Radial Distribution Function of Hard Spheres

The third-order density correction term g3(r) in the density series for the radical distribution function g(r) is evaluated for a system of hard spheres over all distances. Both the Monte Carlo method (r2σ, σ=sphere diameter) and the analytic method (r≥2σ) are used to calculate the graph integrals appearing in g3(r). In carrying out the Monte Carlo integrations, calculations are simplified if gn(r) is represented by a sum of modified doubly rooted graph integrals instead of the Mayer—Montroll doubly rooted graphs. The modified doubly rooted graphs contain both Mayer f functions and f̃ functions (f̃=f+1). Advantages of this representation are twofold; (a) reduction in the number of graph integrals appearing in gn(r), and (b) the strong dependence of the graph integrals both on distance and dimension for the hard-core potentials. The analytic expression of g2(r) is obtained for r≤σ, by calculating the first-order density correction term of the triplet distribution function. The above results are compared with the corresponding expressions derived by using three approximate integral equations for g(r): the Percus—Yevick equation, the convolution-hypernetted-chain equation, and the Born—Green—Yvon equation. The fifth virial coefficient B5 for hard spheres is calculated from the g3(r) data. Use of the virial theorem and the Ornstein—Zernike relation gives the following values of B5: B5/(B2)4 (virial)=0.1105±0.0004,B5/(B2)4 (Ornstein−Zernike)=0.1106±0.0011. We also compare the truncated density series of g(r) with the molecular-dynamics data of Alder and Wainwright and the g(r) obtained from the Percus—Yevick, the convolution-hypernetted-chain, and the Born—Green—Yvon integral equations at one-half the close-packed density.