Basic Statistical Mechanics

In this chapter we introduce distribution functions for molecular momenta and positions. All equilibrium properties of the system can be calculated if both the intermolecular potential energy and the distribution functions are known. Throughout, we shall make use of the ‘rigid molecule’ and classical approximations. In the rigid molecule approximation the system intermolecular potential energy u(rNωN ) depends only on the positions of the centres of mass rN ≡ r1 . . . rN for the N molecules and on their molecular orientations ωN ≡ ω1 . . . ωN; any dependence on vibrational or internal rotational coordinates is neglected. In the classical approximation the translational and rotational motions of the molecules are assumed to be classical. These assumptions should be quite realistic for many fluids composed of simple molecules, e.g. N2 , CO, CO2 , SO2 CF4 , etc. They are discussed in detail in §§ 1.2.1 and 1.2.2; quantum corrections to the partition function are discussed in §§ 1.2.2 and 6.9, and in Appendix 3D. In considering fluids in equilibrium we can distinguish three principal cases: (a) isotropic, homogeneous fluids (e.g. liquid or compressed gas states of N2 , O2 , etc. in the absence of an external field), (b) anisotropic, homogeneous fluids (e.g. a polyatomic fluid in the presence of a uniform electric field, nematic liquid crystals), and (c) inhomogeneous fluids (e.g. the interfacial region). These fluid states have been listed in order of increasing complexity; thus, more independent variables are involved in cases (b) and (c), and consequently the evaluation of the necessary distribution functions is more difficult. For molecular fluids it is convenient to introduce several types of distribution functions, correlation functions, and related quantities: (a) The angular pair correlation function g(r1r2 ω1 ω2). This gives complete information about the pair of molecules, and arises in expressions for the equilibrium properties for a general potential.

Introduction

The application of statistical mechanics to the study of fluids over the past fifty years † or so has progressed through a series of problems of gradually increasing difficulty. The first and most elementary calculations were for the thermodynamic functions (heat capacities, entropies, free energies, etc.) of perfect gases. These properties are related to the molecular energy levels, which for perfect gases can be determined theoretically (by quantum calculations) or experimentally (by spectroscopic methods, for example). For simple molecules (CO2 , CH4 , etc.) the energy levels, and hence the thermodynamic properties, can be determined with great accuracy, and even for quite complex organic molecules it is now possible to obtain thermodynamic properties with satisfactory accuracy. With the advent of digital computers it became possible to calculate thermodynamic properties for a wide variety of substances and temperatures, and several useful tabulations of perfect gas properties now exist. Having successfully treated the perfect gas, it was natural to consider gases of moderate density, where intermolecular forces begin to have an effect, by expanding the thermodynamic functions in a power series (or virial series) in density. Although the mathematical basis for a theoretical treatment of this series was laid by Ursell in 1927, it was not exploited until ten years later, when Mayer re-examined the problem. Since that time a great deal of effort has been put into evaluating the virial coefficients that appear in the series for a variety of intermolecular force models. As the expressions for the virial coefficients are exact, they provide a very useful means of checking such force models by comparison of calculated and experimental coefficients. While the theory of dilute gases at equilibrium is essentially complete, this is far from being the case for all dense gases and liquids. The virial series cannot be applied directly to liquids. As an alternative to the ‘dense gas’ approach to liquids, there were early attempts to treat liquids as disordered solids by using cell or lattice theories; these were popular from the mid-1930s until the early 1960s.

Perturbation Theory

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.

Intermolecular Forces

Knowledge of the intermolecular potential for simple (i.e. monatomic) molecules has increased greatly in recent years. For polyatomic molecules, on the other hand, such knowledge is still rather meagre, and much more is needed. One needs to know (i) what is the pair potential? (ii) how important are the triplet and other multibody potentials in liquids? These multibody potentials have been studied very little for polyatomic liquids (see refs. 28-35 and §§ 1.2.3, 2.10, and 4.10), and are usually taken into account, if at all, by an effective pair potential. There have been, especially at short range, relatively few theoretical evaluations of the pair potential for diatomic or polyatomic molecules (see, e.g. refs. 21-7 and 40-55a). The most reliable existing knowledge has been obtained from binary collision experiments, or, for the longrange part of the potential, from measurements of properties of single molecules. Examples include molecular beam scattering, induced birefringence, pressure and dielectric virial coefficients, and collision-induced absorption (including gas dimer spectra, which can also be studied by beam resonance spectroscopy) which yield values for the parameters (e.g. Lennard-Jones constants, polarizabilities, dipole moments, quadrupole moments, octopole moments, etc. - see also Appendix D) occurring in the expressions for the intermolecular potentials. The shape of the repulsive core of the potential can be inferred approximately from the molecular structure and charge density as determined experimentally, for example by electron and X-ray diffraction or by quantum calculations. As an example of the last point we show in Fig. 2.1 a contour map for the theoretically calculated charge density of N2 , the prototype molecule for simple nonpolar molecular fluid studies. Over 95 per cent of the total electronic charge is contained within the outermost (0.002 au) contour, and the dimensions of this contour are sometimes used to define a theoretical size of the N2 molecule. The dimensions shown on Fig. 2.1 agree roughly with dimensions obtained experimentally from Lennard- Jones diameters in gases (virial coefficients and viscosity) and so-called van der Waals radii from X-ray diffraction studies of solids.

Integral Equation Methods

In this chapter we describe some of the integral equation methods which have been devised for calculating the angular pair correlation function g(rω1ω2) and the site-site pair correlation function gαβ( r ) for molecular liquids. These methods are in the main natural extensions of methods devised for calculating the pair correlation function g(r) for atomic liquids. They can be derived from infinite-order perturbation theory (an example is given in § 5.4.8), whereby one partially sums the perturbation series of Chapter 4 to infinite order usually with the help of diagrams, or graphs, but alternative methods of derivation are also available, e.g. functional expansions. The original integral equation theories are in a certain sense more complete than perturbation theories, in that the full correlation function g (or gαβ) is calculated, whereas in perturbation theory one calculates the correction g — g0 to the reference fluid value g0. On the other hand the perturbation theory approximations are controlled; one can estimate the error by calculating the next term. I t is extremely difficult to estimate a priori the error in integral equation approximations, since certain terms are neglected almost ad hoc. Their validity must therefore be a posteriori, according to agreement with computer simulation results (or, less satisfactorily, with experiment). Of particular interest are theories which are a combination of perturbation theory and an integral equation, which tend to have some of the advantages of both approaches (see also §5.3.1). An example is the GMF/SSC theory of §5.4.7. The structure of the integral equation approach for calculating g(r ω1 ω2) is as follows. One starts with the Ornstein-Zernike (OZ) integral equation (3.117) between the total correlation function h = g — 1 and the direct correlation function c, which we write here schematically as . . . h = c + pch (5.1) . . . or, even more schematically, as . . . h = h[c], (OZ) (5.2) . . . where h[c] denotes a functional of c.