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We study the interaction energy of a system of N particles with embedded point dipoles in a cube, this cube being duplicated indefinitely throughout space. We are able to evaluate exactly this energy for arbitrary positions and orientations of the N particles, to provide an effective hamiltonian suitable for the study of the statistical mechanics of dipolar systems by both rigorous methods and computer simulation techniques. We describe methods for the rapid numerical evaluation of the hamiltonian and how it may be employed for the tasks outlined above.

Fast moment methods in computational statistical mechanics allow the computation time for a time step of a molecular dynamics algorithm of N charged particles to grow linearly with N. The method is developed and optimized, with timing estimates, for any potential which has a Taylor series about a non-zero interparticle separation. Some examples, are given with a discussion of their errors.

A geometry for simulation of polar surfaces of ionic crystals is described. The simulation cells form a two-dimensional periodic array with one side bounded by the polar surface of a large piece of rigid ionic crystal. To simulate a polar surface in this way, the surface must carry excess charge to ensure that the cells are a model of the surface of a piece of crystal in bulk thermodynamic equilibrium. The electrostatic energy of the simulation cells is presented as a sum of convergent two dimensional lattice sums. The cancellation of divergences implicit in this energy because of the excess charge distributions is described in detail.

We investigate the effects of a change in external dielectric constant ε' on the properties of a system of charges in periodic boundary conditions. The mean square dipole moment fluctuations of a configuration are shown to depend strongly on ε' in the thermodynamic limit. Simulations of 216 restricted primitive electrolyte particles with ε' = 1 and ε' = ∞ at η = 0·35 and Γ = 2, 5 and 10 are reported and obey the formula for the change in mean square dipole moment fluctuations rather well.

We report molecular dynamics simulations of 64 model ethylene oxide (oxirane) molecules at a mass density of 0·9 g cm-3 and a temperature in the vicinity of 260 K. We obtained a dielectric constant of 12·6 ± 0·05 which compares well with the experimental value of 13·9 at 272·15 K.

A method of solution of the mean spherical approximation (MSA) is given for systems of hard spheres whose interaction potential is of the form [image omitted] outside the hard core. The method thus allows solution of the MSA for a wide range of potentials. Details are given for the cases [image omitted] and [image omitted] +φ2 exp (-μ2r). The method is also used to study the MSA for systems in which the interparticle potential is of the form [image omitted] The difficulties of an exact solution for such systems are pointed out and an approximate method of solution developed. The limit μ →0 is considered. Details are given for the case [image omitted]

The RISM equation for diatomic symmetric molecules is solved using a Weiner-Hopf factorization technique. This procedure introduces a function Q(r) which is related to both the site-site total correlation function h(r) and the site-site direct correlation function c(r). Determining Q(r) effectively solves the RISM equation. We show that Q(r) can be written as an infinite sum, for r greater than the hard sphere diameter R, and using this form, solve for Q(r) when 0 < r < R. To generate a solution, this infinite series must be truncated. A method to determine the coefficients in the truncated series is outlined.

A graphical decomposition theorem is proven which allows an exact summation of the one-particle distribution function ρ1(1). Using the superposition approximation, a monolayer density profile and adsorption isotherm are calculated numerically for krypton at 100 K. The monolayer profile illustrates the structure due to hard-sphere geometrical exclusion in the adsorbate. The adsorption isotherm is compared with one derived using the Percus-Yevick approximation.

We introduce a model hard-sphere solid with correlation functions in excellent agreement with hard-sphere solid correlation functions calculated from simulations. The model solid is of independently distributed Gaussian particles centred at the points of a face-centred cubic lattice. We generalize the model to any lattice and use it to calculate contributions to the internal energy due to long-ranged potentials.

We analyse a simple model for the correlation functions of the hard-sphere solid. The result is in agreement with the parametrization introduced by Weis and Kincaid and Weis. By transforming this result we show singularities in the structure factor which result from the long-range part of g(r).

We consider coulombic interactions between particles confined to a region between two half spaces of a dielectric constant different to that of the region in between and with periodic boundary conditions with square unit cells imposed along the surfaces of the plane slab shaped region. We find two representations for the resulting lattice sums and discuss the construction of an algorithm which can optimize numerical performance (in terms of computer time) by using the two representations in different circumstances. Analysis of the resulting algorithm for the coulombic interactions shows that the computation time for the optimized algorithm for an N particle system should grow as N5/3.

Methods for obtaining a static dielectric constant ε from measurements of mean square dipole moment in a simulation are discussed. The simulations are seen as models of a macroscopic experiment to measure the dielectric constant. Such an experiment is usually carried out on a macroscopic sample of fixed shape. This sample is modelled by an array of copies of a microscopic simulation sample, the array having the appropriate shape. The different shapes correspond to different macroscopic experiments. The periodic boundary condition simulation sample hamiltonians appropriate to some different macroscopic geometries are described and appropriate fluctuation formulae for the dielectric constant obtained. The formulae are illustrated by simulations of fairly weakly dipolar molecules (which are rigid models of methyl chloride molecules) using samples of 27, 64, 125 and 216 molecules. All sample sizes give the same dielectric constant to within the error in the simulations, but the structure obtained for 27 molecules seems slightly distorted by small sample sizes. The measured wavevector dependent dielectric tensor ε(k) appears able to be extrapolated to k = 0 to agree with the values of ε obtained.

We use the Percus-Yevick approximation to study indirect correlation functions in the sticky hard-sphere model introduced by Baxter. The model has a critical point below which there is a liquid gas transition. We illustrate the changes in the structure of the correlation function as the density increases on the critical isotherm and on supercritical and subcritical isotherms. We also examine the local effects on this structure of introducing a large isolated sticky particle. For the gas part of the subcritical isotherm, such a particle can act as a condensation nucleus. We examine the dependence of the structure changes on the radius of the large particle.

Some of the problems associated with Monte Carlo and molecular dynamics simulations of three dimensional ionic and dipolar systems are discussed, with emphasis on the use of periodic boundary conditions (PBC). It is shown that analogous problems may arise in two dimensional systems provided that the interactions are two dimensional electrostatic interactions, that is, interactions derived from the two dimensional Laplace equation. The PBC hamiltonian is evaluated by considering the appropriate two dimensional lattice sums, and a computable form for the effective pair interactions in PBC developed. The idea of an external dielectric constant is introduced and its effects included in the PBC hamiltonian. Formulae for evaluating the dielectric constant from a simulation with any external dielectric constant are given. Perturbation formulae showing the effects on the structure and mean square dipole moment of a dipolar system which are caused by a change in external dielectric constant are derived. A formula for the shift in mean square dipole moment of an ionic system is also developed. The problem of interpreting results from such simulations is discussed briefly.

The structure and dielectric properties of the hard sphere dipolar fluid, calculated using both perturbation theory and computer simulation, are reported. Both methods predict a short ranged hΔ(r), a much lower dielectric constant than previously obtained, and non-zero higher spherical harmonic coefficients of the angular correlation function.