Towards an exact orbital-free single-particle kinetic energy density for the inhomogeneous electron liquid in the Be atom

TL;DR

This paper develops an exact expression for the single-particle kinetic energy density of the Be atom's inhomogeneous electron liquid, linking it to electron density and potential, and simplifies it to depend only on a single variable for the spherical case.

Contribution

It combines previous theoretical frameworks to express the kinetic energy density in terms of electron density and potential, then simplifies this to depend solely on a single variable for the Be atom.

Findings

The kinetic energy density can be expressed in terms of electron density and potential.

For the Be atom, the ratio t(r)/n(r) depends only on n'(r)/n(r).

High-order gradients are unnecessary for this spherical case.

Abstract

Holas and March (Phys. Rev. A51, 2040 (1995)) wrote the gradient of the one-body potential V(r) in terms of low-order derivatives of the idempotent Dirac density matrix built from a single Slater determinant of Kohn-Sham orbitals. Here, this is first combined with the study of Dawson and March (J. Chem. Phys. 81, 5850 (1984)) to express the single-particle kinetic energy density of the Be atom ground-state in terms of both the electron density n(r) and potential V(r). While this is the more compact formulation, we then, by removing V(r), demonstrate that the ratio t(r)/n(r) depends, though non-locally, only on the single variable n'(r)/n(r), no high-order gradients entering for the spherical Be atom.