# Allowed permutation symmetry in atomic and molecular wavefunctions. Simple examples

## TL;DR

This paper explores how permutation symmetry in the symmetric group relates to the antisymmetry requirement of wavefunctions in multi-electron systems, using simple models to illustrate the compatibility of symmetry species with antisymmetry.

## Contribution

It analyzes the compatibility of permutation symmetry species with the antisymmetry principle in atomic and molecular wavefunctions, providing simple illustrative examples.

## Key findings

- Certain symmetry species are compatible with antisymmetry
- Hamiltonian invariance under permutations influences wavefunction symmetry
- Simple models demonstrate symmetry-antisymmetry relationships

## Abstract

It is well known that the allowed wavefunctions for an $N$-electron system should be antisymmetric with respect to the permutation of any pair of electron labels. On the other hand, the Hamiltonian for such system is invariant under any permutation of electron labels and, consequently, its eigenfunctions are basis for the irreducible representations of the symmetric group $S_N$.Here, we investigate which symmetry species of the $S_N$ group are compatible with the antisymmetry principle. We illustrate the conclusions by means of simple $N$-particle one-dimensional models with harmonic interactions.