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WIREs Comput Mol Sci
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The XYG3 type of doubly hybrid density functionals

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Doubly hybrid (DH) functionals have emerged as a new class of density functional approximations (DFAs), which not only have a nonlocal orbital‐dependent component in the exchange part, but also incorporate the information of unoccupied orbitals in the correlation part, being at the top rung of Perdew's view of Jacob's ladder in DFAs. This review article focuses on the XYG3 type of DH (xDH) functionals, which use a low rung functional to perform the self‐consistent‐field calculation to generate orbitals and densities, with which a top rung DH functional is used for final energy evaluation. We will discuss the theoretical background of the xDH functionals, briefly reviewing the adiabatic connection formalism, coordinate scaling relations, and Görling–Levy perturbation theory. General performance of the xDH functionals will be presented for both energies and structures. In particular, we will present the fractional charge behaviors of the xDH functionals, examining the self‐interaction errors, the delocalization errors and the deviation from the linearity condition, as well as their effects on the predicted ionization potentials, electron affinities and fundamental gaps. This provides a theoretical rationale for the observed good performance of the xDH functionals. WIREs Comput Mol Sci 2016, 6:721–747. doi: 10.1002/wcms.1274 This article is categorized under: Electronic Structure Theory > Density Functional Theory
(a) A representative AC curve, where Uxc,λ = Ex + Uc,λ. Ex, and Ec are given by the integrals, as indicated. (b) FCI/u‐aug‐cc‐pCVQZ values of Uc,λ (in a.u.) for H2 at different internuclear separations R.
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Mean absolute deviations (MAD, in eV) for the derivative gaps, ionization potentials and electron affinities as compared to the experimental ones. The deviation from the straight line behavior is measured by the differences between the integer gaps and the corresponding derivative gaps.
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Deviations from the corresponding linear interpolations for the carbon atom with methods of (a) BLYP, PBE, B3LYP, PBE0, HF‐BLYP, HF‐PBE and (b) HF, xDH‐PBE0, XYGJ‐OS, XYG3, B2PLYP. Note that the scale changes for the vertical coordinates from (a) to (b). In panel (a), PBE curves overlap with BLYP curves, while HF‐PBE curves overlap with HF‐BLYP curves.
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Fractional charge behaviors for the carbon atom with methods of (a) BLYP, PBE, B3LYP, PBE0, HF‐BLYP, HF‐PBE and (b) HF, xDH‐PBE0, XYGJ‐OS, XYG3, B2PLYP. The exact straight lines, denoted as Ref, are obtained using the experimental IP and EA of the carbon atom.
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Functional performances on structures, including covalent bondings, nonbonded interactions, and transition state structures, for the hybrid GGAs (B3LYP and PBE0), DH functionals (B2PLYP, xDH‐PBE0, XYGJ‐OS, and XYG3) and MP2.
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Functional performances on energies, including heats of formation (HOF), bond dissociation enthalpies (BDE), reaction barrier heights (RBH), and nonbonded interactions (NBI) for the GGA functionals (PBE and BLYP), hybrid GGAs (PBE0 and B3LYP) and DH functionals (MC3BB, B2PLYP, XYG3, XYGJ‐OS, xDH‐PBE0, and PBE‐ACDH).
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