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B. The Instantaneous Vibrational Friction and the Instantaneous Normal Modes of the Solvent

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.



Copyright © 2001 by Taylor & Francis Group, LLC



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







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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160



Stratt



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







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







j



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