A new algorithm is presented which enables the generation of approximate ground state electron densities and approximate ground state energy differences of isoelectronic processes for interacting and non-interacting systems. This is accomplished starting exclusively from knowledge of the following: one isoelectronic ground state 'seed' density, (rho)(,A); the external potential, v(,A), which generates (rho)(,A); and the external potential, v(,B), for which the ground state energy, E(,B), and density, (rho)(,B), are sought. Use is made of a matching theorem in conjunction with a variation of a Legendre transform idea. Also important in the algorithm is the utilization of many pathways from v(,A) to v(,B). Furthermore, knowledge of the exact energy, E(,A), enables the approximation of E(,B) using either of two energy difference formulae which are approximations to the exact integrated Hellmann-Feynman energy difference. The virial theorem and Legendre transform constraints of the algorithm are tested for non-interacting one and ten Coulomb and harmonic oscillator potentials. It is found that increasingly better densities can be generated by simply increasing the number of constructed potentials between v(,A) and v(,B). The algorithm is tested using non-interacting one- and four- electron pathways in which v(,A) and v(,B) are Coulomb potentials and the intermediate potentials are Yukawa potentials. Densities for these Yukawa potentials are generated with the algorithm by starting with reasonably good first approximations to the Yukawa densities and using the algorithm to improve upon these solutions. Typically the energies from the densities adjusted to adhere to the virial theorem and many pathway constraints of the algorithm are improved relative to the unconstrained results anywhere from a factor of 2 to a factor of (TURN)50. Within exchange only density functional theory this algorithm also provides for a means of computing the non-interacting kinetic energy functional T(,s) (rho) using just densities and one-body potentials
