The description of the physical and chemical properties of matter is a central issue that has occupied the minds of scientists since the age of the ancient Greeks. In their route to dissect matter down to what cannot be divided any further, they coined the term atom, the indivisible. Matter became then a collection of atoms. More than twenty centuries had to pass until the development of a more precise concept of atom, thanks, amongst others, to the systematic studies of Mendeleyev and the establishment in 1869 of the periodic table of the elements (Mendeleyev, 1869). The discovery of the electron in 1897 and the first modern model of the atomic structure by Sir Joseph Thomson were soon refined by his student, Sir Ernest Rutherford, who in 1910 showed that an atom was made of a positively charged small nucleus and a number of negatively charged electrons that neutralize the nuclear charge. Much in the spirit of planetary systems, and drawing from the analogy between gravitational and electrostatic interactions, scientists in the beginning of the twentieth century built an image of the atom that consisted of a number Z of electrons – of elementary charge – e – orbiting around the nucleus of charge Ze.

A number of experimental observations, though, were incompatible with this idea of orbiting electrons. In particular, according to the successful electromagnetic theory, charged electrons in orbital (radially accelerated) motion should radiate energy, thus decelerating and eventually collapsing onto the nucleus. Clearly, such a picture would imply that matter is essentially unstable, in flagrant contradiction with our everyday experience of the very existence of matter.

Parallel to the approaches described in the previous chapter, a different line of thought drove L. H. Thomas and E. Fermi to propose, at about the same time as Hartree (1927–1928), that the full electronic density was the fundamental variable of the many-body problem. From this idea they derived a differential equation for the density without resorting to one-electron orbitals (Thomas, 1927; Fermi, 1928). The original Thomas–Fermi approximation was actually too crude, mainly because the approximation used for the kinetic energy of the electrons was unable to sustain bound states. However, it set up the basis for the later development of density functional theory (DFT), which has been the way of choice in electronic structure calculations in condensed matter physics during the past twenty years, and, recently, it also became accepted by the quantum chemistry community because of its computational advantages compared to post-Hartree–Fock methods of comparable quality.

This chapter is organized as follows: we first give an account of Thomas–Fermi theory and then present a modern approach to DFT that takes into account the formal properties of density functionals. We then move into the core of DFT by stating the basic theorems and developing the mathematical framework of orbitalbased (Kohn–Sham) DFT. Next, we describe the most common approximations to exchange and correlation. A detailed analysis of these approximations and a general idea of which kind of systems can be safely treated within them is deferred to Chapter 5.

Thomas–Fermi theory

Thomas (1927), and independently Fermi (1928), gave a prescription for calculating the energy of an electronic system exclusively in terms of the electronic density.

Very often when a postgraduate student begins a research activity in a new field, she or he is presented with a partial view of the big picture, according to the arena where the research group carries out its activity. For the student this implies narrowing the focus to a class of systems such as molecules, surfaces, liquids, defects, or magnetic systems, and to concentrate on a particular theoretical approach and one or a few specific computational techniques out of the many possibilities available. All this is perfectly understandable and reasonable because it is very difficult to absorb rapidly the knowledge accumulated during many decades, since the pioneering work of Hartree, Fock, Slater, Thomas, Fermi, Bloch, Dirac, and Wigner in the twenties and thirties, until the most recent developments in areas such as electronic correlation. This knowledge is built up during many years of practice in the field, participating in schools, conferences, and workshops that promote exchange and collaboration between the different subareas, discussing advantages and disadvantages of different approaches with colleagues around the world, and keeping up with the latest developments in the literature.

Although many excellent books and reviews are available in areas such as density functional theory, solid state physics, quantum chemistry, electronic structure methods, and Car–Parrinello simulations, it is desirable to have all this reference material condensed in a single book that may be used by a fresh graduate student, by a postdoc who is moving into the field, or by a researcher with experience in one or a few subareas who wants to broaden her or his horizons.

Once the level of theory (DFT-LDA, Hartree–Fock, or other) has been chosen, the differences between electronic structure methods are essentially due to the choice of basis set. Pseudopotentials may or may not be part of the package. The main difference is the replacement of the bare Coulomb potential of the nucleus by a softer potential and some technical issues regarding the angular dependence of the pseudopotential, but the abundance of electronic structure methods in the market is mostly due to the quest for the ultimate basis set.

The central and computationally most intensive aspect of an electronic structure calculation is the self-consistent solution of the one-electron eigenvalue equation. This involves the calculation of the Kohn–Sham or the Fock matrix elements and the corresponding energy. In addition, geometry optimization and molecular dynamics simulations require the calculation of forces on the nuclear degrees of freedom. In solid-state applications, the optimization of lattice parameters and constant-pressure molecular dynamics simulations require also the calculation of the stress tensor.

In this chapter we shall describe how the Hamiltonian and the total energy are calculated in practice, in the most widely used methods. We start in Section 9.1 by introducing the KKR method as an approach derived from multiple scattering theory, where basis sets expansions are bypassed by using Green's function techniques. Section 9.2 is devoted to describing the most relevant aspects of allelectron schemes based on augmentation spheres. These are considered amongst the most accurate approaches because they do not approximate the behavior of core electrons through pseudopotentials, and the basis sets are flexible and adjust themselves according to the eigenvalue energies.

This introductory chapter has two goals. The first one is usual for books of this kind and is aimed at providing the reader with a brief outline of the background history of the subject, its main content, form of presentation, and correlation with other related subjects. The second goal is to give this introduction in a way which allows the reader to get a general (although maybe rather rough and superficial) impression of the whole subject and its possibilities, a very brief insight into this trend without reading the corresponding chapters or sections. This is done keeping in mind that quite a number of physicists, chemists, and biologists, who at present are not engaged in the use of the Jahn–Teller effect, may be interested to know in general the status quo in this field and make a fast choice of the parts of it they may be interested in. In other words, the introduction is aimed at giving a very brief qualitative description of the main features of the Jahn–Teller effect theory in a way useful also for the reader who has no intention to read the whole book or its parts. The applications of the theory to chemical problems and molecular systems are given in Chapter 7, while the Jahn–Teller effect in specific solid-state problems is considered in Chapter 8.

This chapter is to introduce the reader to JT problems that may be considered as “less traditional” in the sense that they do not follow exactly from the original (“classical”) JT theorem. Indeed, the PJTE considered below takes place in nondegenerate states, while the RTE is relevant to linear molecules, which are exceptions in the original JT formulation. The product JTE is an approximation to both JT and PJT problems. All these effects are inalienable components of the JT trend as a whole.

Two-level and multilevel pseudo JT (PJT) problems. Uniqueness of the PJT origin of configuration instability and its bonding nature

Among the JT vibronic coupling effects the pseudo JT (PJT) effect occupies an outstanding place. Indeed, while the occurrence of the JT and RT effects is limited to polyatomic systems in degenerate electronic states, the PJT effect may take place, in principle, in any system without a priori limitations. This circumstance enlarges essentially the possible subjects of the JT vibronic coupling effects, transforming the JTE theory into a general tool for better understanding and solving problems of structure and properties of molecules and crystals. The waiver of the degeneracy restrictions together with the proof of the uniqueness of the PJT origin of structural instabilities of polyatomic high-symmetry configurations elevates the JT effect theory to a general approach to molecular and crystal problems.

The PJT effect emerges directly from the basic equations (2.6) for the vibronically coupled electronic states.

With this chapter we begin the applications of the general theory of the JTE to all-range spectroscopy, molecular structure, and solid-state physics (Chapters 6–8). Among them the JTE in spectral properties is the most significant due to the sensitivity of spectra to changes in electronic structure and vibronic coupling. On the other hand, the influence of vibronic coupling on spectra is very specific; it depends on both the system parameters and the spectral range under consideration. Still there are some general features common to all systems with the same JT problem.

This chapter is devoted to such a general theory of the JTE in spectroscopy. More particular problems are considered in Chapters 7 and 8, together with the corresponding specific systems, but the separation of the general theory from more system-oriented questions is to a large extent conventional. Obviously, there is also substantial overlap with calculations of vibronic states presented in Chapter 5: spectroscopy is inalienable from energies and wavefunctions of the system.

Electronic transitions

Optical band shapes

Optical band shapes with the JTE were subjected to multiple investigations by many authors (see the reviews [6.1–6.10] and references therein).

The term band shape means an envelope of elementary transitions between vibronic states, each of which has a specific width so that the vibronic lines merge into a continuous band.

The JTE in local properties of solids

Local properties in crystals are strongly affected by JT vibronic coupling phenomena, quite similar to molecular systems discussed in the previous chapter. There is no sharp border between clusters and coordination systems, considered in Sections 7.6.1 and 7.6.2, and local formations in crystals (which are often modeled by clusters). Differences occur for impurity centers for which the JT problem is in fact a multimode one (Sections 3.5 and 5.5) with a very large (infinite) number of JT active modes that are significantly involved in the vibronic coupling. Another important crystal problem emerges when there are multiple JT impurity centers at sufficiently small intercenter distances with sufficiently strong interaction between them. In the limit of an infinite number of such JT centers that occupy regular positions in the lattice we come to the cooperative JTE (CJTE) considered in Sections 8.2 and 8.3. The JTE in superconductivity and colossal magnetoresistance is discussed in Section 8.4.

Impurity centers in crystals

We start with a single JT center in a regular crystal in the form of an impurity center or a lattice defect that has localized electronic states. The latter may be either orbitally degenerate and subject to the JTE (when the crystal field of the environment has at least one axis of symmetry of third or higher order), or pseudodegenerate with a PJTE (for which there are no symmetry restrictions, but the ground state should be force-equilibrated in the reference configuration, Section 4.1).

The Jahn–Teller effect (JTE) is one of the most fascinating phenomena in modern physics and chemistry. It emerged in 1934 in a discussion between two famous physicists, L. Landau and E. Teller, and grew into a general tool for understanding and an approach to solving molecular and crystal problems, which is applicable to any polyatomic system. The first formulation of this effect as instability of molecular configurations in electronically degenerate states proved to be the beginning of a whole trend which rationalizes the origin of all possible instabilities of high-symmetry configurations, and the peculiar nuclear dynamics resulting from these instabilities as well as the origins of all structural symmetry breakings in molecular systems and condensed matter.

Intensive development of the JTE theory began in the late 1950s together with a wave of main applications to spectroscopy, stereochemistry, and structural phase transitions, which lasted a couple of decades. The next significant resurgence of interest in the Jahn–Teller effect is related to the late 1980s and is still continuing. It was triggered by one of the most important Nobel Prize discoveries in physics of our times inspired by the Jahn–Teller effect: the high-temperature superconductivity. As explained by the authors of this discovery, “the guiding idea in developing this concept was influenced by the Jahn–Teller polaron model” (J. G. Bednorz and K. A. Müller, in Nobel Lectures: Physics, Ed. G. Ekspong, World Scientific, Singapore, 1993, p. 424).