A. Vibrational Energy Relaxation and Vibrational Friction
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Mechanisms of Vibrational Relaxation
153
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
1
D
T1
1
ÁR ω0
(3)
where ÁR ω is the cosine transform of the vibrational friction
1
ÁR ω D
dt cos ωt Á t
(4)
0
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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(5)
154
Stratt
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
6
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
(7)
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
t
Eu t D
dt0 Fext t0 v t0
(8)
0
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
1
2
ω0 2 x2
xFext q
9
Then Equations (6) become the equations of motion for a forced harmonic
oscillator, letting us solve exactly:
t
x t D x 0 t C 1/ ω0
dt0 sin ω0 t
t0 Fext q t0
0
0
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
155
total energy transfer in terms of the time evolution of the external force.Ł
t
Eu t D
dt1 v 0 t1 Fext t1
0
C
t
1
t1
dt1
dt2 cos ω0 t1
t2 Fext t1 Fext t2
10
0
0
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 dhEu t i/dt D
t!1
1
1
dt cos ω0 t CFF t Á
0
0
0
0 Fext
ti
CFF t D hFext
1
CFF
R ω0
11
12
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:
0
0
t i D hv 0 t i hFext
t iD0
hv 0 t Fext
0
0
0
0
t1 Fext
t2 i D hFext
0 Fext
t2
hFext
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
(13)
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
Ł
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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156
Stratt
seen to be proportional to the value of the friction at the natural vibrational
frequency:
lim hEu t i
t!1
1
dhEu t i/dt D lim d lnhEu t i/dt D
t!1
1
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
friction.
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
Ł
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
157
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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