Local-resolvent formulation and assessment of linear-scaling Kohn-Sham calculations

Abstract

Kohn-Sham density-functional theory is expensive because it needs orbitals that extend over the whole system. For insulators this can be avoided by using only the diagonal of the one-electron resolvent, $g(\mathbf{r},z)=\langle\mathbf{r}|(z-\hat h)^{-1}|\mathbf{r}\rangle$, at a few complex energies $z$. We call the scheme local-resolvent density-functional theory (LRes-DFT). The density and the band energy are linear in $g$, so neither orbitals nor a kinetic-energy functional are needed. The complex energies are $M$ equally spaced points on a circle that encloses the occupied levels, offset by half a spacing so that no point lies on the real axis. Summing the resolvent over them is equivalent to occupying each level with $1/(1+u^M)$, where $u$ is the level's energy measured from the circle's center in units of its radius. This is a smooth step that is near 1 inside the circle and near 0 outside, and it stays between 0 and 1 for every $M$. If the circle moves with the chemical potential, forces at fixed electron number are exact. Gauss-Legendre and Zolotarev rules need fewer points when the gap is small. In three dimensions we use eigenpairs of small windows around each molecule instead of resolvents. This is divide-and-conquer (DC) with a different occupation function, and it is as accurate as DC with Fermi-Dirac occupations. For H$_2$ molecular crystals of up to 1024 electrons it agrees with Kohn-Sham to $2.5\times10^{-7}$ Ha per electron. On one core it is slower than dense diagonalization below about 500 electrons and 3.5 times faster at 1024 electrons. It loses accuracy for disordered or compressed systems and for diffuse basis functions.