Inverting the external-potential-to-electron density map in systems of electrons by minimization of a least-squares fit functional for analytic potentials

In a system of electrons, there is a map connecting any external potential v with its electron density rho(v). In this work, we describe a procedure for inverting this potential-to-density map, so that potentials (if any) corresponding to a target density rho(t) can be obtained. We give the trial external potential v(alpha), an analytic expression depending on a number of parameters alpha = (alpha(1), horizontal ellipsis ) and then minimize the least-squares integral integral(rho(alpha) - rho(t))(2) dr by the conjugate gradient method, where rho(alpha) is the density corresponding to v(alpha). The implementation takes advantage of the analytic nature of v(alpha). The procedure can be applied to any system and quantum chemistry model, and works both for ground and excited states, as well as for ensembles of states. The method is tested with some excited states of the particle-in-a-box model, confirming the lack of a Hohenberg-Kohn theorem for excited states. It is also applied to the first singlet excited state of the helium atom, where, apart from the nucleus-electron attraction potential, some generalized external potentials are found.