Density Functional Theory

Density functional theory (DFT) is based on the Hohenberg-Kohn theorem [188], which states that the exact energy of a system can be expressed as a functional depending only on the electron density. In the Kohn-Sham implementation of DFT [189], this density is expressed in terms of Kohn-Sham orbitals {ψ_i}:

(1)

similarly to the density expression used for Hartree-Fock SCF calculations. For simplicity, we consider only closed shell systems in this overview of the method.

The Kohn-Sham orbitals are expressed as a linear combination of basis functions , and the coefficients for this expansion are solved iteratively using a self-consistent field method, as for Hartree-Fock. However, DFT includes exchange and/or correlation density functionals within the Fock matrix used for the SCF procedure. For DFT calculations, the Hartree-Fock exchange term K_ij in the Fock matrix is replaced by the exchange-correlation potential matrix elements :

(2)

where is an exchange-correlation functional and γ is .

The exchange-correlation functional is usually separated into exchange and correlation functional components that are local or non-local in the density:

(3)

Under the local density approximation (LDA), the non-local functionals and are ignored; when either or both of these terms are included, the generalized gradient approximation (GGA), also known as the non-local density approximation (NLDA), applies. The local and non-local exchange and correlation functionals available within Jaguar are described in Density Functional Theory and its references.

The electronic ground state energy E₀ is given by

(4)

(in Hartree atomic units), where V_nuc is the nuclear potential and J is the Coulomb potential. Therefore, for a given exchange-correlation functional, it is possible to solve iteratively for Kohn-Sham orbitals and the resulting density ρ to yield a final DFT energy.

A more detailed description of density functional theory can be found in Refs. [190] and [191].