Variational method for excited atomic states

An analysis is presented of the extremum properties of energy functionals for the excited states of many-electron systems, in particular, atoms, in the case when there exist low-lying states of the same symmetry as the excited state under consideration. Two theorems are proved concerning the relationship between the upper bound on the eigenvalues corresponding to the excited states and the extremum properties of the energy functional determined by variational test functions which depend on parameters. In this context, different variants of the one-electron approximation used in the excited-state calculations are considered: the method of obtaining self-consistent solutions with one-electron functions orthogonal within the configurations (the Hartree-Fock method for configurations); the frozen atomic core approximation for the excited configuration; the method of nonorthogonal orbitals in the excited configuration; and the approximation of the frozen ionic core. Cases are identified and reasons given for the possible “collapse” of the excited state energy level to an unjustifiably low value of the energy.