Schur–Weyl Duality in Molecules

Abstract

Both, the intra-molecular dynamics of a single molecule and the inter-molecular dynamics of, e.g., reacting molecules, are strongly influenced by the symmetry of the respective nuclear spin states. Due to Paulis exclusion principle it restricts the allowed ro-vibrational molecular states. In this chapter, the unification of the nuclear spin angular momentum symmetry and the permutational symmetry of the nuclei is shown. Traditional approaches to find these symmetries become very impracticable when it comes to molecules with a large number of identical nuclei or a large individual spin quantum number. With the Schur–Weyl duality theorem, this complication is solved, it actually provides a simple rule to find both symmetries in a coherent manner. Furthermore, the use of the duality theorem is simplified by using the ingenious technique of Young tableaus, which are graphical depictions of irreducible representations. With this new technique, a few examples of calculating the nuclear spin symmetries are shown, including the famous H\(_{5}^{+}\) complex and its fully deuterated isotopologue, D\(_{5}^{+}\).