Hard fluid model for solvent-induced shifts in molecular vibrational

J. Phys. Chem. 1993,97, 2295-2306

2295

Hard Fluid Model for Solvent-Induced Shifts in Molecular Vibrational Frequencies Dor Ben-Amotz’ Department of Chemistry, Purdue University, West Lafayette, Indiana 47907

Dudley R. Herscbbach Department of Chemistry, Harvard University, Cambridge, Massachusetts 021 38 Received: September 17, 1992; In Final Form: October 30, 1992

A model for packing forces in liquids is derived by generalization of exact results in the continuum and dense hard sphere fluid limits. The parameters of this ‘hard fluid” model are fixed by the equation of state of the corresponding mixed hard sphere fluid. Results are generalized to include the very accurate Boublik-MansooriCarnahanStarling-Leland (BMCSL) equation as well as the Percus-Yevick, scaled-particle-theory, and BoublikKolafa equations of state. Predicted hard sphere cavity distribution functions, chemical potentials, and solvation forces are compared with computer simulations and previous analytical approximations. The application to the prediction of solute vibrational frequency shifts in molecular liquids is illustrated by comparison with data obtained from high-pressure Raman spectra for iodine and pyridine in various solvents.

In section I1 we review the definition of the hard sphere cavity I. Introduction distribution function and previous models. Section I11 introduces Recognition of the importance of packing forces in liquids is our hard fluid model and presents comparisons with the results a guiding principle of current approaches to liquid theory.14 The of previous analytical and numerical approximations. Section common assumption of these approaches is that short-range IV briefly describes the experimental methods used to measure repulsive forces largely determine the zero-order structural and vibrational frequency shifts at high pressure and reports data for dynamicpropertiesof liquids. The effects of these repulsive forces I2 in three solvents and pyridine in four. The hard fluid predictions are often conveniently approximated using hard-sphere reference for solvent-induced changes in molecular vibrational potentials systems. The contributions from longer ranged attractive forces are evaluated and compared with experimental vibrational are typically treated as van der Waals mean fields4~S*7~8 or more frequency shifts. In section V we assess the results and examine general perturbative (inverse temperature) e ~ p a n s i o n s . 2 ~ ~ * ~ ~ ~ aspects that foster the remarkable predictive accuracy of the Current applications of hard sphere fluids in modeling real hard fluid model. Details and generalizations are relegated to liquids range from the calculation of distribution functions,l.I0 the Appendix. equations of state,*J1J2and vapor-liquid e q ~ i l i b r i a , to ~ ~the -~~ calculation of s o l ~ b i l i t y , ~chemical J~ reactivity,I7-l9 and spec11. Hard Sphere Cavity Distribution Function troscopic property7sz0changes induced by solvation. All these are linked by their reliance on hard sphere fluid reference systems The hard sphere cavity distribution function pertains to a pair in the calculation of repulsive contributions to real liquid of hard sphere solute atoms (‘cavities”) dissolved in a hard sphere properties. fluid solvent. It provides a measure of packing forcesexperienced A significant limitation to the more widespread use of hard by the diatomic solute as a function of the interatomic separation sphere models for liquid-phasechemistry is the lack of convenient of the solute pair (bond length). When the atomic separation algorithms for calculating certain key statistical parameters. For becomes larger than the averagediameter of the two solute atoms, example, the calculationof radial and cavity distribution functions the cavity distribution function becomes identical to the radial for hard sphere systems often requires numerical integration or distribution function for the two atoms. In fact, the entire cavity extrapolationfrom tables ofderived parameter~.l~~2~+22 Knowledge distribution function can be viewed as simply the radial distribution of these distribution functions is essential to the prediction of function for two hard sphere atoms which are allowed to chemical potentials, packing forces, and equilibrium coninterpenetrate each other. It is as if the interaction between the ~tants.~J~J~ two specified hard spheres had been turned off, while retaining This paper presents a new model for the cavity distribution their interactions with all the solvent spheres. functions (and the radial distribution function near contact Relation to Chemical Potential. The practical utility of the separation) in mixed hard sphere fluids of arbitrary composition cavity distribution function in chemical calculations derives from and density. Termed the ‘hard fluid” model, it provides a closed its use in approximating repulsive contributions to the chemical analytical form which can easily be incorporated into practical potential of real molecules in solution. Consider a hard sphere calculations. Results obtained with this model are shown to diatomic of bond length r12and diameters U I and u2 dissolved in compare favorably with independent numerical and analytical a fluid solvent, possibly a mixture, composed of hard spheres of estimates of neat and mixed hard sphere fluid distribution diameters ui and partial densities pi. The excess (relative to an functions, chemical potentials, and solvation forces. ideal gas system) chemical potential change, Al,rHS(rl2;u1,u2,ui,pi), The intimate connection between pressure-inducedvibrational associated with the formation of the solvated diatomic from two frequency shifts and solute-solvent interaction potentials has long separated atoms in the same solution is related in a simple way been r e c o g n i z ~ d . ~ . Few ~ ~ - ~previous ~ studies have, however, to the cavity distribution function, y ~ ~ ( r 1 2 ; u I , u 2 , u i , p i ) . focused on using experimental frequency shifts to critically test HS solvation models. In this work high-pressure Raman studies of A ~ H S ~ r l Z ; ~ I , u=2 -k,T , ~ i , ~lny,, i ) ( r 1 2 ; ~ I , u 2 ~(~l ai), ~ i ) I2 and pyridine dissolved in a variety of solvents are compared This excess chemical potential change represents the reversible with hard fluid model predictions for the effects of repulsive and work done in bringing the pair of hard spheres together to form attractive solvation forceson molecularpotential energy surfaces. 0022-3654193 12091-2295SO4.0O10

0 1993 American Chemical Societv

2296 The Journal of Physical Chemistry, Vol. 97, No.

Ben-Amotz and Herschbach

IO, 1993

the diatomic, and is thus also equal to the repulsive contribution to the potential of mean force exerted by the solvent on the solute,

The excess chemical potentialchangeis by definition the difference between the excess chemical potential of the diatomic, pHS,and the two separated atoms from which it is formed, p k and

For notational brevity we now leave implicit the dependence on solute and solvent diameters and densities (and thus y;’(O) represents a cavity distribution function for two like cavities, of diameter ui, at zero separation). The essential result of the hard fluid model, derived in the Appendix, is an explicit analytical expression for the distribution functions appearing in the above equations. In evaluating the repulsive contribution to solvation-induced vibrational frequency shifts, the key quantity is the solvation mean forceexerted along the bond axis of the hard sphere diatomic solute, given by

where the derivativesare evaluated at the equilibriumseparation, r12= re. In section IV we illustrate how these relations are used to approximate repulsive contributions to the chemical potentials and solvent mean forces in real liquid systems. There we also specify procedures for estimating the effective hard sphere diameters of the solvent molecules and solute atoms (or pseudoatoms) from independently available data. Previous Models. For the low-density regime, exact cluster expansions for thecavity distribution function of neat hard sphere fluids have long been known.3’ Approximate virial expansions for mixed hard sphere fluids have been proposed and shown to compare favorably with simulation res~lts.3~ At liquid densities, however, the convergence of such density expansions becomes dubious, and no exact theory for y y f ( r 1 2 exists. ) Various high-density approximations to y y f ( r 1 2 have ) been proposed. We review briefly seven such models. These vary widely in scope, predictive accuracy, and practical utility. (1) The Percus-Yevick approximation yields closed forms for yyf(r12)in neat39and mixed40hard sphere fluids. This model is very good for large separations, corresponding to rlz/u12 > 1.3, where 1712 = (a1 u2)/2 represents the contact separation for two cavities of diameters ul and 172. However, the Percus-Yevick approximation fails by orders of magnitude for r12 near zero. (2) Grundke and Henderson22 proposed a semiempirical approach to model y y f ( r I 2 more ) accurately near zero separation, based on a truncated power series for the cavity distribution function,

+

The four coefficients in this expression were evaluated using the CarnahanStarling hard sphere liquid equation of state, or its generalization to mixtures,4143in combination with the VerletWeis semiempirical radial distribution function model.2’ The equation of state determines three parameters: the magnitude and slope of y y ! ( r I 2at) zero separation as well as its magnitude at the contact separation. The fourth parameter is determined

in a less straightforward manner, by using the Verlet-Weis radial distribution function model to estimate the slope of y y ! ( r 1 2at) contact. ( 3 ) Pratt, Hsu, and Chandler19 extended the GrundkeHenderson results to the case of a diatomic at infinite dilution in a dense hard sphere fluid, although only homonucleardiatomic cavities were treated. These prescriptions prove to be quite accurate in predictingyFs(r,,) inside the contact separation, but fail rapidly in the radial distribution function region beyond contact (as noted in section 111). In addition, the practical utility and scope of these models are reduced by the need to rely on semiempirical numerical results for the fourth parameter. (4) A simpler but less accurate approach to modeling yy: ( r lI )was taken by Ballance and Speedy,44who proposed a threeparameter form for the cavity distribution function, InbEYr, ,)/Y~:(o)I = A V ~ I)(al , + a2rl (5) Here AYdenotes the excess volume excluded from the solvent by a diatomic of bond length rl I relative to that excluded by one atom of the diatomic. The values of yy:(O), a,, and a2 in eq 5 are, again, fixed by exact thermodynamic relations to the hard sphere fluid equation of state. The resulting expression is exact at small rll and at low density. The practical value of this approximation is that it has one less adjustable parameter than the Grundke-Henderson and Pratt-Hsu-Chandler expressions and so does not need to call on the semiempirical Verlet-Weis results. This model thus leads to a closed form solution for the cavity distribution function of a homonuclear solute in a neat hard sphere fluid. ( 5 ) Labik and co-workers38 started with expressions derived from a virial approximation in modeling the chemical potential of a hard diatomic at high density. This approach again avoids use of the Verlet-Weis approximation and is more accurate at moderatelylarger12and highdensitythan the BallanceandSpeedy expression.45 (6) B ~ u b l i khas ~ ~developed a more accurate approximation for the chemical potential of a hard diatomic solute, based on an extension of the scaled particle theory. This model, like the Labik and the BallanceSpeedy approximations,is expressed in a closed analytical form. Moreover, Boublik extended his model to fluids composed of a mixture of hard spheres and generalized convex hard bodies. One of the few shortcomings of this model is that it does not give thermodynamicallyconsistentvalues for the radial distribution function at contact and in the large solute limit (as discussed in section 111). (7)Finally, the hard fluid model used in this workoffers many of the advantages of the best of the above models. Like the BallanoeSpeedy, Labik, and Boublik models, the hard fluid model offers the practical advantages of a closed analytical form. Furthermore, in its most general form (see Appendix), the hard fluid model can be used to predict the heteronuclear cavity distribution function for cavities dissolved at arbitrary dilution in a solvent composed of an arbitrary mixture of hard spheres. Less general hard fluid results, pertaining to an infinitely dilute heteronuclear cavity pair in a single component hard sphere fluid, have previously been pre~ented.~’For an infinitely dilute homonuclear cavity pair in a single component solvent, these reduce to results derived independently by Yushimura and Nakahara.47 In section I11 we compare several of the above approximations with computer simulationsand the hard fluid model. The global accuracy of the hard fluid model proves to be at least as good as that of Boublik’s model, and in fact gives slightly more accurate values for the radial distribution function at contact and in the limit of a large solute diameter (or small solvent diameter). 111. Hard Fluid Model

The hard fluid model for y y ! ( r 1 2is) motivated by exact results in the continuum and ideal gas solvent limits. In a continuum

The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2297

Model for Shifts in Molecular Vibrational Frequencies solvent at pressure P the excess chemical potential change associated with the formation of a homonuclear hard sphere diatomic of bond length ~ I (I l a l ) from two separate spheres of diameter uI is given exactly by Ap""'(r,,) = -P(r/l2){2ul3

- 3a12rll+ rlI3)

+

uI3

U;

+

+

~

~

Monte Cirlo data 0 Torrie-Pstey A Barker-Henderson TheoreticII model8 Hard Fluid . ' Boublik BOublik Ballance-Speedy - Grundke-Henderson

.

(6)

This equation simply states that the reversible work in forming a hard diatomic from two infinitely separated hard spheres in a continuum solvent is equal to PAV, where AV is the difference in volume between the bound (overlapping) pair of cavities and the dissociated cavities. For a heteronuclearhard sphere diatomic of bond length r12(where I(UI - 4 / 2 1 5 r12I( U I+ 42)/2) and diameters u1 and u2 in a continuum solvent the corresponding result is

- 3-(a12 2 ~

l " l " " i " ' " " "

)

Again, this simply expresses the PAVwork associated with the formation of a diatomic in a hydrodynamic continuum, where AVis equal to the total volume of the diatomic solute minus the volume of the two isolated solute atoms. The above expressions have been derived simply by geometrical determination of AVas a function of ul, u2, and r12(the AVvalues are equivalent to those derived from previous expressions for the total volume of a diatomic as a function of the bond l e ~ ~ g t h ) . ~ ~ , ~ ' Equations 6 and 7 are exact not only for a continuum solvent but also for a solvent composed of hard spheres, in the low density limit. In this latter case the diameters uIand u2 must be replaced by those representing the volume excluded to the solvent centers by the solute diatomic. Accordingly, then ui becomes uis= ut uswhere usisthesolvent diameter. In thecaseofeq6 theresulting expression is closely related to a well-known low-density result for the radial (and cavity) distribution function of a single component hard sphere f l ~ i d . ~ ~ , ~ ~ The essential approximation used in deriving the hard fluid model involves assumingthat the functional dependenceof ApHS(r12) on r12 at arbitrary density has the same general form as in the above continuum and low-density expressions. Thus, the excess chemical potential change for the formation of a hard sphere diatomic from two separate hard spheres dissolved in a hard sphere fluid solvent is assumed to be expressible as

-

Density, p* = 0.8 08

10

00

10

05

20

15

Ill '0s

r Figure ~ 1. ~Distribution function for two identical hard sphere cavities in a hard sphere fluid of the same diameter (ui = u2 = us) as a function of the ratio of the cavity separation rI I to the contact radius us,for a reduced density of pu,3 = 0.8: points, from Monte Carlo simulations of Torrie and Pateys2 and Barker and Hender~on;~) solid curve, from hard fluid model; dot-dashed curve, from Boublik dotted curve, from Ballance and Speedy;44dashed curve, from Grundke and Henderson.22

+

- A C L ~ ~ ( ~ I ~ ) / KAI2 B T+=BI2rl2 + C12(rlJ3 + D12(1/rl2) (8) With this assumption, thecoefficients ,412, Biz, CIZ,and D12, and therefore the excess chemical potential of a diatomic, may be determined directly from the equationof state of the corresponding hard spherefluid (as shown in the Appendix). For a homonuclear diatomic, only three coefficients are nonzero, A II , BII , and CII; these are fixed using the very accurate@-5'Boublik-MansooriCarnahan-Starling-Leland42~~3(BMCSL) mixed hard sphere fluid equationof state, which is a generalization of the CarnahanStarling4I equation for single component fluids. For a heteronuclear diatomic, one additional constraint must be used to determine the fourth coefficient. In this case the heteronuclear B12coefficient is assumed to be equal to the arithmetic mean of the two corresponding homonuclear Bitcoefficients(which is again an exact result in the continuum and low-density limits). In the Appendix we give explicit expressions for A12,B12, CIZ,and 0 1 2 along with generalizationsof the results to other equation of state models. Equation l a relates the hard fluid cavity distribution function yrf(r12), directly to the excess chemical potential change in eq 8. Comparisonwith Simulation Results. Figure 1 compares four models for yis(rl I )in a neat hard sphere fluid with Monte-Carlo computer simulation result^.^^^^^ Corresponding results for the

Density, p* =0.8

-

-101

1

1

1

1

1

1

1

1

1

1

1

1

1

1

~

Ben-Amotz and Herschbach

2298 The Journal of Physical Chemistry,Vol. 97,No.10, 1993

a* lol = 0.794.

Yonle Carlo dab 0 Lsblk-Smith

-

Theoralleal Models

6 -2

Boubllk

-- - -Hard Fluld,BMCSL Percus-Yevick

Y

.

Labik-Smith, Monte Carlo Hard Fluid model

I

0

..... Scsled Pankle Theory

J

0

-2

p* = 0.7

Y

s L

-6

0.0

0.5

1.o

1.5

2.0

-8 I 1 2 '012

0.0

Figure 3. Distribution function for two nonidentical hard sphere cavities in a hard sphere fluid (ut = 0.693~1;U I = a,) as a function of the cavity a2)/2, separation 1 1 2 normalized to theseparation at contact, 4 1 2 = (01 for a reduced solvent density of pur) = 0.9. Points are from Monte Carlo simulationsof LabikandSmith,45 thedot-dashedcurveisfromthe Boublik model,& and the solid curve is from the hard fluid model (derived from the BMCSL equation of state). The long- and short-dashed curves are hard fluid results derived from the Percus-Yevick and scaled-particletheory equations of state, respectively (see Appendix for details). The logarithmic ordinate scale has the argument normalized to thedistribution function at zero separation. Notice that the slope of the distribution function is zero at small separations, where the smaller solute cavity (diameter a2) is embedded within the larger cavity (01).

+

diameters not equal to each other and/or to the solvent. Labik and Smith4s have recently provided simulation data for some such hard sphere fluid systems. In these simulations, one of the diatomic atoms has the same volume as the solvent and the other varies between l / j and 3 times the solvent volume. The reduced density range is wide, 0.3 Iper: S 0.9. Unfortunately, the results were reported only in graphical form (and attempts to acquire tabulated data from the authors were not successful),so manual digitization had to be used in order to compare with analytical model predictions. The most demanding test of the models is obtained using highdensity simulation data, since all become exact at low density. Figure 3 compares the hard fluid (solid curve) and Boublik (dotdashed curve) models with high-density simulation data for yHS(r,,) out to more than 1.8 times the contact value. Note that y Ius2( r I 2 )becomes constant at small cavity separations because the smallercavity then becomes imbedded within the larger cavity. Both the hard fluid and Boublik models give excellent agreement with the simulation results. The remaining two curves in Figure 3 (long-dashed and short-dashed) are discussed in the Appendix. Labik and Smith also generated data for the heteronuclear cavity distribution function in a binary mixture of hard spheres as a function of the mole fraction.4s In these simulations the reduced density of the fluid was held fixed at 0.7,within the liquid density range. Figure 4 compares the hard fluid model with these simulation results. Again the agreement is excellent, andextendsout well beyond thecontact separation (rl2/u12 = 1). Boublik's model gives essentially the same results.45 Clearly these models can be used to quantitatively predict cavity distribution functions in nondilute systems. Macroscopic Solute Limit. It is instructive to compare the predictions of the hard fluid and Boublik models in the macroscopic solute limit. As a spherical solute cavity becomes infinitely large its excess chemical potential, firs,should approach the continuum value, ficont= PAV, where AV is the volume of the solute. In order to examine the approach to this continuum limit, we plot the ratio of the excess chemical potential to PAV as a function of the inverse of the solute cavity diameter, ulrnormalized to the solvent diameter, us. This ratio should approach unity as us/uI approaches zero, i.e., as the solute diameter approaches infinity. Figure 5 shows the predictionsof both the hard fluid and Boublik models at three values of the reduced density, p*pu>. Again the

1.o

0.5

r 12

1.5

'012

Figure 4. Distribution function for two nonidentical hard sphere cavities in a hard sphere solvent (01 = a,; a2 = 0.794,), as in Figure 3 but for various mole fractions XI of the larger sphere, with the total reduced density heldfixed,p*pla13 +p2q3 = 0.7. Theabscissascaleisnormalized to the cavity separation at contact, a12 = (a1 4 / 2 .

+

p* = 0.1

0.5

- Hard Fluid model ..... Boublik model

-

1.o

0.0 ' ' ' ' ~ ' ' ' ' ~ ' ' ' ' 1 " ' ' ' ~ " ' ~ ' ' ' ' 0.0 0.5 1.0 1.5 2.0 2.5 3.0

4 Figure 5. Ratio of the excess chemical potential of a hard spherical solute to thePAVworkrequired tocreateaspherical cavity ina continuum solvent at an external pressure P,where the cavity volume is equal to the ~ . abscissa is the inverse solute's excluded volume A V = ( 4 / 3 ) ~ a , ~The of the solute diameter, al, normalized to the solvent diameter, a,. Curves are shown for three reduced densities: full curves, from hard fluid model; dashed curves, for the Boublik

two models are very similar, although the Boublik model slightly undershoots the exact continuum limit as u S / qapproaches zero. We note that the hard fluid results for the chemical potential of a spherical solute are implicit in the BMCSL hard sphere fluid equation of state, as detailed in the Appendix, and do not rely on approximations made to obtain eq 8. This produces more accurate large solute limit results than Boublik's model, although neither is rigorously exact in this limit.57 In summary, both the hard fluid and Boublik models are quite successful in predicting chemical potentials for hard diatomics in hard sphere fluids. The Boublik model is slightly less accurate when extrapolated to the large solute limit and does not yield thermodynamically consistent values for the radial distribution function at contact. On the other hand, the Boublik model has greater scope. It has been generalized to include convex hard bodies and three-cavity distribution functions, although the accuracy of these may not be as good as that obtained for hard sphere two-cavity distribution functions.46

IV. Application to Vibrational Frequency Shifts When a gas-phase molecule enters a solution, its vibrational frequenciesusually shift downward, because at ordinary densities attractive solute-solvent interactions tend to dominate. These shifts can be offset and reversed by compressing the solution to reduce the mean distance between molecules and thereby enhance

Model for Shifts in Molecular Vibrational Frequencies

The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2299

TABLE I: Experimental Frequency Shifts (at 25 f 2 "C) gas to liquid shiftsb (cm-I, i 0 . 3 ) P(kbar, i0.3) 12-methylcyclohexane 0.0 I1 atm) 1.1 3.3 5.2 7.2 9.3 10.2 11.6 12-n-hexane 0.0 (1 atm) 0.3 0.7 5.0 7.8 9.9 11.8 I2-n-octane 0.0 ( atm) 0.2 2.6 3 .O 5.9 pyridine-n-octane 0.0 ( atm) 0.4 2.1 2.2 3.7 4.7 5.9 pyridine-isooctane 0.0 (1 atm) 0.7 2.7 3.6 5.2 7.6 pyridine-CH2C12 0.0 (1 atm) 0.3 3.0 5.3 8.6 0.0 (1 atm) pyridine-acetone 1.5 3.6 6.1 8.7

pa

Av

Av

Av

( n ~ n - ~ )( 1 4 ) ( 2 4 ) ( 3 4 ) 4.697 5.10 5.49 5.71 5.87 6.00 6.05 6.12 4.583 4.77 4.95 5.78 6.03 6.17 6.27 3.686 3.77 4.28 4.33 4.59 3.687 3.84 4.21 4.22 4.41 4.50 4.59 3.633 3.89 4.29 4.40 4.54 4.70 9.423 9.67 10.92 1 1.49 12.04 8.151 9.04 9.71 10.22 10.58

-1.9 -1.5 -0.8 -0.1

0.8 1.7 1.7 2.2 -1.3 -1.1 -0.9 0.7 1.9 2.8 3.4 -1.6 -1.5 -0.5 -0.2 1 .o -2.1 -2.3 -1.1 -1.1 -0.2 -0.2 0.3 -2.0 -1.4 -0.2 -0.7 0.7 1.1 -1.2 -1.1 0.7 2.1 3.8 -1.9 -1.2 -0.2 0.7 1.7

-3.8 -2.9 -1.8 -0.5 1.2 2.6 3.0 3.8

-6.1 -4.7 -2.3 -0.6 1.6 3.8 5.9 6.0

Densities calculated using the Carnahan-Starling-van der Waals equation of state for pure solvent, P / p k T = (1 r ) 72 - $)/(l - 7)) - ~(?,/T)I),where r ) = (7r/6)pus3, using usand i svalues in Table I1 (ref 8). Gas-phase frequencies are taken from refs 58 and 59.

+ +

repulsive interactions. The pressure dependence of vibrational frequencies thus offers information about the solutesolvent interaction potential. Much data has accumulated exhibiting vibrational frequency shifts with solvation or c o m p r e ~ s i o n . ~However, ~ - ~ ~ little of it permits comparisonsof theory with experiment for a given solute molecule, both as a function of density and in different solvents. For this purpose we have measured frequency shifts (relative to gas-phasevalues)58.59 as a function of pressure for iodine in three solvents and pyridine in four. The pyridine shifts pertain to the symmetric ring breathing mode (usually designated VI), which is quasidiatomic in the sense that the frequency is related to the square root of the bond stretching force constant in the same way as for a diat0mic.3~Table I lists experimentalpressures,densities, and frequency shifts, and Table I1 gives pertinent solute and solvent parameters. Experimental Details. The experimental conditions and techniques used to measure vibrational frequency shifts at high pressure have been described previously.31 Briefly, Raman backscattering measurements are performed using a lOO-mW, 514nm CW argon ion laser and a Merrill-Bassett diamond anvil The sample pressure is measured using the ruby fluo-

rescence technique: and the CarnahanStarling-van der Waals equation of state is used to convert sample pressures to densities.8 The ambient temperature in these studies, and that at which the pressure measurements were performed, was 25 f 2 "C. During the I2 resonance Raman measurements the local solvent temperature was somewhat higher as a result of resonant absorption of the excitation laser light. However, laser intensity studies3' indicate that the reported frequencies are temperature independent to within the estimated experimental frequency measurement accuracy of f0.3 cm-I. Thus, the measured vibrational frequencies are taken to represent those at 25 "C. The solutes and solvents used in these studies were obtained from Aldrich at reagent or higher gradeof purity, and used without further puri.fication. Solute concentrations are 0.01-0.02 mol/L for 1 2 solutionsand from 20: 1 to 50: 1 volume dilution for pyridine solutions. Studies of pyridine in octane at both 20:l and 50:l dilution reveal no measurable effect of concentration on solute vibrational frequency in this concentration range. Vibrational FrequencyShift Model. The prototype theoretical treatments of solvent-inducedvibrationalfrequency shifts consider a diatomic oscillator immersed in a benign ~ o l v e n t . ~The > ~net ~-~~ frequency shift, Av = AYR PYA,is the resultant of a positive contribution from repulsion and a negative one from attraction. Since the shifts are small, typically only of the order of 0.156, perturbation methods are usually adequate. Thus, only the leading terms need be retained in expansionsof the vibrational potential UOof the isolated (gas-phase) solute oscillator,

+

and its interaction with the solvent, described by the potential of mean force,

Accordingly, the frequency shift can be expressed in terms off and g, the quadratic (harmonic) and cubic (anharmonic) vibrational force constants of the gas-phase solute molecule, and F and G, the analogous linear and quadratic coefficients of the potential of mean force. The resulting perturbation expression26 for the frequency shift can be written as

-

This gives the gas to liquid frequency shift Av for the u = 0 u = nvibrational transition, normalized to YO,the isolated molecule (gas-phase) frequency. The gas to liquid shift is predicted to be proportional to n, the vibrational quantum number (to second order in perturbation theory).26 The major dependence of the shift on the interaction with the solvent is simply proportional to F, the first derivative of the potential of mean force with respect to rI2. The contribution of the second derivative, G, is relatively small and will in fact be neglected in this work (as described in the following subsection). If instead of the cubic approximation of eq 9, the oscillator potential is represented by a Morse function, the only change is to multiply the two terms within the square brackets of eq 11 by additionalf a c t o r ~ . ~ O However, ? ~ ~ these factors are each very nearly unity (typically within l%), except when the vibrational amplitude becomes unusually large, as is the case for hydrogen vibrations. Evaluation of Repulsive Shifts. The repulsive contribution to the vibrational frequency can be computed from eq 11 using the hard fluid model to evaluate the F and G coefficients of the mean force potential. From eq 3, we obtain F FHSas the first derivative of the chemical potential, - k T l n y ~ ~ ( r , (eq ,) la), and with eq 8 have

-

Two different approximations to the quadratic repulsive solvation force have been employed in previous work,

2300 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993

Ben-Amotz and Herschbach

TABLE II: Molecular Parameters Solute Parameters

iodine (12) pyridine ( C S H ~ N )

213.3 991.4

2.666 1.380

4.0 3.4

4.0 3.4

0.0122 0.0146

4.0489 -0.0385

11.26 9.5'

0.00 2.19

Solvent Parameters attractive shift coefficients8 n-hexane methylcyclohexane n-octane isooctaneh CH2C12 acetone

T(OC)

uJ(A)

25 25 25 25 25 25

5.96 6.02 6.55 6.54 4.62 4.85

7/(K)

2008 2327 2397 2268 1816 1815

' a (A-3)

11.9 12.9 15.6 15.6 6.5 6.4

p P (D)

Ca(12) (cm-'/nm-))

-0 -0 -0 -0

C,(CsHsN) (cm-'/nm-')

-0.68 -0.86 -0.94

-1.19 -1.14 -0.48 -0.60

1.60 2.88

a Vibrational resonance frequencies, Uvib. and bond lengths, re,in the low-densityvapor phase (refs 58 and 59). Atomic and pseudoatomic hard sphere diameters, q,u2, are estimated using liquid X-ray data (see text). Force constants are obtained from vibrational frequencies and extended Badger's rule correlations (ref 77). Average polarizability calculated using ref 78. Polarizabilities (a [(3/4r)(M/p)(n2 - l)/n2 2)]) and dipole moments taken from ref 79. /Solvent diameters, us,and attractive force parameters, 'T~,obtained from the CarnahanStarling-van der Waals equation of state (ref 8). 8 Attractive force parameters areobtained from vapor-liquid vibrational frequency shiftsat 1 atm (see text, section IV). 2,2,4-Trimethylpentane.

-

+ *

Reduced solvent density, p'= po; 1 .o

1.2

1.1

1.3

I, / Methylcyclohexane 0

The first is a mean field a p p r ~ x i m a t i o which n ~ ~ equates GHSto half the second derivative of the mean force potential. This approximation ignores contributions to GHSfrom fluctuations in the solvent force.j2 The second approximation, suggested by O ~ t o b y ~ * and ~ ~ derived J ~ J ~ from v ~ ~binary collision theory, relates GHSto FHSthrough a repulsive force range parameter L 0.014(u1 u2 2uJ. For a homonuclear diatomic, GOxtoby and Gmean field are of opposite sign, and in either case contribute relatively weakly to the predicted frequency shift. Thus, for homonuclear (or nearly homonuclear) diatomics we find that GHSmay reasonably be taken to be zero.

+ +

The quantitative significance of this approximation is illustrated in Figure 6c (and discussed in section IV, Vibrational Frequency Shifts: Iodine). For heteronuclear diatomics, on the other hand (particularly those such as hydrides with very different atomic diameters), Gmean field and Coxtoby may have the same sign (positive), and contribute more significantly to the predicted frequency shift. Furthermore, since G, field represents a rigorous lower bound to the true G H S ,it~may ~ not be appropriate to set GHSequal to zero in such cases. This unique repulsive solvation force behavior, along with the apparently anomalous attractive frequency shift behavior of heteronuclear hydrogen-containing vibrations (as discussed in the next subsection), raises interesting issues whose resolution must be left to future studies. Unless otherwise stated, all the repulsive vibrational frequency shift predictions reported in this work are derived directly from eq 11 with F = F H Sand G = GHS= 0, evaluated at r12= re, the equilibrium bond length of the diatomic oscillator. These computations, in principle, involve no independently adjustable parameters. Aside from vibrational potential of the isolated solute molecule, the quantities required are the solvent density and temperature as well as the effective hard sphere diameters of the solute and solvent. The latter can be accurately estimated from equation of state data* and/or liquid X-ray structure data65-67 if available, or otherwise from molecular volume increment correlation^.^^.^^ In practice, very accurate estimates of the effective hard sphere diameter of the solvent (within 1%) are required in order to obtain quantitatively meaningful frequency shift predictions. In the absence of accurate independent measurements, the solvent diameter may be taken as an adjustable

-

0 Av (1 c 0)

0 Av (3 c 0)

5

0

1%;

I

I

I

I

I

I

s

I

Repulsive Force Models GMW Field a,= &02 A

I

I .

a

I

.

d

...

5.0 5.5 6.0 Solvent density, molecules/nm3 Figure 6. (a) Vibrational frequency shifts (gas to liquid) for 12 solute in methylcyclohexane solvent, as functions of the solvent density. Experimental points (from Table I) and solid curves obtained from hard fluid model (parameters in Table 11) are shown for the fundamental transition and first two overtones; for clarity, data for the overtones are shifted upward by 3 and 6 cm-I, respectively. Only one experimental frequency shift point (solid point) is used to constrain the theoretical predictions (by fixing the attractive force parameter C, in eq 14). Panels b and c represent the same fundamental frequency shift data shown in panel a, along with theoretically calculated curves illustrating the sensitivity of the hard fluid model to nonlinearity in the density dependence of the attractive frequency shift, alternative approximations for the quadratic solvation force coefficient, G. and the solvent hard sphere diameter, us (see text, section IV, Vibrational Frequency Shifts: Iodine, for discussion).

parameter in the model. This may in fact offer a convenient experimental procedure for determination of the effective hard sphere diameters of polyatomic m0lecules.2~ Evaluationof Attractive Shifts. For the attractive contribution to the vibrational frequency shift, we again employ a simple van der Waals mean field appro~imation.~J3J~ This approximation implies that the attractive shift is linearly proportional to the

Model for Shifts in Molecular Vibrational Frequencies

The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2301

solvent density,

A v A ( n C 0 ) = nCAp

(14)

The coefficient CA depends on the differential attractive solvation energy of the solute in the ground and excited vibrational states. This energy can in principle be expressed as a function of dispersiveand multipolarsolventdlute interactionsconvoluted by the solvent-solute radial distribution function. Such calculations involve additional and rough approximation^.^^ Instead, we follow the procedure of Schweizer and Chandler7 and determine CA empirically from the measured gas to liquid frequency shift at 1 atm. Once this is done the frequency shifts at arbitrary density, for both the fundamental and higher overtones, can be calculated. Aside from this choice of CA,there are no other adjustable parameters. In systemswith strong attractive interactions the linear density dependence predicted by the van der Waals approximation may break down. Analysis of pressure-induced shifts for several CH and OH hydrogen stretching vibrations indicates the attractive contributionshave an approximatelyquadratic rather than linear dependence on the solvent density.28bJ0J3 However, a linear dependence has been found in previous studiesof vibrational shifts for N2 as well as polyatomic C-C, C - C , C - 0 , and m N v i b r a t i o n ~ , ~ , 3 ~and J ~ .even ~ 6 ~for the C-H stretches of methane and ethane at low to moderate densities.34 Estimatesof Atomic and Molecular Mameters. Effectivehard sphere diameters for solvent molecules can often be obtained with an accuracy of 1% from equation of state correlations,8 and we use this method here. Solute atom diameters can also be obtained from such correlations in indirect ways. One procedure is to choose the atomic, or pseudoatomic, diameters so as to reasonablyrepresent the local sizeof thesolutemolecule on either side of the vibrating bond of i n t e r e ~ t . ~ JAnother ~ - ~ ~ procedure employs an isolated bond model in which solvent forces on each bond in the polyatomic are treated independently and summed using a normal mode ~ e i g h t i n g .Fortunately, ~ the predicted frequency shifts are less sensitive to the precise details of the solute model than they are to the solvent molecular diameter and density. Typically a 5-10% change in the solute pseudoatom diameter has the same effect as a 1-2% change in the solvent diameter or density. Here we obtain the solute pseudoatomic, diameters from RISM theoretical analysisof liquid X-ray s t r u ~ t u r e d a t a Theiodine .~~~ atom hard sphere diameters used to model 12 were determined by extrapolation of X-ray results for Br2,67by assuming that the ratio of the hard sphere diameters of iodine and bromine is the same as that derived from independent van der Waals diameter estimates.68 For the polyatomicsymmetric ring breathing mode of pyridine, we approximate the vibration as a pseudodiatomic with diameters appropriate for a CH group.69 These CH diameters are again taken from RISM analysis of X-ray diffraction data, in this case for liquid benzene.65.66 Table I1 includesthe solvent and solute diameters used in our calculations. Vibrational Frequency Shifts:Iodine. Figure 6a comparesour model calculations with the experimental frequencyshifts for the fundamental and first two overtones of I2 dissolved in methylcyclohexane as a function of the solvent density. The density, obtained from the CarnahanStarling-van der Waals parameters for the pure solvent,8is plotted in both molecular number density units (lower abscissa scale) and reduced units (upper scale). For clarity, the gas to liquid shifts of the first and second overtones are displaced upward by 3 and 6 cm,-l respectively. Before this displacement all three curves crossed at a reduced density near pu,3 = 1.2, where all the gas to liquid frequency shifts go through zero. Such a common crossing point for the fundamental and overtone vibrational shifts is predicted by eq 11; it occurs at a density such that the repulsive and attractive contributions, AVRand AvA, are equal in magnitude but opposite in sign. At other densities the

-

frequency shifts observed for the fundamental and the successive overtones are accurately in the ratio 1:2:3, again as predicted by eq 11. This is an important test of the basic approximations adopted in the model. As long as a second-order perturbation treatment is adequate, the shifts will be proportional to the quantum number n of the oscillator, provided it is sufficient to retain in eqs 9 and 10 just the first two terms in the expansions of the molecular vibrational potential and the potential of mean force. The attractive force shift coefficient CAof eq 14was determined from a single experimental gas to liquid shift value-that of the fundamental at 1 atm of pressure (solid point to Figure 6a). No other adjustable parameters or experimental frequency shift information was used in the hard fluid model calculations. The sensitivity of the theoretical predictions to the density-dependent form of the attractive contribution is illustrated by the dashed curves in Figure 6b, which compares the fundamental frequency shift data with predictions obtained by replacing the density p in eq 14 byp2(dot-dashes) or by p l / 2 (dashes), respectively. Clearly neither of these replacements fits the experimental results as well as the linear density-dependent attractiveshift model (solid curve). The sensitivity of the results to variation in the repulsive solvation force constant, GHS,is illustrated in Figure 6c. The mean field approximation (long dashes), defined in eq 13a, produces results that are indistinguishable from those obtained assuming GHS = 0, on the scale of this figure. The GOx,oby approximation (short dashes), defined in eq 13b, predicts a larger repulsive force shift, which does not agree as well with the experimentaldata. However, iftheeffective hard spherediameter of the solvent is decreased by only 2% (near the estimated uncertainty of the 6.02-A diameter derived from high-pressure compressibility data),8 the GOxtobypredictions are brought into agreement with experiment, and with the GHS= 0 predictions. This sensitivity to the solvent diameter is clearly illustrated by the dot-dashed curve in Figure 6c which represents the Gontoby predictionsusing a solvent hard sphere diameter of 5.74 A (about 5% smaller than 6.02 A) obtained from a rough hard sphere analysis of the solvent’s self-diffusion ~oefficient.~O It thus appears that, within the uncertainty in reasonable estimates of the effective hard sphere diameters of solvent molecules, alternative models for GHScannot be distinguished using the present experimental results. On the other hand, the results of this and other recent ~tudies’3.3~ suggest that the hard fluid model (with G = G,,, field) can be used to accurately predict pressure-induced vibrational frequency shifts of diatomic and polyatomic solute normal modes (with the exception of some hydrogen vibrations, as discussed above in the Evaluation of Attractive Shifts subsection), as long as the solvent diameters used as input to the hard fluid model are those obtained from solvent compressibility data, analyzed using the CarnahanStarling-van der Waals equation of state.8 The success of this combination of theoretical approximations and empirical diameters is perhaps linked to the fact that the CarnahanStarlingvan der Waals equation is very similar in spirit (and underlying approximations) to the hard fluid model. Thus, these two models are able to reproduce both high-pressure compressibility and vibrational frequency shift data with the same set of solvent hard sphere diametem8 Repulsionvs Attraction. Figure 7 comparesthe frequency shifts of the I2fundamental with n-hexane and n-octane as solvents. To illustrate clearly the competitionbetween repulsive and attractive forces, we show curves obtained from the hard fluid model for the entire gas to liquid density range. Again, the CAparameter for each solvent was determined from the frequency shift at 1 atm, the only experimentalpoint used in deriving the model results. At low densities, the frequncy shifts are negative (red shifts), since the attractive forcesare dominant. As the density increases, the shifts become more negative, reach a negative maximum, and then abruptly climb through zero and become strongly positive (blue shifts) when the repulsive forces become dominant.

Ben-Amotz and Herschbach

2302 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 l2 / n-Hexane 0 Ip / n-Octane Hard Fluid model 6.55 A uWer 5.96 A, 0

r

-

E 0

I

I I

'

I

-

-

E,

I I I ,

I

1

2

1

'

1

'

' 3

m "

1 '

I

4

a

'

"

I

5

"

'

I

a

0

6

Solvent density, molecules/nm3 Figure 7. Vibrational frequency shifts (gas to liquid, fundamental transition) for 12 solute in n-hexane and n-octane solvents, with curves obtained from the hard fluid model. Asterisks mark points at which the reduced density of the solvent, pas3,is equal to 1. In each case the lowest density experimental point is used to constrain the theoretical predictions (by fixing the attractive force parameter CA in eq 14). No other parameters are adjusted in order to obtain agreement with the highdensity frequency shift measurements.

The maximum red shift occurs at about the same 'most comfortable" value of the reduced density, pus3= 0.8, for both solvents. Likewise, the frequency shifts pass through zero, indicating that the attractive and repulsive forces cancel out, at about the same *balance" value of the reduced density, pus3 1.2, for both solvents (as in Figure 6). The steep onset of the blue shift, which begins near a reduced density of pus3 1 (marked by the asterisks in Figure 7), is thus a direct measure of the solvent diameter. It occurs a t a lower density in n-octane than in n-hexane as a consequence of the larger size of the n-octane molecule. Thegood quantitativeagreement between the observed frequency shifts and the model predictions in the high-density regime again confirms the utility of the hard fluid model, when employing solvent hard sphere diameters derived from highpressure equation of state data.* The magnitudes of the attractive forcecoefficients CAcorrelate roughly with the polarizability cx of the solvent molecules, as seen in Table 11. This is consistent with the expectation that dispersion forces should chiefly govern the attractive interaction between nonpolar m0lecules.~J3~~~ This correlation suggests that CAvalues for I2 in alkane solvents can be estimated (to within about 10%) from CA zs 0.061a, in the units of Table 11. Plots of gas to liquid frequency shifts vs the full range of density, as in Figure 7,make evident the competition between attractive and repulsive forces. This is not apparent in the customary presentation of vibrational frequency shifts, which display displacements from the ambient liquid rather than the gas-phase reference frequency as a function of pressure rather than density. Such plots are typically linear over the experimental range. Figure 8 illustrates this for the same data shown in Figure 7. The points for I2 in n-hexane and n-octane now fall on a common line, in contrast to thedistinct curves of Figure 7. Such linearly pressuredependent, and nearly solvent-independent, vibrational shift behavior is suggestive of that expected in a continuum solvent, in which no molecular information about solvent-solute interactions is retained (as discussed further below). On the other hand, vibrational frequency shifts are in fact very sensitive to solvent size and the strength of long-range attractive interactions. This sensitivity is clearly evident in Figure 7 as well as in the solid and dashed lines in Figure 8. These lines show results obtained for I2 in hexane using, respectively, the same parameters as in Figure 7 (and Table 11) or adjusting the solvent diameter, us= Ohex, upward by 0.5%, a change sufficient to give very closeagreement with theexperimental points. Similar results (not shown) were obtained for the octane case by adjusting us

-

-

2

4

6

8

1

0

1

2

Pressure, kbar Figure 8. Vibrational frequency shifts (relative to ambient liquid at 1 atm) for I2 solute, plotted versus pressure. Experimental points are the same as those in Figure 7 (Table I). The solid line is from the hard fluid model for I? in n-hexane with the same parameters (Table 11) as in Figure 7 . The dashed line shows the result with the hexane diameter changed from 5.96 to 5.99 A. The dot-dashed line is from the continuum model, as described in the text (section IV, Repulsion vs Attraction).

downward by 0.5%. Such adjustments are within the -1% accuracy of the solvent diameters obtained from equation of state data analysis.* The linearity exemplified in Figure 8 suggests invoking a simple continuum model, in which the force on the solute is related to thePAVworkassociatedwithstretching thesolutebond immersed in the solvent. It was just this notion which led us to develop the hard fluid model as a generalization of continuum and ideal gassolvent relations. To illustrate why a substantial development beyond a simplecontinuum solvation model is required, we include in Figure 8 a line (dot-dashed) derived from eq 11 by taking F equal to the bond length derivative of PAV (and G = 0), where P is equal to the experimental pressure and AV is equal to the change in volume excluded to the solvent as the solute bond stretches. Not only is the slope of the continuum predictions in these high-pressure liquids much larger than the experimental one, but the continuum model leaves us with no clue as to the origin of the red gas to liquid shifts typically observed for molecular vibrations. Thus, although a simple continuum solvation model may offer a useful heuristic view of solute-solvent interactions, a more sophisticated statistical mechanical model is clearly necessary in order to extrezt meaningful molecular information, pertaining to the delicate balance of attractive and replusive solvent-solute interactions, from experimental vibrational frequency shifts. Vibrational Frequency Shifts:Pyridine. A more severe test of the hard fluid model is posed by frequency shift results for a polyatomic solute. Figure 9 displays our data for the symmetric ring breathing mode of pyridine in four solvents, using the same format as in Figure 7. Again, the density dependence exhibits the attractive and repulsive components of the shift, and the onset of a strong repulsive force induced blueshift occurs near a reduced density of pu,3 = 1, marked by asterisks in Figure 9. The attractive force coefficient can again be related to the polarizability of the two alkane solvents, CA = 0 . 0 7 4 ~(in ~ the units of Table 11). Interestingly, even in CHzClz, a strongly dipolar and much less polarizable solvent, the attractive frequency shift scales with solvent polarizability with nearly the same coefficient as in the alkane solvents, CA= 0 . 0 7 5 ~ ~Only . in the most dipolar solvent, acetone, is this coefficient significantly larger, CA = 0.094a, suggesting a dependence of the attractive force shift on more than just the polarizability of the solvent. This behavior is consistent with results recently reported for acetonitrile in various solvents, which shows a similarly weak dependence of CA on the solvent dipole Caution should be exercised, however, in making too much of slight deviations from a simple

Model for Shifts in Molecular Vibrational Frequencies

Solvent density, molecules/nm3 Figure 9. Vibrational frequency shifts (gas to liquid) for the symmetric ring breathing fundamental mode of pyridine solute in four solvents, with curves obtained from the hard fluid model. Asterisks mark points at which the reduced density of the solvent, pu,3, is equal to 1. In each case the lowest density experimental point is used to constrain the theoretical predictions (by fixing the attractive force parameter CAin eq 14). No other parameters are adjusted in order to obtain agreement with the high-density frequency shift measurements.

correlation between CAand a,since the nominal values of CAare uncertain by 10%or more,29as they depend strongly on the value adopted for us (and the latter is uncertain by at least 1%). The number density at which repulsive forces become dominant is more than twice as high in the smaller polar solvents as in the larger alkanes. When normalized to the solvent size, however, these systems become much more similar, although the repulsive forces actually become dominant at slightly lower reduced densities for the polar systemsthan for the alkanes. For example, the net zero shifts occur at a balance reduced density of pus3= 1.2 in the alkane solvents, but at a lower density of pus3= 1.1 in the polar solvents. Likewise, the maximum negative shifts occur at a most comfortable reduced density of pus3= 0.9 in the alkane solvents but at pus3 = 0.7 in the polar solvents.

-

V. Discussion The hard fluid model for the two-cavity distribution function is simple to implement, even for nonidentical cavities at arbitrary dilution in a hard sphere solvent, with an arbitrary number of components of different diameters. This generality, formulated in the Appendix, is a significant advantage over most other available analytical models for ycs(r12)(as discussed in sections I1 and 111). The hard fluid model is also quite accurate, as shown by comparison with Monte Carlo simulations (Figures 14), out to solute atom separations greater than 1 . 5 ~ ~50% 2 , beyond the contact distance. Only the model of B ~ u b l i krivals ~ ~ the hard fluid model in accuracy and generality. The hard fluid model, however, gives better values of y 3 r 1 2 )at contact (which are, by construction, self-consistent with the equation of state in the hard fluid model), and in the large solute limit (Figure 5 ) . Applications to the analysis of solvent-induced vibrational frequency shifts give quite satisfactory results. For molecules with quasidiatomic modes, the treatment outlined in eqs 9-14 offers a practical means to calculate the solvation forces and consequentshifts. There are two distinct parts to the treatment: mechanical and statistical. The mechanical part is a secondorder perturbation treatment of a perturbed cubic anharmonic oscillator; this was long ago provided by Drickamert3in a classical version and by Buckingham26in a quantum version, and later refined by Dijkman and van der Maas.62 The essential statistical part requires evaluating the mean force potential by means of the two-cavity distribution function; this is of recent vintage, due

The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2303

chiefly to Chandler and co-worker~.~J~ However, the two-cavity distribution function employed was restricted to a pair of identical spheres at infinite dilution in a single component solvent (a, = u2 f us). The hard fluid model for the two-cavity distribution has now considerably enlarged the scope of the treatment. It is remarkable that such a simple treatment gives a good account of the density dependenceof vibrational frequency shifts, as illustrated by the comparisons with our experimental data for frequency shifts of iodine (Figures 6-8) and pyridine (Figure 9 ) and other recent r e s ~ l t s . ~Although ~ J ~ ~the ~ ~treatment ~ has one adjustable parameter, the attractive force coefficient CAof eq 14,this is fixed by a datum at 1 atm. Thus, the efficacy of the model in accurately describing the density dependence at high pressures is significant. The model relies on the hard sphere cavity distribution function in approximating the packing forces in real liquids, and makes use of effective diameters obtained from data entirely independent of the frequency shifts. It is heartening that these ingredients serve so well, although the performance of the model indeed seems better than could be expected from the input approximations. Here we comment on four aspects which are pertinent to other anticipated applications of the hard fluid model. Nonsphericity. The good agreement between observed and predicted frequencyshifts is not restricted to moderately compact or nearly spherical solvents. For instance, methylcyclohexane and isooctane (2,2,4-trimethylpentane)are considerably more compact than n-octane, yet for all these solvents the repulsive packing forces seem to be modeled equally well by hard spheres. Apparently, the effective hard sphere diameters obtained from equation of state correlations8 include the major effects of nonsphericity. This suggests that the role of these effects must be quite similar in both the solvent-inducedfrequency shifts and the equation of state. High-Density Limit. The quantitative success of the hard fluid model at the highest experimental pressures (P 1 10 OOO atm) is curious. At such high pressures the reduced densities, p* = psus).of the hard sphere fluid reference systems become quite high, typically approaching, or even exceeding, the random closepacking (RCP) density?* p * ~ ~=p 1.23, although usually remaining below that for a face-centered-cubic (FCC) closepacked lattice, p*Fcc = 1.41. Such extremely high densities are well above those at which the BMCSL equation of ~tate,4*.~3 which we use in evaluating the hard fluid model coefficients of eq 8 (as detailed in the Appendix), can reliably be tested against computer simulations. In any case, it is certain that at these high densities the BMCSL equation underestimates repulsive force pressures and chemical potentials. This is most clearly evident at p * ~ c p= 1.23 where the compressibility factor 2 = P/pkT of the hard spheresystem should approach infinity while the BMCSL equation predicts Z = 43 for a neat hard sphere fluid at this density. On the other hand, real molecules are expected to have lower packing energies and pressure at high density than hard spheres. This is because nonspherical and softly repulsive molecules can pack more efficiently than hard spheres. Thus, the success of the hard fluid model in predicting packing forces in real liquids appears to be due, at least in part, to a fortuitous agreement between the BMCSL equation of state and repulsive contributionsto solvation energetics in real liquids. Similar conclusions can be drawn from the remarkable success of the CarnahanStarling-van der Waals equation of state, a close relative of the BMCSL equation, in predicting high-pressure compressibilities for a wide range of molecular fluids.* Pseudocontinuum Aspects. In many applications,particularly those involving the solvationof macroscopicobjects, a real liquid can reasonably be modeled as a hydrodynamic continuum. In fact a significant degree of umemoryn of the macroscopic hydrodynamic properties of liquids appears to remain on the molecular scale.ls73 Such a memory is evident in the linear pressure

Ben-Amotz and Herschbach

2304 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 dependence of vibrational frequency shifts (Figure 8), which is consistent with that anticipated from a simple continuum PAV argument. A simple continuum model cannot, however, quantitatively account for a pressure derivative of the frequency shift, as illustrated in Figure 8. The reason for this deserves emphasis. It can be inferred from the hard sphere chemical potential results shown in Figure 5 , which clearly illustrate the negativedeviations from continuum PAY predictions for the chemical potential of a solute sphere of comparable size to the solvent, u,/u1 = 1. In other words, the effective microscopic pressure on a hard sphere solute dissolved in a dense solvent is smaller than that which would be predicted for a solute of the same excluded volume, dissolved in a continuumsolventat the same macroscopic pressure. As the density ( p * ) decreases the solute chemical potential approachesPAVpredictions (see Figure 5 ) . This may reasonably be attributed to a reduction in the shielding effects of a solvent cage as the solvent cage disintegrates. Thus, the qualitative explanation for the smaller slope of observed vibrationalfrequency shifts than that predicted by the continuum model is that the shieldingeffects of a solvent cage increase with increasing pressure. In other words, increased shielding at high pressure leads to a reduced pressure derivative of the frequency shift. The often observed near linearity of the total (repulsive plus attractive) frequency shifts with pressure is strikingly reminiscent of the behavior expected in a continuum solvent. From a molecular, perturbed hard sphere, point of view it must arise from a nearly exact cancellation of nonlinearities in the pressure dependencies of repulsive and attractive contributions. Note that although the van der Waals model predicts a linear density dependence for the attractive frequency shift, the pressure dependence of the attractive shift is nonlinear as a result of the decreasing compressibility of liquids with increasing pressure. Again, fortuitous cancellations appear to be conspiring to make molecular behavior in complex fluids appear simple. Some insight into this intriguing behavior is afforded by noting that the slope of the vibrational frequency shift represents the difference between the chemical potential (partial molar Gibb's free energy) of solvation for the first excited and ground vibrational states of the solute. Thus, the pressure derivativeof the frequency shift at constant temperature is related in a simple way to the volume change of the system (soluteplus solvent) upon vibrational excitation, AV,ib,

As pointed out in early work by B~ckingham?~ the influence of nonequilibrium solvation could in principle be observed by comparing Stokes and anti-Stokes vibrational frequencies. Buckingham's estimates, based on inhomogeneous line widths of vibrational transitions, suggest that nonequilibrium solvation should contribute less than 0.1 cm-'to the gas to liquid frequency shifts of I2or pyridine in solution. Such shifts are smaller than our estimated frequency error of f0.3 cm-I. These estimates along with the clear success of the hard fluid model in predicting high-pressure gas to liquid frequency shifts appear to justify the neglect of nonequilibrium solvation effects in modeling vibrational energetics. Other Applications. Special cases of the general hard fluid model results presented in the Appendix have recently been applied to other frequency shift33~3~+56b and statistical thermodynamic c a l ~ u l a t i o n s .The ~ ~ ~hard ~ ~ ~fluid distribution function for two heteronuclearcavities dissolvedat infinite dilution in a hard sphere solvent has been used to predict repulsive frequency shifts for three normal modes of acetonitrile (CH3CN) dissolved in various solvent^.)^ Good agreement with high-pressure shifts of the CC and CN stretch vibrations was obtained, while the CH stretch appeared to have a nonlinearly density-dependent attractive frequency shift, as found for some other hydrogen-stretching vibrations.30 Lower pressure supercritical fluid vibrational frequency shift studies of both the CC and CH stretch modes of ethane as well as the N N and C H stretch modes of nitrogen and methane have been found to agree with the hard fluid model predi~tions.3~J6b Furthermore, the hard fluid cavity distribution function has been used in calculations of Lennard-Jones fluid thermodynamicpropertiesand radial distribution functionsS5and the prediction of reaction volumes and free energies for model diatomic dissociationreactions, as a function of the so1vent:solute size ratio, density, pressure, and temperature.s6a These have, again, been found to be in good agreementwith availablecomputer simulation measurements.

Acknowledgment. This work was supported by the Exxon Education Foundation and a Presidental Young Investigator Award from the National Science Foundation (for D.B.-A,). The use of experimental Raman and Ruby fluorescence equipment at the Exxon Research and Engineering Co. in Annandale, NJ, in the laboratory of Hubert King Jr. are gratefully acknowledged, as are useful discussions with Harry Drickamer, Mitch Zakin, and Rich LeSar. Appendix: Hard Fluid Cavity Distribution Function

where h is Plank's constant and cis the speed of light. Thus, the fact that vibrational frequency shifts are typically found to scale linearly with pressureut liquiddensities indicates that the volume change of the system upon vibrational excitation of the solute is nearly pressureindependentunder these conditions. On the other hand, below liquid densities vibrational frequencies typically red shift with increasingp r e s ~ u r e , 2 ~ reflecting , 3 ~ , ~ ~ a slight contraction of the system volume upon vibrational excitation. This change in the sign of AVvib arises from the dominance of attractive solvation forces which, at low density, favor the more polarizable excited vibrational state. Nonequilibrium Solvation. The hard fluid model is an equilibrium model for solvation energetics. Serious questionsmay be raised concerning the appropriateness of such an equilibrium model for vibrational frequency shift calculations.~5 There is no reason to expect solvent structure to remain in equilibriumaround the solute upon vibrational excitation, at least not on the time scale of an infrared absorption or Raman scattering event. On the other hand, the change in the equilibrium bond length of a solute upon vibrational excitation is typically of the order of 0.01 A, and so it may not be unreasonable to anticipate relatively little change in the solvation energy due to nonequilibrium solvation effects.

The equation of state of a fluid composed of a mixture of hard spheres with total number density p , containing n componentsof diameters ui and mole fractions xi,can be expressed in the following general form:

with

This expression includes, as special cases, the very a c c ~ r a t e ~ ~ - s ' BMCSL equation of state42.43 (which is used as the basis of the hard fluid model results reported in this work), the Percus-Yevick equation of state (derived using the virial theorem),75the scaledparticle-theoryequation of state (equivalent to the Percus-Yevick compressibility e q ~ a t i o n ) , ~and 5 *the ~ ~recently proposed BoublikKolafa equation (whose predictions are very similar to those of the BMCSL e q u a t i ~ n ) . ~The ~ . ~corresponding ~ am,D,, and ym coefficients are given in Table 111.

Model for Shifts in Molecular Vibrational Frequencies

The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2305

TABLE III: Coefficients for Hard Sphere Fluid Equations of State equation ao a1 a2 Bo 81 82 YO Y I 72 BMCSL Percus-Yevick scaled particle theory Boublik-Kolafa

1 - 2 1 1 - 2 1 1 -2 1 1 -2 1

3 - 3 0 3 - 1

3 - 3 0 3 - 3 3 -3 0 3 0 3 -3 0 3 -21,

0 0 0 -21,

hard sphere fluid:

These integrals are the same for all equation of state models of the form of eq Al, and the first five can be expressed as I l ( € ) = q - - l )1

In addition to determining the pressuretemperature-volume relationship of a fluid, the equation of state of a mixed hard sphere fluid also constrains certain fixed points of the two-cavity distribution function.22~~6 These points pertain to two distances of separation of the pair of cavities: (1) when the cavities are in contact, so r12= u12,and (2) when the smaller cavity is contained within the larger one, so r12 I(112, with the distances given by

Thefued pointsspecify thevalueof thecavitydistribution function at contact, y12(u12),as well as its value and logarithmic slope, a(ln y(r12))/ar12,at zero separation, Le., where r12Ia12. Thus, the equation of state of a mixed hard sphere fluid can be used as a starting point in the development of general models for the two-cavity distribution f~nction.1~~22 Here we present explicit relationships between the fixed points of the cavity distribution function and equations of state of the form of eq Al, and use these to define the hard fluid model for y12(r12). The value of the cavity distribution function at contact can be expressed as

(1 -€I3

The density-independent c k coefficients depend on the equation of state and are given, in terms of the coefficients in Table 111 (with ai = Bi = yi = 0 whenever i < 0 or i > 2), by c k

=

+ (cyk - a k - l ) + ((bk-1 - @ k - 2 ) f 2 b i + 2

2

- 8k-2)fI + 3(Yk-2 - Yk-3)f2 i‘i + ([(4 - k)ak-l + W f o + [ ( 5 - w o k - 2 + (k- 1)Bk-IIf2fI + [(6 - k h k - 3 + (k - 2 ) ~ ~ - ~ 1 f ~ (~~1 1g 0i 3) where Bk = [4!/[k!(4 - k)!]](-l)k are binomial coefficients (defined such that Bk = 0 when k > 4) and S;.= ti/&are density((16k-I

with sij = uiuj/(ui+ uj) and the coefficient C equal to 2 , 3 , 0 , or independent ratios involving weighted sums of the diameters (see (7 - &)/3 for the BMCSL, scaled-particle-theory, Percuseq Yevick, or Boublik-Kolafa equations, r e s p e ~ t i v e l y . ~ ~ , ~ ~ , ~ ~ Alb). Only the first three ck’s are nonzero in the scaledparticle-theory equation while all of the first five Ck’s are nonzero The derivativeof the solute two-cavity distribution function at in the Boublik-Kolafa equation. For the BMCSL equation only zero separation is related to the value at solvent-solute contact the first four Cis are nonzero, and these can be expressed as by22 n cl = 1 3f2Uj 3flU; foU; (A1 l a )

+

+ C3 = 3 + 3f2ai + [3fl-

+

+

+ [6fIf2 - 2303~; (A1 lb) 12f22]~;+ [fo-6flf2 + 8f;lu;

C2 = -3 - 6 f 2 ~ j [9f2’ - 6 f I ] ~ ;

where hi, is the Kroneker delta function (hi, = 1 when i = j and 0 otherwise), Pk is the number density of the kth component, and n is the total number of components. The value of the cavity distribution function at zero separation is related to the equation of state through the compressibility equation,22

This zero separation value of the cavity distribution function has a particularly important physical ~ignificance.~~ It is directly related to the excess chemical potential, pys (relative to an ideal gas at the same temperature and density), of an isolated cavity of diameter ui dissolved in a hard sphere fluid, byS = k,T

In yii(0)

(‘46) where yii(0) = yij(rijIaij)with cri < uj (as implied by eq A4, which requires the slope of y12(r12) to be zero whenever the smaller solute cavity is imbedded within the larger solute cavity). For equations of state of the form of eq A1 we can express the zero separation value of the cavity distribution function as a sum, 5

The coefficients c k are density independent, and the functions Ik([) depend only on the packing fraction, t €3, of the mixed

(A1 lc)

c,

= -1

+ 3f;u;

- 2r;u;

(A1 Id)

Armed with these relations between the equation of state models and the values of the cavity distribution function at contact and zero separation, we can readily determine the coefficients Al2, B12, C12, and D12in eq 8 (and eq la) for the cavity distribution function of the hard fluid model. When the two cavity diameters are equal, we assume, in keeping with the exact continuum result expressed in eq 6, that the coefficient 0 1 2 = Dll is identically equal to zero. The remaining three coefficients can therefore be fixed by the three thermodynamic relations involving the cavity distribution function at contact and zero separation. When the two cavity diameters are not equal, an additional constraint must be applied in order to fix the D12coefficient. For this purpose, we invoke another relation which is exact in the continuum and ideal gas limits. This constrains the coefficient 8 1 2 for two unlike cavities to be equal to the mean value of that for two like cavities,

4 , = (41+

(A121 The four coefficients in eq 8 are then uniquely determined by the hard sphere fluid equation of state through the following relations:

2306 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993

Ben-Amotz and Herschbach

(28) (a) Schindler, W.; Sharko, P. T.; Jonas, J. J . Chem. Phys. 1982, 76, 3493. Lee, Y. T.; Wallen, S. L.; Jonas, J. J. Phys. Chem. 1992, 96, 4282. (29) Zakin, M. R.; Herschbach, D. R. J. Chem. Phys. 1986.85. 2376. (30) Zakin, M. R.; Herschbach, D. R. J . Chem. Phys. 1988, 89, 2380. (31) Ben-Amotz, D.; Zakin. M. R.; King, H. E., Jr.; Herschbach, D. R. J . Phys. Chem. 1988, 92, 1392. (A13b) (32) LeSar, R. J . Chem. Phys. 1987,86,4138. (33) Ben-Amotz, D.; Lee, M.-R.; Cho, S.Y.; List, D. L. J . Chem. Phys. 1992, 96, 8781 and references therein. (34) Ben-Amotz, D.; LaPlant, F.; Shea, D.; Gardecki, J.; List, D. In Supercritical Fluid Z’echnology;ACS Symposium Series 488; Bright, F. V., McNally, M. P., Eds.; American Chemical Society: Washington, DC, 1992; p 18. (35) Myers, A. B.; Markel, F. Chem. Phys. 1990, 149, 21. (A13d) (36) Josien, M.-L.; Nelson, R. J . Chem. Phys. 1954, 22, 1169. Bayliss, N. S.; Cole, A. R. H.; Little, L. H. Ausr. J . Chem. 1954,8, 26. Thomas, B. The hard fluid model for the two-cavity distribution function in H.; Orville-Thomas, W. J. J . Mol. Struct. 1969,3, 191. Perrot, M.; Turrell, a mixed hard sphere fluid is obtained by inserting these coefficients G.; Huong, P. V. J . Mol. Spectrosc. 1970, 34,47. Kiefer, W.; Bernstein, H. J. J . RamanSpectrosc. 1973, I , 417. Wright, R. B.; Wang, C. H. J. Chem. into eq 8 (and eq la). This result can be used for heteronuclear Phys. 1973, 58, 2893. Bekarek, V.; Mikulecka, A. Collect. Czech. Chem. diatomic cavities at arbitrary dilution in a hard sphere solvent Commun. 1978,43,2879. Constant, M.; Fauquembergue, R. J . Chem. Phys. with an arbitrary number of components. 1980,72,2459. Tanabe, K.; Hiraishi, J. Ado. Mol. Relax. Interact. Processes 1980, 16, 281. Clouter, M. J.; Kiefer, H.; Jain, R. K. J . Chem. Phys. 1980, All of the cavity distribution function and frequency shift results 73, 673. Schindler, W.; Jonas, J. J . Chem. Phys. 1980, 73, 3547. Put, J.; Maes, G.; Huyskens, P.; Zeegers-Huyskens, Th. Spectrochim. Acta 1981, presented in this work have been obtained using the BMCSL 37A, 699. Persson, I.; Sandstrom, M.;Goggin, P. L. Inorg. Chim. Acta 1987, equation of state parameters (see Table 111). This equation of 129, 183. state is significantly more accurate than the Percus-Yevick or (37) Ree, F. H.; Keller, R. N.; McCarthy, S. L. J . Chem. Phys. 1966,44, scaled-particle-theory equations. The Boublik-Kolafa equation, 3407. (38) Labik, S.; Malijevsky, A.; Nezbeda, 1. Mol. Phys. 1987, 60, 1107. on the other hand, is essentially indistinguishable from the (39) Percus, J. K.; Yevick, G. L. Phys. Rev. 1958, 110, 1 . BMCSL equation.49 The effects of the equation of state models (40) Lebowitz, J. L. Phys. Rev. 1964, 133, A895. on the corresponding hard fluid cavity distribution functions are (41) Carnahan, N.; Starling, K. J . Chem. Phys. 1969,51, 635. (42) Boublik, T. J . Chem. Phys. 1970.53, 471. illustrated in Figure 3, which compares the BMCSL (solid curve), (43) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W., Percus-Yevick (long-dashed curve), and scaled-particle-theory Jr. J . Chem. Phys. 1971,54, 1523. (short-dashed curve) results with Labik and Smith’ss i m ~ l a t i o n s . ~ ~ (44) Ballance, J. A.; Speedy, R. J. Mol. Phys. 1985, 54, 1035. (45) Labik, S.; Smith, W. R. J . Chem. Phys. 1988, 88, 1223. The Boublik-Kolafa hard fluid results (not shown) areessentially (46) Boublik,T.Mol.Phys. 1986,59,775. Notethateq51 ofthisreference indistinguishable from the BMCSL hard fluid and Boublik model should read VHHD= ~ / 1 24 { rather than V,,,, = ~ / l 6 ...). [ results shown in Figure 3. Clearly all the above equations of (47) Yoshimura, Y.; Nakahara, M. Ber. Bunsen-Ges. Phys. Chem. 1986, 90, 58. state produce reasonable cavity distribution function predictions, (48) de Boer, J. Rep. Prog. Phys. 1949, 12, 305. although the BMCSL hard fluid (and Boublik) predictions are (49) Boublik, T.; Nezbeda, 1. Collect. Czech. Chem. Commun. 1986,51, obviously better than the Percus-Yevick and scaled-particle2301. (50) Jackson, G.; Rowlinson, J. %van Swol, F. J . Phys. Chem. 1987.91. theory results. 4907. (51) Kranendonk, G. T.; Frenkel, D. Mol. Phys. 1991, 72, 715. (52) Torrie, G.; Patey, G. N. Mol. Phys. 1977, 34, 1623. References and Notes (53) Barker, J. A.; Henderson, D. Mol. Phys. 1971, 21. 187. (54) Barker, J. A.; Henderson, D. Annu. Rev. Phys. Chem. 1972, 23, ( I ) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; 2554. Academic Press: New York, 1986; see also work cited therein. (55) de Souza, L. E. S.; Ben-Amotz, D. Mol. Phys., in press. (2) Chandler, D.; Weeks, J. D.; Andersen, H. C. Science 1983,220,787. (56) (a) Ben-Amotz, D. J . Phys. Chem., this issue. (b) Devendorf, G. S.; (3) Barker, J. A.; Henderson, D. Rev. Mod. Phys. 1976, 48, 587. Ben-Amotz, D. J . Phys. Chem., this issue. (4) Longuet-Higgins, H. C.; Widom, B. Mol. Phys. 1964, 8, 549. (57) Waisman, E.; Henderson, D.; Lebowitz, J. L. Mol. Phys. 1976, 32, (5) Snider, N. S.; Herrington, T. M. J . Chem. Phys. 1967, 47, 2248. 1373. Staveley, L. S. K. J . Chem. Phys. 1970, 53, 3136. (58) Kiefer, W.; Bernstein, H. J. J . Raman Specrrosc. 1973, I , 417. (6) Zwanzig, R. W. J . Chem. Phys. 1954, 22, 1420. (59) Wong, K. N.; Colson, S. D. J . Mol. Spectrosc. 1984, 104, 129. (7) Schweizer, K. S.; Chandler, D. J . Chem. Phys. 1982, 76, 2296. (60) Merrill, W.; Bassett, H. J. Rev. Sci. Instrum. 1974, 45, 290. (8) Ben-Amotz, D.; Herschbach, D. R. J . Phys. Chem. 1990, 94, 1038. (61) Barnett, J. D.; Block, S.; Piermarini, G. Rev. Sci. Instrum. 1973, I , (9) Alder, B. J.; Hecht, C. E. J . Chem. Phys. 1969,50,2037. Alder, B. 44. J.; Young, D. A.; Mark, M. A. J . Chem. Phys. 1972, 56, 2013. (62) Dijkman, F. G.; van der Maas, J. H. J . Chem. Phys. 1977,66,3871. (IO) Llano-Restrepo, M.; Chapman, W. G.J. Chem. Phys. 1992,97,2046. (63) Oxtoby, D. J. Chem. Phys. 1979, 70, 2605. ( I I ) Johnston, K. P.; Eckert, C. A. AIChE J . 1981, 27, 773. (64) de Souza, L. E. S.; Guerin, C. B. E.; Ben-Amotz, D.; Szleifer, I. (12) Gubbins, K. E. AIChE J . 1973, 19, 684. Manuscript in preparation. ( I 3) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. Eng. (65) Lowden, L. J.; Chandler, D. J . Chem. Phys. 1974, 61, 5228. Chem. Res. 1990, 29, 1709. (66) Narten, A. H. J . Chem. Phys. 1977,67, 2102. (14) Jackson, G.; Chapman, W. G.; Gubbins, K. E. Mol. Phys. 1988.65, (67) Hsu, C. S.; Chandler, D.; Lowden, L. J. Chem. Phys. 1976,14,213. I. (68) Bondi, A. J.Phys. Chem. 1964,68,441. Bondi, A. PhysicalProprties (15) Oellrich, L. R.; Knapp, H.; Prausnitz, J. M. Fluid Phase Equilib. of Molecular Crystals, Liquids and Glasses; John Wiley and Sons: New 1978. 2., .163. . -, . York, 1968. (16) Shucla, K. P.; Lucas, K. Fluid Phase Equilib. 1986, 28, 21 1. (69) Sandroff, C. J.; King, H. E., Jr.; Herschbach, D. R. J . Phys. Chem. (17) Morita, T.; Ladanyi, B. M.; Hynes, J. T. J . Phys. Chem. 1989, 93, 1984, 88, 5647. 1386. (70) Jonas, J.; Hasha, D.; Huang, S. G. J. J. Chem. Phys. 1979,71, 3996. (18) Joslin, C. G.; Gray, C. G.; Chapman, W. G.; Gubbins, K. E. Mol. (71) Schindler, W.; Jonas, J. 1980, 73, 3547. Phys. 1987,62, 843. (72) Song, Y.; Stratt, R. M.; Mason, E. A. J . Chem. Phys. 1987,88,1126. (19) Pratt, L. R.; Hsu,C.S.;Chandler, D.J. Chem. Phys. 1978,68,4202. (73) Alder, B.; Edward, A. Phys. Today 1984, 37, 1 . (20) Dobrosavljevic, V.; Henebry, C. W.; Stratt, R. M. J . Chem. Phys. (74) Buckingham, A. D. Tram. Faraday SOC.1960, 56, 753. 1988, 88, 5781. (75) Lebowitz. J. L.; Rowlinson, J. S . J . Chem. Phys. 1964. 41, 133. (21) Verlet, L.; Weis, J.-J. J . Chem. Phys. 1972, 5, 939. (76) Lebowitz, J. L.; Helfand, E.; Praestgaard, E. J . Chem. Phys. 1965, (22) Grundke. E. W.; Henderson, D. Mol. Phys. 1972, 24, 269. 43, 774. (23) Benson, A. M.; Drickamer, H. G. J . Chem. Phys. 1957, 27, 1164. (77) Herschbach, D. R.; Laurie, V. W. J. Chem. Phys. 1961, 35, 458. (24) Wiederkehr, R. R.; Drickamer, H. G. J . Chem. Phys. 1958,28,31 I . (78) Applequist, J.; Carl, J. R.; Fung, K.-K. J . Am. Chem. SOC.1972.94, (25) Fishman, E.; Drickamer, H. G. J . Chem. Phys. 1956, 24, 548. 2952. (26) Buckingham, A. D. Proc. R . SOC.London, A 1957, 248, 169. (79) Gordon, A. J.; Ford, R. A. The Chemists Companion; John Wiley (27) Schindler, W.; Jonas, J. J . Chem. Phys. 1980, 72, 5019. and Sons: New York, 1972.