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 x HF = − 1 2 ∑ i , j ∬ ψ i ∗ ( r 1 ) ψ j ∗ ( r 2 ) 1 r 12 ψ j ( r 1 ) ψ i ( r 2 ) d r 1 d r 2 {\displaystyle 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.
B3LYP For example, the popular B3LYP (Becke, 3-parameter, Lee–Yang–Parr) exchange-correlation functional is
E xc B3LYP = ( 1 − a ) E x LSDA + a E x HF + b △ E x B + ( 1 − c ) E c LSDA + c E c LYP , {\displaystyle 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}}+cE_{\text{c}}^{\text{LYP}},}
where a = 0.20 {\displaystyle a=0.20} , b = 0.72 {\displaystyle b=0.72} , and c = 0.81 {\displaystyle c=0.81} . E x B {\displaystyle 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 c LSDA {\displaystyle 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.
PBE0 The PBE0 functional mixes the Perdew–Burke–Ernzerhof (PBE) exchange energy and Hartree–Fock exchange energy in a set 3:1 ratio, along with the full PBE correlation energy:
E xc PBE0 = 1 4 E x HF + 3 4 E x PBE + E c PBE , {\displaystyle 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}},}
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