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

where the first three terms are calculated in a usual selfconsistent hybrid Kohn-Sham (KS) calculation, and the last perturbative term is evaluated with the previously obtained orbitals and added a posteriori. The two empirical parameters a x and a c can be determined by fitting to a thermochemistry database. For example, the B2-PLYP double-hybrid approximation [1] is obtained by choosing the Becke 88 (B) exchange functional [4] for E x [n] and the Lee-Yang-Parr (LYP) correlation functional [5] for

and the empirical parameters a x = 0.53 and a c = 0.27 are optimized for the G2/97 subset of heats of formation.

Another approach has been proposed in which the perturbative contribution is evaluated with normal B3LYP orbitals rather than orbitals obtained with the weighted correlation density functional (1 -a c )E c [n] [6,7].

Recently, Sharkas, Toulouse, and Savin [8] have provided a rigorous reformulation of the double-hybrid approximations based on the adiabatic-connection formalism, leading to the density-scaled one-parameter double- * Electronic address: julien.toulouse@upmc.fr † Electronic address: carlo-adamo@chimie-paristech.fr hybrid (DS1DH) approximation

where E c [n 1/l ] is the usual correlation energy functional evaluated at the scaled density n 1/l (r ¯) = (1/l) 3 n(r ¯/l). This reformulation shows that only one independent empirical parameter l is needed instead of the two parameters a x and a c . The connection with the original doublehybrid approximations can be made by neglecting the density scaling

which leads to the one-parameter double-hybrid (1DH) approximation [8]

Equation ( 4) exactly corresponds to the double-hybrid approximation of Eq. ( 1) with parameters a x = l and a c = l 2 . Very recently, Brémond and Adamo [12] have proposed a new class of double-hybrid approximations where the correlation functional is weighted by (1-l 3 ) and the MP2 correlation energy is weighted by l 3 , instead of (1-l 2 ) and l 2 , respectively. Applying this formula with the Perdew-Burke-Ernzerhof (PBE) [13] exchange-correlation density functional, they have constructed the PBE0-DH doublehybrid approximation which performs reasonably well. In this work, we give a rationale for this class of doublehybrid approximations. For this, we start by recalling that the density-scaled correlation functional E c [n 1/l ] tends to the second-order Görling-Levy (GL2) [14] correlation energy when the density is squeezed up to the high-density limit (or weak-interaction limit) 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 [10,11], respectively. which is finite for nondegenerate KS systems. The GL2 correlation energy can be decomposed as (see, e.g., Ref. 15)

where

is the usual MP2 correlation energy expression

with the antisymmetrized two-electron integrals φ i φ j ||φ a φ b , and E ∆HF c is an additional contribution involving the difference between the local multiplicative KS exchange potential vKS x and the nonlocal nonmultiplicative HF exchange potential vHF

In both Eqs. ( 7) and ( 8), φ k are the KS orbitals and ε k are their associated energies, and the indices i,j and a,b stand for occupied and virtual orbitals, respectively. The single-excitation contribution E ∆HF c vanishes for two-electron systems, and in most other cases is negligible [15], so that the GL2 correlation energy is well approximated by just the MP2 contribution (evaluated with KS orbitals),

. This leads us to propose an approximation for E c [n 1/l ] based on a linear interpolation formula

Plugging Eq. ( 9) into Eq. ( 2), we directly arrive at what we call the linearly scaled one-parameter double-hybrid (LS1DH) approximation

with the weights (1 -l 3 ) and l 3 , thus giving a stronger rationale to the expression that Brémond and Adamo have proposed on the basis of different considerations.

Further insight into this approximation can be gained by rewriting Eq. ( 9) in the alternative form

which can then be interpreted as a first-order expansion in l around l = 0 with E c [n] -E MP2 c approximating the third-order correlation energy correction E

In this sense, the LS1DH approximation of Eq. ( 10) can be considered as a next-order approximation in l beyond the usual hybrid approximation. Figure 1 illustrates the different approximations to the density-scaled correlation energy E c [n 1/l ] as a function of l for the He and Be at

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