There are no unfilled shells in unrestricted Hartree-Fock theory.

We prove that in an exact, unrestricted Hartree-Fock calculation each energy level of the Hartree-Fock equation is either completely lled or completely empty. The only assumption needed is that the two-body interaction is| like the Coulomb interaction|repulsive; it could, however, be more complicated than a simple potential, e.g., it could have tensor forces and velocity-dependence. In particular, the HF energy levels of atoms and molecules, often called shells, are never partially lled. The Hartree-Fock (HF) variational calculation provides an approximate determination of ground states and ground state energies of quantum mechanical systems such as atoms, molecules, solids, superconductors, nuclei, etc., and is widely used in physics and chemistry. Physicists often turn to it for qualitative, if not quantitative guidance because particle correlations can sometimes be mimicked by a suitable one-body mean-eld operator. Sometimes HF theory is used as the beginning of a more accurate calculational scheme, but our interest here is on qualitative features rather than numerics. The picture of quantum systems that HF theory yields is one of independent particle levels which are