Density functionals and dimensional renormalization for an exactly solvable model

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Abstract
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 harmonic oscillator potential. Exact expressions are obtained for the ground-state wave function and electron density, the Hartree–Fock solution, the correlation energy, the Kohn–Sham orbital, and, by inversion, the exchange and correlation functionals. These functionals pertain to the ‘‘intermediate’’ density regime (rs≥1.4) for an electron gas. As a test of customary approximations employed in density functional theory, we compare our exact density, exchange, and correlation potentials and energies with results from two approximations. These use Becke’s exchange functional and either the Lee–Yang–Parr or the Perdew correlation functional. Both approximations yield rather good results for the density and the exchange and correlation energies, but both deviate markedly from the exact exchange and correlation potentials. We also compare properties of the Hooke’s Law model with those of two-electron atoms, including the large dimension limit. A renormalization procedure applied to this very simple limit yields correlation energies as good as those obtained from the approximate functionals, for both the model and actual atoms.