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Density-scaled correlation energy E c [n 1/λ] for the He (left) and Be (right) atoms as a function of λ. Accurate calculations (from the parametrizations of Ref. 9) are compared with different approximations: PBE without density scaling [Eq. (3)], PBE with density scaling (from the parametrizations of Ref. 9), and linear interpolation between MP2 (with PBE orbitals) and PBE [Eq. (9)]. The MP2 calculations for He and Be (including core excitations) have been performed with the cc-pV5Z and cc-pCV5Z basis sets (Refs. 10 and 11), respectively.
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Mean absolute errors for the AE6 (left) and BH6 (right) test sets as functions of the parameter λ for the 1DH, DS1DH, and LS1DH approximations with the PBE exchange-correlation density functional. All calculations were carried out with the cc-pVQZ basis set.
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We provide a rationale for a new class of double-hybrid approximations introduced by Brémond and Adamo [J. Chem. Phys.135, 024106 (2011)]10.1063/1.3604569 which combine an exchange-correlation density functional with Hartree-Fock exchange weighted by λ and second-order Møller-Plesset (MP2) correlation weighted by λ3. 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)]10.1063/1.3544215, as approximating the density-scaled correlation functional E c [n 1/λ] by a linear function of λ, interpolating between MP2 at λ = 0 and a density-functional approximation at λ = 1. Numerical results obtained with the Perdew-Burke-Ernzerhof density functional confirms the relevance of this double-hybrid model.
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