Gradient corrections for semiclassical theories of atoms in strong magnetic fields

TL;DR

This paper enhances semiclassical atomic theories in strong magnetic fields by incorporating gradient corrections, analyzing their mathematical properties, and proposing modified functionals for the lowest Landau band with implications for quantum energy estimates.

Contribution

It introduces the von Weizs"acker term into Magnetic TF theory, analyzing its effects and proposing gradient corrections for atoms in strong magnetic fields, especially in the lowest Landau band.

Findings

The von Weizs"acker term produces the Scott correction for magnetic fields up to order Z^2.

Gradient corrections for the lowest Landau band lead to a new functional depending on one-dimensional densities.

The Weizs"acker correction for a one-dimensional Fermi gas has a negative sign, affecting the functional modifications.

Abstract

This paper is divided into two parts. In the first one the von Weizs\"acker term is introduced to the Magnetic TF theory and the resulting MTFW functional is mathematically analyzed. In particular, it is shown that the von Weizs\"acker term produces the Scott correction up to magnetic fields of order $B \ll Z^2$, in accordance with a result of V. Ivrii on the quantum mechanical ground state energy. The second part is dedicated to gradient corrections for semiclassical theories of atoms restricted to electrons in the lowest Landau band. We consider modifications of the Thomas-Fermi theory for strong magnetic fields (STF), i.e. for $B \ll Z^3$. The main modification consists in replacing the integration over the variables perpendicular to the field by an expansion in angular momentum eigenfunctions in the lowest Landau band. This leads to a functional (DSTF) depending on a sequence of one-dimensional densities. For a one-dimensional Fermi gas the analogue of a Weizs\"acker correction has a negative sign and we discuss the corresponding modification of the DSTF functional.