Schrodinger supergroup

It was sometimes believed, in the scientific literature, that the Molecular Symmetry Group of Longuet-Higgins and the Schrodinger Supergroup of Altmann were isomorphic, i.e., both theories were equivalent [19]. We shall see, however, that is not generally true, especially when some symmetry is retained in the molecule. 

Taking into account that the CH3 — BF2 molecule, in a random configuration, does not possess any symmetry at all, its Schrddinger subgroup, G, is the identity. The Schrodinger supergroup, S, of this molecule takes the form ... 

Inspection of Fig. 2, shows that the equivalence between the Schrodinger Supergroup and the Molecular Symmetry Group operations, for the CH3—BF2 molecule, is easily verified. In Table 2, the two groups appear to be identical. 

I ble 2 Equivalence between the Molecular Symmetry Group and Schrodinger Supergroup operations in the CH3 — BFj molecule. 

Notice that all the groups considered have the same structure, but they are not identical. In particular, the restricted and full NRG are not identical between themselves, or to the Schrodinger Supergroup because Cp. 

In contrast, the Altmann s Schrodinger Supergroup, for the symmetric or-thoformic acid is expressed as [39] ... 

Similarly, because the full NRG considers the external and internal motions as well as the coupling between them, the full NRG does not coincide with the Altmann Schrodinger Supergroup, even when the molecule under study has no symmetry such as in the case of a random orthoformic acid. 

The Altmann s Schrodinger Supergroup for ethane is provided in ref. [19. Considering that ethane in a random conformation possesses still one C3 and three C2 symmetry axes, see Fig. 11, the Schrodinger Group of this molecule is ... 

Taking into account the symmetry point group of the molecule, Z 3/, the Schrodinger Supergroup is expressible as ... 

The symmetric PFs molecule is a clear example where the Schrodinger Supergroup cannot be factorized into two subgroups, as expected by the Alt-mann s theory. In this case, the internal rotations around a single atoms induce overall rotations of the molecule as a whole, i.e. the internal and external motions are not separable. 

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