On the Kinetic Energy Density Functional: The Limit of the Density Derivative Order

TL;DR

This paper establishes that the maximum derivative order for kinetic energy density functionals in orbital-free density functional theory is D+1, where D is the system's spatial dimension, providing a systematic development guide.

Contribution

It proves that the highest relevant density derivative order is D+1 for systems of dimension D, clarifying a long-standing ambiguity in kinetic energy density functional development.

Findings

Maximum derivative order is D+1 for D-dimensional systems.

Imposing finiteness of KED leads to the D+1 limit.

Provides a systematic guide for developing kinetic energy functionals.

Abstract

Within ``orbital-free'' density functional theory, it is essential to develop general kinetic energy density (KED), denoted as $t(\mathbf{r})$. This is usually done by empirical corrections and enhancements, gradient expansions, machine learning, or axiomatic approaches to find forms that satisfy physical necessities. In all cases, it is crucial to determine the largest spatial density derivative order, $m$ in, $t(\mathbf{r})$. There have been many efforts to do so, but none have proven general or conclusive and there is no clear guide on how to set $m$. In this work, we found that, by imposing KED finitude, $m=D+1$ for systems of dimension $D$. This is consistent with observations and provides a needed guide for systematically developing more accurate KEDs.