Visualisation and orbital-free parametrisation of the large-Z scaling of the kinetic energy density of atoms

Abstract

The scaling of neutral atoms to large Z, combining periodicity with a gradual trend to homogeneity, is a fundamental probe of density functional theory, one that has driven recent advances in understanding both the kinetic and exchange-correlation energies. Although research focus is normally upon the scaling of integrated energies, insights can also be gained from energy densities. We visualize the scaling of the positive-definite kinetic energy density (KED) in closed-shell atoms, in comparison to invariant quantities based upon the gradient and Laplacian of the density. We notice a striking fit of the KED within the core of any atom to a gradient expansion using both the gradient and the Laplacian, appearing as an asymptotic limit around which the KED oscillates. The gradient expansion is qualitatively different from that derived from first principles for a slowly-varying electron gas and is correlated with a nonzero Pauli contribution to the KED near the nucleus. We propose and explore orbital-free meta-GGA models for the kinetic energy to describe these features, with some success, but the effects of quantum oscillations in the inner shells of atoms makes a complete parametrization difficult. We discuss implications for improved orbital-free description of molecular properties.

An alternative derivation of orbital-free density functional theory

“…In the limit of an infinite box, the overall electron density tends to zero, meaning that the Thomas-Fermi uniform electron density expression vanishes for this system, as previously mentioned. 32,56,57 Spherical Bessel functions are not typically used for atomic and molecular calculations in DFT; Gaussian basis sets are the standard. One may consider trying to use Gaussians in the present context, but implementing the most scalable or computationally efficient basis set is not the objective of this paper.…”

Asymptotic Analysis of the Pauli Potential for Atoms

“…The primary tool for our study of atomic Pauli potentials is Lieb and Simon's ζ scaling of the kinetic energy of neutral atoms 26,27,42,43,45 . The Lieb-Simon theorem scales the potential and particle number of a system simultaneously:…”

“…[15][16][17] The simplest functional beyond Thomas-Fermi, the gradient expansion (GE), does very well for atoms, but still not so well for molecular binding. Many attempts have been made to build on this foundation to develop semilocal or "single-point" functionals using the local density and its gradient as ingredients, [18][19][20][21][22][23][24] sometimes adding the Laplacian of the density, [25][26][27][28][29] and the electronic Hartree potential. 30 These more complex models generally share the problems of their predecessors, but can be competitive 28 with more expensive empirical nonlocal functionals for some solids.…”

“…26,43 The limiting behavior of total energies is reflected in the kinetic energy density, which is locally approximated by a variant of the gradient expansion in the core region of the atom. 27 The physical property that has not been explored carefully in the Lieb-Simon limit is the Pauli potential. Even though the total Pauli kinetic energy should be well described by TF theory in this limit, the same does not necessarily hold point for point for the potential.…”

Deorbitalization strategies for meta-generalized-gradient-approximation exchange-correlation functionals

“…This numerical scheme matched, to machine-precision, the integrated KE obtained by analytical integration for all eighteen atoms. 44 , the regularized version of the Thomas-Fermi plus Laplacian (TFLreg) mGGA 34 , and the modified VT84F plus Laplacian (MVT84F+L). The PC mGGA uses a modified fourth-order gradient expansion (MGE4) with several appealing features.…”