Nearsightedness

Throwing out k-space

The concept of localization can be imbedded directly into the methods of electronic structure to create new algorithms that take advantage of locality or “nearsightedness” as coined by W. Kohn. As opposed to the textbook starting point for describing crystals in terms of extended Bloch eigenstates, many physical properties can be calculated from the density matrix ρ(r, r′), which is exponentially localized in an insulator or a metal at finite T. For large systems, this fact can be used to make “order-N” or O(N) methods where the computational time scales linearly in the size of the system. There are two aspects of the problem: “building” the hamiltonian and “solving” the equations. Here we emphasize the second aspect, which is more fundamental, and describe representative O(N) approaches that either treat ρ(r, r′) directly or work in terms of Wannier-like localized orbitals.

The reader should be aware (beware) that O(N) methods are under development; there are problems and shortcomings in actual practice.

Classification of crystals and their excitations by symmetry is a general approach applicable to electronic states, vibrational states, and other properties. The first part of this chapter deals with translational symmetry which has the same universal form in all crystals, and which leads to the Bloch theorem that rigorously classifies excitations by their crystal momentum. (The discussion here follows Ashcroft and Mermin, [84], Chs. 4–8.) The other relevant symmetries are time reversal and point symmetries. The latter depend upon a specific crystal structure and are treated only briefly. Detailed classification can be found in many texts, and computer programs that deal with the symmetries can be found on-line at sites listed in Ch. 24.

The subject of this chapter is the role of plane waves and grids in modern electronic structure calculations, which builds upon the basic formulation of Ch. 12. Plane waves have played an important role from the early OPW calculations to widely used methods involving norm-conserving pseudopotentials. Plane waves continue to be the basis of choice for many new developments, such as quantum molecular dynamics simulations (Ch. 18), owing to the simplicity of operations. Efficient iterative methods (App. M) have made it feasible to apply plane waves to large systems, and recently developed approaches such as “ultrasoft” pseudopotentials and projector augmented waves (PAWs Ch. 11) have made it feasible to apply plane waves to difficult cases such as materials containing transition metals. Real-space grids are an intrinsic part of efficient planewave calculations and there is a growing development of real-space methods, including multigrids, finite elements, wavelets, etc.

E Pluribus Unum

The fundamental tenet of density functional theory is that any property of a system of many interacting particles can be viewed as a functional of the ground state density n0(r); that is, one scalar function of position n0(r), in principle, determines all the information in the many-body wavefunctions for the ground state and all excited states. The existence proofs for such functionals, given in the original works of Hohenberg and Kohn and of Mermin, are disarmingly simple. However, they provide no guidance whatsoever for constructing the functionals, and no exact functionals are known for any system of more than one electron. Density functional theory (DFT)would remain a minor curiosity today if it were not for the ansatz made by Kohn and Sham, which has provided a way to make useful, approximate ground state functionals for real systems of many electrons. The subject of this chapter is density functional theory as a methodology for many-body systems; Ch. 7 describes the Kohn–Sham ansatz that replaces the interacting problem with an auxiliary independent-particle problem with all many-body effects included in an exchange–correlation functional; Ch. 8 deals with widely used approximations for the exchange–correlation functional; and Ch. 9 is devoted to solution of the Kohn–Sham independent-particle equations in a general form useful for all Kohn–Sham calculations. Following chapters in this volume are devoted to algorithms for actual calculations, and applications to problems in atomic, molecular, and condensed matter physics.

Many properties of materials – mechanical, electrostatic, magnetic, thermal, etc. – are determined by the variations of the total energy around the equilibrium configuration, defined by formulas such as (2.2)–(2.7). Experimentally, vast amounts of information about materials are garnered from studies of vibration spectra, magnetic excitations, and other responses to experimental probes. This chapter is devoted to the role of electronic structure in providing predictions and understanding of such properties, through the total energy and force methods described in previous chapters, as well as recent advances in efficient methods for calculation of response functions themselves. Through these developments, calculation of full phonon dispersion curves, dielectric functions, infrared activity, Raman scattering intensities, magnons, anharmonic energies to all orders, phase transitions, and many other properties have been brought into the fold of practical electronic structure theory.

Theoretical analysis of the electronic structure of matter provides understanding and quantitative methods that describe the great variety of phenomena observed. A list of these phenomena reads like the contents of a textbook on condensed matter physics, which naturally divides into ground state and excited state electronic properties. The aim of this chapter is to provide an introduction to electronic structure without recourse to mathematical formulas; the purpose is to lay out the role of electrons in determining the properties of matter and to present an overview of the challenges for electronic structure theory.

A density is a field defined at each position r, for example the particle number density n(r), which is a well-defined, experimentally measurable function. It would be desirable to have expressions for other densities, in particular, energy and stress densities. However, energy and stress densities are not unique on a microscopic quantum scale, even though they are the basis of the theory of elasticity on a macroscopic scale. This appendix brings out three points: (1) certain integrals of energy and stress densities are unique and very useful; (2) there are important contributions to the energy or stress density that are completely unique – these include all terms that arise from the fact that electrons are a many-body system of fermions; (3) all other terms that are non-unique can be shown to involve only the single scalar number density – there are different possible choices for these terms, each involving only derivatives of the density n(r) or the classical Coulomb potential VCC(r) which is directly related to n(r). It follows that all the issues of non-uniqueness are exactly the same as in a one-particle problem.

Localized functions afford a satisfying description of electronic structure and bonding in an intuitive localized picture. They are widely used in chemistry and have been revived in recent years in physics for efficiency in large simulations, especially “order-N” methods (Ch. 23). The semi-empirical tight-binding method is particularly simple and instructive since the basis is not explicitly specified and one needs only the matrix elements of the overlap and the hamiltonian. This chapter starts with a definition of the problem of electronic structure in terms of localized orbitals, and considers various illustrative examples in the tight-binding approach. Many of the concepts and forms carry over to full calculations with localized functions that are the subject of the following chapter, Ch. 15.

The fundamental idea of a “pseudopotential” is the replacement of one problem with another. The primary application in electronic structure is to replace the strong Coulomb potential of the nucleus and the effects of the tightly bound core electrons by an effective ionic potential acting on the valence electrons. A pseudopotential can be generated in an atomic calculation and then used to compute properties of valence electrons in molecules or solids, since the core states remain almost unchanged. Furthermore, the fact that pseudopotentials are not unique allows the freedom to choose forms that simplify the calculations and the interpretation of the resulting electronic structure. The advent of “ab initio norm-conserving” and “ultrasoft” pseudopotentials has led to accurate calculations that are the basis for much of the current research and development of new methods in electronic structure, as described in the following chapters.

Many of the ideas originated in the orthogonalized plane wave (OPW) approach that casts the eigenvalue problem in terms of a smooth part of the valence functions plus core (or core-like) functions. The OPW method has been brought into the modern framework of total energy functionals by the projector augmented wave (PAW) approach that uses pseudopotential operators but keeps the full core wavefunctions.

The subject of this appendix is the macroscopic stress that enters mechanical properties of matter in the form of stress–strain relations. The stress tensor is the generalization of pressure to all the independent components of dilation and shear, and the “stress theorem” is the generalization of the virial theorem for scalar pressure to all components of the stress tensor. In condensed matter, the state of the system is specified by the forces on each atom and the stress, which is an independent variable. The conditions for equilibrium are: (1) that the total force vanishes on each atom, and (2) that the macroscopic stress equals the externally applied stress.