  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.
