Sum rules for exchange and correlation potentials

Two recent papers [S. Liu, P. W. Ayers, and R. G. Parr, J. Chem. Phys. 111, 6927 (1999); A. Görling, Phys. Rev. Lett. 83, 5459 (1999)] have stated that ∫∇2νxc(r)dr=4π, where νxc(r) is the exchange-correlation potential of density functional theory. Here, we derive this sum rule and related rules such as ∫∇2νx(r)dr=4π and ∫∇2νc(r)dr=0, where νx(r) and νc(r) are the exchange and correlation components of νxc(r). Using similar methods, we derive the sum rule for the “screening” portion of the exchange-correlation potential and also “generalized” sum rules for νc(r) and the “response” portion of the exchange-correlation potential, vxcresponse(r). From the sum rule for vxcresponse(r), we deduce the asymptotic decay of the density response of the hole-correlation function. We conclude by discussing the probable utility of these results for the development of new exchange-correlation functionals.