Perturbation theory of the pair correlation function in molecular fluids

Abstract

We study the perturbation theory of the angular pair correlation function g(rω 1 ω 2 )in a molecular fluid. We consider an anisotropic pair potential of the form u = u 0 + u a , where u 0 is an isotropic 'reference' potential, and for simplicity in this paper we assume the perturbation potential u a to be 'multipole-like', i.e., to contain no l = 0 spherical harmonics. We expand g in powers of u a about g 0 , the radial distribution function appropriate to u 0 . This series is examined by expanding h a = h−h 0 (where h = g−1) and its corresponding direct correlation function c a in spherical harmonic components. We consider approximate summations of the series in u a that automatically truncate the corresponding harmonic series, so that the Ornstein–Zernike (OZ) equation relating h a and c a can be solved in closed form. We first expand c a = c 1 + c 2 + … where c n includes all terms in c a of order (u a ) n . Taking u a to be a quadrupole–quadrupole interaction, we find that a 'mean field' (MF) approximation c a = c 1 gives rise to only three nonvanishing harmonic components in h a , so that OZ is solved explicitly in Fourier space. The MF solution for multipoles of general order is given in an appendix. Graphical methods are then used to identify the class of all terms in the u a series that are restricted to the harmonic space defined by MF. A portion of this class can be summed by solving OZ with the closure c a = −βg 0 u a + h 0 (h a −c a ), where β = (kT) −1 , h 0 = g 0 −1 This system is designated as generalized MF (GMF), and solved by numerical iteration. Numerical results from MF and GMF are presented for quadrupolar u a , taking u 0 to be a Lennard-Jones potential. Symmetries imposed by the restricted harmonic space are foreign to the full g, yet harmonics within this space are sufficient for evaluation of many macroscopic properties. The results are therefore evaluated in harmonic form by comparison with the corresponding harmonic components of the 'correct' g as evaluated by Monte Carlo simulation.