Approximate normalizations for approximate density functionals

Abstract

It seems self-evident that a density functional calculation should be normalized to the number of electrons in the system. We present multiple examples where the accuracy of the approximate energy is improved (sometimes greatly) by violating this basic principle. In one dimension, we explicitly derive the appropriate correction to the normalization. Beyond one dimension, Weyl asymptotics for energy levels yield these corrections for any cavity. We include examples with Coulomb potentials and the exchange energy of atoms to illustrate relevance to realistic calculations.

Figures (5)

• Figure 1: Bottom: Red is the TF density, green the exact density, and blue the normalization-corrected density for $N=~5$. Top: Percent errors for the TF energy evaluated on each density (See Table S1 in SM).
• Figure 2: Densities for 19 electrons (11 filled shells) in a circular cavity of radius $1$. Green is the exact density, red is TF, and blue is ncTF (Sec. S7 of SM, which includes watson). An analogous phenomenon, also derived from Weyl asymptotics, is observed in CorsoFriesecke.
• Figure 3: Red is energy-minimized TF, green is TF on the exact density, and blue is normalization-corrected TF, for a $1 \times \sqrt2 \times \pi$ box.
• Figure 4: Red is TF, green is TF on the exact density, and blue is ncTF, for a range of rectangular boxes (continuous) and for a circular cavity (discrete) at $N=1000$. The dimensionless $|\partial \Omega|/\sqrt{|\Omega|}$ is proportionsl to $B$ (Tab. \ref{['t:ab']}) and equals $2\sqrt \pi$ for a circle and 4 for a square (see Secs S6 and S7 of SM).
• Figure 5: Percent errors in Bohr atom energies for TF (red), ncTF with $Z$ fixed (blue), and ncTF with $Z=N$ (purple). See Section S8 and Table S2 of SM.