Angular part of trial wavefunction for solving helium Schr\"{o}dinger equation

TL;DR

This paper develops a simplified, geometrically motivated basis set for solving the helium Schrödinger equation, emphasizing angular parts and offering advantages in completeness and ease of generalization.

Contribution

It introduces a new variational basis set focusing on geometric characteristics, simplifying calculations and improving completeness for helium states.

Findings

Basis set is complete for natural L states with L+1 terms.

Basis set is complete for unnatural L states with L terms.

The basis set is easier to use and generalize to multi-particle systems.

Abstract

In this article, the form of basis set for solving helium Schr\"{o}dinger equation is reinvestigated in perspective of geometry. With the help of theorem proved by Gu $et~al.$, we construct a convenient variational basis set, which emphasizes the geometric characteristics of trial wavefuncions. The main advantage of this basis is that the angular part is complete for natural $L$ states with $L + 1$ terms and for unnatural $L$ states with $L$ terms, where $L$ is the total angular quantum number. Compared with basis sets which contain three Euler angles, this basis is very simple to use. More importantly, this basis is quite easy to be generalized to more particle systems.