Equation of states for classical Coulomb systems. The use of Hubbard-Schofield approach

An effective method based on Hubbard-Schofield approach [Phys. Lett. A {\bf 40}, 245 (1972)] is developed to calculate the free energy of classical Coulomb systems. This method significantly simplifies the derivation of the cluster expansion. A diagrammatic representation of the cluster integrals is proposed. Simple rules providing the leading order in density $n$ of each diagrammatic contribution are found. We calculate the $n^3$ contribution and recover the results at the order $n^{5/2}$ obtained by the traditional method of resummation of diverging Mayer bonds.