Delocalization errors in density functional theory are essentially quadratic in fractional occupation number

TL;DR

This paper demonstrates that delocalization errors in density functional theory are essentially quadratic in fractional occupation number, and introduces a simple method to assess and potentially reduce this error across various functionals.

Contribution

It reveals the quadratic nature of delocalization errors in DFT and proposes a new approach for evaluating and tuning functionals to minimize these errors.

Findings

Delocalization error is nearly perfectly linear in fractional electron number.

The quadratic form allows comparison of 47 functionals based on curvature.

Using fractional charge data can help tune functionals to reduce delocalization error.

Abstract

Approximate functionals used in practical density functional theory (DFT) deviate from the piecewise linear behavior of the exact functional for fractional charges. This deviation causes excess charge delocalization, which leads to incorrect densities, molecular properties, barrier heights, band gaps and excitation energies. We present a simple delocalization function for characterizing this error and find it to be almost perfectly linear vs the fractional electron number for systems spanning in size from the H atom to the C$_{12}$H$_{14}$ polyene. This causes the delocalization energy error to be a quadratic polynomial in the fractional electron number, which permits us to assess the comparative performance of 47 popular and recent functionals through the curvature. The quadratic form further suggests that information about a single fractional charge is sufficient to eliminate the principal source of delocalization error. Generalizing traditional two-point information like ionization potentials or electron affinities to account for a third, fractional charge based data point could therefore permit fitting/tuning of functionals with lower delocalization error.