Calculation of High-Order Virial Coefficients with Applications to Hard and Soft Spheres

Abstract

A virial expansion of fluid pressure in powers of the density can be used to calculate a wealth of thermodynamic information, but the Nth virial coefficient, which multiplies the Nth power of the density in the expansion, becomes rapidly more complicated with increasing N. This Letter shows that the Nth virial coefficient can be calculated using a method that scales exponentially with N in computer time and memory. This is orders of magnitude more efficient than any existing method for large N, and the method is simple and general. New results are presented for N = 11 and 12 for hard spheres, and N = 9 and 10 for soft spheres.

Calculation of high-order virial coefficients for the square-well potential

“…The recursive method [12] takes as its starting point the quantity exp(−E / k B T ) for a set of N particles and for all subsets thereof. For the pair-additive square-well potential, this quantity is either zero, if any hard-core overlaps occur in the (sub)set, The integrand depends only on the graph that describes whether each pair separation is in the hard-core region, the well, or the zero-energy long-range region, and not explicitly on the particle positions within these regions.…”

“…Two similar but distinct Monte Carlo methods, designated A and B, are implemented to compute SW virial coefficients within this framework, each following respective approaches analogous to previous work for hard spheres [12,14]. Both execute a repeated process of: (1) generating a configuration at random by placing each particle in succession relative to a previously-positioned particle, starting with the first at the origin; (2) evaluating some simple metrics for the resulting configuration to determine whether to compute f B for it; (3) if so indicated, computing f B using the recursive method, and also computing the total probability w for producing the configuration, considering how step (1) could yield it for all N!…”

“…In Method A [12], the particles are placed in a chain, such that each new particle is inserted relative to the one just preceding it, with probability that is uniform within the sphere for distances r from 0 to λσ , and decreasing proportionally to r −12 for larger distances; this probability tail is needed so that all biconnected diagrams have a nonzero chance of being produced. Dynamically generated lookup tables are used to evaluate the integrand.…”

“…This is partly because high-order virial coefficients are needed to investigate the convergence properties of the series, and to improve the convergence by, for example, resummation via rational functions or other forms [1][2][3][4][5][6][7][8], but such coefficients are difficult to calculate. Much work has been devoted to the hard-sphere model, for which orders N ≤ 10 have been calculated [9][10][11], before a breakthrough in the algorithm was made [12] that gave access to higher orders [12][13][14]. Recently, the algorithm also made possible calculations beyond N = 8 for the Lennard-Jones potential [15].…”

Fourth virial coefficient of additive hard-sphere mixtures in the Percus–Yevick and hypernetted-chain approximations

“…The first four coefficients of pure HS fluids are analytically known 15,16 and accurate numerical evaluations of the 5th to 12th coefficients can be found in the literature. [17][18][19][20][21] Much less information is available for HS mixtures, the results being usually restricted to the binary case. While the second and third virial coefficients are exactly known for additive and nonadditive mixtures with any number of components, [22][23][24][25][26] the fourth to eighth coefficients have been numerically computed for binary mixtures at a number of size ratios and/or nonadditivities.…”