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IV. HOW DOES DIELECTRIC FRICTION EFFECT VIBRATIONAL ENERGY RELAXATION?

IV. HOW DOES DIELECTRIC FRICTION EFFECT VIBRATIONAL ENERGY RELAXATION?

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vibrational energy was, presumably, a kind of dielectric friction paralleling

the dielectric friction thought to govern the translation of ions and the

reorientation of dipoles in polar solvents (58,59).

That electrostatic forces could be crucial to vibrational energy relaxation was amply demonstrated by the liquid water simulations of Whitnell

et al. (34). They noted that since the electrostatic portion of the force

between their solvent and a dipolar solute was linear in the solute dipole

moment, Equations (12) and (13) implied that the electrostatic part of the

friction ought to scale as the dipole moment squared. When they then

found that their entire relaxation rate scaled with the square of the solute

dipole moment, it certainly seemed to be convincing evidence that electrostatics forces were indeed the primary ingredients generating ultrafast

relaxation. Subsequent theoretical work on relaxation rates in such manifestly protic solvents as water and alcohols has largely served to reinforce

this message (37,38,60,61).

But consistent with the overall theme of this chapter, we need to

ask ourselves whether we really have to conclude that the mechanism of

vibrational energy relaxation is fundamentally electrostatic just because we

find the overall relaxation rate to be sensitive to Coulombic forces. Let

us attempt to get at this question through another mechanistic analysis of

the INM vibrational influence spectrum, this time looking at the respective

contributions of the electrostatic part of the solvent force on our vibrating

bond, the nonelectrostatic part (in most simulations, the Lennard-Jones

forces), and whatever cross terms there may be.

The results of such a calculation, shown in Fig. 8 (52), seem to tell a

very different story from the earliest studies. With nondipolar I2 as a solute

and CO2 as a solvent, the complete domination of the solvent response by

the Lennard-Jones forces is impressive, but perhaps not all that startling.

One might surmise that the quadupole-quadrupole forces at work in this

example are a bit too weak to accomplish much. Yet, when we have a

dipolar solute dissolved in the strongly polar solvent CH3 CN, we get almost

the same kind of complete control by Lennard-Jones forces. Electrostatics

now seems totally unimportant.

We can begin to get at this discrepancy if we carry out an equivalent

to the Whitnell-Wilson-Hynes calculation (34,38), looking at the effects of

Figure 8 Vibrational influence spectra for the three systems illustrated in Fig. 3

(52). In each panel the total influence spectrum is compared with the portion of

the spectrum arising from purely Lennard-Jones coupling to the solute (LJ), from

purely electrostatic coupling (elec.), and from cross coupling (LJ-elec.)



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Mechanisms of Vibrational Relaxation



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systematically varying the solute dipole on the vibrational friction. Rather

than monitoring the friction itself, however, it is better for mechanistic

purposes to look just at its form, the normalized friction Á t /Á 0 (Fig. 9)

(62). What we see is quite telling, especially in CH3 CN solutions. The

magnitude of the friction [as measured by the initial friction Á 0 ] is, in

fact, quite sensitive to electrostatics; it changes by more than a factor of 2

in going from the nondipolar “Br2 ” solute to the almost 9 D dipole moment

of the “d8” solute. The detailed time dependence of the vibrational friction,

however, is virtually unchanged in the polar solvent. Hence we can see that

it is the magnitude, not the mechanism, of the vibrational friction that is

controlled by electrostatic forces. As Fig. 8 makes clear, the mechanism of

the relaxation, at least in our example of a polar solvent, does rely almost

completely on the solvent modulating the Lennard-Jones–like forces on the

solute, regardless of the solvent’s polarity.

That there might be such a crucial distinction between magnitude

and mechanism was actually anticipated by several authors (63–65). But

perhaps we can see it a little more clearly by adopting our familiar instantaneous perspective, distinguishing carefully between equilibrium phenomena

that go into selecting a set of instantaneous liquid configurations and the

subsequent dynamics that are what we really mean by the term “mechanism” (62). When we view things in this fashion, the reason why electrostatics forces are important in vibrational relaxation has to do solely

with their equilibrium role. Coulombic forces are sufficiently powerful that

they can position solvents much higher on the steep repulsive wall of the

solute-solvent potential than, say, van der Waals forces. By the same token

though, these forces are not rapidly varying enough to contribute much to

frequency domain friction at anything but the lowest frequencies. In polar

solvents, as in others, it is going to be the most rapidly varying forces, the

sharp repulsive forces, whose time evolution is ultimately responsible for

the relaxation. Polar solvents simply seem to amplify this repulsive-force

friction by encouraging small solute-solvent distances (63–65).

We should probably keep in mind that these conclusions were drawn

from studies of neutral solutes in aprotic solvents, whereas the most striking

experiments have featured ionic solutes in hydrogen-bonding solvents

(5,56,57). Presumably, the interactions with ionic solutes should not change

matters much: the forces are even stronger than with neutrals, but the forces

are even more slowly varying. It is not out of the question, though, that

the directional character of hydrogen bonding makes it somehow unique in

the context of vibrational relaxation (66,67). Further mechanistic analysis

of this point might prove interesting.



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Mechanisms of Vibrational Relaxation



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Figure 9 The normalized vibrational friction felt by a range of diatomic solutes

dissolved in liquid carbon dioxide and liquid acetonitrile (62). The solutes are

meant to represent the nondipolar molecule Br2 itself and two bromine mimics

differing only in the replacement of the bromine quadrupole by permanent dipoles

of different strengths. The “d5” solute has a dipole moment of 5.476 D and the

“d8” solute a dipole moment of 8.762 D. (The notation Ávv emphasizes the fact

that only potential-energy contributions are included in the calculations; centrifugal

force terms are neglected.)



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V. VIBRATIONAL ENERGY RELAXATION AT

HIGH FREQUENCIES



As we noted earlier, the fact that our instantaneous vibrational friction

is built from a set of harmonic solvent modes suggests that it is natural

to think of vibrational energy relaxation as taking place through resonant

energy transfer to the solvent: relatively rapid energy transfer that occurs

when the solute finds a solvent mode matching the solute’s own vibrational

frequency. We did discover in Section III that only a few key atoms in

a mode are involved in providing the gateway necessary to access the

mode, but this observation by itself does nothing to cast doubt on the basic

resonant-transfer paradigm.

However, several items should make us continue to be wary. Most T1 s

for neutral small-molecule vibrational relaxations are not ultrafast events,

nor, for that matter, do they involve vibrational frequencies remotely close

to the few hundred cm 1 bands of typical liquids (5).Ł More typical are the

940 ps it takes to relax the 3265 cm 1 C–H stretch in HCN (5) and the

4.4 ns required to relax the 2850 cm 1 HCl vibration (68) when either one

of these hydrides is dissolved in CCl4 . Even slower relaxations are well

known; the 3 ms relaxation times for O2 ’s 1556 cm 1 vibration in liquid

oxygen, for example, (69) has been the subject of much recent discussion (70–72).

One way to put these problems into perspective is to look at what

might be one of the simplest examples of a high-frequency vibrational

energy relaxation, that of I2 dissolved in Xe, (15,17,73–76) (Fig. 10). The

211 cm 1 vibrational frequency of iodine hardly qualifies it as high by most

standards, but the weak interatomic forces and the high atomic weight

of Xe cause its INM density of states to be increasingly small beyond

120 cm 1 . Indeed, experiments indicate that I2 relaxes quite slowly in Xe,

with vibrational lifetimes on the order of hundreds of ps (15–17,74). The

difficulty with the INM theory we outlined in Section II is that the low

density of solvent modes would lead to a calculated vibrational friction

orders of magnitude lower still. Apparently our harmonic theory is correct

in predicting significant slowdowns once the solute’s vibration is out of

Ł



To simplify our discussion, we will ignore the presence of the high-frequency

intramolecular degrees of freedom present in most molecular solvents, though

they do pose interesting conceptual and computational issues.



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Mechanisms of Vibrational Relaxation



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Figure 10 The situation confronted by a vibrationally excited I2 molecule

dissolved in liquid Xe (76). The solid line is the INM density of states for the

liquid; the arrow indicates the vibrational frequency of I2 .



the solvent band, but it seriously underestimates the actual rates. The true

relaxation mechanism, it would seem, must rely on some fundamentally

anharmonic aspect of the dynamics.

Having to understand the full spectrum of anharmonic possibilities for

liquids would pose a formidable challenge, but for our present purposes we

need only come to grips with the highest-frequency dynamics (14,77). But

high-frequency behavior is something we do understand, at least within

INM theory. The very highest frequency modes inside the INM band

are instantaneous binary modes, modes that, except for some perturbative

corrections, behave as if they involve no more than a pair of atoms at a

time (50). As we look for higher and higher frequencies, it becomes less



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