The influence of abstract group theory on molecular symmetry

8th International Symposium on Quantum Theory and Symmetries

Boyle, L.L.^1

Vine Cottage, Patrixbourne, Kent CT4 5BX, United Kingdom^1

关键词: Abstract groups;    Axis of symmetry;    Character tables;    Infinite groups;    Molecular symmetry;    Number of electrons;    Three-fold symmetry;    Underlying principles;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/512/1/012019/pdf
DOI  :  10.1088/1742-6596/512/1/012019

来源: IOP

【 摘 要 】

It is shown that the identification of the double groups required for treating molecular problems involving an odd number of electrons can be solved using abstract group theory without consideration of explicit matrices as in Pauli's discussion using spin matrices. This presupposes that the character tables are arrays of invariant numbers which alone are sufficient to specify the relevant properties of the spin systems. Consideration of subduction from infinite groups such as SU(2) is thus avoided and the criteria used to identify abstract groups as suitable double groups for the odd-electron spin problem include the need for subduction from finite symmetry groups of higher order as well as the requirement to produce closed shells of electrons in which no two electrons can have the same quantum numbers. Explicit calculations are provided for groups of orders 1, 2 and 4 and these provide the underlying principles for any molecular symmetry group. Novel aspects include the consideration of the separable four-fold degeneracies which occur in molecules having a single four-fold axis of symmetry. It is also shown that the treatment required for threefold symmetries is consistent with the principles derived from the study by abstract group theory of 1-, 2- and 4-fold symmetries.