Model DFT exchange holes and the exact exchange hole: Similarities and differences

Exchange and correlation holes are unique quantum concepts for understanding the nature of electron interactions based on quantum conditional probabilities. Among those, the exact exchange hole is of special interest since it is derived rigorously from first principles without approximations and is often modeled by approximate exchange expressions of density functional theory. In this work, the algorithm for the computation of the spherically averaged exact exchange hole for a given reference point is developed and implemented for molecular orbitals in Gaussian basis functions. The formulas include a novel recursive relation for the spherical average of the Bessel function of the first kind and the asymptotic expressions when the exponential factor of the Bessel function becomes large. This new capability is used to explore the extent to which current popular model exchange holes resemble or differ from the exact exchange hole. Point-wise accuracy of the exchange holes for isolated atoms is important in local hybrid schemes, real-space models of static correlation, and others. We find in this vein that among the models tested here, only the BR89 exchange hole seems more or less suitable for that purpose, while better approximations are still very much on demand. Analyzing the deviations of model exchange holes from the exact exchange hole in molecules such as H₂ and Cr₂ upon bond stretching reveals new aspects of the left-right static correlation.