Delocalization Errors in Density Functional Theory Are Essentially Quadratic in Fractional Occupation Number

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 C12H14 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.