Hybrid functionals

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 the rest of the exchange–correlation energy from other sources (ab initio 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".

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 the rest of the exchange–correlation energy from other sources (ab initio 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 (HF) exchange (also called exact exchange) provides a simple scheme for improving the calculation of 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_\text{x}^\text{HF} = -\frac{1}{2} \sum_{i,j} \iint \psi_i^*(\mathbf r_1) \psi_j^*(\mathbf r_2) \frac{1}{r_{12}} \psi_j(\mathbf r_1) \psi_i(\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, although in the case of the "adiabatic connection functionals" the weights can be set a priori. E_\text{xc}^\text{B3LYP} =(1-a) E_\text{x}^\text{LSDA} + aE_\text{x}^\text{HF} + b\vartriangle E_\text{x}^\text{B} + (1-c)E_\text{c}^\text{LSDA} + c E_\text{c}^\text{LYP} , where a = 0.20 , b = 0.72 , and c = 0.81 . E_\text{x}^\text{B} is a generalized gradient approximation: the Becke 88 exchange functional and the correlation functional of Lee, Yang and Parr for B3LYP, and E_\text{c}^\text{LSDA} is the VWN (Vosko, Wilk, and Nusair 1980 Correlation functional) local spin density approximation to the correlation functional. The three parameters defining B3LYP have been taken without modification from Becke's original fitting of the analogous B3PW91 functional to a set of atomization energies, ionization potentials, proton affinities, and total atomic energies. E_\text{xc}^\text{PBE0} = \frac{1}{4} E_\text{x}^\text{HF} + \frac{3}{4} E_\text{x}^\text{PBE} + E_\text{c}^\text{PBE}, where E_\text{x}^\text{HF} is the Hartree–Fock exact exchange functional, E_\text{x}^\text{PBE} is the PBE exchange functional, and E_\text{c}^\text{PBE} is the PBE correlation functional. E_\text{xc}^{\omega\text{PBEh}} = a E_\text{x}^\text{HF,SR}(\omega) + (1 - a) E_\text{x}^\text{PBE,SR}(\omega) + E_\text{x}^\text{PBE,LR}(\omega) + E_\text{c}^\text{PBE}, where a is the mixing parameter, and \omega is an adjustable parameter controlling the short-rangeness of the interaction. Standard values of a = 1/4 and \omega = 0.2 (usually referred to as HSE06) have been shown to give good results for most systems. The HSE exchange–correlation functional degenerates to the PBE0 hybrid functional for \omega = 0 . E_\text{x}^\text{HF,SR}(\omega) is the short-range Hartree–Fock exact exchange functional, E_\text{x}^\text{PBE,SR}(\omega) and E_\text{x}^\text{PBE,LR}(\omega) are the short- and long-range components of the PBE exchange functional, and E_\text{c}^\text{PBE}(\omega) is the PBE correlation functional.

References

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