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.

The atoms of a crystal execute small, oscillatory motions about their equilibrium positions called lattice vibrations. These vibrations are stimulated by thermal energy or by external agents such as electromagnetic and mechanical forces. As with molecular vibrations, the atomic motions of the lattice can be expressed as linear combinations of the normal modes of motion. Classically, the energy contained in a given normal mode is unrestricted. In quantum theory the energy in a normal mode is quantized in discrete units of ħω. A quantum (ħω)ofenergyin a normal mode of vibration is called a phonon. More loosely, the lattice vibration wave in a crystal is also called a phonon.

Because of the translation symmetry of an (infinite) crystal the normal modes are characterized by a wavevector, k. In the case of lattice vibrations we associate a vector with the physical displacement of each atom from its equilibrium position. The Cartesian components of displacements transform in the same way as the p-orbitals and therefore the application of space-group theory to lattice vibrations is analogous to finding the tight-binding energy bands of a crystal with only p-orbitals on each atom. The method of analysis of lattice vibrations is the same as that employed in Chapter 10 for tight-binding energy bands. Instead of energy bands we obtain “phonon branches”.

In this chapter we illustrate how group theory can be used to find the normal mode frequencies of isotopically substituted molecules. We consider triatomic molecules of the AB2 type with isotopic substitutions for either the A atom or the B atom, or both. In particular, we analyze and derive numerical results for H2O when deuterium or tritium is substituted for hydrogen and when 18O is substituted for 16O. We begin by analyzing the vibrational modes of the AB2 molecule, and then go on to discuss the vibrational modes of isotopically substituted AB2 molecules.

Step 1: Identify the point group and its symmetry operations

A schematic representation of an AB2 molecule is shown in Fig. 2.1(a). In two dimensions the only symmetry operations are the identity, E, and 180-degree rotation about an axis through the A site, bisecting the B–B line. Reflection in the line of the C2 axis gives the same result as C2 in two dimensions. The group is C2. In three dimensions, the elements include C2, a reflection in the plane perpendicular to the plane of the paper and containing the C2 axis, σv, and reflection in the plane of the paper,σ′v. In three dimensions the group is C2v. Since the AB2 molecule has no out-of-plane vibrational modes (Exercise 1.15), we shall use the smaller C2 group for analysis.

Unlike orbital angular momentum, the total spin of a system can be integral or half-integral. Fermions such as electrons, positrons, neutrinos, and quarks possess intrinsic angular momentum or spin with a measurable value of ±1/2 (in units of ħ). Composite particles such as protons and neutrons also have measurable spin of ±1/2 and atomic nuclei can have half-integral spin values (1/2, 3/2, 5/2, …). The “spinor” function for half-integral spin is unusual in that rotation by 2π transforms it into the negative of itself. A rotation by 4π is required in order to transform the spin function into itself. While this may at first glance seem unreasonable, there are simple examples that display this property.

Take a strip of paper and form a Möbius strip by twisting one end 180° and joining it to the other end. Start at any point on the strip and trace a line through 360°. You do not end up at the starting point, but rather on the other side of the paper strip, as shown in Fig. 5.1. Continue tracing along the surface for another 360° and you will return to the original starting point.

Early spectroscopic experiments on hydrogen and hydrogen-like atoms revealed that there were twice as many states as predicted by the solutions of Schrödinger's equation. The idea that an electron could have intrinsic angular momentum (spin) with two possible states was proposed by Kronig, Uhlenbeck, and Goudsmit [5.1] in 1925.

Multilinear algebra is basically just linear algebra with many vector spaces at the same time. The fundamental objects are generalizations of vectors called tensors. Tensors can be viewed from many different perspectives. Mathematicians introduce tensors formally as a quotient of a certain module, while physicists introduce tensors using objects with many indices that transform in a specific way under a change of basis. Each definition is viewed disparagingly by the other camp, even though they are equivalent and both have their uses. We follow a middle approach here.

The tensor product

To start, we define a new kind of vector product, called the tensor product, usually denoted by the symbol ⊗. Given two vectors v and w, we can form their tensor product v ⊗ w. The product v ⊗ w is called a tensor of order 2 or a second-order tensor or a 2-tensor. The order of the factors matters, as w ⊗ v is generally different from v ⊗ w. In fancy language, the tensor product is noncommutative. We can form higher-order tensors by repeating this procedure. For example, given another vector u we can construct u ⊗ v ⊗ w, a third-order tensor. (The tensor product is associative, so we need not worry about parentheses.) Order-0 tensors are just scalars, while order-1 tensors are vectors.

This book offers a concise overview of some of the main topics in differential geometry and topology and is suitable for upper-level undergraduates and beginning graduate students in mathematics and the sciences. It evolved from a set of lecture notes on these topics given to senior-year students in physics based on the marvelous little book by Flanders [25], whose stylistic and substantive imprint can be recognized throughout. The other primary sources used are listed in the references.

By intent the book is akin to a whirlwind tour of many mathematical countries, passing many treasures along the way and only stopping to admire a few in detail. Like any good tour, it supplies all the essentials needed for individual exploration after the tour is over. But, unlike many tours, it also provides language instruction. Not surprisingly, most books on differential geometry are written by mathematicians.

The idea of a differentiable manifold had its genesis in the nineteenth century with the work of Carl Friedrich Gauss and of Georg Friedrich Bernhard Riemann. Gauss was interested in surveying and cartography, which led him to develop the tools of calculus on curved surfaces. His famous theorema egregium, or remarkable theorem, revealed that one could consider the intrinsic properties of a surface independently of the way in which it was embedded in three-dimensional space, and this led him, Riemann, and others, to abstract these concepts even further. Their ideas have had far reaching applications in many areas of mathematics and the natural sciences.

Roughly, an n-dimensional manifold (or n-manifold) can be thought of as a kind of patchwork quilt built from pieces of ℝn. Classic examples of 2-manifolds are the 2-sphere S2 and the 2-torus T2 (see Figure 3.1). Usually one pictures these as living in ℝ3, but one can consider them in their own right just as bits of ℝ2 sewn together in certain ways. The technical definition of a manifold requires considerable background, which we will try to keep to a minimum. First, we need the idea of a topology.

Basic topology*

Consider a basketball. When it is inflated, its surface is a sphere. But when it is deflated its surface is still a topological sphere.

As we have seen, a differentiable or smooth manifold M is just a topological space on which we have the ability to differentiate stuff. This is adequate if one is interested only in differential topology, but to do more we need to introduce additional structure. This additional structure is a geometry. Basically, we want to be able to measure distances on M as well as determine the angles between vectors. Of course, the vectors don't live on M – they live on the tangent spaces to M. What we need is an inner product on the tangent spaces. But we want to be able to differentiate things, so we want the inner product to vary smoothly.

This leads us to the following definition. A smooth inner product or metricg on M is just a smooth map p ↦ gp, where gp is an inner product on TpM. A geometric manifold(M, g) is a manifold M equipped with a smooth inner product g. Two geometric manifolds (M, g) and (N, h) are considered equivalent if they are isometric, meaning there exists a diffeomorphism φ : M → N with φ*g = h.