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

In previous work we have studied the solution of the Ornstein-Zernike equation with a general multiyukawa closure. Here the direct correlation function is expressed by a rapidly converging sum of M (complex) exponentials. For a simple fluid the mathematical problem of solving the Ornstein-Zernike equation 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 scaling matrix Gamma. For only one component this matrix is diagonal and the general solution using the properties of M-dimensional S0(M) Lie group is given. Tn the Mean Spherical Approximation (MSA) the excess entropy is obtained 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 was 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 completely with the early one. Other thermodynamic properties such as the energy equation of state are also obtained, explicitly for 2 and 3 exponentials. The analytical solution of the effective MSA is also obtained from the simple variational 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, which 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.