Inherent restrictions of the Hylleraas-Undheim-MacDonald higher roots, and minimization functionals at the excited states

Abstract

The excited states, being energy saddle points in the Hamiltonian eigenfunction Hilbert space, cannot be computed variationally by minimization of the energy. Thus, functionals (cf. arXiv:0801.3673) are presented, that have local minimum at the bound excited states of a non-degenerate Hamiltonian, allowing the computation at any desired accuracy, by using crude approximations of the lower lying states. They are useful for larger systems, because the higher roots of the standard secular equation (via the Hylleraas-Undheim and MacDonald theorem) have several restrictions (cf. arXiv:0809.3826), which render them of lower quality relative to the lowest root, if the latter is good enough. Preliminary test-results are presented for He $^{1}$S 1s2s.