Background Theory

In Hartree-Fock theory, the energy has the form,

$$E_{HF} = h + J - K + V_{nuc}$$ (4.1)

where $h$ is the one-electron (kinetic plus potential) energy, $J$ is the classical Coulomb repulsion of the electrons, $K$ is the exchange energy resulting from the quantum (fermion) nature of electrons, and $V_{nuc}$ is the nuclear repulsion energy.

In density functional theory, the exact Hartree-Fock exchange for a single determinant is replaced by a more general expression, the exchange-correlation functional, which can include terms accounting for both exchange energy and the electron correlation which is omitted from Hartree-Fock theory. The DFT energy is expressed as a functional of the molecular electron density $\rho({\bf r})$,

$$E_{DFT}[\rho] = T[\rho] + V_{ne}[\rho] + J[\rho] + E_x[\rho] + E_c[\rho] + V_{nuc}$$ (4.2)

where $T[\rho]$ is the kinetic energy, $V_{ne}[\rho]$ is the nuclei-electron interaction, $E_x[\rho]$ and $E_c[\rho]$ are the exchange and correlation energy functionals.

The exchange and correlation functionals normally used in DFT are integrals of some function of the density and possibly the density gradient. In addition to pure DFT methods, DSCF and GRAD modules support hybrid functionals in which the exchange functional includes the Hartree-Fock exchange, e.g. B3-LYP.

With TURBOMOLEversion 5.9, the exact (non-RI) exchange for DFT hybrid functionals and Hartree-Fock can be used in combination with RI-$J$, using the modules RIDFTand RDGRAD. RIDFTwill use techniques that reduce the scaling behaviour of the exchange evaluation. This reduces the computational cost of such calculations, especially for large systems, significantly.