Point transformations applied to density-functional calculations

The method of point transformations enables us to generate an unknown wave function ${\Psi }_{\rho }$ ${(\mathrm{\{}\mathrm{r}}_k$}) from a given electron density ρ(r) based on a reference wave function ${\Psi }_0$ ${(\mathrm{\{}\mathrm{r}}_k$}) initially chosen for the system. The generated wave function ${\Psi }_{\rho }$ ${(\mathrm{\{}\mathrm{r}}_k$}) can be used to evaluate the energy $E_{\rho }$ associated with the given density ρ(r). This procedure constitutes a rigorous density-functional theory, if the density ρ(r) and reference function ${\Psi }_0$ ${(\mathrm{\{}\mathrm{r}}_k$}) are varied. Within the Hartree-Fock framework numerical applications are presented for the ground-state helium atom and its analogs both in position and momentum spaces. Simple position and momentum densities are reported that well reproduce the Hartree-Fock limit energy. A few calculations beyond the Hartree-Fock approximation are also given.