The previous examples on parameter estimation and model comparison have demonstrated the benefits of Bayesian probability theory for quantitative inference based on prior knowledge and measured data. However, Bayesian probability theory is not a magic black box capable of compensating for badly designed experiments. Information absent in the data cannot be revealed by any kind of data analysis. This immediately raises the question of how the information provided by a measurement can be quantified and, in a next step, how to optimize experiments to maximize the information gain. Here one of the very recent areas of applied Bayesian data analysis is entered: Bayesian experimental design is an increasingly important topic driven by progress in computer power and algorithmic improvements [132, 214]. So far it has been implicitly assumed that there is little choice in the actual execution of the experiment, in other words, the data to be analysed were assumed to be given. While this is the most widespread use of data analysis, the active selection of data holds great promise to improve the measurement process. There are several scenarios in which an active selection of the data to be collected or evaluated is obviously very advantageous, for example:

• Expensive and/or time-consuming measurements, thus one wants to know where to look next to learn as much as possible – or when to stop performing further experiments.

Stochastic processes

If a classical particle moves in an external force field F(x) its motion is deterministic and it could be called a ‘deterministic process’. If the particle interacts with randomly distributed obstacles its motion is still deterministic, but the trajectory depends on the random characteristics of the obstacles, and maybe also on the random choice of the initial conditions. Such a motion is an example of a ‘stochastic process’, or rather a ‘random process’.

In a more abstract generalized definition, a stochastic process is a random variable Xξ that depends on an additional (deterministic) independent variable ξ, which can be discrete or continuous. In most cases it stands for an index ξ ∈ N or time t, space x or energy E. An almost trivial ubiquitous stochastic process is given by additive noise η(t) on a time-dependent signal s(t), i.e. Xt = s(t) + η(t). As a consequence, Xt is no longer continuous. The most apparent applications of stochastic processes are time series of any kind that depend on some random impact. A broad field of applications are time series occurring, for example, in business, finance, engineering, medical applications and of course in physics. Beyond time series analysis, stochastic processes are at the heart of diffusion Langevin dynamics, Feynman's path integrals [43], as well as Klauder's stochastic quantization [62], which represents an unconventional approach to quantum mechanics. Here we will give a concise introduction and present a few pedagogical examples.

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.