Computing Correct Truncated Excited State Wavefunctions

TL;DR

This paper shows that standard methods for computing excited state wavefunctions can be incorrect when using truncated expansions, and proposes a more reliable alternative demonstrated on helium excited states.

Contribution

It introduces a new method for computing small truncated excited state wavefunctions that remains accurate regardless of orthogonality constraints.

Findings

Standard methods may produce incorrect wavefunctions with truncated expansions.

The proposed method yields correct and reliable excited state wavefunctions.

Validated on helium excited states using Hylleraas and configuration-interaction expansions.

Abstract

We demonstrate that, if a truncated expansion of a wave function is small, then the standard excited states computational method, of optimizing one root of a secular equation, may lead to an incorrect wave function - despite the correct energy according to the theorem of Hylleraas, Undheim and McDonald - whereas our proposed method [J. Comput. Meth. Sci. Eng. 8, 277 (2008)] (independent of orthogonality to lower lying approximants) leads to correct reliable small truncated wave functions. The demonstration is done in He excited states, using truncated series expansions in Hylleraas coordinates, as well as standard configuration-interaction truncated expansions.