Explicit corrections to the gradient expansion for the kinetic energy in one dimension

TL;DR

This paper develops explicit correction methods to improve the gradient expansion for kinetic energy calculations in one-dimensional systems, enhancing accuracy especially for finite, discrete spectra, with potential applications in density functional theory.

Contribution

It introduces a new mathematical framework that provides explicit corrections to the gradient expansion for kinetic energy in one dimension, improving upon previous approximations.

Findings

Explicit corrections significantly improve accuracy in simple models.

The framework reduces to the gradient expansion for slowly-varying densities.

Corrections can avoid certain singularities in evaluation.

Abstract

A mathematical framework is constructed for the sum of the lowest N eigenvalues of a potential. Exactness is illustrated on several model systems (harmonic oscillator, particle in a box, and Poschl-Teller well). Its order-by-order semiclassical expansion reduces to the gradient expansion for slowly-varying densities, but also yields a correction when the system is finite and the spectrum discrete. Some singularities can be avoided when evaluating the correction to the leading term. Explicit corrections to the gradient expansion to the kinetic energy in one dimension are found which, in simple cases, greatly improve accuracy. We discuss the relevance to practical density functional calculations.