Slope of the Delocalization Function Is Proportional to Analytical Hardness

Conceptual Density Functional Theory (CDFT) has been extended beyond its traditional role in elucidating chemical reactivity to the development of density functional theory methods, e.g., the investigation of the delocalization error. This delocalization error causes the dependence of the energy on the number of electrons (N) to deviate from its exact piecewise linear behavior, an error which is the basis of many well-known limitations of commonly used density-functional approximations (DFAs). Following our previous work on the analytical hardness η^± for pure functionals, we extend its application to hybrid and range-separated functionals. A comparison is made between the analytical hardness and the slope of the delocalization function introduced by Hait and Head-Gordon. Our results show that there is a linear relationship between its slope and the analytical hardness. An approximate scheme is presented to construct the energy vs N curve without fractional occupation number calculations. The extension to densities is discussed.