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

TL;DR

This paper compares the traditional HUM method and a new minimization principle for excited state wave functions, showing the new method's reliability with small truncated expansions especially for large systems.

Contribution

The paper introduces a minimization principle for excited states that yields more accurate wave functions than HUM when using small truncated expansions.

Findings

HUM method may produce incorrect wave functions with small expansions

The new minimization method provides correct, reliable excited state wave functions with small expansions

Demonstrations performed on helium excited states with various truncated series

Abstract

We demonstrate that, if a truncated expansion of a wave function is Large, then the standard excited states computational method, of optimizing one root of a secular equation, according to the theorem of Hylleraas, Undheim and McDonald (HUM), tends to the correct excited wave function, comparable to that obtained via our proposed minimization principle for excited states [J. Comput. Meth. Sci. Eng. 8, 277 (2008)] (independent of orthogonality to lower lying approximants). However, if a truncated expansion of a wave function is Small - that would be desirable for large systems - then the HUM-based methods may lead to an incorrect wave function - despite the correct energy (: according to the HUM theorem) whereas our method leads to correct, reliable, albeit Small truncated wave functions. The demonstration is done in He excited states, using truncated series Small expansions both in Hylleraas coordinates, and via standard configuration-interaction truncated Small expansions, in comparison with corresponding Large expansions. Beyond that, we give some examples of linear combinations of Hamiltonian eigenfunctions that have the energy of the 1st excited state, albeit they are orthogonal to it, demonstrating that the correct energy is not a criterion of correctness of the wave function.