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

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.