Mayer F function In terms of Mayer f function, the first 2 Virial coefficient: 7
Cluster diagram and star diagram Numbered circles represent particles whose coordinates we integrate over the bond between pairs of circles means particles are connected by a Mayer f function. All the Virial coefficients can be expressed in terms of a special set of cluster, These are called doubly connected, or star diagram. 3 1 2 Cluster diagrams 3 1 3 2 Doubly connected Star diagram 1 3 2 1 2 Not doubly connected Not star diagram Note: There are 2 topologically distinct cluster diagrams for 3 particles, there us 1 star diagram and 3 which is not 8
All star diagram for j+1 particles. (The sum of all products of f functions that connect 1, 2, …, j+1 molecules such that the cluster are connected so that the removal of any point, along with all lines to that point, still results in a connected graph) Star diagram for 4 particles: 1 2 2 3 4 4 3 1 1 4 2 4 3 2 9
A. Hard sphere potential: Very important theoretically, can get exact results! Starting point for ##### very important for liquids �� /2 10
B. Square well potential For square well potential: 11