Irreducible bases in icosahedral group space

TL;DR

This paper explicitly calculates irreducible bases in the icosahedral group space, facilitating symmetry-adapted basis construction for systems with I or Ih symmetry, demonstrated through Hückel Hamiltonian examples for Carbon-60 and Carbon-240.

Contribution

It provides a general method to compute irreducible bases in icosahedral symmetry, simplifying the analysis of symmetry-adapted wavefunctions in complex molecular systems.

Findings

Explicit irreducible bases for icosahedral group space are derived.

Symmetry-adapted bases can be easily constructed for I and Ih symmetric systems.

Hückel Hamiltonian submatrices for Carbon-60 and Carbon-240 are recalculated using these bases.

Abstract

The irreducible bases in the icosahedral group space are calculated explicitly by reducing the regular representation. The symmetry adapted bases of the system with {\bf I} or {\bf I}$_{h}$ symmetry can be calculated easily and generally by applying those irreducible bases to wavefunctions of the system, if they are not vanishing. As examples, the submatrices of the H\"{u}ckel Hamiltonians for Carbon-60 and Carbon-240 are re-calculated by the irreducible bases.