Density Functional Theory is Not Straying from the Path toward the Exact Functional

TL;DR

This paper challenges previous claims about density functionals diverging from the path to exactness by analyzing errors in energies and densities, revealing linear relationships and proposing a new measure of functional exactness.

Contribution

It introduces a new approach to evaluate density functionals based on errors in energies and densities for the same systems, clarifying their true performance and relationships.

Findings

Different functionals show linear error relationships between ho and E[ ho].

Ranking of functionals based on previous studies breaks down under this analysis.

A new measure of 'exactness' is proposed using the product of errors in E[ ho] and ho.

Abstract

Recently (Science, 355, 6320, 2017, 49-52) it was argued that density functionals stray from the path towards exactness due to errors in densities (\rho) of 14 atoms and ions computed with several recent functionals. However, this conclusion rests on very compact \rho\ of highly charged 1s2 and 1s22s2 systems, the divergence is due to one particular group's recently developed functionals, whereas other recent functionals perform well, and errors in \rho\ were not compared to actual energies E[\rho] of the same distinct, compact systems, but to general errors for diverse systems. As argued here, a true path can only be defined for E[\rho] and \rho\ for the same systems: By computing errors in E[\rho], it is shown that different functionals show remarkably linear error relationships between \rho\ and E[\rho] on well-defined but different paths towards exactness, and the ranking in Science, 355, 6320, 2017, 49-52 breaks down. For example, M06-2X, said to perform poorly, performs very well on the E,\rho\ paths defined here, and local (non-GGA) functionals rapidly increase errors in E[\rho] due to the failure to describe dynamic correlation of compact systems without the gradient. Finally, a measure of "exactness" is given by the product of errors in E[\rho] and \rho; these relationships may be more relevant focus points than a time line if one wants to estimate exactness and develop new exact functionals.