Systematic corrections to the Thomas-Fermi approximation without a gradient expansion

TL;DR

This paper introduces a novel method to improve the Thomas-Fermi approximation for fermion densities by relating it to the unitary evolution operator and using Suzuki-Trotter factorization, avoiding gradient expansions.

Contribution

It proposes a new hierarchy of inhomogeneity corrections to the Thomas-Fermi approximation based on operator factorization, providing a more accurate and systematic approach.

Findings

Achieved satisfactory accuracy in benchmark cases with known densities.

Developed a fourth-order leapfrog algorithm for classical equations of motion.

Demonstrated the method's potential for improved density approximations.

Abstract

We improve on the Thomas-Fermi approximation for the single-particle density of fermions by introducing inhomogeneity corrections. Rather than invoking a gradient expansion, we relate the density to the unitary evolution operator for the given effective potential energy and approximate this operator by a Suzuki-Trotter factorization. This yields a hierarchy of approximations, one for each approximate factorization. For the purpose of a first benchmarking, we examine the approximate densities for a few cases with known exact densities and observe a very satisfactory, and encouraging, performance. As a bonus, we also obtain a simple fourth-order leapfrog algorithm for the symplectic integration of classical equations of motion.