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# Local hybrid functionals: Theory, implementation, and performance of an emerging new tool in quantum chemistry and beyond

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The state of the art in the development, implementation, validation, and application of local hybrid functionals with position‐dependent exact‐exchange admixture is reviewed comprehensively. Starting from the general formulation of a local hybrid functional, possible constructions of local mixing functions to determine the position dependence are scrutinized in terms of their exact properties as well as their desirable features. The gauge problem pertaining to the ambiguity of exchange‐energy densities is discussed. “Smoking guns” for the gauge problem are identified, and the first calibration functions to improve the match of the exact and semi‐local exchange‐energy densities are compared. In view of the appearance of nonstandard exact‐exchange integrals, a crucial aspect is the efficient implementation of local hybrid functionals. This has been achieved recently using a resolution of the identity and/or semi‐numerical techniques, both for ground‐state energies and gradients, and for linear‐response time‐dependent density functional theory. The validation and early applications of local hybrids are reviewed, including thermochemistry, reaction barriers, structures and vibrational frequencies, electric and magnetic properties, as well as electronic excitation spectra. This article is categorized under: Electronic Structure Theory > Density Functional Theory Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry
Real‐space behavior of the Kümmel‐local mixing function (LMF) along the bond axis of the NO molecule with qualitative assignment of different regions in real space (see text)
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Mean absolute errors (MAEs) of the Lh12ct‐SsirPW92 local hybrid (in eV) for the singlet and triplet excitations of the Thiel's test set as well as for the core, Rydberg and charge‐transfer excitations of the Tozer test sets compared to selected common exchange‐correlation functionals
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Mean absolute errors (MAEs) of two local hybrids (in kcal/mol) for several subsets of GMTKN30 that are known to depend particularly on weak noncovalent interactions. The figure is similar to figure 4 of Reference and based on data from that work but with a different uncalibrated local hybrid. Results for the uncalibrated LSDA‐based Lh12ct‐SsirPW92 local hybrid (red, light dashed line without and solid line with D3 dispersion corrections) and the calibrated Lh14t‐PBE‐CG functional (green, light dashed line without and solid line with D3 dispersion corrections) are compared
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Dissociation curve of the argon dimer calculated with Hartree‐Fock (HF), SVWN, and the Lh12ct‐SsirPW92 local hybrid (black, blue, and green curve, respectively). Spurious nondynamical correlation of the local hybrid functional, denoted by ΔENDC, is illustrated by the light‐green‐filled section, while the deviation of the SVWN functional from the HF reference, denoted by ΔEXC, is represented by the cyan filling
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Spatial behavior (lengths given in Bohr) of the unscaled t‐local mixing function (LMF) in the molecular plane of the B2H6 molecule
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Spatial behavior of different local mixing functions (LMFs) in the bonding region along the molecular axis of the stretched NO molecule. Parametrizations from the corresponding references have been used for the s‐LMF, the z‐LMF, the PSTS‐LMF, the Kümmel‐LMF, and the common t‐LMF
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Spatial behavior of different LMFs along the molecular axis of the NO molecule. Parametrizations from the corresponding references have been used for the Kümmel‐local mixing function (LMF), the Θ‐LMF I, and the Θ‐LMF II
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Spatial behavior of different local mixing functions (LMFs) along the molecular axis of the NO molecule. Parametrizations from the corresponding references have been used for the s‐LMF, the z‐LMF, the PSTS‐LMF, and the common t‐LMF
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