Analytical solution of the Yukawa closure of the Ornstein-Zernike equationIV: the general 1-component case

In previous work we have studied the solution of the Ornstein-Zernike equat ion with a general multiyukawa closure. Here the direct correlation functio n is expressed by a rapidly converging sum of M (complex) exponentials. For a simple fluid the mathematical problem of solving the Ornstein-Zernike eq uation is equivalent to finding the solution of a linear algebraic equation of order M. The solution for the arbitrary case is given in terms of a sca ling matrix Gamma. For only one component this matrix is diagonal and the g eneral solution using the properties of M-dimensional S0(M) Lie group is gi ven. Tn the Mean Spherical Approximation (MSA) the excess entropy is obtain ed and expressed as a sum of 1-dimensional integrals of algebraic functions . We remark that the general solution of the M exponents-1 component case w as found in our early work (Blum, L., and Hoye, J. S., 1978, J. stat. Phys. , 19, 317) in implicit form. The present explicit solution agrees completel y with the early one. Other thermodynamic properties such as the energy equ ation of state are also obtained, explicitly for 2 and 3 exponentials. The analytical solution of the effective MSA is also obtained from the simple v ariational form for the Helmholtz excess free energy Delta A partial derivative[beta Delta A(Gamma)]/partial derivative Gamma = 0, where Delta A(Gamma) = Delta E(Gamma) - T Delta S(Gamma), where both the excess energy Delta E(Gamma) and the excess entropy Delta S( Gamma) are functionals of Gamma, which opens interesting possibilities that are discussed elsewhere. We remark that this is a non-trivial property, wh ich is certainly true for the MSA (Chandler, D., and Andersen, H. C., 1972, J. chem. Phys., 57, 1930). It implies cross-derivative properties for the closure equations, which have been verified in all cases.