Deriving approximate functionals with asymptotics

TL;DR

This paper explores the mathematical foundations of density functional theory, proposing new asymptotic methods to derive more accurate approximate functionals with potential for significant improvements in electronic structure calculations.

Contribution

It introduces a unified asymptotic framework for analyzing and improving density functional approximations, connecting gradient expansions with hyperasymptotics and generalizing the Euler-Maclaurin formula.

Findings

Errors as small as 10^{-32} Hartree in simple models

New asymptotic techniques can significantly enhance functional accuracy

A generalized Euler-Maclaurin formula broadens theoretical understanding

Abstract

Modern density functional approximations achieve moderate accuracy at low computational cost for many electronic structure calculations. Some background is given relating the gradient expansion of density functional theory to the WKB expansion in one dimension, and modern approaches to asymptotic expansions. A mathematical framework for analyzing asymptotic behavior for the sums of energies unites both corrections to the gradient expansion of DFT and hyperasymptotics of sums. Simple examples are given for the model problem of orbital-free DFT in one dimension. In some cases, errors can be made as small as 10$^{-32}$ Hartree suggesting that, if these new ingredients can be applied, they might produce approximate functionals that are much more accurate than those in current use. A variation of the Euler-Maclaurin formula generalizes previous results.