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A. Vibrational Energy Relaxation and Vibrational Friction

A. Vibrational Energy Relaxation and Vibrational Friction

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Mechanisms of Vibrational Relaxation


friction appears on the right-hand side of Equation (1) says that it leads

to an effective force proportional to the mode velocity v, but opposing

it — much as one would expect from some sort of frictional drag. The

fact that this drag has a time delay, that the drag at time t results from a

velocity at an earlier time t0 [which makes Equation (1) a generalized rather

than an ordinary Langevin equation], might seem a bit of a complication,

but it too is eminently reasonable. One can think of any motion of the

solute mode, v, as perturbing the solvent away from its preferred parts of

phase space. The solvent, in its best LeChatelier fashion, reacts to restore

the status quo by evolving in such a way as to penalize any subsequent

motion of the mode — that is, it generates a frictional drag. However, in

any genuinely molecular picture, the effects of this solvent back-reaction

cannot be instantaneous; it has to have a time lag commensurate with the

time scales on which the solvent moves. Much of our study of vibrational

relaxation can therefore be interpreted as an investigation into just what

these time scales are.

This conceptual link between the solvent vibrational friction and

vibrational energy relaxation is actually mirrored by an important practical connection. Within the rather accurate Landau-Teller approximation,

(29,33,34), the rate of vibrational energy relaxation for a diatomic with

frequency ω0 and reduced mass is given by





ÁR ω0


where ÁR ω is the cosine transform of the vibrational friction


ÁR ω D

dt cos ωt Á t



In other words, the ability of the solvent to absorb a quantum of energy h¯ ω0

(or its classical equivalent) is determined quite literally by the ability of the

solvent to respond to the solute dynamics at a frequency ω D ω0 . One can

derive this relation quantum mechanically by assuming that the solvent’s

effect on the solute can be handled perturbatively within Fermi’s golden

rule (1), but it is actually more general than that. Perhaps it is worth pausing

to see how the same basic result appears in a purely classical context.

Quite generally we can imagine the Hamiltonian for our system as a

sum of Hu , a Hamiltonian for the solute vibration, Hv , a Hamiltonian for

the solvent, and Vc , the piece of the potential energy coupling the two:

H D Hu p, x C Hv p, q C Vc x, q

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Here x is the solute coordinate, the vector q denotes all the solvent coordinates, and p and p are the corresponding momenta. Taking V D Vu x C

Vv q C Vc x, q to be the total potential energy, we know that Hamilton’s

equations, the classical equations of motion, tell us that we can always find

x t by solving

dp/dt D ∂V/∂x

dx/dt D p


provided we can simultaneously solve a mole of analogous equations for

the solvent dynamics. However, if we define the vibrational energy of the

solute alone to be

Eu t D Hu p t , x t

D p2 /2

C Vu x


these equations also tell us that the rate of change of the vibrational energy

is given by

dEu t /dt D p/ dp/dt C ∂Vu /∂x dx/dt

D dx/dt ∂Vc /∂x

Thus, the total amount of energy transferred into the vibration between time

0 and time t is


Eu t D

dt0 Fext t0 v t0



the work done on our solute by the “external” (solvent) force Fext D

∂Vc /∂x (35).

Now suppose we specialize to a harmonic solute and make the approximation that the vibrational amplitude is small enough that the coupling to

the solvent can be taken to be linear in the vibrating coordinate:

Vu D

Vc D



ω0 2 x2

xFext q


Then Equations (6) become the equations of motion for a forced harmonic

oscillator, letting us solve exactly:


x t D x 0 t C 1/ ω0

dt0 sin ω0 t

t0 Fext q t0



with x t what the vibrating coordinate would be in the absence of any

solvent coupling. Hence, writing v 0 D dx 0 /dt, we find we can express the

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Mechanisms of Vibrational Relaxation


total energy transfer in terms of the time evolution of the external force.Ł


Eu t D

dt1 v 0 t1 Fext t1







dt2 cos ω0 t1

t2 Fext t1 Fext t2




Unfortunately, evaluating this formula exactly would still require that

we know the fully coupled solute-solvent dynamics because it calls for

Fext t D Fext q t , but since the solvent perturbs the solute vibration only

weakly, a perturbative treatment suffices (just as it does quantum mechan0

ically). To leading order, Fext t D Fext

t , what the solvent force would be

if the solute’s vibrational mode were held fixed. Thus, the average rate of

solute-solvent energy transfer in the steady state is

lim dhEu t i/dt D




dt cos ω0 t CFF t Á




0 Fext


CFF t D hFext



R ω0



proportional to the cosine transform of autocorrelation function for this

frozen-mode force (36). Note that in writing Equation (11) we made use of

the statistical independence of the solute and solvent degrees of freedom

in the absence of coupling and we took advantage of the invariance of

equilibrium behavior to anything but time intervals:



t i D hv 0 t i hFext

t iD0

hv 0 t Fext





t1 Fext

t2 i D hFext

0 Fext



t1 i

The frozen-mode force correlation function CFF t not only closely

resembles the vibrational friction [Equation (2)], it is often a rather accurate way of calculating it in practice (29,32). One reason for this fortunate circumstance is that in typical molecular vibrations the vibrational

frequency is so large that the solvent hardly sees the effects of the dynamics

on the forces (32). If we take this identification for granted, however,

CFF t D kB T Á t


and remember that the equilibrated solute vibrational energy will be kB T, we

reach our final destination: the rate constant for vibrational energy transfer is


A related expression for energy transfer appears when one studies the underdamped (energy diffusion) limit of chemical reaction dynamics in liquids. See,

for example, Reference 11.

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seen to be proportional to the value of the friction at the natural vibrational


lim hEu t i



dhEu t i/dt D lim d lnhEu t i/dt D



ÁR ω0

in perfect agreement with Equation (3), the Landau-Teller formula.Ł

B. The Instantaneous Vibrational Friction and the Instantaneous

Normal Modes of the Solvent

The problem before us now is not simply to evaluate the vibrational friction numerically for realistic examples of vibrational relaxation in liquids;

Equation (13) has already been applied in the literature to a wide variety of

interesting cases (33,34,37,38). What we are concerned with here is finding

out what the specific molecular events are that contribute to this vibrational


Having posed the problem in this fashion, we must deal with the

knotty question of what it is that would constitute a satisfactory answer.

The orientations, locations, and even identities of the participating solvent

molecules are constantly changing. So how can there be any kind of definitive mechanism to find? The short answer is that for a general liquid process

there is no such detailed mechanism, at least not one with any claim to

generality. Ultrafast processes, however, are a different story — and vibrational friction is, in fact, an ultrafast process.

Actually, this last statement might seem quite counterintuitive, especially for small solutes. The vibrational energy relaxation of diatomics and

triatomics, for example, can be quite slow when judged by the ps and

sub-ps time scales of intermolecular motion. Though ps relaxation times

have been seen (39,40), there are also some well-known examples of T1 s

in the µs and even ms ranges (5). Yet, the vibrational friction apparently

begins its work very quickly, even in these slowest of cases. As one can

see from the behavior of a typical vibrational friction (Fig. 1) (Y. Deng

and R. M. Stratt, unpublished), the solvent retains its memory of the solute

dynamics for only a short time, so the time lag between solute motion and


This derivation is largely meant to be a schematic way of helping us see how the

vibrational friction influences the rate of solute-solvent energy transfer. Notice,

however, that we never specified the actual initial conditions of an experiment

(in particular whether the solute was to be initially hot or cold with respect to

the surrounding solvent). Without such a specification we cannot predict the net

sign of the energy flow.

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Mechanisms of Vibrational Relaxation


Figure 1 Vibrational friction on a symmetrical linear triatomic molecule dissolved

in high-density supercritical Ar. The figure compares the differing frictions felt by

the symmetrical and asymmetrical stretching modes of the triatomic.

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