Understanding and Reducing Errors in Density Functional Calculations

We decompose the energy error of any variational density functional theory calculation into a contribution due to the approximate functional and that due to the approximate density. Typically, the functional error dominates, but in many interesting situations the density-driven error dominates. Examples range from calculations of electron affinities to preferred geometries of ions and radicals in solution. In these abnormal cases, the error in density functional theory can be greatly reduced by using a more accurate density. A small orbital gap often indicates a substantial density-driven error.

Figure 1

Errors in ground-state energies of two-electron ions as a function of nuclear charge: PBE energies evaluated on exact [53, 54] (solid lines) and Hartree-Fock (dotted lines) densities [55].

Figure 2

Exact [Quantum Monte Carlo (QMC) calculations [53, 54]] and PBE radial densities for ${\mathrm{H}}^{-}$ $(Z=1)$ and He $(Z=2)$.

Figure 3

Energy of NaCl as a function of Na-Cl distance in several calculations, and the PBE HOMO-LUMO gap.

Figure 4

Potential energy surface scans for the $\mathrm{HO}·{\mathrm{Cl}}^{-}$ complex along $\mathrm{Cl}\mathrm{O}\mathrm{H}$ angle of 0° (hydrogen-bonding structure, black lines) and 60° [hemibonding structure, gray (red) lines] using various methods; $R$ is the $\mathrm{Cl}\mathrm{O}$ separation.