Hybrid Density Functionals

A familiar theme when accessing the accuracy of exchange-correlation functionals is that density based functionals will overestimate a quantity which HF theory will underestimate (for example, bond lengths [100]). With this in mind, Becke [119] has argued that the exact exchange-correlation functional must include a fraction of HF exchange. Initially Becke [120] proposed a functional that consisted of 50% HF exchange and 50% density-based exchange-correlation. This functional was quickly replaced by the three parameter mix denoted B3P
$$\begin{align}E_{xc}^{B3P} = E_{xc}^{LSDA} + a_{0}(E_{x}^{HF}-E_{x}^{D30})+a_{x}(E_{x}^{B88}-E_{x}^{D30})+a_{c}E_{c}^{PW91} \end{align}$$
where E_c^PW91 is the gradient correction of the Perdew-Wang correlation functional [107,108] (often used in conjunction with E_x^PW91). The three parameters a₀=0.20, a_x=0.72 and a_c=0.81 were determined by minimizing the atomization energies, ionization energies, electron affinities and proton affinities of the G2 dataset.

An alternative functional has been proposed, using E_c^LYP instead of E_c^PW91 [121]. B3LYP has the form
$$\begin{align}E_{xc}^{B3LYP} = E_{xc}^{LSDA} + a_{0}(E_{x}^{HF}-E_{x}^{D30})+a_{x}(E_{x}^{B88}-E_{x}^{D30})+a_{c}(E_{c}^{LYP}-E_{c}^{VWN}) \end{align}$$
with the same parameters as in B3P. B3LYP shows surprising accuracy for thermochemistry, structures and spectroscopic properties of first row molecules [122]. The high accuracy of B3LYP has made it perhaps the most popular functional of modern density functional theory.