A discrete atomic level which falls within the energy bands of crystal persists in the adsorbed atom as a virtual level with fractional occupancy determined by the level's position with respect to the Fermi level. But in the Hartree-Fock theory of atoms, the orbital energies, and their occupancies are related through the Coulomb and exchange interactions of the electrons. Consequently, the position of a virtual level with respect to the Fermi level cannot be known until its occupancy is known, and vice versa. There is therefore a self-consistent problem to be solved before the electronic structure of the adsorbed atom (i. e., its level densities, and occupancies) can be determined. The solutions to this problem are discussed for alkali, hydrogen, and transition series atoms on metals. The theory of the electronic structure of impurity atoms, and vacancies in a crystal surface is briefly presented