Adiabatic connection in density-functional theory: Two electrons on the surface of a sphere

Abstract

Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr\"odinger equation of this helium-type two-electron system is solved here accurately for different values $\ensuremath{\alpha}∊\mathbb{R}$ of a constant that is multiplied to the electron-electron repulsion ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}$. The correlation structure in the resulting wave functions is analyzed for different values of $\ensuremath{\alpha}$. The asymptotic limits $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional ${E}_{\mathit{xc}}[\ensuremath{\rho}]$ of the exchange-correlation energy in the real system with $\ensuremath{\alpha}=1$ is tested against the exact adiabatic connection in density-functional theory.