A first order derivation of the differential equations for the effective pair potentials

The effective potential of a system is assumed to be the sum of the effective pair potentials between constituent particles of that system. Differential equations for the effective pair potentials are needed to compute the effective pair potentials as a function of distance between pair particles. A new method of deriving a set of differential equations for effective pair potential equations are obtained as the sum of a power series expansion in density. If we take only the first order in the expansion, we get differential equation which are quite similar to those obtained by the previous methods. The higher powers of the density term are proved to be small when the density of the system is low or when no distinct structures, such as molecules or crystalline structures, are present. The problem of clustering encountered in the previous methods may be resolved by the inclusion of the higher powers of the density term. (author)