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

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