Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfaction

Abstract

We construct a nonlocal density functional approximation with full exact exchange, while preserving the constraint-satisfaction approach and justified error cancellations of simpler semilocal functionals. This is achieved by interpolating between different approximations suitable for two extreme regions of the electron density. In a "normal" region, the exact exchange-correlation hole density around an electron is semilocal because its spatial range is reduced by correlation and because it integrates over a narrow range to −1. These regions are well described by popular semilocal approximations (many of which have been constructed nonempirically), because of proper accuracy for a slowly-varying density or because of error cancellation between exchange and correlation. "Abnormal" regions, where nonlocality is unveiled, include those in which exchange can dominate correlation (one-electron, nonuniform high-density, and rapidly-varying limits), and those open subsystems of fluctuating electron number over which the exact exchange-correlation hole integrates to a value greater than −1. Regions between these extremes are described by a hybrid functional mixing exact and semilocal exchange energy densities locally, i.e., with a mixing fraction that is a function of position r and a functional of the density. Because our mixing fraction tends to 1 in the high-density limit, we employ full exact exchange according to the rigorous definition of the exchange component of any exchange-correlation energy functional. Use of full exact exchange permits the satisfaction of many exact constraints, but the nonlocality of exchange also requires balanced nonlocality of correlation. We find that this nonlocality can demand at least five empirical parameters, corresponding roughly to the four kinds of abnormal regions. Our local hybrid functional is perhaps the first accurate fourth-rung density functional or hyper-generalized gradient approximation, with full exact exchange, that is size-consistent in the way that simpler functionals are. It satisfies other known exact constraints, including exactness for all one-electron densities, and provides an excellent fit to the 223 molecular enthalpies of formation of the G3/99 set and the 42 reaction barrier heights of the BH42/03 set, improving both (but especially the latter) over most semilocal functionals and global hybrids. Exact constraints, physical insights, and paradigm examples hopefully suppress "overfitting".

Testing the Jacob's ladder of density functionals for electronic structure and magnetism of rutileVO2

“…Less or no exact exchange is needed in metals, and more than 25% is needed in large-gap insulators. The fraction of exact exchange should vary locally throughout any system, as it does in a local hybrid functional [65]. The all-electron 3p and 3d orbitals for a free V +4 ion, in arbitrary units, are also shown.…”

Comparison of the performance of exact-exchange-based density functional methods

“…5,6,11 Perdew et al have addressed the same problem in a somewhat different manner, based on sophisticated real-space analysis of the exact-exchange hole combined with model DFT exchange-correlation hole. 12,13 By extending the Tao-Perdew-Staroverov-Scuseria (TPSS) XC scheme to a new generalized form, they thoroughly explored in their PSTS model, the abnormal regions in real space, where nondynamic correlation effects are locally dominant. 12 Based on this analysis, they proposed a real space correction to the energy functional that takes into account nondynamic correlation effects.…”

“…12,13 By extending the Tao-Perdew-Staroverov-Scuseria (TPSS) XC scheme to a new generalized form, they thoroughly explored in their PSTS model, the abnormal regions in real space, where nondynamic correlation effects are locally dominant. 12 Based on this analysis, they proposed a real space correction to the energy functional that takes into account nondynamic correlation effects. The PSTS functional satisfies all known exact scaling constraints that a given functional should obey.…”

“…Also the exact-exchange energy density is incorporated as an extra variable, which is a signature of the hyper-GGA level of the theory. 12 The presence of this variable considerably increases the complexity and the cost of these methods. The calculation of the exact-exchange energy density is prohibitively expensive given that thousands of grid points are used for each atom.…”

Kinetic and Exchange Energy Densities near the Nucleus

“…Finally, we remark that the exact nuclear behavior can be reproduced only by high level functionals, which include the exact-XE density, such as the optimized-effective potential (OEP) method [82][83][84] or hyper-GGAs methods [85][86][87], which are significantly more expensive than the proposed Equation (53).…”