Apparent violation of the sum rule for exchange-correlation charges by generalized gradient approximations

The exchange-correlation potential of Kohn-Sham density-functional theory, vXC(r), can be thought of as an electrostatic potential produced by the static charge distribution qXC(r) = -(1∕4π)∇(2)vXC(r). The total exchange-correlation charge, QXC = ∫qXC(r) dr, determines the rate of the asymptotic decay of vXC(r). If QXC ≠ 0, the potential falls off as QXC∕r; if QXC = 0, the decay is faster than coulombic. According to this rule, exchange-correlation potentials derived from standard generalized gradient approximations (GGAs) should have QXC = 0, but accurate numerical calculations give QXC ≠ 0. We resolve this paradox by showing that the charge density qXC(r) associated with every GGA consists of two types of contributions: a continuous distribution and point charges arising from the singularities of vXC(r) at each nucleus. Numerical integration of qXC(r) accounts for the continuous charge but misses the point charges. When the point-charge contributions are included, one obtains the correct QXC value. These findings provide an important caveat for attempts to devise asymptotically correct Kohn-Sham potentials by modeling the distribution qXC(r).