When the electronic many-body problem is solved at any level of theory, its solution provides energies and atomic forces that allow for the determination of structural and thermodynamic properties. Forces also open the road to molecular dynamics simulation and the computation of dynamical properties. In addition, and at variance with classical force fields, any such theory would give access to electronic properties such as electronic energy levels or bands, electronic density, charge transfer, and, in principle, also optical and electronic transport properties.

It is clear that the simpler the treatment of the electronic variables, the more efficient the calculation. Therefore, larger systems can be studied, and longer MD simulations permit us to accumulate more reliable statistics and give access to dynamical phenomena occurring on longer time scales. If the information arising from the electronic degrees of freedom is not strictly necessary, then a strategy that removes the electrons altogether out of the picture by means of effective interatomic potentials (classical force fields) is a winning strategy. However, as long as electrons are treated explicitly, the determination of interatomic potentials requires the solution of the Schrödinger equation, and this is far more expensive than just replacing distances and angles in an explicit formula.

Linear scaling with the system size in the solution of Hartree–Fock or Kohn– Sham equations can only be achieved for very large systems. But even in that limit, when using an atom-centered basis set, the calculation of the matrix elements and the diagonalization of the Hamiltonian matrix require a major computational effort; the latter scales as N3.

The wave functions for free electrons in a periodic crystal can be expanded in plane waves (PWs). If the potential due to the atoms is neglected, then PWs are the exact solution. If the potential is reasonably smooth, then it can be treated as a perturbation, thus leading to the so-called nearly-free electron model (see Section 6.2). The potential originated in the atomic nuclei, however, is far from smooth. In the simplest case of hydrogen the potential is –1/r, which diverges at the origin. The 1s wave function does not diverge, but it exhibits a cusp at the origin, and decays exponentially with distance. For heavier atoms the wave functions associated with core states are even steeper. Therefore, a PW expansion of the wave functions in a real crystal is a rather hopeless task, because the number of PW components required to represent such steep wave functions is huge. However, it would be desirable to retain the simplicity of the PW approach. Slater (1937) suggested a possible solution to this problem, where the PW expansion was augmented with the solutions of the atomic problem in spherical regions around the atoms, and the potential was assumed to be spherically symmetric inside the spheres, and zero outside (APW method).

In order to overcome this shape approximation of the potential, Herring (1940) proposed an alternative method consisting of constructing the valence wave functions as a linear combination of PW and core wave functions. By choosing appropriately the coefficients of the expansion, this wave function turns out to be orthogonal to the core states. Hence the name of orthogonalized plane wave (OPW) method.

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.

LEDs can be used for either free-space communication or for fiber communication applications. Free-space communication applications include the remote control of appliances such as television sets and stereos, and data communication between a computer and peripheral devices. LEDs used in optical fiber communication applications are suited for distances of a few km and bit rates up to about 1 Gbit/s. Most fibers used with LEDs are multimode (step-index and graded-index) fibers. However, some applications employ single-mode fibers.

LEDs for free-space communication

Free-space communication LEDs are commonly made with GaAs or GaInAs active regions and are grown on GaAs substrates. The GaInAs layer is pseudomorphic, i.e. sufficiently thin that it is coherently strained, and no dislocations are generated. The emission wavelength of GaAs and coherently strained GaInAs LEDs is limited to wavelengths in the IR ranging from 870 nm (for GaAs active regions) to about 950 nm (for GaInAs active regions).

The wavelength of free-space communication LEDs is in the infrared so that the light emitted is invisible to the human eye and does not distract. Since free-space communication usually involves transmission distances of less than 100 m, the transmission medium (air) can be considered, to a good approximation, to be lossless and dispersionless.

The total light power is an important figure of merit in free-space communication LEDs, so that the internal efficiency and the extraction efficiency need to be maximized. The emission pattern (far field) is another important parameter.