  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 ac of second-order M{\o}ller-Plesset
(MP2) correlation energy is not necessarily equal to the square of the fraction
ax of Hartree-Fock exchange. More specifically, it is shown that ac \leq ax2, 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
{\lambda}1 variant, the corresponding double hybrid approximation has been
tested on a small set consisting of H2, N2, Be2, Mg2, and Ar2. Three
conventional double hybrids (B2-PLYP, B2GP-PLYP, and PBE0-DH) have been
considered. Potential curves obtained with {\lambda}1- and regular double
hybrids can, in some cases, differ significantly. In particular, for the weakly
bound dimers, the {\lambda}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.
