Schrödinger-Poisson model for very-high-pressure cold helium

An alternative numerical solution of the Hartree-Fock integro-differential equations is reported that consists of reformulating the self-consistent ansatz as a nonlinear set of coupled Schr\"odinger and Poisson equations. In the simple case of helium, this yields an amazingly simple dynamical system whose statistical properties are constrained by the virial theorem. This approach leads to an equation of state for helium at zero temperature and very high densities, covering the whole range of astrophysical interest, up to the pressure ionization phase transition that is predicted around 44 Mbar by the present model.