B. The Instantaneous Vibrational Friction and the Instantaneous Normal Modes of the Solvent
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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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158
Stratt
the solvent response is remarkably brief. This brevity is in itself a clue that
the critical solvent motions we are looking for cannot be all that complex,
but the key point for us is that a short-time treatment of the liquid dynamics
may be all that we need to perform the analyses we want to do.
At short enough times, the dynamics of liquids really is quite simple.
There is little opportunity for the basic liquid structure to change over the
course of a few hundred fs, so we can regard the molecules of our liquid
as essentially vibrating in place (or more correctly, beginning to vibrate
in place). The time evolutions of each of the 3N degrees of freedom of
the liquid (with N the number of solvent atoms) are thus described by the
3N normal modes q˛ t , ˛ D 1, . . . 3N , the instantaneous normal modes
(INMs) of the instantaneous liquid conﬁguration (41,42). Of course, any
given experiment is likely to sample an ensemble of such conﬁgurations,
meaning that our formulas will have to be averaged over the equilibrium
distribution of conﬁgurations, but it will turn out to be a particular advantage
for us to being able to think about the speciﬁcs of molecular mechanisms
one conﬁguration at a time.
To see how these harmonic solvent modes translate into vibrational
friction, (43-46) consider how the correlation function for the solvent force
on the frozen mode [Equation (12)], behaves at short times (47). The solvent
modes themselves, q˛ t , are the displacements of the liquid along the 3N dimensional eigenvectors e˛ of each mode. Literally, if the 3N -dimensional
vector giving the position of every atom in the liquid at time t is R(t), the
displacement from the time zero conﬁguration R 0 is the sum
Rt
R0 D
˛
q˛ t e˛
(14)
Thus, at the shortest times, the evolution of the solvent force is just a linear
function of these displacements.
0
0
Fext
t ³ Fext
0 C
0
c˛ Á ∂Fext
/∂q˛
˛
c˛ q˛ t
qD0
0
D ∂Fext
/∂R
15
R0
Ð e˛
16
where the coefﬁcients c˛ describe the efﬁciency with which each mode
˛ modulates the force. On taking advantage of the independence of the
modes, we see that the autocorrelation function of the time derivatives of
the force, which is just the negative of the second derivative of the force
autocorrelation function we want, becomes simply
0
0
d2 CFF t /dt2 D h[dFext
/dt]0 [dFext
/dt]t i D
with v˛ t D dq˛ /dt being the mode velocities.
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c2 v˛
˛ ˛
0 v˛ t
17
Mechanisms of Vibrational Relaxation
159
The reason we work with the derivatives is related to the fact that the
mode displacements are measured from their instantaneous, t D 0, position
rather than some hypothetical harmonic minimum (48). We know that the
global potential energy surface of a liquid is far from harmonic, so such a
minimum would be a rather unphysical construct. What happens instead is
that our modes do obey simple harmonic dynamics, but subject to somewhat
unusual initial conditions (49):
q˛ t D [f˛ /ω˛2 ] 1 cos ω˛ t C v˛ 0 sin ω˛ t
v˛ t D v˛ 0 cos ω˛ t C [f˛ /ω˛ ] sin ω˛ t
18
19
Here the ω˛ are the mode frequencies (technically, the ω˛2 are eigenvalues of
the mass-weighted dynamical matrix evaluated at the instantaneous liquid
conﬁguration), and v˛ 0 and f˛ are the initial velocities, dq˛ /dt, and the
initial forces, ∂V/∂q˛ , along the modes. We can avoid being tripped up by
our ignorance of the distribution of f˛ s (and take advantage of knowing that
the v˛ 0 s are governed by a mass-weighted Maxwell-Boltzmann distribution) by making use of Equation (17) (47). On substituting Equation (19)
and using the facts that
hv˛ 0 f˛ i D 0, hv2˛ 0 i D kB T
we obtain
d2 CFF t /dt2 D kB T
c2
˛ ˛
cos ω˛ t D kB T
dω
vib
ω cos ωt
implying, from Equations (4) and (13), that our desired frequency-domain
vibrational friction itself is just
ÁR ω D
/2
vib
ω /ω2
(20)
with what we shall call the inﬂuence spectrum for vibrational relaxation
the equilibrium distribution of INM frequencies of the solvent weighted by
the ability of each solvent mode to promote vibrational relaxation (50):
vib
ω D
c2 υ
˛ ˛
ω
ω˛
(21)
C. Deducing Molecular Mechanisms from
Instantaneous-Normal-Mode Theory
Equations (20) and (21) help us in a number of ways. For one thing, the
average vibrational inﬂuence spectrum is itself going to be interesting from
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a mechanistic perspective. Its frequency range will provide our basic explanation as to why solvents have the natural time scales they do. Comparisons between the inﬂuence spectra for different kinds of solute relaxation
processes will also be informative in letting us see the differences (and
surprising similarities) between solvent responses to different kinds of
solute motions. Nor, for that matter, are we limited to looking at average
solvent responses. Instead of taking an average over liquid conﬁgurations,
we can examine the response from any instantaneous conﬁguration R0 :
[
vib
ω ] R0 D
c2
˛ ˛
R0 υ ω
ω˛ R0
(22)
and ask mechanistic questions stripped of the conﬁguration-to-conﬁguration
inhomogeneous broadening that makes the dynamics seem so chaotic (50).
Even at the level of the averaged solvent response there is some fairly
detailed analysis we can pursue. Because we know the molecular identity
of each normal mode associated with a given instantaneous conﬁguration
by virtue of knowing its eigenvector e˛ , we can project out of the inﬂuence
spectrum the contributions of any desired subset of solvent molecules or
geometries of solvent motion (49). The conﬁgurational average will then tell
us how whether our candidate set of degrees of freedom — our candidate
mechanism — is actually an important ingredient in the relaxation (51).
To pick an example, suppose we want to the look at the portion of
our inﬂuence spectrum arising from the ﬁrst solvent shell around the solute
(the shell being deﬁned however we care to). One can demonstrate quite
formally that the projected inﬂuence spectrum will resemble the full inﬂuence spectrum, but with different weightings for the individual modes (51):
proj
vib
ω D
˛
2
[cproj
˛ ] υ ω
ω˛
(23)
While the unprojected weighting deﬁned by Equation (16) can be written
(for a solution of rigid molecules) in terms of explicit components of the
mode eigenvectors as
c˛ D
j
Dx,y,z, ˆ
0
∂Fext
/∂rj
R0
e˛
j
with j D 1, . . . , N labeling the molecules of the solution and
denoting
their center of mass translational (x,y,z) and reorientational ˆ degrees of
freedom, the ﬁrst-shell projected weightings will include out of each mode
only the degrees of freedom of the molecules in the ﬁrst solvent shell:
c1st
˛
shell
D
jD1st shell
Dx,y,z, ˆ
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0
∂Fext
/∂rj
R0
e˛
j