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V. W(CO)6 IN SUPERCRITICAL ETHANE

V. W(CO)6 IN SUPERCRITICAL ETHANE

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688



Skinner et al.



theoretical complications of a polyatomic solute in a polyatomic solvent:

intramolecular vibration-vibration energy transfer, as well as intermolecular

vibration-vibration, vibration-rotation, and vibration-translation relaxation

pathways. We have not attempted to provide a general solution for this

problem. In fact, in what follows we implicitly assume that in this system

the dominant pathways involve only other intramolecular vibrations on the

solute and translations. And even so, we pursue a simplified model where

the solute-solvent and solvent-solvent potentials are both isotropic (12).

The Hamiltonian again has the form of Equation (1), and in this case

we will not assume that Hq (the Hamiltonian for the mode that is excited) is

harmonic, but only that Equation (2) applies. The bath involves the other

E as well as all translational

intramolecular coordinates of the solute, Q,

degrees of freedom:

Hb D HQE C T C



ri C

i



s



rij



37



i


E and, as in Section III, T

where HQE is the Hamiltonian associated with Q,

is the translational kinetic energy of the solute and solvent molecules, r

is the solute-solvent pair potential, given by Equation (24), and s r is the

solvent-solvent pair potential, given by Equation (25).

To define the oscillator-bath interaction term, we write the full soluteE , which we take to be

solvent pair potential as r, q, Q





12

6

E

E

q,

Q

q,

Q

E D 4ε q, Q

E 



r, q, Q

38

r

r

Thus we again assume a Lennard-Jones form, where now the well depth and

range parameters depend on the solute’s internal vibrational coordinates.

Without loss of generality we can define these coordinates so that q D

E D 0 corresponds to the minimum in the intramolecular potential. The

Q

solute-solvent potential in Hb above is actually then r Á r, 0, 0 , where

clearly ε Á ε 0, 0 and Á 0, 0 . The oscillator-bath interaction term is

E

[ ri , q, Q



VD



ri ]



39



i



which can be written as

VD



A˛ F ˛

˛



ri

i



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40



VER in Liquids and Supercritical Fluids



689



where ˛ D 1, 2, and

E D 4ε q, Q

E

A1 q, Q

E D 4ε q, Q

E

A2 q, Q

F˛ D



E

q, Q



12



E

q, Q



6



41



f˛ ri



42

43



i



f1 r D



12



44



r

6



f2 r D



45



r



To calculate the VER rate from state j1i to state j0i for the mode

with coordinate q, we use Equation (4):

k1!0 D



1

h¯ 2



1

1



dt eiω10 t hV10 t V01 0 i



46



In this case, because the bath includes intramolecular coordinates, the

evaluation of this formula, which involves products of translational and

vibrational time-correlation functions, is quite complicated (12). For a polyatomic solute with a large enough number of vibrational modes, we argue

that the time-correlation functions for translations decay on the time scale

of the inverse of the characteristic frequencies of the translational bath,

which is much slower than the decay of time-correlation functions for the

intramolecular vibrations. Therefore we can replace the translational timecorrelation functions by their initial values, which we evaluate classically.

The upshot is that the rate constant can be written as (12)

k1!0 D



2

1 hF1 i



C



2

2 hF2 i



C



12 hF1 F2 i



47



where

hF˛ Fˇ i D



dEr f˛ r fˇ r g r

C



2



dEr1 dEr2 f˛ r1 fˇ r2 g r1 g r2 gs r12



48



is the solvent number density, f1 r and f2 r are given in Equations (44)

and (45), and g r and gs r are the solute-solvent and solvent-solvent radial



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690



Skinner et al.



distribution functions, as discussed earlier. The (temperature-dependent)

E

constants 1 , 2 , and 12 are related to matrix elements involving A1 q, Q

E

and A2 q, Q [see Equations (41) and (42)], but since we do not have

explicit models for those, we treat these constants as adjustable parameters.



Figure 3 Solvent-induced VER rate for the asymmetrical CO stretch mode of

W(CO)6 in supercritical ethane at 307.15 K (critical temperature D 305.33 K) as a

function of density. The solid diamonds are the experimental points, and the theory

is given by the open circles.



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691



The experiments by Fayer and coworkers measure VER rates of the

asymmetrical T1u CO stretching mode of W(CO)6 over a wide range

of densities and temperatures in supercritical ethane (8–11) (as well as

in other solvents that we do not consider herein). We use the following

˚

interaction parameters for this system (12): εs /k D 233 K, s D 4.24 A,



Figure 4 Solvent-induced VER rate for the asymmetrical CO stretch mode of

W(CO)6 in supercritical ethane at the critical density (6.87 mol/L) as a function

of temperature. The solid diamonds are the experimental points, and the theory is

given by the open circles.



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692



Skinner et al.



˚ The other quantities we need before we

ε/k D 500 K, and D 5.50 A.

can undertake a fit of the data are solute-solvent and solvent-solvent radial

distribution functions. In Section III we obtained these distribution functions using the HMSA integral equation theory. Unfortunately, however,

this and other popular integral equation closures either do not converge

or are very inaccurate for infinitely dilute mixtures near the solvent’s critical point (24). Since we are particularly interested in this region, we have

been forced instead to calculate the radial distribution functions from Monte

Carlo simulations for each density-temperature pair studied experimentally.

There are two sets of experimental data. The first is a study of the

VER rate constant as a function of density at a constant temperature of

307.15 K (a little less than two degrees above the critical temperature)

(8,10,11). The data for the solvent-induced contribution to the rate constant

(the gas-phase IVR contribution has been subtracted) are shown in Fig. 3.

In Fig. 4 are shown experimental data for the solvent-induced VER rate

constant at the critical density as a function of temperature (9). Our theoretical fits to the experimental data are also shown in Figs. 3 and 4. As

discussed, for each density-temperature pair, the relevant radial distribution

functions were obtained from simulation and the averages hF21 i, hF22 i, and

hF1 F2 i were calculated. A global least-squares fit to both sets of data gives

1

1

1

1 D 0.0104 ns , 2 D 0.00181 ns , and 12 D 0.00862 ns . As seen in

Fig. 3, the fit to the density-dependent data is very good. In Fig. 4 one sees

that the theory reproduces the interesting trend that the rate decreases and

then increases with increasing temperature, but that theory and experiment

are not in quantitative agreement.

VI. CONCLUSION



In this chapter we have reviewed the general theory of vibrational energy

relaxation for a single oscillator coupled to a bath, and we have discussed

the application of these results to three specific systems: iodine in xenon,

neat liquid oxygen, and W(CO)6 in ethane. In the first case the bath is the

translations of the solute and solvent molecules, in the second case it is the

translations and rotations of solute and solvent molecules, and in the third

case it is the solute’s other intramolecular vibrations and the translations

of solute and solvent molecules.

As discussed above, the conventional approach to VER in liquids

involves a classical molecular dynamics simulation of the solute (with

one or more vibrational modes constrained to be rigid) in the solvent.

The required time-correlation functions are computed classically and then



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multiplied by a quantum correction. The major limitation in this approach

is uncertainty in the appropriate quantum correction factor. We have shown

that different choices can lead to wildly different results for high-frequency

oscillators (6). Although we have made some suggestions about the best

way to proceed, at this point this remains an unsolved problem.

One aspect of the last set of experiments on W(CO)6 in supercritical

ethane that we have not yet discussed involves the possible role of “local

density enhancements” in VER and other experimental observables for

near-critical mixtures. The term local density enhancement refers to the

anomalously high solvent coordination number around a solute in “attractive” (where the solute-solvent attraction is stronger than that for the solvent

with itself) near-critical mixtures (24,25). Although Fayer and coworkers

can fit their data with a theory that does not contain these local density

enhancements (10,11) (since in their theory the solute-solvent interaction

has no attraction), based on our theory, which is quite sensitive to shortrange solute-solvent structure and which does properly include local density

enhancements if present, we conclude that local density enhancements do

play an important play in VER and other spectroscopic observables (26) in

near-critical attractive mixtures.

ACKNOWLEDGMENT



The authors are grateful for support from the National Science Foundation

(Grant Nos. CHE-9816235 and CHE-9522057).



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