Rationale for a new class of double-hybrid approximations in density-functional theory

We provide a rationale for a new class of double-hybrid approximations introduced by Br\'emond and Adamo [J. Chem. Phys. 135, 024106 (2011)] which combine an exchange-correlation density functional with Hartree-Fock exchange weighted by $\l$ and second-order M{\o}ller-Plesset (MP2) correlation weighted by $\l^3$. We show that this double-hybrid model can be understood in the context of the density-scaled double-hybrid model proposed by Sharkas et al. [J. Chem. Phys. 134, 064113 (2011)], as approximating the density-scaled correlation functional $E_c[n_{1/\l}]$ by a linear function of $\l$, interpolating between MP2 at $\l=0$ and a density-functional approximation at $\l=1$. Numerical results obtained with the Perdew-Burke-Ernzerhof density functional confirms the relevance of this double-hybrid model.