Testing exchange – correlation functionals at fractional electron numbers

Dependence of the ground-state energy and density on the fractional electron number should be linear, while that of the chemical potential and Fukui function should be constant within the region between integers. In practical calculations with commonly used approximate functionals, results deviate from these ideal conditions. Four indicators are defined here to measure such deviations: from the global and local linearity condition, and from the global and local constancy condition. These indicators are used to test the performance of DFT method with five exchange–correlation functionals, and also the Hartree–Fock method, on a set of high-symmetry atoms: He, Li, Be, Na, Mg. The rCAM-B3LYP functional, having all four indicators small, is found to be the best functional for fractional electron number applications.