This chapter introduces the interactions between particles, a key input to the computer simulations described later in the book. Molecular level and fully atomistic interactions are described, having in mind particles forming liquid crystals phases. The empirical level models discussed comprise purely repulsive hard anisotropic particles (ellipsoids, spherocylinders) and attractive-repulsive (uniaxial and biaxial Gay–Berne type) ones. Expressions for electrostatic interactions and in particular charge, dipole and quadrupole ones are derived and typical values for some common mesogens provided. Dispersion interactions, molecular polarizability and chiral interactions are then introduced via quantum mechanical perturbation theory. Since liquid crystals are also formed by colloidal suspensions, dispersive interactions and Hamaker constants are briefly discussed, as well as model potentials for water useful for lyotropic systems, micelles and membranes.

Off-lattice models both based on purely repulsive or attractive-repulsive Gay–Berne models allow us to simulate liquid crystal phases with some positional as well as orientational order. This chapter summarizes simulation results for anisotropic particles of elongated or discotic shape of the two types either pristine or decorated with charges, dipoles and quadrupoles. Beyond showing the effect of key molecular features (e.g. aspect ratios) on morphologies and phase diagrams, applications specific to liquid crystals, like the calculation of elastic constants and the simulation of a TN LCD, are reported. Tapered, bowlic and biaxial GB type single particle systems as well as more complex ones based of multi-particle mesogens (banana phases, polymers, elastomers) are discussed.

Quaternions and their use in treating the orientation of arbitrary rigid particles and their rotations are introduced. Explicit expressions needed to convert between quaternions and Euler angles representations are given.

A list of the acronyms and mathematical symbols employed in the book.

A minimal introduction to Nuclear Magnetic Resonance and to the two main types of contributions (dipolar and quadrupolar) to the spin Hamiltonian employed in studies of liquid crystals and obtainable from computer simulations.

This chapter starts with the equations of motion for atomistic systems and their time integration, including the multiple time step methods taking care of different timescales. Systems of rigid anisotropic particles are also discussed with the help of the quaternion formulation, avoiding spurious singularities. Constant temperature and constant pressure methods are considered. A summary of available molecular dynamics packages is provided.

An introduction to the most important liquid crystal types and their physical properties and applications, that could also serve as a self-contained undergraduate course. Examples of the chemical structures of mesogens, i.e. molecules yielding the various liquid crystals phases, are given, providing a preliminary information needed for modelling and simulation.

A summary of classical concepts from thermodynamics concerning phase transitions and a survey of phase transitions for a variety of thermotropic and lyotropic systems. The Landau theory for first- and second-order transitions applied to nematics and smectics is discussed in detail. Classic lattice models (Ising, Heisenberg, XY, Lebwohl–Lasher, …) and critical exponents, often associated with certain liquid crystals transitions, are introduced. Examples of experimental phase diagrams for thermotropic and lyotropic liquid crystals are shown.

A brief derivation of the Taylor expansion for a scalar function depending on a vector, with application to the inverse distance dependence in the Coulomb expression for electrostatic interactions.

Common units of measurement used in computational quantum chemistry and computer simulations and conversion factors.

The Dirac delta generalized function with its definition, representations and properties, and in particular its expansion in an orthogonal basis set, as employed in the text to obtain order parameters are introduced.

A summary of essential definitions and useful formulas for vector, matrices and orthogonal basis sets employed in the book.

The preface presents the general plan of the book, its raison d’être and origin.

The dynamic evolution of a classic molecular system is first introduced formally via Liouville equation. Then, single-particle time correlation functions and their general properties (like short and long time limits) for orientational correlation functions (OCF) in liquid crystals are discussed and translational and orientational diffusion coefficients are introduced. The link between OCFs, obtainable from computer simulations, and experiments is established with Linear Response theory and examples from dielectric relaxation, ionic and thermal conductivity, viscosities are presented, with reference to the literature. The rotational diffusion equation in an anisotropic fluid, normally employed to analyze experiments is also introduced, with explicit expressions derived for Fluorescence Depolarization.

This chapter provides a state-of-the-art summary of atomistic simulations illustrating their ability to predict, often within experimental error, physical properties, morphologies and phase transition temperatures for low-molar-mass thermotropics, like the cyano-biphenyls so commonly used in experiment. An illustration of current achievements for various thermotropic liquid crystals (based on rod-like and disc-like mesogens) is given. Selected results for phospholipid based lyotropics (micelles and membranes) are also shown.

Polarized Optical Microscopy (POM) is a standard experimental technique for the characterization of liquid crystal textures and their topological defects. This appendix describes how to produce POM images to be compared with experiment starting from computer simulations.

This chapter develops the essentials of the two main types of approximate molecular theories for liquid crystals, originating from various modifications and extensions of the original theories of Maier and Saupe and of Onsager. Even though quite different, both theories are essentially of the Mean Field type, and obtain the anisotropic potential acting on a single particle by the effect of all the others in the system. A selection of results for the two approaches is presented.

An introduction to the Markov processes and their asymptotic properties of key importance in the Metropolis Monte Carlo simulation method.