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II. COULOMBIC FORCE EFFECTS ON VET

II. COULOMBIC FORCE EFFECTS ON VET

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bond, taken fixed in length. We will often just call this the force (power)

spectrum. (One could also term it the vibrational frequency–dependent friction, a term we like less, since it is associated with a generalized Langevin

equation and Equation (1) has a validity far beyond that of the latter.) As a

side note, the demonstration that these two routes were in good agreement

was the first explicit simulation demonstration that the classical LT formula

gives correct results (for the classical relaxation).

By the artificial simulation device of alternately removing and

doubling the assigned point charges on the C and Cl moieties, it was shown

that Coulomb effects are decisive: T1 shrinks from ¾100 to ¾5 ps as the

charges are turned on to their assumed standard value, and this time declines

to 1.4 ps as those charges are doubled. This latter behavior is in fact just the

behavior expected from a simple scaling of the charges for the amplitude

of the fluctuating force in the LT formula. The importance of the Coulomb

forces was further established in extensive studies of both the diatomic

vibrational phase distributions as well as the surrounding water molecule

spatial distributions sampled in the nonequilibrium dynamics. For example,

it was shown that the range of the Coulomb forces was critical in allowing

the transfer, that water molecules in and beyond the first solvent shell were

involved, and that there was significant participation of water molecules

in locations perpendicular to the C–Cl bond, a feat that would be fairly

Herculean for short-ranged non-Coulomb forces. It is also worth pointing

out that the shape of the force spectrum in the neighborhood of the CCl

frequency is rather different in the absence versus presence of the Coulomb

forces.

While a short-range isolated binary collision (IBC) picture, in its most

extreme form — with the emphasis on “collision” — is on its face ludicrous for long-range force interactions, the corresponding isolated binary

interaction (IBI) idea (5) could be examined. The IBI concept is that it is

only the direct interaction with the solute of a single solvent molecule at

a time that determines the VET; this is of course also a key ingredient in

the IBC picture, but in the IBI there is no a priori specification of exactly

what the interaction force is. If the Fourier component of the fluctuating

force tcf is decomposed as

ˆ Fˆ ω i D

hυFυ



hυFˆ i υFˆ i ω i C

i



hυFˆ i υFˆ j ω i



2



i6Dj



where the sums are over the solvent molecules interacting with the solute,

the first contribution is a self-contribution — which would be the only term

retained in the IBI approximation — while the second cross-correlation



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term indicates the importance of nonbinary interaction forces for the VET.

With the Coulomb forces turned off, the IBI approximation was found to

be nearly perfect, while its quality wanes as the charges are increased: for

example, the IBI approximation gives a T1 about two thirds of the full

value for the standard CH3 Cl charge case. While it would be hyperbolic

to claim this as a “breakdown” of an IBI picture, it seems fair to say that

an IBI approach, while valuable, is clearly missing a significant part of the

picture. (See also Ref. 21 and Chapter 4.)

Finally, it was shown in various ways that it is the water librational

motions that are important in the VET and that these involve coupled water

molecular motions, since there is a significant contribution from non-IBI

terms here. In view of the remarks above about the shape of the force spectrum itself differing in the absence and presence of the solute charges, and

the validity of the IBI perspective in the absence of charges, the implication

is that for the hypothetical no charge CCl vibration at the same frequency,

the librations would still be important for the VET, but they would involve

only pair effects for the VET and would perforce interact significantly more

feebly with the mode.

Among the signatures, alluded to above, of the importance of the

solvent librations was the interpretation of the deuterium isotope effect

by which the simulated VET is slowed, via a shift of the solvent D2 O

librational band away from the CH3 Cl frequency, while the H2 O librational

band is well overlapped with that frequency. While one might think that

this direction of isotope effect arising from the librations is obvious for

any solute vibration frequency in this frequency range, we need to add a

cautionary note. In an LT formula study of the ClO ion, of frequency

713 cm 1 , in aqueous solution, Lim et al. (22) found a negligible solvent

H2 O/D2 O isotope effect, in contrast to the CH3 Cl findings, but in agreement

with these authors’ experimental results (22). Even further, it seems from

Fig. 9 of Lim et al. that the direction of the isotope effect would invert

for hypothetical ClO molecular ions of lower and higher frequency than

the 713 cm 1 value! It seems that such behavior arises from the fact that

the red shift upon deuteration of the solvent librational band in the force

spectrum will only slow the VET if the spectrum is monotonic in frequency

in the relevant frequency region — as it is in Ref. 19 — in contrast with

Ref. 22, where it is not. In this connection, it should be recalled that it is

the dynamical solvent librations occurring in the presence of the solute that

are germane, and these can certainly be different for different solutes. It

should also be borne in mind that not even the equilibrium static structures

of neat H2 O and D2 O liquids are the same (23).



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Before returning to the main Coulomb force theme, it is important

to stress that in LT formula simulations focused on the force on the solute

vibrational mode, any energy transfer to implicated solvent modes, e.g.,

the water librations above, is actually inferred, rather than directly demonstrated. That is to say, such transfer — which is of course just vibrationvibration (VV) transfer — is not directly probed. We return to this issue at

the conclusion of this chapter.

Two subsequent simulation studies for low-frequency vibration

systems clearly show that the strong and striking dominance of Coulomb

force effects found for CH3 Cl in water is by no means so clear-cut (or

even true) in low-frequency diatomic systems. In the first of these, related

to experiments by the Barbara group (24,25), Benjamin and Whitnell (26)

found that for the diatomic I2 of frequency ¾115 cm 1 , with a vibrational

relaxation time of about 1 ps, the presence of Coulomb forces accelerated

the VET in water by about a factor of 4, a noticeable but somewhat muted

effect considering that one is comparing an ion to a neutral (with the same

frequency). The authors noted the importance of the fact that the short-range

non-Coulomb forces themselves are quite efficient at the low I2 frequency.

The Coulomb forces are even more subdued in the LT formula simulations by Gnanakaran and Hochstrasser (27) for HgI, of frequency 125 cm 1 ,

in ethanol. At the same frequency, the charges only reduced the relaxation time from 3 to 2 ps. Further, by examining the force spectrum as

a function of frequency, it was shown that the Coulomb force contribution at the HgI frequency is negligible compared to the short-range

non-Coulomb force contribution. In extended calculations where the spectrum of forces was examined as a function of frequency, the Coulomb

force contribution only becomes the uniformly dominant one above about

1200 cm 1 , although it becomes transiently dominant in a frequency range

near 700 cm 1 . Returning to the actual HgI frequency case, the authors

noted an indirect Coulomb force effect, in which the Coulomb force brings

the polar solvent molecule closer into the diatomic, thereby increasing the

repulsive non-Coulomb forces. It is of interest to note that a strong qualitative similarity was observed between the OKE spectrum of pure ethanol

and the force spectrum on the HgI bond fixed in the solvent. More such

comparisons would help clarify to what degree pure solvent aspects are

directly correlated to the motions of the solvent molecules in the presence

of the solute that are key for VET.

The indirect Coulomb force effect noted above had also been pointed

out in a LT formula simulation study by Bruehl and Hynes (28) for a model

AH Ð Ð Ð B hydrogen-bonded complex in a model polar CH3 Cl solvent for a



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low-frequency AB vibration of the complex in the range 100–300 cm 1

(since the AH Ð Ð Ð B complex was not intended to be any particular molecular

system, a representative range of frequencies for this and other vibrations

was considered). As in both studies above, the non-Coulomb forces dominate the VET in this low-frequency region, but there is nonetheless an

important indirect Coulomb effect, tightening the local solvent cage around

the solute and thereby inducing a more effective non-Coulomb force for

the VET.

From a general point of view, it is mildly ironic that in both the justdescribed AB vibration and the HgI examples, the direct slowly varying

Coulomb force impact on the VET is unimportant in this low-frequency

region; the reason would appear to be that in this region, a standard perspective is evidently correct, even the 100–300 cm 1 frequency range being

sufficiently high that only the repulsive non-Coulomb forces vary rapidly

enough to be effective.

The story changed when the higher frequency vibrations of the

AH Ð Ð Ð B complex were examined (28). For the H-bending vibration in

the 1000–1700 cm 1 range, the direct Coulomb force was essentially

entirely responsible for the VET, a phenomenon in part due to the

fact that the light H motion is a dominant component in the bend

coordinate, and in the model the H had exclusively Coulomb interaction

with the CH3 Cl molecules. Finally, for the proton stretch vibration in

the 2500–3500 cm 1 range, the situation proved to be rather complex,

involving a combination of non-Coulomb, Coulomb, and quite important

cross-correlation effects, all shifting as a function of the stretch frequency.

This complexity defies a simple summary here, but a few points can be

noted. At the highest frequencies ¾3000–3500 cm 1 the direct Coulomb

forces are not important, probably due to the lack of any real possibility of

involvement of strongly coupled librational overtones in the model CH3 Cl

solvent (which might not be the case in, e.g., H2 O). Nonetheless, the

Coulomb/non-Coulomb force cross correlation — whose physical origin

was described (28) — remains quite significant and is in competition with

the dominant direct non-Coulomb force effects. In the lower frequency

range ¾2500 cm 1 , all the force correlation spectra — non-Coulomb,

Coulomb, and cross — are comparable in magnitude.

This model study — which included neither the charge shifting or

intramolecular coupling effects of the sort described elsewhere in this

chapter — suggests a quite rich variety to be sorted out in more realistic

molecular studies. An important general lesson of the hydrogen-bonded



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complex study is that the impact of Coulombic forces is in general likely

to be decidedly mode-specific.

Since experimental results were available for the high-frequency

¾2080 cm 1 diatomic CN in water (as opposed to CH3 Cl) (17), with

an estimated T1 value of some 25 ps, an MD study was undertaken by

Rey and Hynes (29) to clarify the role of Coulomb forces for VET in this

accesible case. The charge distribution of CN in the solvent was modeled

by a negative charge on N and a finite dipole located on the C site (30).

The equilibrium solvent structure about this ion involved greater solvation

number on the N end compared to the C end, a result consistent with some

small cluster calculations (31). Since the frequency shift from the vacuum

and the anharmonicity in the CN bond are both relatively small (29), the

static vibrational aspects of the ion are evidently fairly “clean.”

Analysis of the bond force power spectrum revealed the rather remarkable feature that the Coulomb force effects dominated the VET, with a

small contribution from the Coulomb/non-Coulomb cross correlation. This

central result is in stark contrast to the common thought that the slowly

varying Coulomb forces could not fail to be ineffective at high solute vibrational frequencies. In this context, and in this system, it is not the case that

Coulomb force effects are absolutely large at high frequencies — they are

not; rather they simply decline less rapidly than do the non-Coulomb forces

[a situation totally reversed in the artificial case of lower 200–300 cm 1

solute frequencies, all other things remaining the same]. The only possible

accepting modes in the rigid water molecule solvent of the simulation

would be combinations of the water librations, a feature supported in a

subsequent and entirely different analysis (32) not addressing the Coulomb

force. In a paper essentially confirming the original experimental estimation of T1 for this system, Hochstrasser and coworkers (33) suggested that

a water libration-bend combination band — a band apparent, though weak,

in experimental water spectra (34) and absent in most water simulation

models, and certainly absent in the rigid model used in Ref. 29 — could

play some role in the relaxation, a suggestion supported by some model

calculations (32).

The CN problem deserves reinvestigation, not only to establish

dynamically the involvement of the named solvent molecule combination

bands and to confirm in this context the Coulomb force dominance for

the VET, but also to explore the possible role in the VET of any counter

ions in the CN first solvation shell at higher concentrations of the solute

(17). The latter issue, which is in principle ubiquitous in ionic solute VET

studies, has yet to receive any theoretical attention. Some improvement is



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also in order on the quantitative side: no available calculation is closer to

the experimental result than to within a factor of 2 (29,32).

All the above discussion has focused on the situation in which the

electrical charges in the vibrationally relaxing solute are fixed. Interesting

and important additional effects for VET can arise when instead those

charges vary strongly through a variation of the molecular bond length(s)

and/or the surrounding polar solvent molecule configurations, an aspect

suggested in Ref. 28. An early indication of the potential importance of such

effects arose in the Cl C CH3 Cl ! ClCH3 C Cl SN 2 reaction in water

(12,35): the solvent generalized frictional force exerted on the (unstable)

antisymmetric stretch reaction coordinate was completely dominated by the

“polarization” or “charge shift” (or “flow”) force associated with the rapid

shift of the negative charge from one chlorine to the other as a function of

the antisymmetric stretch.

One illustration of the impact of such a charge shifting force in

the vibrational relaxation context arises in the I2 system. On the experimental side, Lineberger and coworkers (36) found strikingly rapid relaxation of highly vibrationally excited I2 in clusters. (The high vibrational

energy molecular ions were produced via initial electronic excitation of

bound ground state I2 and subsequent transition to high vibrational energy

portions of the ground electronic state surface.) This rapid relaxation feature

was also subsequently seen in corresponding experimental solution studies

(25,37). Since the I2 molecular anion provides a relatively simple vehicle

for describing a charge shifting force and its consequences, we focus our

initial discussion on it.

Briefly, I2 can be described in a two valence bond (VB) state

scenario corresponding to the two charge localized structures I Ð Ð Ð I and

I Ð Ð Ð I (25,38). At any given I-I nuclear separation (r) and arrangement of

the surrounding solvent molecules, the ground electronic state will be some

mixture of these structures. In particular, the resonance electronic coupling

mixing these two localized states is a strong, approximately exponential,

function of r. At large r, this coupling is small, and I2 closely resembles a

(weak coupling, electronically nonadiabatic) outer sphere electron transfer

reaction system. The two charge localized structures are separately stable;

they are separated by an activation barrier in a solvent polarization coordinate associated with the cost to rearrange the solvent molecules between

the two asymmetrical arrangements, appropriate to equilibrium solvation

of the two electronic structures I Ð Ð Ð I and I Ð Ð Ð I. On the other hand, at

smaller separations r, the electronic coupling is large, and overcomes the

preference of the solvent for the more favorable (from the solvent’s point



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of view) interaction with localized charge distributions. Now the electronic

adiabatic ground state is completely delocalized, schematically represented

by I 1/2 I 1/2 .

The above considerations indicate that at some intermediate value of

r, the I2 system on its way to form the completely equilibrated ground state

will experience a significant charge flow, as charge localized I2 converts to

charge-delocalized I2 . Associated with this shift is a corresponding force

that potentially can be quite effective in the transfer of vibrational energy.

The analytic theory (38) of this force for I2 , based on the above

ideas, was followed by an MD simulation (39) involving the molecular

level transcription and showing that the charge shifting or polarization force

could indeed lead to quite rapid vibrational energy transfer from I2 to the

surrounding solvent, in reasonable agreement with the companion experiments (37) on I2 in water. It should be stressed that this VET is of an

unusual character little resembling the more stately progress familiar in VET

at low excitation energy. In particular, significant energy is transferred on a

time scale that is short compared to a vibrational period. Nonetheless, this

type of behavior might be prevalent in a number of high-energy systems,

(22,40,41),Ł although the phenomenon might be convoluted with dynamics

related to the conversion from the excited electronic state to the ground

electronic state (39,42).

In the I2 system, the charge shifting force is only important for

VET at higher internal energies, for the reasons discussed above. A recent

study (43) of the azide ion N3 clearly shows the charge shift force at

work in low-energy relaxation. In particular, it was concluded that in this

system the charge shifting force associated with the antisymmetric stretch

¾2050 cm 1 is dominant in the v D 1 ! v D 0 vibrational relaxation of

this ion in water. This effect, together with intramolecular effects to be

discussed in Section III, suffices to reduce the calculated relaxation time

by about an order of magnitude below a previous simulation, not including

those effects (44). The final result of 0.87 ps is in good agreement with

experiment (1.2 ps) (18). While the authors did not explicitly invoke any

picture involving VB resonance structures for the charge shift force, the

simple three VB state picture of the azide ion described by Pauling (45)

might give useful insight here.

It seems clear that charge shifting force effects on VET should be

quite common in a variety of systems, although whether this effect is

Ł



This same large energy transfer effect has been found to occur in simulations

of I2 in some CO2 clusters (R. Parson, personal communication).



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important at low energies will depend on molecular details of the system.

Certainly there is a good chance for such effects in some energy range for

any molecule in which there should be important contributions of several

VB structures to the electronic structure. There are many significant candidates here, as indicated in our concluding section.



III. SOLUTE INTRAMOLECULAR EFFECTS ON VET



As indicated in Section I, our second major topic is the involvement of

intramolecular solute effects in VET, i.e., the involvement of internal energy

transfer pathways within the vibrationally excited molecule. This statement,

however, is not very precise and could in principle include a wide range of

behavior including for example, IVR not even involving interaction with the

surrounding solvent molecules (46), or solvent-induced transfer of energy

between solute modes that are not themselves intramolecularly coupled

(47–51). Recent studies of azulene in several solvents by Heidelbach et al.

(51) can be consulted for some important insight on the cornucopia of

possibilities. The limit of interest in the present chapter is the one in which

those pathways are not open to the excited molecule in the absence of the

solvent interactions. Even more specifically, we have in mind the case where

the excited vibrational mode in the solute would in the isolated molecule

be anharmonically coupled (more properly, “nonlinearly”) to another mode

within the same molecule, but the energy gap between the two modes is

so far off resonance that transfer between the modes would be effectively

feeble. On the other hand, in the presence of the interaction with the solvent,

transfer to the surroundings can occur to take care of the deficit gap.

A number of examples in the gas phase are known, some of which

are summarized in Refs. 1, 52, and 53. In solution, there is much early

experimental work on VET for the CH stretch fundamental in assorted

molecules, which invoked involvement of a Fermi resonance between the

CH stretch and the overtone of the CH bend (54); reviews of this work can

be found in Refs. 1, 55, and 56.

Here we focus on the case of the excited OH vibration in the HOD

molecule in liquid D2 O solvent, the first solution example, of which we are

aware, studied in detail via MD by the present authors (57). This VET was

first studied experimentally by Vodopyanov (58) and by Graener et al. (59)

in 1991, an effort that has continued to the present (60–62), with, as we will

see later, evolving results. For the moment, we quote the initial experimental

result of 8 š 2 ps (59) as a guideline for our discussion. Beyond the obvious



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general interest in the water system, no simple isolated binary collision

picture for VET would be expected to apply to this strongly hydrogenbonded liquid system, even whether the VET pathway in solution will be

the same as in the gas phase.

The basic formulation of this solution phase problem is in the

Landau-Teller type language originally generally formulated by Oxtoby

(53). As applied to the HOD in D2 O system, the system Hamiltonian can

be written in the form H D HHOD C Hcoupling C HBath . In the semiclassical

approach adopted, a quantum representation of the vibrational HOD modes

is retained, while the remaining coordinates are described classically. HHOD

is the quantum mechanical (anharmonic) Hamiltonian for the vibrational

motions of the HOD molecule; it includes the static effect of the solvent

and, as a consequence, can be expressed in terms of the normal modes

characteristic of the molecule in solution. Such an approach is directly

related for example to that of Berkowitz and Gerber (63) in their theoretical

study of vibrational relaxation in matrices, where the molecule effective

Hamiltonian included the static perturbation from the lattice. Although not

stressed inordinately hereafter, it is clear that such renormalization effects

will always have to be taken into account, or at least examined, when

the vibrationally relaxing solute is in fairly strong interaction with the

surrounding environment.

HBath represents the classical Hamiltonian for the D2 O solvent

together with the translational and rotational contribution of the HOD

molecule (see below). Hcoupling , the nonequilibrium coupling of the

molecular vibrational coordinates to the rest of degrees of freedom,

is split into several contributions: Hcoupling D HV B C HCor C HCen . Here

HV B represents the coupling of the molecular vibrational coordinates to

the surrounding solvent molecules and turns out to be, not surprisingly,

one of the main factors in the VET. In addition, the influence of the

anharmonicity of the intramolecular potential of HOD will prove to be

critical. The remaining contributions to Hcoupling involve vibration-rotation

couplings, which, while not the focus in this chapter, are briefly included

for completeness. HCor represents the Coriolis coupling between normal

modes, which — as known from simple classical arguments (64) — can be

especially strong between bending and stretching for the water molecule.

Finally, HCen represents centrifugal coupling, i.e., the effect that the

variation of moment of inertia with the vibrational motion may have on

the relaxation.

Since the vibrational relaxation rate is evidently some 2 orders of

magnitude smaller than the OH vibrational frequency, it was reasonable



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