Minimization principle for non degenerate excited states (independent of   orthogonality to lower lying known approximants)
  The computation of small concise and comprehensible excited state wave functions is needed because many electronic processes occur in excited states. But since the excited energies are saddle points in the Hilbert space of wave functions, the standard computational methods, based on orthogonality to lower lying approximants, resort to huge and incomprehensible wave functions, otherwise, the truncated wave function is veered away from the exact. The presented variational principle for excited states, Fn, is demonstrated to lead to the correct excited eigenfunction in necessarily small truncated spaces. Using Hylleraas coordinates for He 1S 1s2s, the standard method based on the theorem of Hylleraas - Unheim, and MacDonald, yields misleading main orbitals 1s1s' and needs a series expansion of 27 terms to be corrected, whereas minimizing Fn goes directly to the corect main orbitals, 1s2s, and can be adequately improved by 8 terms. Fn uses crude, rather inaccurate, lower lying approximants and does not need orthogonality to them. This reduces significantly the computation cost. Thus, having a correct 1st excited state {\psi}1, a ground state approximant can be immediately improved toward an orthogonal to psi_1 function. Also higher lying functions can be found that have the energy of psi_1, but are orthogonal to psi_1. Fn can also recognize a flipped root in avoided crossings: The excited state, either flipped or not, has the smallest Fn. Thus, state average is unnecessary. The method is further applied via conventional configuration interaction to the three lowest singlet states of He 1S. 
Naoum C. Bacalis
physics.chem-ph quant-ph