Perturbation Theory

Abstract

In perturbation theory one relates the properties (e.g. the distribution functions or free energy) of the real system, for which the intermolecular potential energy is u(rN ωN), to those of a reference system where the potential is u0(rNωN), usually by an expansion in powers of the perturbation potential u1 ≡ u — u0. The first-order, second-order, etc. perturbation terms then involve both u1 and the distribution functions for the reference system. In the sections that follow we first briefly discuss the historical background of perturbation theory (§ 4.1). As a simple example we then derive the u-expansion for the free energy (§ 4.2). This is followed by general expansions for the angular pair correlation function (§ 4.3) and the free energy (§ 4.4). The expansions developed in these latter sections are for an arbitrary reference system and arbitrary perturbation parameter. We next consider some particular choices of reference system and perturbation parameter. We first consider the u-expansion (§ 4.5) further, for a potential having both attractive and repulsive parts, and also the f-expansion (§ 4.6) which uses a different reference fluid. This is followed by a description of methods for expanding the system properties for an anisotropic repulsive potential about those for a hard sphere potential (§4.7), and then the expansion for a general potential (attractive and repulsive parts) about a non-spherical reference potential (§ 4.8). We also discuss two approximation methods based on perturbation theory, the effective central potential method (§ 4.9), and generalized van der Waals models (§4.11). Non-additive potential effects are discussed in §4.10. Perturbation theories have been the subject of reviews for both atomic and molecular liquids. Perturbation expansions are closely related to inverse temperature expansions, which have been a standard technique since the earliest days of statistical mechanics. As examples, we mention some early work on gases (virial coefficients, dielectric and Kerr constants), and solids (lattice phonon specific heat, polar lattice thermodynamics, electric and magnetic susceptibility properties, alloy order-disorder transitions, ferromagnetism, and diamagnetism). In considering perturbation theory for liquids it is convenient first to discuss the historical development for atomic liquids.