Exact hydrogenic density functionals

Exact density functionals for hydrogenic atoms are discussed, continuing the original work of Gill and Pople [Phys. Rev. A 47, 2383 (1993)]. An exchange-correlation functional whose potential exactly cancels the Coulomb potential does give the exact density, although the potential, denoted $-v_J^Z\mathrm{}\left(r\right),$ differs significantly from the conventional BLYP [A. D. Becke, Phys. Rev. A 38, 3098 (1988); C. Lee, W. Wang, and R. G. Parr, Phys. Rev. B 37, 785 (1988)] potential. Raising $-v_J^Z\mathrm{}\left(r\right)\mathrm{}$ by the hardness of the system leads to very good agreement with BLYP in the energetically important regions, although asymptotically the BLYP potential vanishes rapidly while the shifted potential approaches the hardness of the system. These observations support the theory of Perdew, Parr, Levy, and Balduz [Phys. Rev. Lett. 49, 1691 (1982)] that exact exchange-correlation potentials do not vanish asymptotically. Examples of functionals that give exact hydrogenic densities and energies, with nonvanishing asymptotic potentials, are presented. The computed asymptotic potentials agree well with theoretical predictions, and arise naturally from the requirement that the functional be approximately applicable to all systems, and not just those with one electron.