Since the early work of Waisman [Waisman E., Mol. Phys. 25 (1973) 45] the solution of the Ornstein-Zernike equation with an exponential closure has been a very useful tool for the study of both simple and complex fluids. There is a number of problems, ranging from engineering applications to biological research, polymers, colloidal systems, water and ionic solutions which can be formulated as closures of some kind of either scalar or matrix Ornstein-Zernike (OZ) equation.
Our aim is to show that a general GMSA with an arbitrary number of exponentials that can be solved explicitly can be used to write a functional for the Helmholtz free energy A in terms of a scaling matrix Γ. This functional is the ring sum of an equivalent MSA problem (EMSAP), which, in turn, can be derived using extensions of recently derived functionals for the entropy.
The central assumption is that the direct correlation function can be expanded aswhere the short ranged part of the direct correlation function c^sr_ij(r) is a functional of the pair correlation function h^sr_ij(r) which is derived from generalized free energy functionals A(h). Then the general solution is given in terms of a scaling matrix Γ. When this matrix is diagonal explicit results of the thermodynamic functions are obtained. The general form of the Helmholtz free energy is obtained by thermodynamic integration.