Reformulation of the Virial Series for Classical Fluids

The usual graphical representation of the virial coefficients is reformulated in terms of graphs containing not only Mayer f functions, but also f̃ functions $\text{[f̃≡f+1≡}\exp \mathrm{(-\phi /kT)]}$ . This reformulation has three main advantages: (1) The number of integrals of topological graphs contributing to the virial coefficients is reduced; this simplifies numerical calculations. (2) In Mayer's formulation none of the star integrals contributing to the virial coefficients (for hard potentials, at least) could be ignored; each made a nonnegligible contribution. In the new formulation (again, for hard potentials) many integrals make negligible (or even zero) contributions; the extensive cancellation of positive and negative terms found in Mayer's formulation is reduced. (3) Several new ways of summing the virial series by successive approximation are suggested by the new formulation. One such way is worked out, in the first three approximations, for gases of hard parallel squares and cubes; the third approximation reproduces the first five virial coefficients exactly. The reformulation is not restricted to the virial series alone. We also generalize our treatment to the radial distribution function. It can be applied to any series whose coefficients are integrals of graphs.