A relation between the potential of mean force at zero separation and the excess chemical potential is derived for ``hard'' molecules. Application to hard spheres shows that of the Percus—Yevick, Kirkwood, convolution, and Born—Green—Yvon integral equations, only the Kirkwood equation gives the correct third viral coefficient.
The first seven virial coefficients for hard parallel lines, squares, and cubes, as derived from approximations of the ring and watermelon type, are compared with the exact coefficients. These approximations give no useful information as to the sign or magnitude of the virial coefficients.A Cartesian distribution function depending upon only one space coordinate arises naturally for the line, square, and cube molecules. The first four terms of the exact number density expansion of this function are presented and compared with results obtained by iteration from the Percus-Yevick, Kirkwood, and convolution integral equations. The Percus-Yevick equation * This work was supported by a grant from the Alfred P. Sloan Foundation. t Present address: Lawrence Radiation Laboratory, Livermore, California.t Alfred P. Sloan Foundation Fellow. . 4 In general, we use r as the argument of an angle-dependent function, and r as the argument of a function of one space coordinate only. We call g(r) the "radial distribution function" in this paper; for the potentials which we use, g(r) depends upon both angle and distance. The function g(r) introduced in (2) is, for the potentials which we use, identical with g(r) for .p(r) =0 (r large), but finite and nonzero, unlike g(r), for .p(r) = co (r small). g (r) may be thought of as a "radial distribution function" for two particles which interact normally with particles 3 .. . N, but not with each other [>(rI2) =0].
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