DFT

From Semiconductor Spectroscopy and Devices

To determine the structure and properties of an assembly of atoms accurately, we need to solve the quantum mechanical Schrödinger equation of the system. It is impossible to give an exact solution for anything more complicated than a hydrogen atom, so progress can only be made if a set of approximations are made:

• Born and Oppenheimer argued that, since the nuclear masses are so much larger than those of the electrons, then we can treat the nuclear motion classically and reduce the Schrödinger equation to one only involving the electrons moving in a potential of fixed nuclear sites.
• The Schrödinger equation of the electrons in a field of frozen nuclei still can't be solved because of the inter-electron electrostatic Coulomb interactions.
• Neglecting electron-electron interaction term is too extreme to make this theory useful and various attempts have been made to take this interaction into account. There are two principal schemes: Hartree-Fock (HF) and density functional theory (DFT). Both of these replace the electrostatic potential acting on each electron by an average over the potentials experienced by all the electrons.
• In these theories, the effective potential is composed of two terms: the Hartree potential (the is the electrostatic potential from all the electrons) and the exchange-correlation potential (a purely quantum mechanical contribution).
• In HF theory, the exchange-correlation is a complicated function determined by all the orbitals. In DFT, it is rigorously determined only by the total electron density.
• The precise dependence of the exchange-correlation potential on the density is known only for one particular problem: the homogeneous electron gas.
• For any other problem where the electron density varies throughout space, we can assume that the contribution from exchange-correlation potential at point r is given by the homogeneous electron gas value involving the density, n(r), at the same point r. This is called the local density approximation (LDA).
• This is the first successful approximate functional for DFT, and works well for many covalently solids.

LDA-DFT is implemented in many quantum mechanical modeling codes, we use the [AIMPRO] software.