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III. I2 IN LIQUID AND SUPERCRITICAL XENON

III. I2 IN LIQUID AND SUPERCRITICAL XENON

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682



Skinner et al.



oscillator Hamiltonian for the one-dimensional breathing coordinate to be

harmonic:

Hq D



ω02 q2

2



p2

C

2



22



The bath consists of the translational motions of the solute and all the

solvent atoms. Since all potentials are spherically symmetrical, we write

Hb D T C



ri C

i



s



rij



23



i


where T is the translational kinetic energy of the solute molecule and the

solvent atoms, i and j label solvent atoms, ri is the distance between the

solute and solvent atom i, rij is the distance between solvent atoms i and

j, r is the solute-solvent pair potential, and s r is the solvent-solvent

pair potential. Assuming all interactions to be of the Lennard-Jones form,

we have

r D 4ε

s



r D 4εs



12



6



r



24



r

s



12



s



r



6



25



r



To obtain the oscillator-bath interaction term, we argue that the

solute’s instantaneous size depends linearly on the breathing coordinate

q multiplied by a dimensionless coefficient ˛. The latter is treated as the

single adjustable parameter in the theory, which should on physical grounds

be less than but on the order of unity. This leads to (2)

V'



qF



26



with

FD



f ri



27



i



and

fr D



12ε˛



12



2



r



6



r



28



In the iodine experiments the solute is first excited electronically and

then enters the ground electronic state with many quanta of vibrational

excitation (1). The subsequent vibrational relaxation therefore involves



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VER in Liquids and Supercritical Fluids



683



the whole manifold of vibrational states. Since in our model Hq is

harmonic, we use Equations (12) and (14), together with the “standard”

semiclassical approximation scheme from Equation (17), to calculate the

(single-exponential) VER rate. [Note that we did this calculation (2) before

we discovered (5,6) how poor the standard approximation scheme generally

is, but this is not a serious issue in this case since (as we will see) the system

is nearly classical h¯ ω0 ' kT , and the frequency-domain approximation

schemes of Equations (17), (18), and (21) all give essentially the same

ˆ ω0 ' G

ˆ cl ω0 .] This gives

result that G

ˆ cl ω0

1

tanh ˇ¯hω0 /2 G

D

T1

h¯ ω0



29



Finally, again using the facts that Gcl t is even in time and that one can

subtract any constant from Gcl t within the (finite-frequency) Fourier transform without affecting the result, we obtain

1

2 tanh ˇ¯hω0 /2

D

T1

h¯ ω0



1



dt cos ω0 t C t



30



0



where

C t D hF t 0 icl



hFi2cl



31



Note that for ˇ¯hω0 × 1 this becomes the Landau-Teller result of

Equation (13).

To proceed further we now consider analytic approximations to C t .

We first expand C t in powers of t:

C t D C 0 [1



At2 C Bt4 . . .]



32



and derive (2) approximate but accurate analytic expressions for C 0 , A,

and B that depend on the previously defined f r , r , and s r , and

on g r and gs r , the solute-solvent and solvent-solvent radial distribution

functions, respectively. To obtain these distribution functions we use the

HMSA integral equation approach (20,21). We next approximate the actual

time correlation function by the ansatz (22)

C t D C 0 cos bt / cosh at



33



and choose the parameters a and b so as to reproduce exactly the short-time

expansion of Equation (32) through order t4 . This expression for C t can

be Fourier transformed analytically, leading to an analytic expression for

T1 (2).



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684



Skinner et al.



Figure 1 1/T1 vs. T for I2 in xenon at a density of D 3.0 g/cm3 . The open

diamonds are the experimental points, and the solid circles (with connecting lines)

are from theory.



We then use this theory to analyze the experimental data of Paige and

Harris (1), who studied the VER rate of I2 in liquid Xe at 280 K for a variety

of solvent densities from 1.8 to 3.4 g/cm3 and at a density of 3.0 g/cm3

for several temperatures from 253 to 323 K (at the higher temperatures in

this range Xe is supercritical). We take the following Lennard-Jones param˚ εs /k D 221.7 K, and s D 3.930 A.

˚

eters: (2) ε/k D 349 K, D 4.456 A,

The vibrational frequency of iodine is ω0 /2 c D 214.6 cm 1 , which in



Copyright © 2001 by Taylor & Francis Group, LLC



VER in Liquids and Supercritical Fluids



685



temperature units is 309 K, showing that for this system h¯ ω0 ' kT. For

each density and temperature pair we calculate T1 as described above,

and then make a global comparison to all of the experimental data using

˛ as a fit parameter, obtaining ˛ D 0.7. In Figs. 1 and 2 we compare

theory and experiment at constant density and temperature, respectively.

The agreement between theory and experiment at constant density is excellent. The agreement at constant temperature is satisfactory, although the

theoretical density dependence is clearly steeper than that observed experimentally.



Figure 2 1/T1 vs. for I2 in xenon at T D 280 K. The open diamonds are the

experimental points, and the solid circles (with connecting lines) are from theory.



Copyright © 2001 by Taylor & Francis Group, LLC



686



Skinner et al.



IV. NEAT LIQUID O2



We next consider a more challenging problem, that of VER in neat liquid

oxygen (where now the solute and solvent molecules are both oxygen). This

is more challenging for two reasons: (1) Whereas in the last problem VER

was dominated by the vibration-translation mechanism, in this case since

the solvent is molecular, and one can anticipate that VER will have contributions from both vibration-translation and vibration-rotation channels. This

means that rotations of the solvent will have to be considered explicitly,

and, for example, we will not be able to use the analytical results discussed

above. (2) Whereas the last problem was nearly classical kT ' h¯ ω0 , in

this case, because of the much higher vibrational frequency of oxygen, one

is deeply in the quantum regime h¯ ω0 × kT , and so the issue of the most

appropriate semiclassical approximation scheme becomes very important.

The Hamiltonian for the vibration of a “tagged” (solute) oxygen

molecule is taken to be harmonic, as in Equation (22). The bath

Hamiltonian involves the translations and rotations of all (solute and

solvent) oxygen molecules (4):

L2

P2i

C i

2M

2I



Hb D

i



C



jEri˛



Erjˇ j



34



i


where i and j index the molecules (i D 0 is the solute and i D 1, 2, . . .

E i is the momentum of the ith molecule, M is the

are solvent molecules), P

E i is the angular momentum of the ith molecule, and I

molecular mass, L

is the moment of inertia. ˛, ˇ D 1, 2 index the two sites (atoms) of each

molecule, and Eri˛ is the position of site ˛ on atom i. Thus the potential

energy involves the site-site pairwise potential, r , which is taken to be

of the Lennard-Jones form in Equation (24). The oscillator-bath interaction

has the form V D qF, and F has two contributions, from the centrifugal

and potential forces on the oscillator, respectively (4):

FD



1

L20

C

Ire

2



E ˇ Ð ˆrˇ

F



35



ˇ



where re is the separation of the sites in a molecule (the bond length),

Eˇ D

F

˛



i



Eri˛

jEri˛



Er0ˇ

Er0ˇ j



0



jEri˛



Er0ˇ j



36



is the force on site ˇ of the tagged diatomic, and ˆrˇ is the unit vector

pointing from the center of mass of the tagged diatomic to site ˇ.



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VER in Liquids and Supercritical Fluids



687



In the oxygen VER experiments (3) the n D 1 vibrational state of a

given oxygen molecule is prepared with a laser, and the population of that

state, probed at some later time, decays exponentially. Since in this case

h¯ ω0 × kT, we are in the limit where the state space can be truncated to two

levels, and 1/T1 ' k1!0 . Thus the rate constant k1!0 is measured directly

in these experiments. Our starting point for the theoretical discussion is then

Equation (14). For reasons discussed in some detail elsewhere (6), for this

problem we use the Egelstaff scheme in Equation (19) to relate the Fourier

transform of the quantum force-force time-correlation function to the classical time-correlation function, which we then calculate from a classical

molecular dynamics computer simulation. The details of the simulation are

reported elsewhere (4); here we simply list the site-site potential parameters

˚ and the distance between

used therein: ε/k D 38.003 K, and D 3.210 A,

˚

sites is re D 0.7063 A.

Because of numerical problems emanating from the inherent statistical

noise in Gcl (t), it is impossible to evaluate the Fourier transform in

Equation (19) at the required oxygen frequency of ω0 /2 c D 1552.5 cm 1 .

We have used several methods to extrapolate the numerical results to high

frequency, including the use of the ansatz in Equation (33) (4,23). Here,

however, we simply assume that the rate would follow an exponential

energy gap law (2), k1!0 / e ω0 , and therefore we perform a linear

extrapolation on a log plot (6).

The experimental rate at 70 K is k1!0 D 360 s 1 (3). Our theoretical

rate, obtained from the classical force-force time-correlation function from

the molecular dynamics simulation together with the Egelstaff approximation and the extrapolation described above gives k1!0 D 270 s 1 , in fine

agreement with experiment (considering the extremely slow time scale). It

is important to understand, however, that the theoretical error bar on this

number is probably in the neighborhood of one order of magnitude, due

to uncertainties in the potential, in the extrapolation, and in the semiclassical approximation scheme. One might note that the popular “standard”

approximation scheme of Equation (17) produces a number that differs

from experiment by over five orders of magnitude (6).

V. W(CO)6 IN SUPERCRITICAL ETHANE



Our last specific system involves the solute W(CO)6 in supercritical ethane.

In the experiments (8–11) a particular vibrational mode of the solute is

excited to n D 1, and the population of this level is subsequently probed

as a function of time. VER in this system in principle embodies all the



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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



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