Summary of Electronic Structure Methods
Authors: van Schilfgaarde Mark 
Published: October 2012

Most physical properties of interest in the solid state are governed by the electronic structure—that is, by the Coulombic interactions of the electrons with themselves and with the nuclei. Because the nuclei are much heavier, it is usually sufficient to treat them as fixed. Under this Born‐Oppenheimer approximation, the Schrödinger equation reduces to an equation of motion for the electrons in a fixed external potential, namely, the electrostatic potential of the nuclei (additional interactions, such as an external magnetic field, may be added).

Once the Schrödinger equation has been solved for a given system, many kinds of materials properties can be calculated. Ground‐state properties include the cohesive energy, or heats of compound formation, elastic constants or phonon frequencies, atomic and crystalline structure, defect formation energies, diffusion and catalysis barriers and even nuclear tunneling rates, magnetic structure, work functions, and the dielectric response. Excited‐state properties are accessible as well; however, the reliability of the properties tends to degrade–or requires more sophisticated approaches–the larger the perturbing excitation.

Because of the obvious advantage in being able to calculate a wide range of materials properties, there has been an intense effort to develop general techniques that solve the Schrödinger equation from “first principles” for much of the periodic table. An exact, or nearly exact, theory of the ground state in condensed matter is immensely complicated by the correlated behavior of the electrons. Unlike Newton's equation, the Schrödinger equation is a field equation; its solution is equivalent to solving Newton's equation along all paths, not just the classical path of minimum action. For materials with wide‐band or itinerant electronic motion, a one‐electron picture is adequate, meaning that to a good approximation the electrons (or quasiparticles) may be treated as independent particles moving in a fixed effective external field. The effective field consists of the electrostatic interaction of electrons plus nuclei, plus an additional effective (mean‐field) potential that originates in the fact that by correlating their motion, electrons can avoid each other and thereby lower their energy. The effective potential must be calculated self‐consistently, such that the effective one‐electron potential created from the electron density generates the same charge density through the eigenvectors of the corresponding one‐electron Hamiltonian.

This article resources the states of models electronic structure theory.

The other possibility is to adopt a model approach that assumes some model form for the Hamiltonian and has one or more adjustable parameters, which are typically determined by a fit to some experimental property such as the optical spectrum. Today such Hamiltonians are particularly useful in cases beyond the reach of first‐principles approaches, such as calculations of systems with large numbers of atoms, or for strongly correlated materials, for which the (approximate) first‐principles approaches do not adequately describe the electronic structure. In this article, the discussion is limited to the first‐principles approaches. Local‐density approximation is powerful, but has limitations. By comparing it with other approaches strengths and weakness are elucidated.