# On the permutation symmetry of atomic and molecular wavefunctions

## TL;DR

This paper investigates the permutation symmetry of atomic and molecular wavefunctions, analyzing a new approach that constructs antisymmetric functions based on permutation properties, with insights from exactly solvable models.

## Contribution

It provides a theoretical analysis of wavefunction symmetry, challenging assumptions about permutation properties in many-electron systems and illustrating with solvable models.

## Key findings

- Eigenfunctions are not necessarily symmetric or antisymmetric under particle exchange.
- The approach based on products of space and spin functions with opposite symmetry is analyzed.
- Exactly solvable models demonstrate the symmetry properties of eigenfunctions.

## Abstract

In this paper we analyze a recently proposed approach for the construction of antisymmetric functions for atomic and molecular systems. It is based on the assumption that the main problems with Hartree-Fock wavefunctions stem from their lack of proper permutation symmetry. This alternative building approach is based on products of a space times a spin function with opposite permutation symmetry. The main argument for devising such factors is that the eigenfunctions of the non-relativistic Hamiltonian are either symmetric or antisymmetric with respect to the transposition of the variables of a pair of electrons. However, since the eigenfunctions of the non-relativistic Hamiltonian are basis for the irreducible representations of the symmetric group they are not necessarily symmetric or antisymmetric, except in the trivial case of two electrons. We carry out a simple and straightforward general analysis of the symmetry of the eigenfunctions of the non-relativistic Hamiltonian and illustrate our conclusions by means of two exactly-solvable models of $N=2$ and $N=3$ identical interacting particles.