Rigorous formulation of two-parameter double-hybrid density-functionals.

A two-parameter extension of the density-scaled double hybrid approach of Sharkas et al. [J. Chem. Phys. 134, 064113 (2011)] is presented. It is based on the explicit treatment of a fraction of multideterminantal exact exchange. The connection with conventional double hybrids is made when neglecting density scaling in the correlation functional as well as second-order corrections to the density. In this context, the fraction a(c) of second-order Møller-Plesset (MP2) correlation energy is not necessarily equal to the square of the fraction a(x) of Hartree-Fock exchange. More specifically, it is shown that a(c)≤a(x)(2), a condition that conventional semi-empirical double hybrids actually fulfill. In addition, a new procedure for calculating the orbitals, which has a better justification than the one routinely used, is proposed. Referred to as λ(1) variant, the corresponding double hybrid approximation has been tested on a small set consisting of H(2), N(2), Be(2), Mg(2), and Ar(2). Three conventional double hybrids (B2-PLYP, B2GP-PLYP, and PBE0-DH) have been considered. Potential curves obtained with λ(1)- and regular double hybrids can, in some cases, differ significantly. In particular, for the weakly bound dimers, the λ(1) variants bind systematically more than the regular ones, which is an improvement in many but not all cases. Including density scaling in the correlation functionals may of course change the results significantly. Moreover, optimized effective potentials based on a partially-interacting system could also be used to generate proper orbitals. Work is currently in progress in those directions.