Quantum correction factors for multiphonon processes in condensed
phase vibrational relaxation
Binny J. Cherayil
Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore-560012, India
An influence functional approach is used to determine the role of multiphonon processes in the rates
of vibrational relaxation. Relaxation is considered to occur between a pair of coupled harmonic
oscillators, representing an excited and a receiving mode on a single polyatomic solute, and a
collection of independent harmonic oscillators, representing a solvent reservoir. The interaction
between the oscillator pair in the solute is arbitrary and left unspecified, while interactions between
solute and solvent are taken to be linear in the solute coordinates but quadratic or cubic in the
solvent coordinates. The nonlinearities allow vibrational relaxation to occur through multiple
excitations of phonons. Transitions rates for such multiphonon processes are derived, as are
quantum corrections to the corresponding classical force correlation functions. The quantum
correction factors are also shown to emerge directly from certain terms in the real part of the
influence functional.
I. INTRODUCTION
Calculations of the relaxation rates of vibrationally excited solutes in liquids or other dense media often treat the
solute quantum mechanically and the solvent classically
whenever a fully quantum mechanical approach is impractical. Although this semi-classical approximation is not always
accurate, especially at low temperatures, where the inherent
quantum mechanical nature of the system tends to be most
strongly manifested, its reliability can be improved by the
application of a so-called quantum correction factor
共qcf兲.1–10 For vibrational modes that are coupled linearly 共in
the bath coordinates兲 to a harmonic bath, this factor can be
determined exactly.1 The linear coupling case is, however,
limited as a description of relaxation involving, say, high
frequency vibrational modes, where the process is typically
accompanied by the excitation of more than one normal
mode 共phonon兲 of the solvent, and is induced by nonlinear
couplings between solute and solvent. If the relaxation rate
for a process dominated by multiphonon excitation is derived
from a classical description of the solvent, the quantum correction factor can no longer be calculated exactly. Unfortunately, even the available approximate correction factors1–10
are not all mutually consistent, reflecting the apparent lack of
systematic and well-defined methods to calculate them. The
goal of the present article is therefore to develop such methods from general principles.
Our approach is based on the influence functional formalism of Feynman and Vernon,11 which was very recently
used by Shiga and Okazaki12 to address precisely this issue
of multiphonon relaxation in the condensed phase, including
the question of quantum corrections to classical rate expressions. However, no attempt was made in this latter work to
actually calculate qcf’s, since it was felt that in the absence
of a well-motivated approximation scheme, the calculation
was unlikely to be useful. This may or may not be the case,
but we nevertheless explore the implications of an approxi-
mation applied by Egorov and Skinner13 to solids that Shiga
and Okazaki considered of doubtful relevance to liquids, and
so ignored. The calculations of Shiga and Okazaki were also
concerned only with the problem of a single solute oscillator
in a bath of solvent oscillators, and thus were effectively
constrained to model the relaxation of diatomics alone. But a
number of recent experiments have been able to probe the
vibrational dynamics of polyatomic solutes in polyatomic
solvents.14 For such systems the foregoing influence functional approach must be generalized, and a very simple generalization is considered here: it is to treat the dynamics of a
pair of coupled solute oscillators, one representing the vibrational mode that is excited, and the other representing a
mode on the same molecule into which a fraction of the
excess energy may be deposited, the rest of the energy being
dissipated to the bath through phonon excitation. In this way,
one captures the rudiments of the intramolecular relaxation
pathways that may be additionally available to polyatomics
but not to diatomics. The overall approach in these calculations thus follows the methodolgy of Ref. 12 共which in turn
invokes the work described in Ref. 15 on quantum field
theory兲, but the inclusion of a second internal mode in the
calculations is a sufficient complication that separate treatment is warranted. For this reason, too, the calculations are
presented in sufficient detail that they can be read on their
own.
Despite their utility in problems that deal broadly with
the general phenomenon of ‘‘decoherence,’’ 15–17 functional
integral techniques have not been widely used to study vibrational relaxation. For the most part, they are not needed if
the results of low-order perturbation theory provide an adequate first approximation.18 The great advantage of the
functional integral approach is that it is well-suited for nonperturbative calculations,11 but this advantage is not exploited in the present calculation. Shiga and Okazaki have
already indicated the directions along which a nonperturba-
tive calculation of the vibrational relaxation rate could be
expected to proceed.12 There is, however, another reason for
using this approach: it is formulated around the density matrix, which is a convenient starting point for treating condensed phase vibrational dephasing,19,20 a subject that will be
taken up in future publications.
A review of the approach is presented in the following
section; it is elaborated around a model of a pair of coupled
harmonic oscillators 共representing the polyatomic兲 that interacts nonlinearly with a collection of independent harmonic
oscillators 共representing the normal modes of the bath兲. The
spectral properties of the bath are contained in a quantity
called the influence functional. As its name suggests, the
influence functional determines the extent to which the properties of the bath influence the dynamics of the excited state
solute. As such, it is the key element of the present analysis,
and its calculation is therefore central to the objectives outlined earlier. However, it is not an easy object to calculate,
and we do so only approximately, following the approach
originally introduced by Hu et al.,15 and adopted later by
Shiga and Okazaki.12 The formalism is slightly generalized
here to treat two coupled oscillators in a condensed phase
environment. Although the calculations are relatively
straightforward, they are sufficiently lengthy that they are
relegated to two mathematical appendices so as not to obscure the main threads of the theoretical development, which
are presented in Sec. III, where state-to-state transition rates
are calculated along the lines discussed by Feynman and
Hibbs.11 These results are then used in Sec. IV to derive
expressions for the quantum correction factors, which are the
main results of the article. A brief comparison of our expressions with others in the literature is presented in Sec. V,
which also presents some broad conclusions.
II. BACKGROUND AND REVIEW
As a prototype of a polyatomic solute in a solvent, we
consider a pair of coupled harmonic oscillators interacting
with a bath of N independent harmonic oscillators. The
coupled oscillator pair represents both the high frequency
mode that is initially excited on the polyatomic by, say, a
pump pulse, and a nearby mode on the same molecule 共the
receiving mode兲 that receives some fraction of the energy
dissipated during the course of relaxation. The coupling between the two is arbitrary, and is not specified. The N independent oscillators represent the modes of the bath, which
receive the remainder of the vibrational energy. The
Hamiltonian H for such a system is given by
H⫽H X ⫹H B ⫹H XB ⬅ 共 H 1 ⫹H 2 ⫹H 12兲 ⫹H B ⫹H XB ,
共1兲
where
H ␣ ⫽ 21 m ␣ ẋ ␣2 ⫹ 21 m ␣ ␻ ␣2 x ␣2 ,
␣ ⫽1,2
H 12⫽␭G 共 x 1 ,x 2 兲 ,
N
共3兲
and
兺
共5兲
n⫽1,2,3... .
Here, x 1 , x 2 and R k , and ẋ 1 , ẋ 2 and Ṙ k are the displacements and velocities of, respectively, the excited mode, the
receiving mode and the kth normal mode of the bath 共the
symbols are understood to refer to three-dimensional vectors兲; m 1 and m 2 are the reduced masses of the coupled
oscillator pair, and ␻ 1 and ␻ 2 are their vibrational frequencies; the corresponding upper case symbols refer to the parameters of the bath; ␭ is the coupling constant between the
oscillator pair, and G(x 1 ,x 2 ) is some arbitrary function of
their coordinates; q is a collective coordinate of the bath that
is some linear combination of its normal modes, and is defined as
N
q⫽
兺
k⫽1
共6兲
c kR k ,
where the c k are unknown expansion coefficients; and A (n)
and B (n) are the coupling constants between the solute modes
and the collective coordinate of the bath. The interaction
term in Eq. 共5兲 is a generalization of the nonlinearities studied in Ref. 15, where the analog of H XB was given by
兺 k A (n) x 1 R k . In condensed phases, especially solids, it is the
variable q rather than the normal mode displacement R k that
is expected to be the relevant descriptor of collective
excitations.13
The influence functional approach begins with the following expression for the density operator ␳ (t) at time t: 11,21
␳ 共 t 兲 ⫽e⫺iHt/ប ␳ 共 0 兲 eiHt/ប .
共7兲
Using
the
definition
具 x 1 ,x 2 ,R 兩 ␳ (t) 兩 y 1 ,y 2 ,Q 典
⫽ ␳ (x 1 ,y 1 ;x 2 ,y 2 ;R,Q;t) in the coordinate representation,
where x 1 and y 1 are the displacement coordinates of oscillator 1, x 2 and y 2 are coordinates of oscillator 2, and R and Q
stand for the coordinates of the bath modes 共i.e., R
⬅ 兵 R 1 ,...R N 其 and Q⬅ 兵 Q 1 ,...Q N 其 ), it can be shown that
␳ 共 x 1 ,y 1 ;x 2 ,y 2 ;R,Q;t 兲
⫽
冕
dx 1⬘ dy 1⬘ dx 2⬘ dy 2⬘ dR ⬘ dQ ⬘ K 共 x 1 ,x 2 ,R,t;x 1⬘ ,x 2⬘ ,R ⬘ ,0兲
⫻ ␳ 共 x ⬘1 ,y 1⬘ ;x 2⬘ ,y 2⬘ ;R ⬘ ,Q ⬘ 兲 K * 共 y 1 ,y 2 ,Q,t;y ⬘1 ,y ⬘2 ,Q ⬘ 兲 ,
共8兲
where, in terms of path integrals,
K(x 1 ,x 2 ,R,t;x 1⬘ ,x 2⬘ ,R ⬘ ) is given by
the
propagator
K 共 x 1 ,x 2 ,R,t;x 1⬘ ,x 2⬘ ,R ⬘ 兲
⫽
N
1
1
H B⫽
M Ṙ 2 ⫹
M ⍀ 2R 2 ,
2 k⫽1 k k 2 k⫽1 k k k
兺
共2兲
H XB ⫽A (n) x 1 q n ⫹B (n) x 2 q n ,
冕
x1
x 1⬘
D关 x 1 共 t 兲兴
冋
⫻exp
共4兲
with
冕
x2
x 2⬘
D关 x 2 共 t 兲兴
冕
R
R⬘
D关 R 共 t 兲兴
i
兵 S 关 x 1 ,x 2 兴 ⫹S B 关 R 兴 ⫹S I 关 x 1 ,x 2 ,R 兴 其
ប
册
共9兲
1
S 关 x 1 ,x 2 兴 ⫽ m 1
2
冕
t
0
1
⫹ m2
2
⫺␭
冕
0
N
S B关 R 兴 ⫽
t
1
M
2 k⫽1 k
兺
F 关 x 1 ,y 1 ;x 2 ,y 2 兴
d ␶ 关 ẋ 1 共 ␶ 兲 2 ⫺ ␻ 21 x 1 共 ␶ 兲 2 兴
冕
t
0
⫽
d ␶ 关 ẋ 2 共 ␶ 兲 2 ⫺ ␻ 22 x 2 共 ␶ 兲 2 兴
冕
t
0
S I 关 x 1 ,x 2 ,R 兴 ⫽⫺A (n)
⫺B (n)
d ␶ 关 Ṙ k 共 ␶ 兲 2 ⫺⍀ 2k R 2k 共 ␶ 兲兴 ,
共11兲
冕
冕
t
0
t
0
d ␶ x 1共 ␶ 兲 q 共 ␶ 兲 n
d ␶ x 2共 ␶ 兲 q 共 ␶ 兲 n,
n⫽1,2,3... .
The propagator K * (y 1 ,y 2 ,Q,t;y ⬘1 ,y ⬘2 ,Q ⬘ ) is the complex
conjugate of the propagator K(x 1 ,x 2 ,R,t;x ⬘1 ,x 2⬘ ,R ⬘ ) with
the 兵 x 其 coordinates replaced by 兵 y 其 coordinates and the R
coordinates by Q coordinates. Here D关 x 1 (t) 兴 represents the
functional measure on the space of x 1 trajectories, and the
limits x ⬘1 and x 1 define the end-point conditions x 1 (0)⫽x ⬘1
and x 1 (t)⫽x 1 . The path integrals over x 2 and R, and over
y 1 ,y 2 and Q are defined similarly.
Defining a bath averaged density of states as
˜␳ 共 x 1 ,y 1 ;x 2 ,y 2 ;t 兲 ⬅
冕
R
R⬘
D关 R 共 t 兲兴
冕
R
Q⬘
冋
D关 Q 共 t 兲兴 exp
册
i
兵S 关 R 兴
ប B
共16兲
All of the information relevant to condensed phase vibrational relaxation is contained in the influence functional
F, which describes the extent to which the presence of solvent modifies the dynamical behavior of the isolated oscillator pair. When there is no coupling whatever between solute
and solvent, F⫽1. Departures of F from unity indicate the
involvement of the solvent in the rate of transition between
different eigenstates of the unperturbed solute.
At present, the influence functional can be calculated
exactly only when the coupling between the solute and solvent is linear in their respective coordinates. When the interactions between them involve nonlinearities, F can no longer
be calculated exactly, but it can be obtained approximately
by means of a cumulant expansion around the exact bilinear
coupling results, as first suggested by Hu et al.15 in the context of quantum field theory. The procedure is rigorous and
systematic, and can be carried through to arbitrary order of
approximation in the cumulant expansion. The details of the
calculation, though important, are largely standard, so they
are not discussed in the main part of the article, but they can
be found in the appendices. The following section makes use
of the results presented there to calculate transition rates between different eigenstates of the oscillator pair.
III. TRANSITION RATES
dR ␳ 共 x 1 ,y 1 ;x 2 ,y 2 ;R,R;t 兲 , 共13兲
and assuming separability of the initial equilibrium density
matrix, which implies
␳ 共 x 1⬘ ,y 1⬘ ;x 2⬘ ,y 2⬘ ;R ⬘ ,Q ⬘ 兲 ⫽ ␳ 共 x 1⬘ ,y 1⬘ ;x 2⬘ ,y 2⬘ 兲 ␳ B 共 R ⬘ ,Q ⬘ 兲 ,
共14兲
it is readily shown that
If it is assumed that the system 共oscillators 1 and 2兲 is
initially in the state characterized by some set of quantum
numbers collectively denoted I 共this state being the eigenstate of the unperturbed system兲, and if it is further assumed
that the state is occupied with probability 1 to reflect typical
initial conditions in pump-probe experiments, then given the
influence functional, one may calculate the probability that
the system then makes a transition to a final state 兩 N 典 in the
time interval t. The initial condition is expressed as
␳ 共 0 兲 ⫽ 兩 I 典具 I 兩 .
˜␳ 共 x 1 ,y 1 ;x 2 ,y 2 ;t 兲
共17兲
Thus in coordinate representation,
冕
dx ⬘1 dy ⬘1 dx ⬘2 dy 2⬘
⫻
冕
x ⬘2
冕
共10兲
共12兲
x2
dR ⬘ dQ ⬘ dR
⫺S B 关 Q 兴 ⫹S I 关 x 1 ,x 2 ,R 兴 ⫺S I 关 y 1 ,y 2 ,Q 兴 其 ␳ B 共 R ⬘ ,Q ⬘ 兲 .
d ␶ G 共 x 1 共 ␶ 兲 ,x 2 共 ␶ 兲兲 ,
and
⫽
冕
D关 x 2 共 t 兲兴
册
冕
冕
y2
y ⬘2
x1
x ⬘1
D关 x 1 共 t 兲兴
冕
y 1⬘
冋
D关 y 2 共 t 兲兴 exp
y1
␳ 共 x ⬘1 ,y ⬘1 ;x ⬘2 ,y 2⬘ 兲 ⫽ 具 x ⬘1 ,x ⬘2 兩 ␳ 共 0 兲 兩 y 1⬘ ,y 2⬘ 典
D关 y 1 共 t 兲兴
⫽ 具 x 1⬘ ,x ⬘2 兩 I 典具 I 兩 y ⬘1 ,y ⬘2 典
⫺S 关 y 1 ,y 2 兴 其 F 关 x 1 ,y 1 ;x 2 ,y 2 兴 ␳ 共 x 1⬘ ,y 1⬘ ;x 2⬘ ,y ⬘2 兲 ,
where the quantity
functional—is given by
⫽ ␺ I 共 x ⬘1 ,x 2⬘ 兲 ␺ I* 共 y ⬘1 ,y ⬘2 兲 ,
i
兵 S 关 x 1 ,x 2 兴
ប
F 关 x 1 ,y 1 ;x 2 ,y 2 兴 —the
共15兲
influence
共18兲
where ␺ I (x 1⬘ ,x ⬘2 ) is the solution to the Schrödinger equation
for the coupled oscillators in the absence of solvent 关which is
in general unknown unless G(x 1 ,x 2 ) happens to be bilinear
in x 1 and x 2 .兴 The probability P N (t) that the system is in the
state 兩 N 典 at time t is
P N 共 t 兲 ⫽ 具 N 兩˜␳ 共 t 兲 兩 N 典 ,
共19兲
which, introducing complete sets of states, can be written as
P N共 t 兲 ⫽
冕
dx 1 dx 2 dy 1 dy 2 ␺ N* 共 x 1 ,x 2 兲
⫻˜␳ 共 x 1 ,y 1 ;x 2 ,y 2 ;t 兲 ␺ N 共 y 1 ,y 2 兲 .
共20兲
Equation 共15兲 defining ˜␳ (x 1 ,y 1 ;x 2 ,y 2 ;t) is substituted into
Eq. 共20兲, and as in Feynman and Hibbs,11 the influence functional is then linearized by expanding to first order in the
function ␣ j , which is related to the mean thermal occupation
number, and is defined in Eqs. 共A7兲–共A9兲. 共This procedure is
analogous to the conventional perturbation approach based
on the Schrödinger equation.兲 Each term in the expansion
共including the leading order term of unity兲 is the product of
two factors, each of which is of the general form
冕
dx 1 dx 2 dx 1⬘ dx 2⬘
冕
x1
x 1⬘
D关 x 1 兴
⫻ ␹ * 共 x 1 ,x 2 兲 Ae
iS/ប
冕
x2
x 2⬘
and the parameter ␥ mn is related to the coupling constants
between the solute oscillators and the collective coordinate
of the bath, and is defined by
2
2 2
(1) 2
(2) 2
␥ mn
⫽c m
c n 关共 A (2) x NI
兲 ⫹ 共 B (2) x NI
兲
(1) (2)
⫹2A (2) B (2) x NI
x NI 兴 .
Similarly,
␰ 共 x 1⬘ ,x 2⬘ 兲 ,
␣ ⫽1,2,
共22兲
␣ ⫽1,2,
共23兲
where
N
dx 1 dx 2 ␺ N* 共 x 1 ,x 2 兲 x ␣ ␺ I 共 x 1 ,x 2 兲 ,
␣ ⫽1,2.
共24兲
N
兺 兺
m⫽1 n⫽1
2
␥ mn
4M m M n ⍀ m ⍀ n
冕 ␶冕
t
t
d
0
0
d ␶ ⬘关 ␣ m共 ␶ ⫺ ␶ ⬘ 兲
* 共 ␶ ⫺ ␶ ⬘ 兲 ␣ n* 共 ␶ ⫺ ␶ ⬘ 兲
⫻ ␣ n 共 ␶ ⫺ ␶ ⬘ 兲 e⫺i ␻ 0 ( ␶ ⫺ ␶ ⬘ ) ⫹ ␣ m
⫻ei ␻ 0 ( ␶ ⫺ ␶ ⬘ ) 兴 .
t
0
0
d␶d␶⬘
⫹6 ␣ l 共 0 兲 ␣ m 共 0 兲 ␣ n* 共 ␶ ⫺ ␶ ⬘ 兲兴 ei ␻ 0 ( ␶ ⫺ ␶ ⬘ ) 其 ,
共27兲
where
2
2 2
(1) 2
(2) 2
␴ lmn
⬅c 2l c m
c n 关共 A (3) x NI
兲 ⫹ 共 B (3) x NI
兲
(1) (2)
⫹2A (3) B (3) x NI
x NI 兴 .
共28兲
If now the rate of transition k (n) between the two states
兩 I 典 and 兩 N 典 is defined as
1 (n)
PN 共 t 兲,
t→⬁ t
k (n) ⫽ lim
共29兲
it is readily shown that
k (2) ⫽Re
兺m 兺n
2
␥ mn
M mM n⍀ m⍀ n
冕
⬁
0
dt ␣ m 共 t 兲 ␣ n 共 t 兲 e⫺i ␻ 0 t ,
共30兲
where Re denotes real part. When Eq. 共A9兲 is substituted into
the above expression, and the resulting integral evaluated
according to11
P N (t) is calculated in this way for different values of n,
the number of phonons involved in the process. For n⭓2, the
probability is denoted P N(n) (t). P N(2) (t) and P N(3) (t) are given
in particular by
P N(2) 共 t 兲 ⫽
t
⫹6 ␣ l 共 0 兲 ␣ m 共 0 兲 ␣ n 共 ␶ ⫺ ␶ ⬘ 兲兴 e⫺i ␻ 0 ( ␶ ⫺ ␶ ⬘ )
共21兲
( ␣ ) i(E N ⫺E I ) ␶ ⬘ /ប
e
,
具 I 兩 y ␣ 共 ␶ ⬘ 兲 兩 N 典 * ⬅y NI
冕冕
⫻ 兵 关 4 ␣ l共 ␶ ⫺ ␶ ⬘ 兲 ␣ m共 ␶ ⫺ ␶ ⬘ 兲 ␣ n共 ␶ ⫺ ␶ ⬘ 兲
and
冕
兺
* 共 ␶ ⫺ ␶ ⬘ 兲 ␣ *n 共 ␶ ⫺ ␶ ⬘ 兲
⫹ 关 4 ␣ l* 共 ␶ ⫺ ␶ ⬘ 兲 ␣ m
where A is some functional of the oscillator coordinates.
This expression defines—in the notation of Feynman and
Hibbs—the transition element 具 ␹ 兩 A 兩 ␰ 典 S . 共The sign of S in
the above equation may be positive or negative; the transition element with negative S is the complex conjugate of the
transition element with positive S.兲 When A is unity, the
transition element vanishes by virtue of the orthonormality
of the eigenfunctions of the isolated coupled oscillator problem. For other values of A, the transition element may be
expressed in terms of various overlap integrals, using the
general results,
(␣)
⬅
x NI
2
␴ lmn
3ប
32 l,m,n M l M m M n ⍀ l ⍀ m ⍀ n
P N(3) 共 t 兲 ⫽
D关 x 2 兴
( ␣ ) ⫺i(E N ⫺E I ) ␶ /ប
e
,
具 N 兩 x ␣ 共 ␶ 兲 兩 I 典 ⬅x NI
共26兲
共25兲
Here ប ␻ 0 is the energy difference E I ⫺E N between the initial
and final states of the coupled oscillator pair; the functions
␣ i ( ␶ ⫺ ␶ ⬘ ), as stated before, are defined in Eqs. 共A7兲–共A9兲;
冕
⬁
0
dte⫾i( ␻ ⫾ ␯ )t ⫽⫾iPP
1
⫹␲␦共 ␻⫾␯ 兲,
␻⫾␯
共31兲
where PP stands for principal part, it is found that
k (2) ⫽ ␲
兺m 兺n
2
␥ mn
关共 1⫹n m 兲共 1⫹n n 兲 ␦ 共 ␻ 0
M mM n⍀ m⍀ n
⫺ 共 ⍀ m ⫹⍀ n 兲兲 ⫹n m 共 1⫹n n 兲 ␦ 共 ␻ 0 ⫺ 共 ⍀ n ⫺⍀ m 兲兲
⫹n n 共 1⫹n m 兲 ␦ 共 ␻ 0 ⫺ 共 ⍀ m ⫺⍀ n 兲兲 ⫹n n n m ␦ 共 ␻ 0
⫹ 共 ⍀ m ⫹⍀ n 兲兲兴 ,
where n k ⬅1/关 exp(␤ប⍀k)⫺1兴, with ␤ ⫽1/k B T.
The transition rate k (3) is similarly found to be
共32兲
k (3) ⫽
3␲ប
2 l,m,n
兺
再
2
␴ lmn
关共 1⫹n l 兲共 1⫹n m 兲共 1⫹n n 兲 ␦ 共 ␻ o ⫺ 共 ⍀ l ⫹⍀ m ⫹⍀ n 兲兲
M lM mM n⍀ l⍀ m⍀ n
⫹n l 共 1⫹n m 兲共 1⫹n n 兲 ␦ 共 ␻ o ⫺ 共 ⍀ m ⫹⍀ n ⫺⍀ l 兲兲 ⫹n m 共 1⫹n l 兲共 1⫹n n 兲 ␦ 共 ␻ o ⫺ 共 ⍀ l ⫹⍀ n ⫺⍀ m 兲兲
⫹n n 共 1⫹n l 兲共 1⫹n m 兲 ␦ 共 ␻ o ⫺ 共 ⍀ l ⫹⍀ m ⫺⍀ n 兲兲 ⫹n l n m 共 1⫹n n 兲 ␦ 共 ␻ o ⫺ 共 ⍀ n ⫺⍀ l ⫺⍀ m 兲兲
⫹n l n n 共 1⫹n m 兲 ␦ 共 ␻ o ⫺ 共 ⍀ m ⫺⍀ l ⫺⍀ n 兲兲 ⫹n m n n 共 1⫹n l 兲 ␦ 共 ␻ o ⫺ 共 ⍀ l ⫺⍀ m ⫺⍀ n 兲兲
冎
3
⫹n l n m n n ␦ 共 ␻ o ⫹⍀ l ⫹⍀ m ⫹⍀ n 兲兲 ⫹ 共 1⫹2n l 兲共 1⫹2n m 兲 兵 共 1⫹n n 兲 ␦ 共 ␻ 0 ⫺⍀ n 兲 ⫹n n ␦ 共 ␻ 0 ⫹⍀ n 兲 其 兴 .
2
IV. CORRECTIONS TO CLASSICAL TRANSITION
RATES
In this section, approximate quantum correction factors
are derived for systems in which the interactions between
solute and solvent are quadratic or cubic in the solvent collective coordinate. Before turning to these results, however,
we provide here, for the sake of completeness, and for purposes of illustration, the expression k (1) for the transition rate
between the given initial and final state of a system characterized by bilinear coupling between solute and solvent
共which corresponds to a one-phonon process兲:
k (1) ⫽
␲
ប
⫹n m ␦ 共 ␻ 0 ⫹⍀ m 兲兴 ,
共34兲
where
2
(1) 2
(2) 2
(1) (2)
␧m
⫽ 共 A (1) x NI
x NI .
兲 ⫹ 共 B (1) x NI
兲 ⫹2A (1) B (1) x NI
共35兲
If ␻ 0 ⬎0, as we shall assume, only the first term in Eq. 共34兲
survives; moreover, by virtue of the properties of the delta
function, the thermal occupation factor may be taken outside
the summation sign and evaluated at the frequency ␻ o of the
transition. If in addition a spectral density function J (1) ( ␻ ) is
defined by
J (1) 共 ␻ 兲 ⫽
2
␧m
兺m M m ⍀ m ␦ 共 ␻ ⫺⍀ m 兲 ,
共36兲
then Eq. 共34兲 reduces to
␲
k (1) ⫽ 关 1⫹n 共 ␻ 0 兲兴 J (1) 共 ␻ 0 兲 .
ប
共37兲
The general relation between state-to-state transition rates
and force correlation functions has been derived by Bader
and Berne,1 and is given by
k (1) ⫽
f being a force, and the subscript qm denoting quantum mechanical. As mentioned earlier, it is often only the classical
⬘ ( ␻ 0 ), and not its quantum counterpart
friction coefficient ˜␨ cl
˜␨ qm
⬘ ( ␻ 0 ), that is known 共from molecular dynamics simulations, for example兲. A correction factor must therefore be
⬘ ( ␻ 0 ) may be
applied to bring the two into correspondence. ˜␨ cl
derived from the limit ប→0 of Eq. 共38兲 关together with Eq.
共37兲 defining k (1) 兴, which leads to
2 兩 S IN 兩 2
˜␨ ⬘ 共 ␻ 兲 ,
␤ ប 2 关 1⫹e⫺ ␤ ប ␻ 0 兴 qm 0
˜␨ qm
⬘ 共 ␻0兲⫽
1
Re
2k B T
冕
⬁
0
⫽
dte⫺i ␻ 0 t 具 关 f 共 t 兲 , f 共 0 兲兴 ⫹ 典 qm , 共39兲
共40兲
␲
J (1) 共 ␻ 0 兲 ,
␻ 0 兩 S IN 共 0 兲 兩 2
共41兲
where S IN (0) is the leading order ប→0 limit of the matrix
element S IN . From Eqs. 共37兲, 共38兲 and 共41兲, the relation
between the quantum and classical friction coefficients is
seen to be
˜␨ qm
⬘ 共 ␻0兲⫽
冉 冊
兩 S IN 共 0 兲 兩 2 ប ␻ 0
ប␻0
˜␨ ⬘ 共 ␻ 兲 .
coth
2
兩 S IN 兩
2k B T
2k B T cl 0
共42兲
When the system under consideration is a single harmonic
oscillator bilinearly coupled to a set of independent harmonic
oscillators representing the bath, S IN ⫽S IN (0), and Eq. 共42兲
reduces to the well-known relation between the quantum and
classical friction coefficients derived by Bader and Berne.1
For a single phonon process, therefore, one may define a
quantum correction factor Q 1 as the ratio of the quantum to
the classical frequency-dependent friction coefficients. 共This
definition leads to somewhat more compact expressions than
one defined alternatively in terms of state-to-state transition
rates or in terms of the overall population relaxation rate.兲
Thus,
共38兲
where S IN ⬅ 具 ␺ N* (x 1 ,x 2 ) 关 A (1) x 1 ⫹B (1) x 2 兴 ␺ I (x 1 ,x 2 ) 典 , and
⬘ ( ␻ 0 ), in the notation of Bader and Berne, is the
where ˜␨ qm
real part of the Fourier transform of a symmetrized quantum
mechanical force correlation function:
␲␤ប
(1)
共␻0兲
2 coth共 ␤ ប ␻ 0 /2 兲 J
2
ប→0 兩 S IN 兩
˜␨ cl
⬘ 共 ␻ 0 兲 ⫽ lim
2
␧m
兺m M m ⍀ m 关共 1⫹n m 兲 ␦ 共 ␻ 0 ⫺⍀ m 兲
共33兲
Q 1⫽
冉 冊
兩 S IN 共 0 兲 兩 2 ប ␻ 0
ប␻0
.
coth
2
兩 S IN 兩
2k B T
2k B T
共43兲
Correction factors for multiphonon processes are derived in
much the same way. For instance, for systems with quadratic
coupling between solute and solvent 共corresponding to a
two-phonon process兲, the expression for the rate k (2) , in the
absence of phonon absorption, is from Eq. 共32兲 given by
k (2) ⫽ ␲
␥2
so that eventually
mn
共 1⫹n m 兲共 1⫹n n 兲
兺
M
M
m,n
m n⍀ m⍀ n
⫻ ␦ 共 ␻ 0 ⫺⍀ m ⫺⍀ n 兲 .
共44兲
˜␨ qm
⬘ 共 ␻0兲⫽
13
For reasons explained later, Egorov and Skinner have argued that in solid-state multiphonon relaxation, the frequency dependence of the thermal occupation factors is
much less than that of the degeneracy factor associated with
2
␥ mn
, and that it is therefore a reasonable approximation to
take such factors outside the summation sign and to evaluate
them at half the frequency of the transition ␻ 0 . Adopting this
approximation here 共essentially ad hoc兲, and introducing a
new spectral density function J (2) ( ␻ ) defined by
J (2) 共 ␻ 兲 ⫽
兺
m,n
2
␥ mn
␦ 共 ␻ ⫺⍀ m ⫺⍀ n 兲 ,
M mM n⍀ m⍀ n
共45兲
共46兲
⬘ ( ␻ 0 )⫽ ␤ ប 2 关 1⫹exp(⫺␤ប␻0)兴k(2)/2兩 S IN 兩 2 ,
Hence, using ˜␨ qm
we find
˜␨ qm
⬘ 共 ␻0兲⫽
␲␤ប2
关 coth2 共 ␤ ប ␻ 0 /4兲 ⫹1 兴 J (2) 共 ␻ 0 兲 ,
4 兩 S IN 兩 2
共47兲
the classical limit of which is
˜␨ ⬘cl共 ␻ 0 兲 ⫽
and therefore
˜␨ qm
⬘ 共 ␻0兲⫽
4␲
J
␤ ␻ 20 兩 S IN 共 0 兲 兩 2
(2)
共 ␻0兲,
共48兲
冉 冊冋 冉 冊 册
兩 S IN 共 0 兲 兩 2 ប ␻ 0
兩 S IN 兩 2 4k B T
2
coth2
ប␻0
⬘ 共 ␻0兲.
⫹1 ˜␨ cl
4k B T
共49兲
In this two-phonon process, therefore, the quantum correction factor, Q 2 , may be identified as
Q 2⫽
冉 冊冋 冉 冊 册
兩 S IN 共 0 兲 兩 2 ប ␻ 0
兩 S IN 兩 2 4k B T
2
coth2
ប␻0
⫹1 .
4k B T
共50兲
For the case of cubic coupling between solute and solvent,
and under similar approximations,
k
(3)
2
␴ lmn
3␲ប
⫽
共 1⫹n l 兲共 1⫹n m 兲
2 l,m,n M l M m M n ⍀ l ⍀ m ⍀ n
兺
⫻ 共 1⫹n n 兲 ␦ 共 ␻ 0 ⫺⍀ l ⫺⍀ m ⫺⍀ n 兲
⬇
2
␴ lmn
3␲ប
关 1⫹n 共 ␻ 0 /3兲兴 3
2
l,m,n M l M m M n ⍀ l ⍀ m ⍀ n
兺
⫻ ␦ 共 ␻ 0 ⫺⍀ l ⫺⍀ m ⫺⍀ n 兲
⬅
3␲ប
关 1⫹n 共 ␻ 0 /3兲兴 3 J (3) 共 ␻ 0 兲 .
2
共51兲
⬘ ( ␻ 0 )⫽ ␤ ប 2 关 1⫹exp(⫺␤ប␻0)兴k(3)/2兩 S IN 兩 2 , we obtain
With ˜␨ qm
˜␨ qm
⬘ 共 ␻0兲⫽
3␲␤ប3
关 coth3 共 ␤ ប ␻ 0 /6兲
16兩 S IN 兩 2
⫹3 coth共 ␤ ប ␻ 0 /6兲兴 J (3) 共 ␻ 0 兲 ,
⫹3 coth
3
coth3
ប␻0
6k B T
ប␻0
˜␨ cl
⬘ 共 ␻0兲.
6k B T
共53兲
Thus, the three-phonon quantum correction factor Q 3 is
given by
Q 3⫽
冉 冊冋 冉 冊
兩 S IN 共 0 兲 兩 2 ប ␻ 0
兩 S IN 兩 2 6k B T
3
coth3
冉 冊册
ប␻0
ប␻0
⫹3 coth
6k B T
6k B T
.
共54兲
Similar results for still higher order processes may be obtained in the same way.
V. DISCUSSION
it follows that
k (2) ⬇ ␲ 关 1⫹n 共 ␻ 0 /2兲兴 2 J (2) 共 ␻ 0 兲 .
冉 冊冋 冉 冊
冉 冊册
兩 S IN 共 0 兲 兩 2 ប ␻ 0
兩 S IN 兩 2 6k B T
共52兲
Equations 共50兲 and 共54兲 are the sought for qcf’s for quadratic and cubic phonon excitation processes. Expressions
for higher order processes may be obtained similarly. The
correction factors are, of course, approximate, and are derived by making two key assumptions: 共1兲 that thermal occupation factors, by virtue of being much less frequency dependent than the bath spectral density, may be evaluated at a
single frequency, and 共2兲 that this frequency may be determined by dividing the energy deposited into the bath equally
among the phonons that are excited. The second of these
assumptions ensures that the number of phonons excited during the course of relaxation is as small as possible; such an
outcome is generally more likely than one where the number
of excited phonons 共the order of the process兲 is larger. These
same asumptions have been used with some success in the
analysis of multiphonon relaxation in solids,13 where they
appear to be at least partly justifiable. Their application here
to liquid state dynamics is perhaps not as well motivated, but
there are few other approximations that are both physically
meaningful and analytically tractable. The simplest route to
the characterization of multiphonon relaxation rates is
through their temperature dependence, so experiments that
measure relaxation as a function of temperature may provide
direct tests of these approximations.
Other quantum correction factors for multiphonon relaxation are known. For instance, Egorov and Berne3 have investigated the vibrational dynamics of an effectively harmonic model of solute and solvent in which the two
components have identical normal modes and their mutual
interactions are either exponential or power law in the bath
coordinates. Their approximate results are found to be in
good agreement with exact numerical results obtained by
simulation over a fairly wide range of external conditions,
provided the spectral density of the bath is assumed to have
a particular form 共super-ohmic with Gaussian cut-off in the
exponential case, and ohmic with Gaussian cut-off in the
power law case兲. From their results for the exponential coupling case, a quantum correction factor Q EB defined by the
⬘ ( ␻ 0 )⫽Q EB˜␨ ⬘cl ( ␻ 0 ) can be obtained as
relation ˜␨ qm
Q EB⫽
冉 冊
␾ cl
exp共 C qm 共 0 兲兲
cosh共 ␤ ប ␻ 0 /2兲
exp共 C cl 共 0 兲兲
␾ qm
␻ 0 / ␻ ph
,
共55兲
where C qm (0),C cl (0), ␾ qm and ␾ cl are various parameters
that characterize the spectral properties of the bath. An analogous but more complicated expression for the correction factor can be obtained for the power law case. In comparisons
with exact numerical data, the Egorov–Berne results seem to
perform somewhat better than earlier suggested correction
factors.4 – 6 But one of these earlier expressions 共that of
Schofield4兲 is notable for being both simple and accurate; it
is given by
Q S ⫽cosh共 ប ␻ 0 /2k B T 兲 .
␤ 2ប 2␻ 0␻ D
coth共 ␤ ប ␻ D /2兲 ,
4 tanh共 ␤ ប ␻ 0 /2兲
冉
N
␳ B 共 R ⬘ ,Q ⬘ 兲 ⫽
兿
k⫽1
ACKNOWLEDGMENT
The author is grateful for a number of stimulating discussions with Professor Michael Fayer, Stanford University,
and Professor Jim Skinner, University of Wisconsin, during
the preparation of an early draft of this article.
APPENDIX A: EVALUATION OF THE INFLUENCE
FUNCTIONAL
Although F 关 x 1 ,y 1 ;x 2 ,y 2 兴 as defined by Eq. 共16兲 is not
known in general, it can be determined exactly when the
coupling between system and bath is linear 关corresponding to
再
冊
1/2
M k⍀ k
关共 R k⬘ 2 ⫹Q ⬘k 2 兲
2បsinh共 ␤ ប⍀ k 兲
冎
⫻cosh共 ␤ ប⍀ k 兲 ⫺2R k⬘ Q k⬘ 兴 .
共A1兲
Here ␤ ⫽1/k B T, and a product over the Cartesian components of the vectors is understood. The fact the influence
functional is unity when there are no interactions between
system and bath 共i.e., when S I ⫽0兲 suggests treating the nonlinearities in S I 共with n⭓2兲 as a perturbation around the n
⫽1 results. With ␳ B (R ⬘ ,Q ⬘ ) given by Eq. 共A1兲, a perturbation expansion in the nonlinearities then takes the form of a
cumulant expansion:12,15
冋
共57兲
where ␻ D is a Debye cut-off frequency. Direct comparisons
of the above results with those presented in this article are
complicated by the different forms for the solute–solvent
coupling and by different sets of approximations.
Apart from its application to the determination of qcf’s,
the foregoing work is partly intended to lay the foundations
for a more extended nonperturbative treatment of vibrational
relaxation 共and eventually of vibrational dephasing兲. The
present calculations 关principally Eqs. 共32兲 and 共33兲兴, which
employ first order perturbation theory, corroborate the treatment of multiphonon effects discussed by Kenkre et al.22
using methods analogous to those used in solid state phonon
scattering calculations. They are somewhat more general
than these calculations, however, in that the coupled high
frequency modes on the solute that are often suspected of
being implicated in relaxation are treated explicitly by modeling them as coupled harmonic oscillators.
An important feature of the present functional integral
approach is that quantum mechanical time correlation functions of fluctuating forces are actually contained within the
influence functional itself, as briefly described in Appendix
D. This is a reflection of a deeper structure in the theory that
can be manifested in the form of generalized fluctuationdissipation theorems. This aspect of the formalism has been
discussed at some length in Ref. 11.
M k⍀ k
2 ␲ បsinh共 ␤ ប⍀ k 兲
⫻exp ⫺
共56兲
The same factor appears in Eq. 共55兲, so the success of Q S in
reproducing numerical trends suggests that the other two
contributions to the Egorov-Berne formula, Q EB , effectively
cancel. Rostkier-Edelstein et al.9 have also derived approximate correction factors for multiphonon processes. Defining
⬘ ( ␻ 0 )⫽Q REGN˜␨ ⬘cl ( ␻ 0 ), it can be
these factors through ˜␨ qm
shown that for a two-phonon process, for example,
Q REGN⫽
n⫽1 in Eq. 共5兲兴 and when ␳ B (R ⬘ ,Q ⬘ ) describes the density
matrix of a system of N independent oscillators,
F 关 x 1 ,y 1 ;x 2 ,y 2 兴 ⫽exp
册
1
i
兵 具 S X2 典 0 ⫺ 具 S X 典 20 其 ⫹¯ ,
具S 典 ⫺
ប X 0 2ប 2
共A2兲
where S X ⬅S I 关 x 1 ,x 2 ,R 兴 ⫺S I 关 y 1 ,y 2 ,Q 兴 , and the average
具 (¯) 典 0 is defined by
具 共 ¯ 兲典 0⫽
冕
dR ⬘ dQ ⬘ dR
冕
R
R⬘
D关 R 兴
冕
R
Q⬘
D关 Q 兴
⫻eiS 0 /ប 共 ¯ 兲 ␳ B 共 R ⬘ ,Q ⬘ 兲 ,
共A3兲
with S 0 ⬅S B 关 R 兴 ⫺S B 关 Q 兴 . To evaluate the averages in Eq.
共A2兲, which are averages of various powers of the bath normal modes 兵 R 其 and 兵 Q 其 , a functional F 关 U,V 兴 of two fictitious fields U( ␶ ) and V( ␶ ) is introduced. This functional is
defined by
F 关 U,V 兴 ⫽
再兿 冕
k
dR k⬘ dQ ⬘k dR k
冕
R k⬘
i
ប
冕
兺k
t
0
D关 R k 兴
冋 兺冕
册
⫻eiS 0 /ប ␳ B 共 R ⬘ ,Q ⬘ 兲 exp
⫺
Rk
i
ប
k
d ␶ V k共 ␶ 兲 Q k共 ␶ 兲 ,
冕
Rk
Q k⬘
t
0
D关 Q k 兴
冎
d ␶ U k共 ␶ 兲 R k共 ␶ 兲
共A4兲
which is Eq. 共16兲 with n⫽1, B (1) ⫽0, A (1) c k x 1 ( ␶ )
⫽⫺U k ( ␶ ), and A (1) c k y 1 ( ␶ )⫽⫺V k ( ␶ ). The results of Ref.
11 thus allow F 关 U,V 兴 to be determined at once:
再
F 关 U,V 兴 ⫽exp ⫺
兺k d k 冕0 d ␶ 冕0 d ␶ ⬘ 关 U k共 ␶ 兲 ⫺V k共 ␶ 兲兴
t
t
冎
⫻关 ␣ k* 共 ␶ ⫺ ␶ ⬘ 兲 U k 共 ␶ ⬘ 兲 ⫺ ␣ k 共 ␶ ⫺ ␶ ⬘ 兲 V k 共 ␶ ⬘ 兲兴 ,
共A5兲
where
d k⫽
1
4M k ⍀ k ប
共A6兲
second cumulant of S X is considered further. Evaluating this
quantity by the methods described earlier, and substituting
the result into Eq. 共A2兲, one finds that
and
␣ k共 ␶ ⫺ ␶ ⬘ 兲 ⫽
1
ei⍀ k ( ␶ ⫺ ␶ ⬘ )
1⫺e⫺ ␤ ប⍀ k
(2)
F 关 x 1 ,y 1 ;x 2 ,y 2 兴 ⫽F (2)
1 关 x 1 ,y 1 兴 F 2 关 x 2 ,y 2 兴
e⫺ ␤ ប⍀ k ⫺i⍀ ( ␶ ⫺ ␶ )
⬘
⫹
e k
1⫺e⫺ ␤ ប⍀ k
共A7兲
with ␣ *
k ( ␶ ⫺ ␶ ⬘ ) the complex conjugate of ␣ k ( ␶ ⫺ ␶ ⬘ ). Two
other equivalent forms of this function will be used in what
follows:
␣ k 共 ␶ ⫺ ␶ ⬘ 兲 ⫽z k cos共 ⍀ k 共 ␶ ⫺ ␶ ⬘ 兲兲 ⫹i sin共 ⍀ k 共 ␶ ⫺ ␶ ⬘ 兲兲 ,
(2)
⫻F 12
关 x 1 ,y 1 ;x 2 ,y 2 兴 ,
where
冋
2
(2) 2
F (2)
兲
1 关 x 1 ,y 1 兴 ⫽exp ⫺4ប 共 A
共A8兲
⫻
where z k ⫽coth(␤ប⍀k/2), and
␣ k 共 ␶ ⫺ ␶ ⬘ 兲 ⫽ 共 1⫹n k 兲 ei⍀ k ( ␶ ⫺ ␶ ⬘ ) ⫹n k e⫺i⍀ k ( ␶ ⫺ ␶ ⬘ ) ,
t
0
0
d ␶ d ␶ ⬘ 兵 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴
⫺i 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴
册
(2)
⫻I mn
共 ␶ , ␶ ⬘ 兲关 x 1 共 ␶ ⬘ 兲 ⫹y 1 共 ␶ ⬘ 兲兴 其 ,
␦ 4F
␦ U i共 ␶ 兲 ␦ U j共 ␶ 兲 ␦ V k共 ␶ ⬘ 兲 ␦ V l共 ␶ ⬘ 兲
and
冏
.
冋
⫽exp ⫺8ប 2 A (2) B (2)
共A10兲
U,V⫽0
冕冕
t
t
0
0
2 2
d md nc m
cn
兺
m,n
The functional differentiation of Eq. 共A5兲 according to the
prescription above leads to
⫻
具 R i共 ␶ 兲 R j共 ␶ 兲 Q k共 ␶ ⬘ 兲 R l共 ␶ ⬘ 兲 典 0
(2)
⫻R mn
共 ␶ , ␶ ⬘ 兲关 x 2 共 ␶ ⬘ 兲 ⫺y 2 共 ␶ ⬘ 兲兴
⫽ប 4 关 4d k d l ␣ k 共 ␶ ⫺ ␶ ⬘ 兲 ␣ l 共 ␶ ⫺ ␶ ⬘ 兲共 ␦ il ␦ jk ⫹ ␦ ik ␦ jl 兲
⫹4 ␦ i j ␦ kl d j d l ␣ j 共 0 兲 ␣ l 共 0 兲兴 .
d ␶ d ␶ ⬘ 兵 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴
(2)
⫺i 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴 I mn
共 ␶ , ␶ ⬘ 兲关 x 2 共 ␶ ⬘ 兲 ⫹y 2 共 ␶ ⬘ 兲兴 其
共A11兲
Other averages of the bath coordinates are obtained similarly.
共Averages involving combinations of powers of R k and Q k
that have an overall odd exponent are 0 by symmetry.兲 In this
way, the quantities 具 S X 典 0 and 具 S X2 典 0 are calculated for different values of n. The results are then substituted into Eq. 共A2兲
to produce expressions for F 关 x 1 ,y 1 ;x 2 ,y 2 兴 ; these are then
used to determine probabilities of transition between different states of the coupled oscillator pair.
APPENDIX B: QUADRATIC NONLINEARITIES IN THE
SOLVENT MODES
When n⫽2, corresponding to a second power of the solvent collective coordinate in the interaction Hamiltonian, it
can be shown that 具 S X 典 0 , the mean of the interaction action,
is given by
n
冕
t
0
⫹B (2) 关 x 2 共 ␶ 兲 ⫺y 2 共 ␶ 兲兴 其 .
d ␶ 兵 A (2) 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴
共B1兲
Because its dependence on the oscillator coordinates is linear, 具 S X 典 0 merely renormalizes the potential,15 so its contribution to the influence functional is ignored, and only the
册
共B4兲
where
(2)
R mn
共 ␶ , ␶ ⬘ 兲 ⫽ 21 共 z m z n ⫹1 兲 cos共 ␪ m ⫹ ␪ n 兲
⫹ 12 共 z m z n ⫺1 兲 cos共 ␪ m ⫺ ␪ n 兲
共B5兲
and
(2)
I mn
共 ␶ , ␶ ⬘ 兲 ⫽ 21 共 z m ⫹z n 兲 sin共 ␪ m ⫹ ␪ n 兲
⫹ 12 共 z n ⫺z m 兲 sin共 ␪ m ⫺ ␪ n 兲
具 S X 典 0 ⫽⫺2ប 2 兺 d n c 2n ␣ n 共 0 兲
共B3兲
(2)
F 12
关 x 1 ,y 1 ;x 2 ,y 2 兴
具 R i共 ␶ 兲 R j共 ␶ 兲 Q k共 ␶ ⬘ 兲 R l共 ␶ ⬘ 兲 典 0
⫽ប 4
t
2 2
d md nc m
cn
兺
m,n
(2)
⫻R mn
共 ␶ , ␶ ⬘ 兲关 x 1 共 ␶ ⬘ 兲 ⫺y 1 共 ␶ ⬘ 兲兴
共A9兲
where n k ⬅1/关 exp(␤ប⍀k)⫺1兴, the mean thermal occupation
number.
Averages of the bath coordinates may now be obtained
as functional derivatives of F 关 U,V 兴 with respect to U or V.
By way of illustration, one sees from Eq. 共A4兲 that
冕冕
共B2兲
共B6兲
with ␪ i ⫽⍀ i ( ␶ ⫺ ␶ ⬘ ) and z i defined after Eq. 共A8兲. The func(2)
tion F (2)
2 关 x 2 ,y 2 兴 is obtained from F 1 关 x 1 ,y 1 兴 by replacing
(2)
(2)
A by B , x 1 by x 2 and y 1 by y 2 .
APPENDIX C: CUBIC NONLINEARITIES IN THE
SOLVENT MODES
Here n⫽3, corresponding to a third power of the solvent
collective coordinate in the interaction Hamiltonian. For this
case, the influence functional is found to be
(3)
F 关 x 1 ,y 1 ;x 2 ,y 2 兴 ⫽F (3)
1 关 x 1 ,y 1 兴 F 2 关 x 2 ,y 2 兴
(3)
⫻F 12
关 x 1 ,y 1 ;x 2 ,y 2 兴 ,
where
共C1兲
冋
4
(3) 2
F (3)
兲
1 关 x 1 ,y 1 兴 ⫽exp ⫺24ប 共 A
兺 d l d m d n c 2l c m2 c 2n 冕0 冕0 d ␶ d ␶ ⬘ 兵 关 x 1共 ␶ 兲 ⫺y 1共 ␶ 兲兴
l,m,n
t
t
册
(3)
(3)
⫻R lmn
共 ␶ , ␶ ⬘ 兲关 x 1 共 ␶ ⬘ 兲 ⫺y 1 共 ␶ ⬘ 兲兴 ⫺i 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴 I lmn
共 ␶ , ␶ ⬘ 兲关 x 1 共 ␶ ⬘ 兲 ⫹y 1 共 ␶ ⬘ 兲兴 其 ,
and
冋
(3)
F 12
关 x 1 ,y 1 ;x 2 ,y 2 兴 ⫽exp ⫺48ប 4 A (3) B (3)
共C2兲
兺 d l d m d n c 2l c m2 c 2n 冕0 冕0 d ␶ d ␶ ⬘ 兵 关 x 1共 ␶ 兲 ⫺y 1共 ␶ 兲兴
l,m,n
t
t
册
(3)
(3)
⫻R lmn
共 ␶ , ␶ ⬘ 兲关 x 2 共 ␶ ⬘ 兲 ⫺y 2 共 ␶ ⬘ 兲兴 ⫺i 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴 I lmn
共 ␶ , ␶ ⬘ 兲关 x 2 共 ␶ ⬘ 兲 ⫹y 2 共 ␶ ⬘ 兲兴 其 ,
共C3兲
where
(3)
R lmn
共 ␶ , ␶ ⬘ 兲 ⫽ 41 关共 z l z m z n ⫹z l ⫹z m ⫹z n 兲 cos共 ␪ l ⫹ ␪ m ⫹ ␪ n 兲 ⫹ 共 z l z m z n ⫺z l ⫺z m ⫹z n 兲 cos共 ␪ l ⫹ ␪ m ⫺ ␪ n 兲
⫹ 共 z l z m z n ⫺z l ⫹z m ⫺z n 兲 cos共 ␪ l ⫺ ␪ m ⫹ ␪ n 兲 ⫹ 共 z l z m z n ⫹z l ⫺z m ⫺z n 兲 cos共 ␪ l ⫺ ␪ m ⫺ ␪ n 兲兴 ⫹ 23 z l z m z n cos ␪ n
共C4兲
and
(3)
I lmn
共 ␶ , ␶ ⬘ 兲 ⫽ 41 关共 z l z m ⫹z l z n ⫹z m z n ⫹1 兲 sin共 ␪ l ⫹ ␪ m ⫹ ␪ n 兲 ⫺ 共 z l z m ⫺z l z n ⫺z m z n ⫹1 兲 sin共 ␪ l ⫹ ␪ m ⫺ ␪ n 兲
⫹ 共 z l z m ⫺z l z n ⫹z m z n ⫺1 兲 sin共 ␪ l ⫺ ␪ m ⫹ ␪ n 兲 ⫺ 共 z l z m ⫹z l z n ⫺z m z n ⫺1 兲 sin共 ␪ l ⫺ ␪ m ⫺ ␪ n 兲兴 ⫹ 23 z l z m sin ␪ n
共C5兲
with ␪ m ⫽⍀ m ( ␶ ⫺ ␶ ⬘ ) as before.
APPENDIX D: FORCE CORRELATION FUNCTIONS
FROM THE INFLUENCE FUNCTIONAL
F̄ ⬘ 关 x 1 ,x 2 ;y 1 ,y 2 兴
冋
⫽exp ⫺
Expressions for the friction coefficients may be obtained
directly from the expressions for the influence functionals
themselves. To see how this comes about, notice that had the
interactions between the displacements of the solute oscillators and the medium been described in terms of a fluctuating
force f rather than in terms of a collection of N other independent oscillators, Eq. 共5兲 would have been given by
H XB ⫽ax 1 f ⫹bx 2 f ,
共D1兲
where a and b are constants, and the resulting influence
functional F ⬘ 关 x 1 ,y 1 ;x 2 ,y 2 兴 would have assumed the form
冋冕
F ⬘ 关 x 1 ,y 1 ;x 2 ,y 2 兴 ⫽exp
i
ប
t
0
d ␶ 兵 a f 共 ␶ 兲关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴
册
⫹b f 共 ␶ 兲关 x 2 共 ␶ 兲 ⫺y 2 共 ␶ 兲兴 其 .
共D2兲
Further, if the fluctuations in f were governed by a Guassian
probability distribution, such that
具 f 共 ␶ 兲 f 共 ␶ ⬘ 兲典⫽ ␾共 ␶ ⫺ ␶ ⬘ 兲,
共D3兲
冕冕
t
t
0
0
d ␶ d ␶ ⬘ 兵 a 2 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴 ␾ 共 ␶ ⫺ ␶ ⬘ 兲
⫻ 关 x 1 共 ␶ ⬘ 兲 ⫺y 1 共 ␶ ⬘ 兲兴 ⫹b 2 关 x 2 共 ␶ 兲 ⫺y 2 共 ␶ 兲兴 ␾ 共 ␶ ⫺ ␶ ⬘ 兲
⫻ 关 x 2 共 ␶ ⬘ 兲 ⫺y 2 共 ␶ ⬘ 兲兴 ⫹2ab 关 x 1 共 ␶ 兲 ⫺y 1 共 ␶ 兲兴
册
⫻ ␾ 共 ␶ ⫺ ␶ ⬘ 兲关 x 2 共 ␶ ⬘ 兲 ⫺y 2 共 ␶ ⬘ 兲兴 其 .
共D4兲
Aside from the absence of purely imaginary contributions,
this functional is very similar in structure to the functionals
that describe quadratic and cubic nonlinearities in the interactions between solute and solvent 关Eqs. 共B4兲 and 共C3兲兴.
Caldeira and Leggett21 argued on the basis of the analogous
correspondence that obtained for the case of linear solute–
solvent interactions that the real part of the the modedependent portion of the influence functional could be identified with a symmetrized, quantum mechanical time
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from an inspection of the first terms of Eqs. 共B5兲 and 共C4兲
(2)
(3)
and R lmn
兲, such
共i.e., the first terms of the real portions R mn
an identification leads directly to Eqs. 共47兲 and 共52兲 once the
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共D1兲 are suitably adjusted.
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␾ being some known function of its arguments, the force
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