Virial Coefficient - Definition in Terms of Graphs

Definition in Terms of Graphs

The virial coeffcients are related to the irreducible Mayer cluster integrals through

The latter are concisely defined in terms of graphs.

The rule for turning these graphs into integrals is as follows:

1. Take a graph and label its white vertex by and the remaining black vertices with .
2. Associate a labelled coordinate k to each of the vertices, representing the continuous degrees of freedom associated with that particle. The coordinate 0 is reserved for the white vertex
3. With each bond linking two vertices associate the Mayer f-function corresponding to the interparticle potential
4. Integrate over all coordinates assigned to the black vertices
5. Multiply the end result with the symmetry number of the graph, defined as the inverse of the number of permutations of the black labelled vertices that leave the graph topologically invariant.

The first two cluster integrals are

In particular we get

where particle 2 was assumed to define the origin . This classical expression for the second virial coefficient was first derived by L. S. Ornstein in his 1908 Leiden University Ph.D. thesis.