Spherical Density Functional Theory and Atoms in Molecules

It has recently turned out that in molecules and solids a set of spherically symmetric densities determines uniquely the external potential. Here it is shown that any of the spherically symmetric densities obeys a Schrödinger-like differential equation that is equivalent to the Euler equation of this density. The exact effective potential is presented explicitly in terms of wave function expectation values. The spherically symmetric densities reflect several properties of atoms. Though there is some resemblance to the "Atoms in Molecules", this is a different concept.