Construction of the Pauli potential, Pauli energy, and effective potential from the electron density

The Kohn-Sham (KS) one-electron Schrödinger equations assume the existence of a one-body effective potential ${\mathit{v}}_{\mathrm{eff}}$(x), defined to generate the correct electron density ρ(x) of the ground state. This paper returns to the electron-density description of an N-fermion system. It is best thought of as starting from a given ρ(x), ideally to be obtained from diffraction experiments. A method is then set up that focuses predominantly on the way the ‘‘correct’’ ${\mathit{v}}_{\mathrm{eff}}$(x) can be ‘‘recovered,’’ if it exists, from such an experimental density. Certainly the method has associated with it one practical disadvantage in common with the KS procedure; an order of N Euler equations have to be solved, with input information ρ(x), though the ‘‘unknown’’ potential ${\mathit{v}}_{\mathrm{eff}}$(x) does not now appear. In this program, we have found it most helpful to work with the Pauli potential and energy, which enable the N-fermion problem to be converted to a boson problem for the density amplitude [ρ(x)${]}^{1/2}$. The way the above-mentioned Euler equations determine the Pauli potential and energy is worked out explicitly. Examples that embrace the important area of atomic central-field calculations are presented to illustrate the method. As a by-product, the theory developed can afford a direct test as to whether a given electron density is, in fact, representable via a one-body potential ${\mathit{v}}_{\mathrm{eff}}$(x).