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We demonstrate that, if a truncated expansion of a wave function is Large,then the standard excited states computational method, of optimizing one rootof a secular equation, according to the theorem of Hylleraas, Undheim andMcDonald (HUM), tends to the correct excited wave function, comparable to thatobtained via our proposed minimization principle for excited states [J. Comput.Meth. Sci. Eng. 8, 277 (2008)] (independent of orthogonality to lower lyingapproximants). However, if a truncated expansion of a wave function is Small -that would be desirable for large systems - then the HUM-based methods may leadto an incorrect wave function - despite the correct energy (: according to theHUM theorem) whereas our method leads to correct, reliable, albeit Smalltruncated wave functions. The demonstration is done in He excited states, usingtruncated series Small expansions both in Hylleraas coordinates, and viastandard configuration-interaction truncated Small expansions, in comparisonwith corresponding Large expansions. Beyond that, we give some examples oflinear combinations of Hamiltonian eigenfunctions that have the energy of the1st excited state, albeit they are orthogonal to it, demonstrating that thecorrect energy is not a criterion of correctness of the wave function.

Correct Small-Truncated Excited State Wave functions Obtained via  Minimization Principle for Excited States compared / opposed to  Hylleraas-Undheim and McDonald higher roots