Hybrid functional

Hybrid functionals are a class of approximations to the exchange-correlation energy functional in density functional theory (DFT) that incorporate a portion of exact exchange from Hartree-Fock theory with exchange and correlation from other sources (ab initio, such as LDA, or empirical). The exact exchange energy functional is expressed in terms of the Kohn-Sham orbitals rather than the density, so is termed an implicit density functional. One of the most commonly used versions is B3LYP, which stands for Becke, 3-parameter, Lee-Yang-Parr.

Origin

The hybrid approach to constructing density functional approximations was introduced by Axel Becke in 1993. Hybridization with Hartree-Fock (exact) exchange provides a simple scheme for improving many molecular properties, such as atomization energies, bond lengths and vibration frequencies, which tend to be poorly described with simple "ab initio" functionals.

Method

A hybrid exchange-correlation functional is usually constructed as a linear combination of the Hartree-Fock exact exchange functional, $E_x^{\rm HF}$:

$E_x^{\rm HF}=\frac{1}{2}\sum_{i,j}\int\int\psi_i^*(\mathbf{r_1})\psi_j^*(\mathbf{r_1})\frac{1}{r_{12}}\psi_i(\mathbf{r_2})\psi_j(\mathbf{r_2})d\mathbf{r_1}d\mathbf{r_2}$,

and any number of exchange and correlation explicit density functionals. The parameters determining the weight of each individual functional are typically specified by fitting the functional's predictions to experimental or accurately calculated thermochemical data.

B3LYP

For example, the popular B3LYP (Becke, three-parameter, Lee-Yang-Parr) exchange-correlation functional is:

$E_{xc}^{\rm B3LYP} = E_{xc}^{\rm LDA} + a_0 (E_x^{\rm HF} - E_x^{\rm LDA}) + a_x (E_x^{\rm GGA} - E_x^{\rm LDA}) + a_c (E_c^{\rm GGA} - E_c^{\rm LDA}),$

where $a_0=0.20 \,\;$, $a_x=0.72\,\;$, and $a_c=0.81\,\;$ are the three empirical parameters determined by fitting the predicted values to a set of atomization energies, ionization potentials, proton affinities, and total atomic energies; $E_x^{\rm GGA}$ and $E_c^{\rm GGA}$ are generalized gradient approximations: the Becke 88 exchange functional and the correlation functional of Lee, Yang and Parr, and $E_c^{\rm LDA}$ is the VWN local-density approximation to the correlation functional.

HSE

The HSE (Heyd-Scuseria-Ernzerhof) exchange-correlation functional uses an error function screened Coulomb potential to calculate the exchange portion of the energy in order to improve computationally efficiency, especially for metallic systems.

$E_{xc}^{\rm \omega PBEh} = a E_x^{\rm HF,SR}(\omega) + (1-a) E_x^{\rm PBE,SR}(\omega) + E_x^{\rm PBE,LR}(\omega) + E_c^{\rm PBE},$

where $a = \frac{1}{4}$ is the mixing parameter and ω is an adjustable parameter controlling the short-rangeness of the interaction. The HSE exchange-correlation functional degenerates to the PBE0 hybrid functional for ω = 0. $E_x^{\rm HF,SR}(\omega)$ is the short range Hartree-Fock exact exchange functional, $E_x^{\rm PBE,SR}(\omega)$ and $E_x^{\rm PBE,LR}(\omega)$ are the short and long range components of the PBE exchange functional, and $E_C^{\rm PBE}(\omega)$ is the PBE correlation functional.