Minimization principle for non degenerate excited states (independent of orthogonality to lower lying known approximants)

TL;DR

This paper introduces a variational principle, Fn, for computing excited states that does not require orthogonality to lower states, leading to more accurate and computationally efficient wave functions, demonstrated on helium.

Contribution

The paper presents a new variational principle, Fn, for excited states that avoids orthogonality constraints and improves accuracy with less computational effort.

Findings

Fn correctly identifies excited states with smaller basis sets.

The method reduces computational cost compared to traditional orthogonality-based approaches.

Application to helium's excited states demonstrates effectiveness and accuracy.

Abstract

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.