Primitive Period Isometric Transformations

The investigation of the set of distances (dkk (i-) w.r.t. isometric transformations in many cases leads to transformations of the type 

However, NC Xk(if), Zk, Mk and NC Xk( - p), Zk, Mk are not identical, but may be mapped onto each other by a nontrivial element of 0(3). Associating operators PFp with primitive period transformations of type (2.39) allows to express this relation by 

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by 

For SRMs with nontrivial primitive period transformations one has thus to distinguish between two types of internal isometric groups ( ) and JF (jz). The 

At the present time the following two cases concerning the group theoretical 

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The isomorphism strictly holds for SRMs without primitive period isometric transformations only (cf. Sect. 2.2.2). However, as will be shown in Sect. 2.2.2, the group theoretical relations derived in this section also apply for SRMs with primitive period transformations if is replaced by an appropriately extended group Jr. The sets... 

Key For SRMs with primitive period isometric transformations the isomorphisms hold strictly for the representations of y, jr, , r and IF, but / is homomorphic onto... 

In Sect. 2.2.2 we have shown that if a SRM admits primitive period isometric transformations, representations of two groups and may be derived. Extension of a representation of J7 by leads to the corresponding representation of St, whereas extension of the representations of St by g gives those of St. The use of St or depends on the problem to which the isometric group is to be applied, as has been pointed out in Section 2.2.2. In order to simplify the notation we shall for general discussions not distinguish between the representations of St and St. 

Primitive Period Isometric Transformations and the Longuet-Higgins Group... 

It is of interest to point out the role of primitive period isometric transformations (cf. Sect. 2.2.2.) in both the isometric and the Longuet-Higgins groug. According to Eq. (2.41) this type of transformations is represented on the basis X <( ) by... 

Since the dynamical problem (3.10) refers to the LS, the primitive period isometric transformations are to be included in ( ). A proof of this important theorem has been given earlier14 9. 5 H represents symmetry of H w.r.t. to operations of the... 

In the symmetrization process the primitive period isometric transformations Fp play an outstanding role. For all SRMs with r(3 (Fp) e SO(3) these operators are represented in the representation r NCI St by the unit matrix (cf. Sect. 2.3.3). 

According to theorem 2.2.3.1 there belong NCs with Cs or Cj covering symmetry to these fixed points. They are shown schematically in Fig. 7. The role of the primitive period isometric transformation F3 has been discussed in Sect. 2.4 and is illustrated in a suggestive manner by Fig. 2. 

Since the problem of conformational isomerism is entirely a question of relative nuclear configuration, primitive period isometric transformations have to be omitted. 

Analogously for SRMs with proper covering group g) and primitive period transformations, the full isometric group Wig) =Jrg) ( ) is homomorphic to the permutation-inversion group... 

Key All internal isometric transformations, including primitive period transformation are shown... 

This formula shows that the primitive period p of r is equal to w. If we take the domain — n/2 < t < + rr/2, the only nontrivial internal isometric transformation of this SRM is F2 r = -r and therefore, since F2 = E... 

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