Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy

Abstract

We construct a Laplacian-level meta-generalized gradient approximation (meta-GGA) for the non-interacting (Kohn-Sham orbital) positive kinetic energy density τ of an electronic ground state of density n. This meta-GGA is designed to recover the fourth-order gradient expansion τ GE4 in the appropiate slowly-varying limit and the von Weizsäcker expression τ W = |∇n| 2 /(8n) in the rapidlyvarying limit. It is constrained to satisfy the rigorous lower bound τ W (r) ≤ τ (r). Our meta-GGA is typically a strong improvement over the gradient expansion of τ for atoms, spherical jellium clusters, jellium surfaces, the Airy gas, Hooke's atom, one-electron Gaussian density, quasi-two dimensional electron gas, and nonuniformly-scaled hydrogen atom. We also construct a Laplacian-level meta-GGA for exchange and correlation by employing our approximate τ in the Tao, Perdew, Staroverov and Scuseria (TPSS) meta-GGA density functional. The Laplacian-level TPSS gives almost the same exchange-correlation enhancement factors and energies as the full TPSS, suggesting that τ and ∇ 2 n carry about the same information beyond that carried by n and ∇n. Our kinetic energy density integrates to an orbital-free kinetic energy functional that is about as accurate as the fourth-order gradient expansion for many real densities (with noticeable improvement in molecular atomization energies) , but considerably more accurate for rapidly-varying ones.

Kinetic‐energy‐density dependent semilocal exchange‐correlation functionals

“…In general, modeling s KS using density ingredients only (i.e., s and q) is a very difficult problem [60,126] because of its strong nonlocality even for light atoms. [129] In fact, s KS has been evaluated as a functional of the density using a machine-learning technique with about one hundred thousands empirical parameters, for onedimensional systems, [130,131] as well as for real molecules.…”

“…This class of functionals have been also quite largely investigated. [54][55][56][57][58][59][60][61] As previously stated, L-meta-GGA are explicit-density functionals, in contrast to all the functionals which depend on s KS . Due to the aforementioned numerical problems with the Laplacian, L-metaGGAs can, in some cases, be transformed in simple GGAs, eliminating the Laplacian term by an integration by parts (see, e.g., the…”

“…We also remark that, in principle, any KED dependent functional can be transformed into a L-meta-GGA one using an approximate gradientand Laplacian-dependent expression for s KS , despite this is not a simple task in practice. [60,64] In this Review, we will consider all the functionals which depend on s KS : thus we will consider the first five classes, whereas L-meta-GGAs will not be discussed.…”

“…We also remark that, in principle, any KED dependent functional can be transformed into a L-meta-GGA one using an approximate gradientand Laplacian-dependent expression for s KS , despite this is not a simple task in practice. [60,64] In this Review, we will consider all the functionals which depend on s KS : thus we will consider the first five classes, whereas L-meta-GGAs will not be discussed.K-meta-GGAs and LK-meta-GGAs (simply meta-GGAs in the following) are the most promising: in fact functionals which uses both rq and s KS , can be made one-electron self-interaction free and can distinguish the core, valence, and bonding regions of different systems, ranging from atoms to solids. [65][66][67] As a consequence, meta-GGA functionals display a clear advantage with respect to the GGAs.…”

Kinetic and Exchange Energy Densities near the Nucleus

“…At the nucleus of the helium isoelectronic series, F s ≈ F W s ≈ 0.2353, i.e., there is an even more pronounced de-enhancement than in the exchange case. On the other hand, most of the GGA KE functionals have F s ≥ 1, while few semilocal KE functionals recover the VW at the nucleus [59,61,62].…”