  Density functional theory (DFT) is used in thousands of papers each year, yet
lack of universality reduces DFT's predictive capacity, and functionals may
produce energy-density imbalances. The absolute electronegativity (\chi) and
hardness (\eta) directly reflect the energy-density relationship via the
chemical potential dE/dN and we thus hypothesized that they probe universality.
We studied \chi and \eta for atoms Z = 1-36 using 50 diverse functionals
covering all major classes. Very few functionals describe both \chi and \eta
well. \eta benefits from error cancelation whereas \chi is marred by error
propagation from IP and EA; thus almost all standard GGA and hybrid functionals
display a plateau in the MAE at 0.2-0.3 eV for \eta. In contrast, variable
performance for \chi indicates problems in describing the chemical potential by
DFT. The accuracy and precision of a functional is far from linearly related,
yet for a universal functional we expect linearity. Popular functionals such as
B3LYP, PBE, and revPBE, perform poorly for both properties. Density sensitivity
calculations indicate large density-derived errors as occupation of degenerate
p- and d-orbitals causes "non-universality" and large dependency on exact
exchange. Thus, we argue that performance for \chi for the same systems is a
hallmark of universality by probing dE/dN. With this metric, B98, B97-1,
PW6B95D3, APFD are the most "universal" tested functionals. B98 and B97-1 are
accurate for very diverse metal-ligand bonds, supporting that a balanced
description of dE/dN and dE2/dN2, via \chi and \eta, is probably a first simple
probe of universality.
