Definitions

Symmetry operations leave a set of objects in indistinguishable configurations which are said to be equivalent. A set of symmetry operators always contains at least one element, the identity operator E. When operating with E the final configuration is not only indistinguishable from the initial one, it is identical to it. A proper rotation, or simply rotation, is effected by the operator R(φn), which means “carry out a rotation of configuration space with respect to fixed axes through an angle φ about an axis along some unit vector n.” The range of φ is − π < φ ≤ π. Configuration space is the three-dimensional (3-D) space ℛ3 of real vectors in which physical objects such as atoms, molecules, and crystals may be represented. Points in configuration space are described with respect to a system of three space-fixed right-handed orthonormal axes x, y, z, which are collinear with OX, OY, OZ (Figure 2.1(a)). (A right-handed system of axes means that a right-handed screw advancing from the origin along OX would rotate OY into OZ, or advancing along OY would rotate OZ into OX, or along OZ would rotate OX into OY.) The convention in which the axes x, y, z remain fixed, while the whole of configuration space is rotated with respect to fixed axes, is called the active representation.

Crystallographic magnetic point groups

Because the neutron has a magnetic moment, neutron diffraction can reveal not only the spatial distribution of the atoms in a crystal but also the orientation of the spin magnetic moments. Three main kinds of magnetic order can be distinguished. In ferromagnetic crystals (e.g. Fe, Ni, Co) the spin magnetic moments are aligned parallel to a particular direction. In antiferromagnetically ordered crystals, such as MnO, the spins on adjacent Mn atoms are antiparallel, so there is no net magnetic moment. In ferrimagnetic crystals (ferrites, garnets) the antiparallel spins on two sublattices are of unequal magnitude so that there is a net magnetic moment. In classical electromagnetism a magnetic moment is associated with a current, and consequently time reversal results in a reversal of magnetic moments. Therefore the point groups G of magnetic crystals include complementary operators ΘR, where Θ is the time-reversal operator introduced in Chapter 13. The thirty-two crystallographic point groups, which were derived in Chapter 2, do not involve any complementary operators. In such crystals (designated as type I) the orientation of all spins is invariant under all R ∈ G. In Shubnikov's (1964) description of the point groups, in which a positive spin is referred to as “black” and a negative spin as “white,” so that the time-reversal operator Θ induces a “color change,” these groups would be singly colored, either black or white.

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.