Definitions

Group theory is a very broad field of study. We shall look only at a narrow part of the field. We are concerned here with the application of group theory to the analysis of physical and chemical systems. For our discussions a group consists of elements (or operators) that mathematically represent operations that leave a system in an equivalent state. For our purposes group multiplication is the sequential application of symmetry operations or the multiplication of square matrices representing two symmetry operations. The point group of interest in the analysis of atoms, molecules, and solids is the covering group, which consists of the elements (or operators) of rotation, reflection, and inversion under which the atom, molecule, or solid remains invariant. For crystalline solids the group (space group) is enlarged to include rotations, reflections, and inversion combined with translations under which the crystalline solid remains invariant.

A group is a collection of distinct elements that possess the following four characteristics.

1. Closure. The product of any two elements is an element of the group. If A and B ∈ G and AB = C, then C ∈ G (the symbol ∈ means “belongs to” or “is a member of the set that follows”).

2. Every group must contain the identity element, E, which commutes with all elements of G: EA = AE = A for all A ∈ G.

3. Elements of the group obey the associative law: A(BC) = (AB)C.

4. Each element has an inverse. If A ∈ G, then A−1 ∈ G, where AA−1 = A−1 A = E.

The majority of all knowledge accumulated in physics and chemistry concerning atoms, molecules, and solids has been derived from applications of group theory to quantum systems.

My (T.W.) first encounter with group theory was as an undergraduate in physics, struggling to understand Wigner's Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (1959). I felt there was something magical about the subject. It was amazing to me that it was possible to analyze a physical system knowing only the symmetry and obtain results that were absolute, independent of any particular model. To me it was a miracle that it was possible to find some exact eigenvectors of a Hamiltonian by simply knowing the geometry of the system or the symmetry of the potential.

Many books devote the initial chapters to deriving abstract theorems before discussing any of the applications of group theory. We have taken a different approach. The first chapter of this book is devoted to finding the molecular vibration eigen-values, eigenvectors, and force constants of a molecule. The theorems required to accomplish this task are introduced as needed and discussed, but the proofs of the theorems are given in the appendices. (In later chapters the theorems needed for the analysis are derived within the discussions.) By means of this applications-oriented approach we are able to immediately give a general picture of how group theory is applied to physical systems. The emphasis is on the process of applying group theory.

In Chapter 9 nearly free-electron energy bands were discussed. In this chapter we discuss the energy bands of a crystal for which the electronic states are derived from atomic-like orbitals. For molecules this approach is the LCAO model. Applied to crystalline solids, it is referred to as the tight-binding model. In either case the wavefunctions are linear combinations of atomic-like orbitals. We shall analyze the energy bands of a class of perovskite oxides known as the d-band perovskites.

The structure of the ABO3 perovskites

We consider the cubic ABO3 oxides having the perovskite structure with the transition metal B ions. Examples include insulators, such as SrTiO3, BaTiO3, and KTaO3, and metals, such as NaWO3 and ReO3 (a perovskite without an A ion). The structure of these oxides is shown in Fig. 10.1. Many oxides (e.g., BaTiO3 and SrTiO3) undergo structural phase transitions at lower temperatures, leading to non-cubic lattices and ferroelectric and piezoelectric properties. Here we are concerned with the cubic phase.

As illustrated in Fig. 10.1, the B ions are at the center of a cube, the oxygen ions are on the faces of a cube, and the A ions are on the corners of a cube.

The five ions of the unit cell are labeled A, B, 2, 3, and 5 in Fig. 10.1. The origin is taken at the B site and the oxygen ions are located at a(1, 0, 0), a(0, 1, 0), and a(0, 0, 1).

The structure of a “single-walled” carbon nanotube (SWCNT) can be formed by rolling up a graphene sheet into a seamless tube as shown in Fig. 14.1. Multi-walled carbon nanotubes consist of two or more concentric SWCNTs or scroll-wrapped graphene sheets. Typical SWCNTs have diameters around 2 nm, but the range is from 0.7 to 50 nm. Some tubes are open at the ends and some are capped with a bucky hemisphere.

Nanostructures are produced by many different methods, including arc discharge, chemical vapor deposition, laser ablation, and flame burning of carbon-containing gases and solids. Some of these methods are becoming commercially practical for producing large quantities of carbon nanotubes.

Carbon nanotubes are of great scientific and technological interest because of their unique properties [14.1]. The sp2 carbon–carbon bond is stronger than the sp3 carbon–carbon bond of diamond. The tensile strength of an SWCNT is 10 to 100 times stronger than that of steel. Perhaps the most important potential application is in electronics, since SWCNTs can be grown as metallic threads that can carry an electrical current density that is 1,000 times greater than that of copper. The energy gaps between the valence and conduction bands in SWCNTs range from 0 to 2 eV. Semiconducting nanotubes have been fabricated as transistors that operate at room temperature and are capable of digital switching using just a single electron.

Splitting of d-orbital degeneracy by a crystal field

As we saw in the previous chapter the s- and p-orbital degeneracies are unaffected when an atom or ion is placed in a site of octahedral symmetry. On the other hand, the d-and f-orbital degeneracies are changed. Transition metal ions of Ti, Fe, Ni, and Co, for example, have 3d electrons in their outer unfilled shells and exist as positively charged ions in solids and molecular complexes. Most frequently, the transition metal ion is coordinated with six neighboring ligands at a site of octahedral symmetry. The second most common situation is tetrahedral coordination with four neighboring ligands. Many of these transition metal solids and molecular complexes are colored and many are magnetic. The colors are attributed to vibronic (electronic plus vibration) transitions between the d-orbital groups that are split in energy by the non-spherical potential of the ligands. When the ligand orbitals are included in determining the splitting, the procedure is called ligand-field theory. Splitting due to adjacent ligands is discussed in Chapter 6.

Crystal-field theory was developed by Bethe [4.1] and Van Vleck [4.2] to explain the optical spectra of transition metal complexes and to understand their magnetic properties. In its simplest form the crystal-field model represents the ligands surrounding a metal ion as point charges that interact with the transition metal ion only through an electrostatic potential.

Vectors

A vector in n-dimensional space is specified by n components that give the projections of the vector onto the n unit vectors of the space: V = (v1, v2, v3, …, vn). Two vectors are equal if all of the components are equal. The components may be real or complex numbers. Vectors obey the following rules:

(r and s are any real or complex numbers). The inner product or (Hermitian) scalar product of two n-dimensional vectors U and V is

The magnitude, or length, of V is |V| = (V, V)1/2. It is a positive number. It is zero if and only if V = 0, and V = 0 if and only if vi = 0 for all i. A “normalized” vector has |V|= 1.

The scalar product, (U, V)/|V|, is the projection of U onto V, and the cosine of the angle, ø, between two vectors is

If cos ø = 1, U and V are parallel vectors. If cos ø = 0, U and V are orthogonal. The scalar product of two vectors is unchanged if both vectors are subjected to the same symmetry operation. For example, if U and V are subjected to a rotation R or operator PR,

Aset of m normalized, mutually orthogonal, n-dimensional vectors, U1, U2, …, Um, is an orthonormal set.

In this chapter we illustrate the solution of a simple physical problem in order to familiarize the reader with the procedures used in the group-theoretical analysis. Since this is the initial chapter, we shall give nearly all of the details involved in the analysis. Some readers familiar with group theory may find that the discussion includes too much detail, but we would rather be clear than brief. In later chapters less detail will be required since the reader will by then be familiar with the analysis method.

Theorems from group theory are stated and discussed when employed in the analysis, but the proofs of the theorems are not presented in this chapter. Readers interested in the proofs can find them in Appendix B or refer to a number of excellent standard group-theory texts [1.1].

The procedures employed in this chapter are simple but somewhat tedious and not the most efficient way to analyze the simple example discussed. However, these procedures will prove extremely valuable when we are faced with more complex problems. Therefore the reader is encouraged to work through the details of the chapter and the exercises at the end of the chapter.

In-plane molecular vibrations of squarene

As an introductory example we consider a fictitious square molecule we shall call “squarene”. The squarene molecule, shown in Fig. 1.1, lies in the plane of the paper with identical atoms at each corner of a square.

Definitions

A crystal or crystalline solid is an ordered array of atoms, molecules, or ions whose pattern or lattice is repeated periodically. A “single crystal” is perfectly ordered. Most crystalline solids are polycrystalline, meaning that they are composed of many small single crystals with defective, bounding surfaces between them. Large single crystals occur in nature, but often it is necessary to prepare them by special crystal-growth methods in laboratories. Single crystals are the preferred form for studying the intrinsic properties of a crystalline material.

Theoretical analysis of single-crystal properties usually assumes the crystalline structure is infinite or imposes periodic boundary conditions requiring the wave-function to repeat itself after a sufficiently large number of chemical units. This is a reasonable approximation, since a cubic-centimeter crystal contains on the order of 1022 to 1024 repeated units.

To sharpen our description of a lattice some definitions are useful.

1. • Bravais lattice. A Bravais lattice is a space-filling array of points generated by three primitive lattice vectors, a, b, and c. The vectors of the infinite set of vectors {R(l, m, n)} terminate on the Bravais lattice points. These vectors are R(l, m, n) = la + mb + nc, where l, m, and n are positive or negative integers or zero. Instead of the vectors a, b, and c, a Bravais lattice can be specified by the magnitudes of the primitive vectors, a = |a|, b = |b|,and c =|c|, and the angles between them, α, β, and γ.

Much of what we understand about the chemistry and optical properties of molecules has come from theoretical studies of very simple, empirical models. In most cases the theoretical models employ such drastic approximations that one may wonder why the results have any relevance at all to actual molecular systems. The success of these models may be attributed almost entirely to their use of group-theoretical concepts. In many cases symmetry is the dominant factor determining the electronic structure of a molecule. While the models are crude approximations, the general structure imposed by symmetry is usually exact and often independent of the details of the model employed. As a result many of the features have a much deeper truth than the model from which they are derived.

In this chapter we discuss the use of the LCAO method (linear combinations of atomic orbitals) to analyze the electronic structure of molecules. The term “atomic orbitals” is used loosely to mean one-electron orbitals whose angular functions are the spherical harmonics. The precise specifications of the radial parts of the orbitals are not needed for our discussion.

N-electron systems

It is generally assumed that the electronic states of a molecule or solid can be calculated for fixed positions of the nuclei. The electron's velocity is very large compared with the speed of vibratory motion, so that in effect the electrons instantly readjust to any motion of the nuclei. This assumption is referred to as the Born–Oppenheimer approximation.