Minimizers for open-shell, spin-polarised Kohn–Sham equations for non-relativistic and quasi-relativistic molecular systems

We study the open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic N-electron Coulomb systems, that is, systems where the kinetic energy of the electrons is given by either the non-relativistic operator -?xn or the quasi-relativistic operator v-a-²?xn + a-4 - a-². For standard and extended Kohn-Sham models in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge Ztot of K nuclei is greater than N - 1. For the quasi-relativistic setting we also need that Ztot is smaller than a critical charge Zc = 2a-¹p-¹.