Spherical densities and potentials in exactly solvable model molecules

A recently initiated variant of density functional theory utilizes a set of spherically symmetric densities instead of the density. The exact functionals are unknown in the new theory akin to the standard density functional theory. In order to test approximate functionals exactly solvable models are introduced. A harmonic molecular ion, the analogue to the hydrogen molecule ion and a harmonic two-electron molecule showing analogy to the hydrogen molecule are proposed. It has been found that the wave function and the density can be given analytically. The exact spherical densities and the effective potentials of the Euler equations also have analytical form. It has been shown that the models can be easily extended to several "nuclei."