Permutation inverse

Relation Between the Isometric Group and the Permutation-Inversion Group (Longuet-Higgins Group) of Nonrigid Molecules... 

In this section a relation of the isometric group approach to the permutation-inversion group of nonrigid molecules, introduced by Longuet-Higgins7, will be established. Such an interrelation is obtained in a natural way, if the isometric transformations... 

Table 4. Relation between the isometric group and the permutation-inversion group...

It is of the same form as r NC1) JF and forms the analogue of the Longuet-Higgins permutation-inversion operations associated with covering symmetry operations of a SRM. 

The semidirect product of the two groups (2.90) and (2.92) may be considered as the analogue of the permutation-inversion groups1 for molecules with non-trivial covering groups ( )... 

Therefore, SRMs with ( ) = Cj, for which the permutation-inversion group is always isomorphic to r(NCl) J the relation between SfSt and the isometric group ( ) is only a homomorphism... 

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... 

The permutation-inversion group of this SRM is only homomorphic to (r), since F NCIl(Fp) = l 2

Hougen s Approach and the Longuet-Higgins Permutation-Inversion Group... 

Figure 8.15. Correlation diagram between levels of a rigid rotor K = 0 (water dimer with Cs symmetry in the nontunneling limit), a rotor with internal rotation of the acceptor molecule around the C2 axis (permutation-inversion group G ), and group G16. The arrangement of levels is given in accordance with the hypothesis by Coudert et al. [1987], The arrows show the allowed dipole transitions observed in the (H20)2 spectrum. The pure rotational transitions E + (7 = 0) - E (J = 1) and E (7 = 1) <- E + (/ = 2) have frequencies 12 321 and 24 641 MHz, respectively. The frequencies of rotationtunneling transitions in the lower triplets AI (7 = 1) <- A,+ (7 = 2) and A," (7 = 3) <- A,+ (7 = 4) are equal to 4863 and 29 416 MHz. The transitions B2(7 = 0)<- B2(7 = l) and BJ(7 = 2) <- B2 (7 = 3) with frequencies 7355 and 17123 MHz occur in the higher multiplets.

The knowledge of the permutation-inversion group allows us to determine the intensity pattern that should be observed in the microwave experiments. It can be... 

In relation to permutation inversion symmetry species the superscript + or — may be used to designate parity. 

Stone applied the theory of Longuet-Higgins to deduce the character tables for the multiple internal rotation in neopentane and in octahedral hexa-ammonium metallic complexes [6]. Dalton examined the use of the permutation-inversion groups for determining statistical weights and selection rules for radiative processes in non-rigid systems [7]. Many applications of the Molecular Symmetry Groups have been reviewed later by Bunker [8,9]. 

In this paper, both theories will be briefly reviewed presenting their differences. From these comparisons, the Non-rigid Molecule Group (NRG) will be stricktly defined as the complete set of the molecular conversion operations which commute with a given Hamiltonian operator [21]. The operations of such a set may be written either in terms of permutations and permutations-inversions, just as in the Longuet-Higgins formalism, or either in terms of physical operations just as in the formalism of Altmann. But, the order, the structure, the symmetry properties of the group will depend exclusively on the Hamiltonian operator considered. 

In Fig. 2, the twelve possible isoenergetic configurations that can be obtained by permutations and permutations-inversions of the hydrogen and fluorine atoms, are presented. In this figure, the identical particles, the protons... 

In order to achieve this comparison between the Longuet-Higgins and Altmann s theories, and the present NRG one, let us consider again the orthoformic acid molecule, but in a random configuration such cis upper one shown in Fig. 10. From this configuration, it is easily seen that six different isoenergetic structures may be derived from one of them, for example, by permutations and permutations-inversions of the identical nuclei ... 

Each of these subgroups are isomorphic to C3 . The Molecular Smmetry Group for ethane is identical to that of acetone, because the number of feasible permutations and permutations-inversions of identiccil particles is the same in both systems, and isomorphic to the full NRG Groups. 

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