Laplacian‐based models for the exchange energy

Abstract

Recent Quantum Monte Carlo data for the exchange‐correlation energy density of pseudopotential systems strongly suggest the value of using the Laplacian of the density as a variable for constructing first‐order corrections to the local density approximation of density functional theory. We report on an exchange functional built on these observations and extended to the all‐electron case. The model keeps the typical properties of constraint‐based generalized gradient approximations (GGAs) and also has a finite‐valued potential at the nucleus, unlike the GGA. Problems with oscillatory behavior in the potential due to higher order derivatives are controlled by a curvature minimization constraint. The results are tested against exact potentials for the He and Ne atom. A combination of gradient and Laplacian as suggested by a gradient expansion of the exchange hole gives the best overall results. © 2012 Wiley Periodicals, Inc.

Kinetic and Exchange Energy Densities near the Nucleus

“…This issue has been considered by Tao [49], who constructed an exchange functional with the correct XE density at the nucleus, using the single inhomogeneity parameter proposed by Becke [43] Q B = 1 − τ τ HEG + 5s 2 3 + 10 3 q with τ HEG = (3/10)(3π 2 ) 2/3 ρ 5/3 . Further improvements on the development of Laplacian-dependent exchange functionals have been found by Cancio et al [50].…”

“…x F x (s, q, α, x u ) (53) and using the generalized Kohn-Sham method [81], the XE potential can be computed in the same manner, as has been shown in Equation (50), and will differ from a regular meta-GGA potential only due to an extra integration. 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).…”

Kinetic‐energy‐density dependent semilocal exchange‐correlation functionals

“…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…”

“…It may produce strong and unphysical oscillations of the XC potential, due to the term r 2 ð oðqexcÞ or 2 q Þ present in the functional derivative of any Laplaciandependent functional (see Appendix C). This issue has been recently investigated and partially solved, [59] but still poses a major limitation for the use of the Laplacian of the density as main quantity in functional development.…”

“…It may produce strong and unphysical oscillations of the XC potential, due to the term r 2 ð oðqexcÞ or 2 q Þ present in the functional derivative of any Laplaciandependent functional (see Appendix C). This issue has been recently investigated and partially solved, [59] but still poses a major limitation for the use of the Laplacian of the density as main quantity in functional development.Note also that, if a functional has an energy density which is linear in the Laplacian, then the Laplacian can be integrated out by parts using the relation [32,53] : ð d 3 r f½qðrÞ r 2 qðrÞ52 ð d 3 r jrqðrÞj 2 df½q dq ðrÞ :In the case of an exchange enhancement factor linear in the reduced Laplacian q, from Equation 15 we obtain:Note that Equations 15 and 16 are valid under the assumption that the density vanishes at infinity (e.g., for finite systems). …”