X-ray crystallography of 3-D bulk materials opened entire new fields of discovery in the twentieth century, from elemental crystals to DNA molecules. Similarly, the determination of atomic-scale structure of 2-D surfaces of condensed matter has achieved fundamental new understanding and generated powerful techniques that helped spawn new areas of research, from catalysis to nanotechnology, for the twenty-first century. Specific examples include, among many others: new catalysts; various new carbon structures (such as buckminsterfullerenes, nanotubes and graphene); quantum dots used in optoelectronic displays (including television displays); molecules allowing electron transport and switching; nanoparticles enabling targeted drug delivery; and nanomachines for future manufacturing and medical applications.

LEED has found widespread application in surface science, since the LEED experiment can be performed in a small laboratory and LEED systems are commercially available. A main advantage compared to surface X-ray diffraction is that on the LEED screen most of the 2-D diffraction pattern is visible, thus allowing a quick and comprehensive overview of the symmetry and to some extent about the degree of ordering of the surface under examination. A LEED system is therefore included in most UHV chambers to control the quality of the surface preparation for a wide range of surface studies. A qualitative interpretation of the diffraction pattern is the most common use of LEED: it allows the identification of the surface unit cell, the estimation of the degree of ordering and the identification of different surface phases in adsorption systems (and thereby often a check on adsorbate coverage). The diffraction pattern thus reflects the translational symmetry and the crystalline order of the surface.

As mentioned in Section 2.2, a kinematic, that is, single-scattering, theory of LEED cannot describe experimental intensities with an accuracy that is sufficient to determine atomic positions and other non-structural information about surfaces. This degree of accuracy requires the inclusion of multiple scattering at a level of sophistication that is similar to that of electronic band structure calculations; in fact, some early versions of LEED theory employed methods of 3-D band structure theory, such as Bloch waves and pseudopotentials. However, the goal of surface structure determination by iterative optimisation of atomic positions with lower-dimensional periodicity and sometimes large 2-D unit cells requires very efficient calculational schemes of the multiple scattering of electrons.

This appendix introduces the classification of physical properties according to their behaviour under rotation (scalars, vectors, and in general tensors of a certain rank). Cartesian and spherical representations are presented and explicit expressions for the tensors of second and fourth rank needed when studying liquid crystals are given.

The application of molecular and mesophase symmetries to identify the minimal set of order parameters required to treat single-particle properties for different mesophases and mesogens or solutes is discussed. The use of symmetry to build rotational invariants (Stone S functions) to describe pairwise properties is described and explicit expressions of invariants are provided for uniaxial and biaxial particles.

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.