Orbital Functionals in Static and Time-Dependent Density Functional Theory

Density functional theory (DFT) is among the most powerful quantum mechanical methods for calculating the electronic structure of atoms, molecules and solids [1, 2, 3]. In this introduction we present a brief overview of DFT, putting particular emphasis on the appearance of orbital functionals. To describe the electronic many-body system, we have to deal with the Hamiltonian $$ \hat H = \hat T + {\hat W_{C1b}} + \hat V$$ (1) where $$ \hat T = \sum\limits_{i = 1}^N {\left( { - \frac{{{h^2}}}{{2m}}\nabla _i^2} \right)}$$ (2) denotes the kinetic-energy operator, $$ {\hat W_{C1b}} = \frac{1}{2}\sum\limits_{\begin{array}{*{20}{c}} {i,j = 1} \\ {i \ne 1} \end{array}}^N {\frac{{{e^2}}}{{|{r_i} - {r_j}|}}} $$ (3) represents the Coulomb interaction between the electrons, and $$ \hat V = \sum\limits_{i = 1}^N {v({r_i})} $$ (4) contains all external potentials acting on the electrons, typically the Coulomb potentials of the nuclei.