Computing an orthonormal basis of symmetric or antisymmetric hyperspherical harmonics

TL;DR

This paper introduces a numerical method for constructing orthonormal bases of symmetrized hyperspherical harmonics, with algorithms for transformation coefficients and state counting, demonstrated on systems with up to five bodies.

Contribution

It presents new algorithms for calculating transformation coefficients and directly determining independent symmetric and antisymmetric hyperspherical states.

Findings

Algorithms successfully applied to systems with up to five bodies.

Efficient computation of transformation coefficients between different hyperspherical harmonic sets.

Method enables accurate expansion of A-body wave functions with symmetry considerations.

Abstract

A numerical method to build an orthonormal basis of properly symmetrized hyperspherical harmonic functions is developed. As a part of it, refined algorithms for calculating the transformation coefficients between hyperspherical harmonics constructed from different sets of Jacobi vectors are derived and discussed. Moreover, an algorithm to directly determine the numbers of independent symmetric hyperspherical states (in case of bosonic systems) and antisymmetric hyperspherical-spinisospin states (in case of fermionic systems) entering the expansion of the A-body wave functions is presented. Numerical implementations for systems made with up to five bodies are reported.