DENSITY FUNCTIONALS AND DIMENSIONAL RENORMALIZATION FOR AN EXACTLY SOLVABLE MODEL

We treat an analytically solvable version of the ''Hooke's Law'' model for a two-electron atom, in which the electron-electron repulsion is Coulombic but the electron-nucleus attraction is replaced by a harmoni c oscillator potential. Exact expressions are obtained for the ground- state wave function and electron density, the Hartree-Fock solution, t he correlation energy, the Kohn-Sham orbital, and, by inversion, the e xchange and correlation functionals. These functionals pertain to the ''intermediate'' density regime (r(s) greater-than-or-equal-to 1.4) fo r an electron gas. As a test of customary approximations employed in d ensity functional theory, we compare our exact density, exchange, and correlation potentials and energies with results from two approximatio ns. These use Becke's exchange functional and either the Lee-Yang-Parr or the Perdew correlation functional. Both approximations yield rathe r good results for the density and the exchange and correlation energi es, but both deviate markedly from the exact exchange and correlation potentials. We also compare properties of the Hooke's Law model with t hose of two-electron atoms, including the large dimension limit. A ren ormalization procedure applied to this very simple limit yields correl ation energies as good as those obtained from the approximate function als, for both the model and actual atoms.