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II. GENERAL THEORY OF VIBRATIONAL ENERGY RELAXATION

# II. GENERAL THEORY OF VIBRATIONAL ENERGY RELAXATION

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678

Skinner et al.

where kn!m is the time-independent rate constant for making a bathinduced transition from oscillator state jni to state jmi. While a detailed

discussion of the validity of this equation is not warranted here, sufﬁce it

to say that if (1) the oscillator-bath coupling is sufﬁciently weak (which

means that one can use lowest-order perturbation theory to calculate the

rate constants and that the “bath correlation time” is much shorter than the

inverse of these rate constants), and (2) the equations of motion for the diagonal and off-diagonal density matrix elements are uncoupled (either because

of special symmetry properties of the problem or because the off-diagonal

elements decay to zero sufﬁciently rapidly), then the master equation is

valid. We will simply assume this to be the case in what follows.

The rate constants in the master equation can be derived from Fermi’s

golden rule, with the result that (14)

kn!m D

1

h¯ 2

1

1

dt eiωnm t hVnm t Vmn 0 i

4

where ωnm D En Em /¯h, hÐ Ð Ði D Trb [e ˇHb ... ]/Trb [e ˇHb ], Vnm D

hnjVjmi, and for any bath operator O, O(t) D eiHb t/¯h Oe iHb t/¯h . It is easy

to show that these rate constants so deﬁned satisfy the all-important

(without it the system would not reach equilibrium) property of detailed

balance, such that (15)

km!n D e

ˇ¯hωnm

kn!m

5

In general the relaxation to equilibrium of E(t) is nonexponential,

since the rate matrix in the master equation has an inﬁnite number of (in

principle) nondegenerate eigenvalues if there are an inﬁnite number of states

jni. There are, however, two instances where the relaxation is approximately

exponential. In the ﬁrst instance one assumes that the initial nonequilibrium

state has appreciable population only in the ﬁrst two oscillator eigenstates,

and further that k1!0 × k1!m and k0!1 × k0!m for m ½ 2. If one neglects

terms involving these small rate constants, the master equation reduces to

a pair of coupled rate equations for a two-level system:

P˙ 0 t D k0!1 P0 t C k1!0 P1 t

k1!0 P1 t

P˙ 1 t D k0!1 P0 t

6

7

It is well known that for this pair of equations both populations P0 (t) and

P1 (t) relax to equilibrium exponentially, with the same relaxation time, T1 ,

where 1/T1 D k0!1 C k1!0 . It follows then that the nonequilibrium energy

Copyright © 2001 by Taylor & Francis Group, LLC

VER in Liquids and Supercritical Fluids

679

obeys

E˙ t D

1

[E t

T1

Eeq ]

8

that is, it also relaxes to equilibrium with the same time constant. Note that

using detailed balance one can write

1

D [1 C e

T1

ˇ¯hω10

]k1!0

9

Since the assumptions leading to the truncation of the state space to two

states also typically imply that e ˇ¯hω10 − 1, to an excellent approximation

one can then write 1/T1 ' k1!0 .

The second instance leading to exponential decay of E t follows from

a more detailed speciﬁcation of the model. The oscillator-bath coupling term

V is a function of oscillator and bath coordinates and can be expanded in

powers of q. Matrix elements between oscillator states jni and jmi of the

zeroth-order term in this expansion will vanish, and so this term does not

contribute to the rate constants. The leading order term, then, is ﬁrst order

in q. Deﬁning the bath operator F by F D ∂V/∂q jqD0 , the rate constants

are approximately given by

kn!m D

1

jqnm j2

h¯ 2

1

dt eiωnm t hF t F 0 i

10

We next assume that the oscillator is harmonic, so that En D h¯ ω0 n C 12 . It

p

is also well-known that qnm / υm,nš1 , and that qn,n 1 D n¯h/2 ω0 , where

is the oscillator mass. Therefore, all rate constants vanish except between

neighboring pairs of levels, and kn!n 1 D nk0 , where k0 is given by

k0 D

1

2 h¯ ω0

1

1

dt eiω0 t hF t F 0 i

11

The rate constants kn!nC1 are determined by detailed balance. In this special

case one can show (16), perhaps surprisingly, that E t relaxes to equilibrium exponentially, with the single relaxation time T1 given by

1

D [1

T1

e

ˇ¯hω0

]k0

12

which is distinctly different from Equation (9). Of course, if the oscillator

is harmonic and the oscillator-bath coupling is expanded to ﬁrst order in q

(in which case ω10 D ω0 and k1!0 D k0 ), then these two expressions must

agree in the low-temperature ˇ¯hω0 × 1 limit, as is the case.

Copyright © 2001 by Taylor & Francis Group, LLC

680

Skinner et al.

It is now interesting to consider the classical h¯ ! 0 limits of these two

expressions. Actually, in the ﬁrst instance it is inappropriate to take this

limit, since just the opposite limit ˇ¯hω10 × 1 is invoked in truncating

the state space to two levels. However, in the second instance, one can

smoothly take the classical limit, and one ﬁnds that in this case T1 is given

by the usual classical Landau-Teller result:

1

1

1

D

T1

kT

dt cos ω0 t hF t F 0 icl

13

0

where the subscript cl on the angular brackets indicates that this is now

a classical time correlation function. We have used the fact that in the

classical limit any time autocorrelation function is real and even.

There are also situations when one is not in the classical limit, and

so Equation (13) would not seem applicable, and instead one would like to

approximate one of the quantum mechanical expressions for T1 by relating

the relevant quantum time-correlation function to its classical analog. For

the sake of deﬁniteness, let us consider the case where the oscillator is

harmonic and the oscillator-bath coupling is linear in q, as discussed above.

In this case k1!0 can be written as

ˆ ω0

G

2 h¯ ω0

14

dt eiωt G t

15

k1!0 D k0 D

where

ˆ ω D

G

1

1

ˆ ω is the Fourier transform of the

with G(t) D hF(t)F 0 i. That is, G

quantum force-force time-correlation function. We (5,6) and others

(7,14,16–18) have discussed at some length various approximate schemes

ˆ ω to its classical analog

for relating G

ˆ cl ω D

G

1

1

dt eiωt Gcl t

16

with Gcl t D hF(t)F 0 icl . Here we list three such schemes:

1.

The “standard” scheme, where

ˆ ω D

G

2

1Ce

ˇ¯hω

ˆ cl ω

G

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17

VER in Liquids and Supercritical Fluids

2.

The “harmonic” scheme, where

ˆ ω D

G

3.

681

1

ˇ¯hω ˆ

Gcl ω

e ˇ¯hω

18

The Egelstaff scheme (7), where

ˆ ω D eˇ¯hω/2

G

1

1

dt eiωt Gcl

t2 C ˇ¯h/2

2

19

Each of these schemes satisﬁes the important property of detailed balance:

ˆ

G

ω De

ˇ¯hω

ˆ ω

G

20

The standard scheme has been the most popular one for the last 20 years

or so. The harmonic scheme gets its name because if the bath is harmonic

(more precisely, if F can be represented as a linear combination of harmonic

coordinates), then this scheme is exact (16,19). By comparing these schemes

to results from exactly solvable models we have concluded that the standard

scheme is really not very accurate at high frequencies and that the harmonic

and Egelstaff schemes are more promising (5,6). A fourth scheme satisfying

detailed balance,

ˆ ω D eˇ¯hω/4

G

ˇ¯hω

1 e ˇ¯hω

1/2

ˆ cl ω

G

21

may also have some merit (6).

Here is a fascinating result: if the harmonic scheme of Equation (18)

is applied to Equation (14), using Equation (12) for T1 recovers exactly

the classical Landau-Teller result, as recently shown by Bader and Berne

(16). Thus, if the oscillator and bath are both harmonic and are bilinearly

coupled, the exact quantum result and the exact classical result are identical!

This provides some justiﬁcation for using the purely classical result of

Equation (13) even in situations where one clearly is not in the classical

limit.

III. I2 IN LIQUID AND SUPERCRITICAL XENON

With the goal of describing some VER experiments on the solute iodine

in Xe solvent (1), in this section we specialize to the case of a diatomic

solute in an atomic solvent. In fact, we consider a simpliﬁed model where

the diatomic solute is replaced with a “breathing” sphere (2). We take the

Copyright © 2001 by Taylor & Francis Group, LLC

682

Skinner et al.

oscillator Hamiltonian for the one-dimensional breathing coordinate to be

harmonic:

Hq D

ω02 q2

2

p2

C

2

22

The bath consists of the translational motions of the solute and all the

solvent atoms. Since all potentials are spherically symmetrical, we write

Hb D T C

ri C

i

s

rij

23

i

where T is the translational kinetic energy of the solute molecule and the

solvent atoms, i and j label solvent atoms, ri is the distance between the

solute and solvent atom i, rij is the distance between solvent atoms i and

j, r is the solute-solvent pair potential, and s r is the solvent-solvent

pair potential. Assuming all interactions to be of the Lennard-Jones form,

we have

r D 4ε

s

r D 4εs

12

6

r

24

r

s

12

s

r

6

25

r

To obtain the oscillator-bath interaction term, we argue that the

solute’s instantaneous size depends linearly on the breathing coordinate

q multiplied by a dimensionless coefﬁcient ˛. The latter is treated as the

single adjustable parameter in the theory, which should on physical grounds

be less than but on the order of unity. This leads to (2)

V'

qF

26

with

FD

f ri

27

i

and

fr D

12ε˛

12

2

r

6

r

28

In the iodine experiments the solute is ﬁrst excited electronically and

then enters the ground electronic state with many quanta of vibrational

excitation (1). The subsequent vibrational relaxation therefore involves

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