Symmetry pervades many forms of art and science, and group theory provides a systematic way of thinking about symmetry. The mathematical concept of a group was invented in 1823 by Évariste Galois. Its applications in physical science developed rapidly during the twentieth century, and today it is considered as an indispensable aid in many branches of physics and chemistry. This book provides a thorough introduction to the subject and could form the basis of two successive one-semester courses at the advanced undergraduate and graduate levels. Some features not usually found in an introductory text are detailed discussions of induced representations, the Dirac characters, the rotation group, projective representations, space groups, magnetic crystals, and spinor bases. New concepts or applications are illustrated by worked examples and there are a number of exercises. Answers to exercises are given at the end of each section. Problems appear at the end of each chapter, but solutions to problems are not included, as that would preclude their use as problem assignments. No previous knowledge of group theory is necessary, but it is assumed that readers will have an elementary knowledge of calculus and linear algebra and will have had a first course in quantum mechanics. An advanced knowledge of chemistry is not assumed; diagrams are given of all molecules that might be unfamiliar to a physicist.

The book falls naturally into two parts. Chapters 1–10 (with the exception of a few marked sections) are elementary and could form the basis of a one-semester advanced undergraduate course.

Hybridization

In descriptions of chemical bonding, one distinguishes between bonds which do not have a nodal plane in the charge density along the bond and those which do have such a nodal plane. The former are called σ bonds and they are formed from the overlap of s atomic orbitals on each of the two atoms involved in the bond (ss σ bonds) or they are sp or pp σ bonds, where here p implies a pz atomic orbital with its lobes directed along the axis of the bond, which is conventionally chosen to be the z axis. The overlap of px or py atomic orbitals on the two atoms gives rise to a π bond with zero charge density in a nodal plane which contains the bond axis. Since it is accumulation of charge density between two atoms that gives rise to the formation of a chemical bond, σ or π molecular orbitals are referred to as bonding orbitals if there is no nodal plane normal to the bond axis, but if there is such a nodal plane they are antibonding orbitals. Carbon has the electron configuration 1s2 2s2 2p2, and yet in methane the four CH bonds are equivalent. This tells us that the carbon 2s and 2p orbitals are combined in a linear combination that yields four equivalent bonds. The physical process involved in this “mixing” of s and p orbitals, which we represent as a linear combination, is described as hybridization.

Definitions

All crystals and most molecules possess symmetry, which can be exploited to simplify the discussion of their physical properties. Changes from one configuration to an indistinguishable configuration are brought about by sets of symmetry operators, which form particular mathematical structures called groups. We thus commence our study of group theory with some definitions and properties of groups of abstract elements. All such definitions and properties then automatically apply to all sets that possess the properties of a group, including symmetry groups.

Binary composition in a set of abstract elements {gi}, whatever its nature, is always written as a multiplication and is usually referred to as “multiplication” whatever it actually may be. For example, if gi and gj are operators then the product gi gj means “carry out the operation implied by gj and then that implied by gi.” If gi and gj are both n-dimensional square matrices then gi gj is the matrix product of the two matrices gi and gj evaluated using the usual row × column law of matrix multiplication. (The properties of matrices that are made use of in this book are reviewed in Appendix A1.) Binary composition is unique but is not necessarily commutative: gi gj may or may not be equal to gj gi.

An introductory overview of intermolecular forces

Molecular and supramolecular units

The firm establishment of John Dalton's atomic theory in the early nineteenth century ushered in a long period of preoccupation with the nature of molecules and the bond types responsible for molecule formation. By the mid twentieth century, a molecule was commonly defined in operational terms as “the smallest part of a chemical substance that can exist free in the gaseous state, with retention of the composition and chemical properties that are possessed by the gaseous material in bulk,” or in more theoretical terms as “an aggregate of atoms which is held together by relatively strong (valence) forces, and which therefore acts as a unit.”

Let us first seek to give a more rigorous and operational ab initio characterization of such “units.” The important physical idea underlying the above definitions is that of the connecting covalent bonds that link the nuclei. One can therefore recognize that a molecular unit is equivalently defined by the covalent-bond network that contiguously links the nuclei included in the unit. We can re-state the definition of a “molecular unit” in a way that emphasizes the electronic origin of molecular connectivity.