J . Phys. Chem. 1985, 89, 4131-4134
recently Smith et a1.I8 have critically assessed many equations for this purpose, and have chosen several to be among the best. Equation 5 fits the vapor pressures of the 85 liquids, generally within their experimental uncertainties, over the applicable range of eq 1 equally well as several of these better equations when compared on a similar basis. For comparison as a fitting equation only, P, and T , were incorporated as unknowns into the four-parameter form of eq 5, and the resulting fits were compared with those obtained from, among the other equations, the four-parameter forms of the Riedel-Plank-Miller (RPM) and Riedel (R) eq~ations!-’~.~~ The average deviation from the smoothed values for water over the 423-643 K range was 0.035% compared with 0.052% and 0.035% for the RPM and R equations, respectively, although these latter two equations provided a somewhat better fit over the entire range of liquid-phase existence (0.2% vs. 1.1% for the present equation), the objective of their developers. Nevertheless, the observation that eq 5 fits the vapor pressures of numerous dissimilar liquids very well from 1 to 0.6 for T , may support a close adherence of the vaporization properties of liquids to eq 1 over this range, with (18) Smith, B. D.; Muthu, 0.;Dewan, A,; Gierlach, M. J . Phys. Chem. ReJ Data 1982, 1 1 , 941. (19) Miller, D. G. J . Phys. Chem. 1964, 68, 1399.
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possibly some physical significance. Equation 1 thus appears to describe a vaporization ratio of liquids over wide ranges of temperature, and it produces for a liquid at the limit of the critical temperature a property that is equal in value to d In PJd In T , (adopted earlier by Riedel) minus 1 but applied here as a ratio of energy terms (AE/PAV): a critical vaporization ratio. The approach to a temperature-independent proportionality of hE to PAV at T , (or in the very near vicinity of T, because of complications introduced by scaling laws) would appear to be significant. The symmetry of the present equation requires a calculated asymptotic approach of RE to B at T,. For equations that produce more complex relationships for RE14,16-19 an asymptotic approach must be assumed (although reasonable from the fits) as a condition for reducing the number of required parameters.
Acknowledgment. I thank R. H. Busey, H. F. Holmes, Elijah Johnson, M. H. Lietzke, H. F. McDuffie, R. E. Mesmer, and J. E. Ricci for their most helpful reviews of drafts of the manuscript. This work was sponsored by the Division of Chemical Sciences, Office of Basic Energy Sciences, U. S. Department of Energy, earlier under Contract W-7405-eng-26 with Union Carbide Corporation and later under Contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc.
Calculation of the Third Virial Coefficients for Water Using ab Initio Two-Body and Three-Body Potentials G. C. Lie,*+ Department of Chemistry, National Tsing Hua University, Hsinchu, Taiwan
G . Corongiu, and E. Clementi* IBM Corporation, 48B/MS 428, Kingston, New York 12401 (Received: November 26, 1984; In Final Form: May 14, 1985)
A Monte Carlo technique is employed to carry out the calculations of the third virial coefficient for steam using the ab initio two-body configuration interaction (CI) potential of Matsuoka-Clementi-Yohimine (1976) and the three-body Hartree-Fock potential of Clementi-Corongui (1983). The nonadditive three-body interaction is found to contribute somewhat more than 10% to the calculated third virial coefficient for temperatures lower than about 500 K and bring it to slightly better agreement with experiments.
Introduction Since the publication of a CI water-water interaction potential by Matsuoka, Clementi, and Yoshimine (henceforth referred to as MCY potentia1)l in 1976 and its successful application in the simulation of the structure of liquid water by Lie, Clementi, and Yoshimine,* there have been widespread interests in the use of the potential to study structures and dynamics of water, ice, and ~ o l u t i o n s . ~ -Although ~ satisfactory agreement with some experimentally determined properties is often observed in these studies, discrepancies such as internal energy,2 pres~ure,~,’ and density8 for the simulated water and density and oxygen-oxygen distance for the ice crystal structure5 have also been pointed out. These have led to the expectation that since the MCY potential is only the first term in a series expansion of many-body interactions, many-body effects may play a role in reducing the discrepancies. Of all the many-body effects in water, the most important one is of course the three-body nonadditive interaction energy. Our ‘Present address: IBM, 48B/MS 428, P.O. Box 100, Kingston, NY 12401.
0022-3654/85/2089-4131$01.50/0
preliminary results indicate that three-body interaction energy can be as large as 1 kcal/mol,lOJ1to be compared with the MCY two-body interaction energy of about 6 kcal/mol. A full scale search for an analytical representation of the three-body interaction energy has been recently completed. In that study,I2 Clementi (1) 0. Matsuoka, E. Clementi, and M. Yoshimine, J. Chem. Phys., 64, 1351 (1976). (2) G. C. Lie, E. Clementi, and M. Yoshimine, J . Chem. Phys., 64, 2314 (1976). (3) D. C. Rapaport and H. A. Scheraga, Chem. Phys. Lett., 78, 491 (1981). (4) M. Mezei and D. L. Beveridge, J. Chem. Phys., 74, 622 (1981). ( 5 ) M. D. Morse and S. A. Rice, J . Chem. Phys., 76, 650 (1982). (6) K. S. Kim, G. Corongiu, and E. Clementi, J . Biomol. Srruct. Dyn., 1, 263 (1983). (7) R. W. Impey, P. A. Madden, and I. R. McDonald, Mol. Phys., 46,513 ( 1982). (8) J. C. Owicki and H. A. Scheraga, J . Am. Chem. Soc., 9,7403,7414 (1977). (9) E. Clementi, ‘Computational Aspects for Large Chemical Systems”, Springer-Verlag, West Berlin, 1980, Lect. Notes Chem. No. 19. (10) H. Kistenmacher, G. C. Lie, H. Popkie, and E. Clementi, J . Chem. Phys., 61, 546 (1974). (1 1) E. Clementi, W. Kolos, G. C. Lie, and G. Ranghinbo, Znt. J. Quantum Chem., 17, 377 (1980).
0 1985 American Chemical Society
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The Journal of Physical Chemistry, Vol. 89, No. 19, 1985
and Corongiu have computed the Hartree-Fock energy for 173 different geometrical arrangements of three water molecules and fitted the resulting energies to an analytical expression with mean standard deviation of 0.21 kcal/mol. The analytical three-body potential (henceforth referred to as CC) has then been used in a Monte Carlo simulation of liquid water,I2 and improvements, relative to the use of M C Y potential only, have been obtained for the radial distribution functions, X-ray and neutral scattering intensities, and the internal energy. While the effects of the C C potential on the dynamic properties of liquid water are currently under study, we report here its effect on the third virial coefficients of steam calculated from MCY potential and point out the direction to improve the latter potential.
Mathematical Preliminary The interaction energy for an N-body system can in general be written as a sum of various cluster contributions E(1,2,...,N ) = c E 2 ( i j ) + i
