␣2共X1,X2,t − iប␤兲 = W␣1␣1⬘␣2␣2⬘共X1,X2,t兲* ,
共38兲
which are the analogs of Eqs. 共8兲 and 共9兲.
The quantum-classical limit of the transport coefficient
is obtained by evaluating the evolution equation for the matrix elements of W̄ in the quantum-classical limit. This limit
was taken in Ref. 14 and the result was found to be25
⳵ ␣⬘␣ ␣⬘␣
W̄ 1 1 2 2共X1,X2,t兲
⳵t
共34兲
AW共X1兲 =
共36兲
W␣1⬘␣1␣2⬘␣2共X1,X2,t兲
i=1
冕
␣1␣1⬘
␣2␣2⬘
dXiAW
共X1兲BW
共X2兲
i=1
where the matrix elements of W are given by
⬘ 兲其兴.
+ 2i共pc⬘r12 − rc p12
AW共X1兲 =
冕兿
2
=
1
2
兺
␤1⬘␤1␤2⬘␤2
共iL␣⬘␣1,␤⬘␤1共X1兲
1
1
⫻␦␣⬘␤⬘␦␣2␤2 − iL␣⬘␣2,␤⬘␤2共X2兲␦␣⬘␤⬘␦␣1␤1兲
2 2
⫻W̄
2
␤⬘1␤1␤⬘2␤2
2
1 1
共X1,X2,t兲,
共39兲
which has the same structure as Eq. 共20兲. The quantum- classical Liouville operator in the adiabatic basis iL is given by7
iL␣␣⬘,␤␤⬘共X兲 = 关i␻␣␣⬘共R兲 + iL␣␣⬘共X兲兴␦␣␤␦a⬘␤⬘
− J␣␣⬘,␤␤⬘共X兲,
共35兲
␣1␣1⬘
共X1兲 = 具␣1 ; R1兩ÂW共X1兲兩␣⬘1 ; R1典.
where AW
Inserting this expression and its analog for BW共X2兲 into
Eq. 共16兲 for t1 = t2 = 0, we have
共40兲
where the classical evolution operator is defined as
iL␣␣⬘ =
⳵
P ⳵ 1 ␣
␣⬘
+ 关FW共R兲 + FW
共R兲兴 ,
M ⳵R 2
⳵P
共41兲
with
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214105-7
J. Chem. Phys. 122, 214105 共2005兲
Transport properties of quantum-classical systems
J␣␣⬘,␤␤⬘共X兲 = −
−
冋
册
1
⳵
P
d␣␤ 1 + S␣␤共R兲
␦ ␣⬘␤⬘
2
M
⳵P
冋
册
1 *
⳵
P *
d␣⬘␤⬘ 1 + S␣⬘␤⬘共R兲
␦␣␤ .
2
M
⳵P
various forms of the evolution equation for W̄ given in Eqs.
共39兲 and 共43兲.
共42兲
Here the frequency ␻␣␣⬘共R兲 ⬅ 关E␣共R兲 − E␣⬘共R兲兴 / ប, the
␣
Hellmann–Feynman force FW
= −具␣ ; R兩⳵V̂W共q̂ , R兲 / ⳵R̂兩␣ ; R典,
the nonadiabatic coupling matrix element is d␣␤
= 具␣ ; R兩ⵜR兩␤ ; R典, and S␣␤ = 共E␣ − E␤兲d␣␤关共P / M兲d␣␤兴−1. In
view of Eqs. 共17兲 and 共19兲 we can also write the following
equivalent forms of the evolution equation:
⳵ ␣⬘␣ ␣⬘␣
W̄ 1 1 2 2共X1,X2,t兲
⳵t
=
兺 iL␣⬘␣ ,␤⬘␤ 共X1兲W̄␤⬘␤ ␣⬘␣ 共X1,X2,t兲
1 1 2 2
1 1
␤1⬘␤1
=−
1 1
兺 iL␣⬘␣ ,␤⬘␤ 共X2兲W̄␣⬘␣ ␤⬘␤ 共X1,X2,t兲.
1 1 2 2
2 2
␤⬘2␤2
2 2
共43兲
The derivation of these expressions follows directly from
taking the quantum-classical limits of Eqs. 共17兲 and 共19兲 and
expressing them in an adiabatic basis.
One may use the different forms of the quantumclassical evolution equation given above to derive different
but equivalent expressions for the transport coefficient in the
quantum-classical limit. For example, a particularly useful
expression for the evaluation of chemical reaction rates when
the observable is a function of only the classical bath coordinates is obtained as follows: We use the first equality in Eq.
共43兲, insert this into Eq. 共36兲, and move the evolution operator iL共X1兲 onto the AW共X1兲 dynamical variable. Next, we use
the second equality in Eq. 共43兲 and formally solve the equation to obtain W̄共X1 , X2 , t兲 = e−iL共X2兲tW̄共X1 , X2 , 0兲. Finally, we
substitute this form for W̄共X1 , X2 , t兲 into Eq. 共36兲 and move
the evolution operator to the dynamical variable BW共X2兲. In
the adiabatic basis, the action of the propagator e−iL共X2兲t on
B̂W共X2兲 is
␣2␣⬘2
共X2,t兲 =
BW
兺 共e−iL共X 兲t兲␣ ␣⬘,␤ ␤⬘BW␤ ␤⬘共X2兲.
2 2
2
2 2
␤2␤⬘2
2 2
共44兲
A. Reaction rate coefficient
As an illustration of the utility of this transport coefficient expression, consider the calculation of the reaction rate
for the interconversion A B between metastable A and B
states. We suppose that the reaction can be characterized by a
scalar reaction coordinate ␰共R兲 which is a function of the
classical bath coordinates. The A and B species operators
may then be defined as N̂A = ␪共␰‡ − ␰共R兲兲 and N̂B = ␪共␰共R兲
− ␰‡兲 where ␰‡ is the location of the free-energy barrier top
along the ␰ coordinate and ␪ is the Heaviside step function.
An example of such a many-body reaction coordinate is the
solvent polarization that may be used to characterize proton
or electron transfer processes in polar solvents. Although the
rate process is intrinsically quantum in character, this reaction coordinate involves only the classical bath degrees of
freedom.
Equation 共45兲, specialized to the time-dependent rate
constant for the reaction, is given by
kAB共t兲 =
␭AB共t兲 =
兺
␣1,␣⬘1,␣2,␣⬘2
冕兿
dXi共iL共X1兲AW共X1兲兲
nAeq ␣ ,␣⬘,␣ ,␣⬘
1 1 2 2
共45兲
which is the quantum-classical analog of Eq. 共24兲. This equation can serve as the basis for the computation of transport
properties for quantum-classical systems. Comparing with
˙
Eq. 共12兲, we see that the Wigner transform of  is replaced
by its analog for a quantum-classical system and the time
evolution of the dynamical variable is given by quantumclassical Liouville evolution. Full quantum effects are described by the initial value of W̄. A number of other equivalent forms may be derived by using combinations of the
dXi共iL共X1兲NA共X1兲兲␣1␣1⬘
i=1
共iL共X1兲NA共X1兲兲␣1␣1⬘ = −
共46兲
P1
ⵜR ␰共R1兲␦共␰共R1兲 − ␰‡兲␦␣1␣⬘ ,
1
M 1
共47兲
the rate coefficient takes the form
1
nAeq ␣
i=1
␣2␣2⬘
⫻BW
共X2,t兲W̄␣1⬘␣1␣2⬘␣2共X1,X2,0兲,
冕兿
Using the fact that
kAB共t兲 =
␣1␣⬘1
兺
2
⫻NB␣2␣2⬘共X2,t兲W̄␣1⬘␣1␣⬘2␣2共X1,X2,0兲.
The result of these operations is
2
−1
⫻
兺
1,␣2,␣2⬘
冕兿
2
dXi␦共␰共R1兲 − ␰‡兲
i=1
P1
共ⵜR1␰共R1兲兲NB␣2␣2⬘共X2,t兲
M
⫻W̄␣1␣1␣2⬘␣2共X1,X2,0兲.
共48兲
The evaluation of the rate constant then entails being able to
evolve the B species variable using quantum-classical Liouville surface-hopping dynamics26 and sampling from a
weight function determined by W̄共X1 , X2 , 0兲, where the delta
function restricts ␰共R1兲 at the barrier top.
If W̄ is approximated by its value in the hightemperature, classical-bath limit14
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214105-8
J. Chem. Phys. 122, 214105 共2005兲
H. Kim and R. Kapral
W̄␣1⬘␣1␣2⬘␣2共X1,X2,0兲
=
e␤共E␣⬘1共R1兲−E␣2⬘共R1兲兲 − 1
␤共E␣⬘共R1兲 − E␣⬘共R1兲兲
1
⫻
2
1 −␤共共P2/2M兲+E 共R 兲兲
␣1⬘ 1 ␦
1
e
␣1⬘␣2␦␣2⬘␣1␦共R12兲␦共P12兲,
ZQ
共49兲
where
ZQ ⬇
兺␣
冕
dRdPe−␤共共P
2/2M兲+E 共R兲兲
␣
,
共50兲
the time-dependent rate coefficient can be written in the
simple form,
kAB共t兲 =
1
兺
nAeq ␣1
⫻
冕
dX1␦共␰共R1兲 − ␰‡兲
P1
␣1
共X1兲.
共ⵜR1␰共R1兲兲NB␣1␣1共X1,t兲␳We
M
共51兲
␣1
−1
Here ␳We
共X1兲 = ZQ
exp关−␤共P21 / 2M + E␣1共R1兲兲兴 is the equilibrium distribution. This equation is amenable to calculation
using rare event sampling methods since the delta function
confines the initial value of the reaction coordinate to the
barrier top. As above, the evolution of the species variable
NB␣1␣1共X1 , t兲 is to be carried out using quantum-classical
Liouville surface-hopping dynamics. This result is the same
as that derived using linear response theory based on the
quantum-classical Liouville equation when the diagonal part
of the equilibrium quantum-classical density is used in the
rate coefficient expression.27 Equation 共48兲 provides a more
general expression for the reaction rate that incorporates bath
quantum effects in the equilibrium structure. The focus then
shifts to the derivation of more accurate analytical expressions for W̄␣1⬘␣1␣2⬘␣2共X1 , X2 , 0兲 or the construction of simulation methods to calculate this quantity.
Employing Eq. 共51兲, quantum-classical dynamics was
used previously to study reactive dynamics in a two-level
quantum system coupled to a nonlinear oscillator, which in
turn was coupled to a harmonic bath.27 Making use of the
new expression for the reaction rate in Eq. 共48兲, along with
an approximate form for W̄ that incorporates quantum dispersion in the reaction coordinate, we have been able to obtain additional quantum effects on the nonadiabatic reaction
rate that are outside the scope of quantum-classical treatments that neglect quantum effects in the equilibrium structure of the bath.28
system as well as that of the bath. Its calculation is a difficult
problem, but more tractable than the simulation of the full
quantum time evolution of the entire system. The harmonic
oscillator results provided insight into its structure, especially into the nature of the coupling of the two phase spaces.
Given the general expressions for the time evolution of
W̄ in Eqs. 共39兲 and 共43兲, one can place the time dependence
on either operator or the spectral density function. This allows one to choose the most convenient strategy for the
evaluation of a specific transport coefficient. One useful
strategy for the calculation of the rates of activated chemical
reactions is to evolve B̂W共X2兲 using quantum-classical evolution and average over the two phase spaces with a weight
determined by the equilibrium spectral density function. In
the limit of a high-temperature bath approximation for
W̄共X1 , X2 , 0兲 we showed that the resulting expression for the
reaction rate can be related to that obtained earlier using
linear response theory based on the quantum-classical Liouville equation.27 The more general expressions obtained in
this paper allow one to incorporate equilibrium quantum bath
effects which are outside the scope of the quantum-classical
linear response results, while still carrying out the dynamics
using quantum-classical surface hopping schemes. Calculations of the reaction rate and other transport properties using
this formalism will be given in the future work.
ACKNOWLEDGEMENTS
This work was supported in part by a grant from the
Natural Sciences and Engineering Council of Canada.
APPENDIX A: ALTERNATIVE PROOF OF W
EVOLUTION EQUATIONS
Another proof of Eqs. 共17兲 and 共19兲 is possible by direct
differentiation of W共X1 , X2 , t兲. The time derivative of Eq. 共7兲
can be expressed as either
1
⳵
W共t兲 =
⳵t
共2␲ប兲2␯ZQ
⫻
冕
ds
再冓
dZ1dZ2e−共i/ប兲共P1·Z1+P2·Z2兲
R1 +
Z1
2
兩Ĥ兩s
冔冓
s兩e共i/ប兲Ĥt兩R2 −
Z2
2
冓 冏 冏 冔
冓 冏 冏 冔
冏 冔冎
冓 冏
⫻ R2 +
V. CONCLUSION
The quantum-classical expressions for transport coefficients derived in this paper, such as Eq. 共45兲, form the basis
for algorithms that can be used to compute these quantities
using surface-hopping methods. The expressions involve a
doubling of the classical phase space with the quantum connectivity between two phase spaces accounted for by the
spectral density W. The initial value of the spectral density
depends on the quantum equilibrium structure of the sub-
i
ប
冕
− R1 +
⫻ R2 +
Z2
2
Z1
2
Z2
2
ˆ
e−共i/ប兲H共t−iប␤兲 R1 −
e共i/ប兲Ĥt R2 −
ˆ
冔
Z1
2
Z2
2
e−共i/ប兲H共t−iប␤兲兩s典具s兩Ĥ R1 −
Z1
2
共A1兲
or
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214105-9
J. Chem. Phys. 122, 214105 共2005兲
Transport properties of quantum-classical systems
1
⳵
W共t兲 =
⳵t
共2␲ប兲2␯ZQ
⫻
i
ប
冕
ds
冕
dZ1dZ2e−共i/ប兲共P1·Z1+P2·Z2兲
再冓 冏
R1 +
Z1
2
冏
e共i/ប兲Ĥt兩s典具s兩Ĥ R2 −
Z2
2
冔
冓 冏 冏 冔
冓 冏 冏 冔
冏 冔冎
冓 冏
⫻ R2 +
− R1 +
⫻ R2 +
Z2
ˆ
e−共i/ប兲H共t−iប␤兲 R1 −
2
Z1
Z2
Z1
2
2
ˆ
Ĥ兩s典具s兩e−共i/ប兲H共t−iប␤兲 R1 −
2
APPENDIX B: ANALYTICAL EXPRESSIONS FOR
CORRELATION FUNCTIONS
Z2
e共i/ប兲Ĥt R2 −
2
have the desired form. A similar set of manipulations can be
carried out to show that the potential-energy parts also have
equivalent forms involving X1 or X2.
Z1
2
If the operators  and B̂ are functions of only p̂ or q̂, we
can further simplify the expression for W. When both  and
B̂ depend only on the position operator, we can integrate
W共x1 , x2 , t兲 over momenta to obtain
.
共A2兲
We will prove that the former expression reduces to Eq. 共17兲
and the latter reduces to Eq. 共19兲. The matrix element
具s兩Ĥ兩Ri ± Zi / 2典 is equal to 兵共−ប2 / 2M兲共⳵2 / ⳵s2兲 + V共s兲其
⫻␦共Ri ± Zi / 2 − s兲. We first examine the kinetic-energy part.
Using this form of the matrix element and noting the general
structure of the terms in Eqs. 共A1兲 and 共A2兲, we are led to
consider integrals of the form,
− iប
2M
冕
dse−共i/ប兲Pi·Zi
冉
冉
Zi
⳵2
−s
2 ␦ Ri ±
⳵s
2
⫻
再
Zi
2
W共p1⬘,p2⬘,t兲 =
=
共A3兲
具q̂iq̂共t兲 j典 =
Substituting these relations into the kinetic matrix elements
in Eqs. 共A1兲 and 共A2兲, we obtain 共⳵ / ⳵t兲W = 共P1 / M兲
⫻共⳵ / ⳵R1兲W and 共⳵ / ⳵t兲W = −共P2 / M兲共⳵ / ⳵R2兲W, respectively, which shows that the free-streaming contributions
Fij共x,y兲 ⬅
冦
共for even i and j兲 x⌫
共for odd i + j兲
0.
册
冕
dr1dr2W共x1,x2,t兲
a
冋
exp
␲冑b
册
a
− 兵共p1⬘2 + p2⬘2兲c1 + p1⬘ p2⬘c2共t兲其 ,
b
which has the same form as Eq. 共B1兲.
From Eq. 共B1兲 for W共r1 , r2 , t兲 we can compute the position correlation function as
Pi
Zi
− iប ⳵
±
g Ri ⫿
.
2M ⳵Ri M
2
=
i
j
⌫ 1+
2
2
1+i
1+j
⌫
2
2
冕
dr1dr2ri1r2j W共r1,r2,t兲
冑b
␲c21
j 3 y2
i
,1
+
; ;
F
1
+
2 1
2
2 2 4x2
2F 1
冉 冊
b
ac1
共i+j兲/2
Fmn共c1,c2兲,
共B3兲
where
冉 冊冉 冊 冉
冉 冊冉 冊 冉
− y⌫ 1 +
冋
a
exp − 兵共r21 + r22兲c1 + r1r2c2共t兲其 ,
b
共B2兲
Zi
⳵
f Ri ±
⳵Ri
2
共for odd i and j兲
a
␲冑b
where b ⬅ c21 − 共1 / 4兲c22共t兲 = 1 + 共1 / 4兲c23共t兲, and the second
equality follows because c22共t兲 + c23共t兲 = 4c21 − 4. Note that the
symmetry relation W共r1 , r2 , t兲 = W共r2 , r1 , t兲 holds. Similarly,
integrating W over position, we find
iPi
Zi
⳵
⫻
−
g Ri ⫿
⳵Zi ប
2
= e−共i/ប兲Pi·Zi
dp1⬘dp2⬘W共x1,x2,t兲
共B1兲
Zi
2iប −共i/ប兲P ·Z ⳵
i i
e
=
f Ri ±
⳵Zi
M
2
再
冕
=
冊
冊
冉 冊
冎冉 冊
冉 冊
冎冉 冊
⫻f共s兲g Ri ⫿
W共r1,r2,t兲 =
1 + i 1 + j 1 y2
,
; ;
2
2 2 4x2
冊
冊
冧
共B4兲
Downloaded 30 Jun 2005 to 142.150.225.29. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
214105-10
J. Chem. Phys. 122, 214105 共2005兲
H. Kim and R. Kapral
Here 2F1 is the hypergeometric function defined by
k
2F1共a , b ; c ; z兲 ⬅ 兺k共共a兲k共b兲k / 共c兲k兲共z / k!兲 and ⌫ is the gamma
29
function. Since Fij共x , y兲 = F ji共x , y兲, one can easily see
具q̂iq̂共t兲 j典 = 具q̂ jq̂共t兲i典. Also, 具p̂⬘i p̂⬘共t兲 j典 = 具q̂iq̂共t兲 j典 leads to
具p̂i p̂共t兲 j典 = 共m␻兲i+j具q̂iq̂共t兲 j典.
When  is a function of p̂ and B̂ is a function of q̂, the
relevant quantity is
W共r1,p2⬘,t兲 =
冋
a
exp
␲冑b⬘
−
册
a 2
兵共r + p⬘22兲c1 + r1 p⬘2c3共t兲其 ,
b⬘ 1
共B5兲
with b⬘ ⬅ c21 − 共1 / 4兲c23共t兲, while when  is function of q̂ and B̂
is function of p̂, we need
W共p1⬘,r2,t兲 =
a
␲冑b⬘
冋
exp −
册
a 2
兵共r + p⬘12兲c1 − r2 p⬘1c3共t兲其 .
b⬘ 2
共B6兲
The analytical solutions for 具p̂iq̂共t兲 j典 or 具q̂i p̂共t兲 j典 have forms
similar to that for 具q̂iq̂共t兲 j典
具p̂iq̂共t兲 j典 =
冑b⬘
␲c21
and
具q̂i p̂共t兲 j典 =
冑b⬘
␲c21
冉 冊
b⬘
ac1
冉 冊
b⬘
ac1
共i+j兲/2
共i+j兲/2
Fij共c1,c3兲
共B7兲
Fij共c1,− c3兲.
共B8兲
For general many-body quantum-classical systems similar reduced forms for W enter the correlation functions if the
operators depend only either on the bath positions or the
momenta.
However,
equations
of
motion
for
␣1⬘␣1␣2⬘␣2
␣1⬘␣1␣2⬘␣2
共R1 , R2 , t兲 or W
共P1 , P2 , t兲 in position or moW
mentum space cannot be obtained in closed form for these
quantities and approximations must be employed. In this
connection methods based on moments may prove useful
since the evaluation of the correlation function involves a
low-order moment of W.30
R. Kubo, Rep. Prog. Phys. 29, 255 共1966兲.
Methods based on quantum mode-coupling techniques 关E. Rabani and D.
Reichman, J. Chem. Phys. 120, 1458 共2004兲; A. A. Golosov, D. R. Reichman, and E. Rabani, ibid. 118, 457 共2003兲兴 and the initial value repre-
1
2
sentation 关M. Thoss, H. Wang, and W. H. Miller, J. Chem. Phys. 114, 47
共2001兲兴 have been used to evaluate quantum correlation functions.
3
P. Pechukas, Phys. Rev. 181, 166 共1969兲.
4
M. F. Herman, Annu. Rev. Phys. Chem. 45, 83 共1994兲.
5
J. C. Tully, Modern Methods for Multidimensional Dynamics Computations in Chemistry, edited by D. L. Thompson 共World Scientific, Singapore, 1998兲, p. 34.
6
R. Kapral and G. Ciccotti, Lect. Notes Phys. 605, 445 共2002兲.
7
R. Kapral and G. Ciccotti, J. Chem. Phys. 110, 8919 共1999兲.
8
I. V. Aleksandrov, Z. Naturforsch. A 36A, 902 共1981兲.
9
V. I. Gerasimenko, Reports of Ukrainian Academy Science 10, 65 共1981兲;
Teor. Mat. Fiz. 150, 7 共1982兲.
10
C. C. Martens and J.-Y. Fang, J. Chem. Phys. 106, 4918 共1996兲; A.
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11
I. Horenko, C. Salzmann, B. Schmidt, and C. Schütte, J. Chem. Phys. 117,
11075 共2002兲.
12
Q. Shi and E. Geva, J. Chem. Phys. 121, 3393 共2004兲.
13
S. Nielsen, R. Kapral, and G. Ciccotti, J. Chem. Phys. 115, 5805 共2001兲.
14
A. Sergi and R. Kapral, J. Chem. Phys. 121, 7565 共2004兲.
15
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16
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共2003兲.
17
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18
Q. Shi and E. Geva, J. Chem. Phys. 108, 6109 共2004兲.
19
S. Bonella and D. F. Coker, LAND-Map, a Linearized Approach to Nonadiabatic Dynamics Using the Mapping Formalism 共to be published兲.
20
Apart from cross-transport coefficients that couple fluxes and forces for
different variables, most often we are interested in situations where B̂ = Â,
and this condition is trivially satisfied.
21
In general, transport coefficients are defined in terms of flux–flux correlation functions involving projected dynamics for which the infinite time
integral is well defined. In practice, projected dynamics is replaced by
ordinary dynamics and the transport coefficient is determined from the
plateau value of the finite-time integral, assuming that a separation of time
scales exists.
22
E. P. Wigner, Phys. Rev. 40, 749 共1932兲; K. Imre, E. Özizmir, M. Rosenbaum, and P. F. Zwiefel, J. Math. Phys. 5, 1097 共1967兲; M. Hillery, R. F.
O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 共1984兲.
23
One may show that the evolution equation involving the differential operator iLW共X兲 is equivalent to the integro-differential equation derived by
S. Filinov, Y. V. Medvedev, and V. L. Kamskyi, Mol. Phys. 85, 711
共1995兲; V. S. Filinov, ibid. 88, 1517 共1996兲; 88, 1529 共1996兲.
24
R. P. Feynman, Statistical Mechanics 共Benjamin, Reading, MA, 1972兲.
25
There is a sign difference in the evolution operator due to a difference in
the definition of W in Ref. 14.
26
D. Mac Kernan, G. Ciccotti, and R. Kapral, J. Phys.: Condens. Matter 14,
9069 共2002兲.
27
A. Sergi and R. Kapral, J. Chem. Phys. 118, 8566 共2003兲.
28
H. Kim and R. Kapral 共unpublished兲.
29
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions
共Dover, New York, 1972兲.
30
I. Burghardt and G. Parlant, J. Chem. Phys. 120, 3055 共2004兲.
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