Faraduy Biscuss. Chem. Suc., 1982, 73, 235-240 Van der Waals Molecules and Condensed Phases BY J. A. BARKER IBM Research Laboratory, 5600 Cottle Road, San Jose, California 95 193, U.S.A. Receitled 1 1 th January, 1982 Spectroscopic studies of Van der WaaIs mofecules have recentIy provided detailed and precise information on the forces between molecules and in particuIar rare-gas atoms. This information can be used to calculate properties of condensed phases, thus providing a stringent test of theoreticd and computational methods. Condensed phases may themselves be regarded as very large Van der Wads molecules, and smaller Van der WaaIs molecules (clusters) play an important role in the nucleation and growth of condensed phases. This paper reviews these questions with particular reference to rare gases.The rare gases play an important role as test cases in many areas of chemical physics, and the use of spectroscopic methods to determine intermolecular potential- energy functions is no exception. Historically the potential-energy function for interaction of two ground-state argon atoms was determined with high accuracy (better than 1 % in the depth of the potential) almost simultaneously by two indepen- dent methods. Barker, Fisher and Watts (BFW) used experimental data on solid argon at 0 K, gaseous argon and liquid argon, together with the assumption that the Axilrod-Muto-Teller triple-dipole interaction was the only significant many-body interaction, to determine a multi-parameter potential function. At almost the same time Maitland and Smith (MS) used a Rydberg-Klein-Rees analysis of the spectro- scopic data of Tanaka and Yoshino on the vibrational levels of the argon dirner together with data on the second virial coefficient to determine another argon-argon potential, which proved to be almost identical with that of BFW.The close agree- ment of these two potentials demonstrated convincingly the value of the spectroscopic data, and incidentally provided strong evidence for the validity of the assumption concerning many-body interactions. Subsequently these potential functions were shown to be consistent with the argon- argon differential scattering cross-section data of Lee and c o ~ o r k e r s , ~ - ~ who also derived a quite similar potential function from their data. More recently Aziz and Chen derived another potential using the rotationally resolved spectroscopic data of Colbourn and Douglas and a fresh analysis of gas-phase data.For consideration of condensed-phase properties at pressures which are not too high the differences between these potentials may be regarded as small, and attention will be confined here to the BFW potential. It should be emphasised that the modern argon-argon potentials differ in depth by ca. 18% from the earlier 6-12 potential. The 6-12 potential may be regarded as qualitatively representing the properties of rare gases but it is quite unsatisfactory quantitatively. Accurate potential functions are also known for other like pairs of rare-gas atoms; details are given in the excellent book by Maitland et aZ.* It is interesting to note that in the determination of an accurate xenon-xenon potential the spectroscopic23 6 VAN DER WAALS MOLECULES AND CONDENSED PHASES data of Freeman et a1.I' played a much more important role than in the argon case, partly because of inconsistencies in the bulk-phase data for xenon.CONDENSED-PHASE PROPERTIES Knowledge of an accurate potential-energy function provides a powerful predictive tool. The properties of the substance in question in gaseous, solid and liquid states can be calculated by appropriate methods. In the case of argon this programme has been carried through. Low-density gas properties (second virial coefficients, gas transport properties) depend only on the interactions of pairs of molecules. They are the subject of another paper in this Discussion, by E.B. Smith, and will not be discus- sed here. At somewhat higher gas densities interactions of three molecules at a time become significant, and this is reflected in the values of the third virial coefficient. A comparison of calculated and experimental third virial coefficients for argon is shown in fig. 1. If only two-body interactions are included the calculated values are smaller 3000 2000 N 0 100 200 300 400 500 temperature/K FIG. 1.-Third virial coefficients of argon. The circles are experimental data of Michels et aZ.I3 The dashed curve is calculated with the BFW potential alone, the dotted curve includes the triple- dipole interaction, the dash-dotted curve adds third-order dipole-quadrupole interactions, the solid curve adds fourth-order dipole interactions. than the experimental values by almost a factor of two at low temperatures. Inclusion of the triple-dipole three-body interaction corrects much of this discrepancy, and the third-order quadrupole and fourth-order dipole terms much of the remainder.Note that there is substantial cancellation between the latter two terms. In the condensed phases this cancellation is even closer, so that their net contribution is very small and will be neglected. In fig. 2-4 a comparison with experimental data of some calculated properties of solid argon at low temperatures is shown; they are the Debye parameter, which is a measure of the specific heat, the pressure-volume relationship at pressures up to 20 kbar,* and the thermal expansion.Fig. 5 compares calculated and experimental * 1 bar = lo5 Pa.J . A . BARKER 237 100 95 90 k4 3 E -E ---. Y cd 85 ' 80 75 70 o Q 2.5 5.0 7.5 10.0 12.5 15.0 TIK FIG. 2.-Debye parameter for argon as function of temperature. potential and triple-dipole interaction. of Finegold and Phillip~.'~ Solid curve, calculated with BFW Circles, experimental values from specific heat measurements 23 22 21 20 19 18 17 16 V/cm3 mol-' FIG. 3.-Pressure-volume data for solid argon near 0 K. and triple-dipole interaction. Solid curve, calculated with BFW potential Squares and circles, experimental measurements (on two different samples) of Anderson and Swen~on.'~238 VAN DER WAALS MOLECULES AND CONDENSED PHASES 0.001 1 5 O.OOO! -u" 4s O.OOO[ I 1 1 I I I TIK FIG. 4.-Low-temperature integrated thermal expansion of solid argon.The solid curve is calculated for the BFW potential and the triple-dipole interaction, the error bars and triangles respectively are experimental data of Peterson et ~ 1 . ' ~ and Tilford and Swen~on.~' 14 - 12- E ' 0 - 2 + l.7 . 0 - % 6 - 4 - 2 - 20.0 22.0 24.0 V/cm3 mol-I FIG, 5.-Pressure-volume relation for solid and fluid argon on the melting line. culated with BFW potential and triple-dipole interaction. et UZ.,'~ Crawford and Daniels l9 and Stishov and Fedositov.20 Solid curves, cal- Circles, experimental data of FlubacherJ . A . BARKER 239 pressures for solid and fluid argon at high temperatures along the melting line, and fig. 6 shows a comparison of the calculated and experimental radial distribution function of liquid argon at 85 K.Agreement between calculated and experimental values is uniformly excellent.; this is a stringent test of both theoretical and experimental 3.0 4.0 5.0 6.0 7 .O RIA FIG. 6.-Radial distribution of liquid argon at 85 K. Solid curve, experimental neutron-diffraction result of Yarnell et al. ;21 circles, calculated with BFW potential and triple-dipole interaction. techniques. Phonon frequencies have also been calculated and compared with values derived from inelastic neutron scattering, again with excellent agreement. These are the vibrational frequencies of the giant Van der Waals molecule which the crystal comprises. These comparisons are intended to show the range of applications of the kinds of potential-energy functions that can be derived from the study of Van der Waals molecules. MOLECULES, CRYSTALLITIES AND DROPLETS The simplest Van der Waals molecules are dimers, but there exist also trimers, tetramers and so on up to macroscopic crystals and liquid droplets.No doubt it is a matter of taste at what point one stops speaking of molecules and uses a more neutral term like cluster. In any event clusters ranging in size from a few to a few hundred atoms are important for the nucleation of condensed phases, and can be observed for example in supersonic nozzle experiments. Garcia and Torroja l2 have made Monte Carlo calculations of the free energy of argon clusters using the 6-12 potential and made comparisons with the results of nucleation experiments. The agreement was not quantitatively satisfactory.In view of what has been said here about the limit- ations of the 6-12 potential this is perhaps not surprising. Etters et aZ.13 have cal- culated the ground-state energies of clusters using the BFW potential with three-body interactions. However, for comparison with a nucleation experiment one needs more than this: one needs the free energy of the clusters as a function of temperature. Calculations of this kind are currently being made in our laboratory by D. Romeu using the Monte Carlo method. As part of this work vibrational frequencies of clusters are being calculated and these results will be published at a later date.240 V A N DER WAALS MOLECULES A N D CONDENSED PHASES J. A. Barker, R. A. Fisher and R. 0. Watts, Mol. Phys., 1971, 21, 657.Y. 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