Hybrid Functionals

An interesting class of functionals are hybrids [80], which combine exact (Hartree-Fock) exchange with conventional GGAs, the general form is,

$$E_{\hbox{{XC}}}^{\hbox{{hybrid}}} = \alpha (E_{\hbox{{X}}}^{\hbox{{HF}}} - E_{\hbox{{X}}}^{\hbox{{GGA}}}) + E_{\hbox{{XC}}}^{\hbox{{GGA}}} \, ,$$ (2.48)

where $E_{\hbox{{X}}}^{\hbox{{HF}}}$ is the Hartree-Fock exchange expression given in (1.26), except Kohn-Sham rather than Hartree-Fock orbitals are used, hence the wording ``exact-exchange''. The coefficient, $\alpha$, that determines the amount of exact-exchange mixing cannot be assigned from first-principles and so is fitted semi-empirically.

The logic behind this prescription was put forward by Becke [80] who noted that the limits of the adiabatic connection integral for the exact exchange-correlation energy (A.13) could be approximated as:

$$E_{\hbox{{XC}}} = \int_0^1 U^{\lambda} \, d \lambda = \frac{1}{2} U^0 + \frac{1}{2} U^1 \, .$$ (2.49)

Since $\lambda=0$ corresponds to the exchange only limit, this could well be described using Hartree-Fock theory, while $\lambda=1$ represents the most local part of the electron interactions, as a result of correlation, and so could be amenable to a local-type density functional treatment. As a result, Becke proposed the so-called half-and-half functional,

$$E_{\hbox{{XC}}} = \frac{1}{2} E_{\hbox{{X}}}^{\hbox{{HF}}} + \frac{1}{2} E_{\hbox{{XC}},\lambda=1}^{\hbox{{DF}}} \, ,$$ (2.50)

where $E_{\hbox{{XC}},\lambda=1}^{\hbox{{DF}}}$ is obtained from a density functional approximation such as the LDA. It later emerged from semi-empirical fits to atomic and molecular data that the optimum amount of exchange mixing should be reduced to $\sim~0.25$, although the precise value to employ depends upon the fitting data [81].

Hybrids give significant improvement over GGAs for many molecular properties, consequently they are a very popular choice of functional in quantum chemistry. Possibly the most widely used hybrid is the B$3$LYP functional proposed by Stevens  et al. [82] which is a generalisation of the B$3$P$86$ form devised by Becke [83]. Hybrids are not generally used in solid state physics because of the difficulty of computing the exact-exchange part within a plane-wave basis set. Nonetheless, hybrid functionals successfully demonstrate the need to incorporate fully non-local information in order to deliver greater accuracy.