Permutation-inversion operation

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 component Mz belongs to the species 4" in the Dah group because fiz is not changed by pure permutations and it changes sign by permutation—inversion operations (Section 4.1). The overall symmetry selection rule therefore allows transitions only between vibration—inversion-rotation states with opposite parity with respect to the operation of inversion (cf. Fig. 6). 

The rotation-contortion-vibration electronic states of the nonrigid TRN and MA molecules are classified according to the G4 molecular symmetry group using the permutation-inversion operations E, P, i, and F- [58]. The G4 group is iso-... 

The subgroup of the complete nuclear permutation-inversion group that contains all feasible permutation-inversion operations. 

Since the global minimum has Ci symmetry the total number of versions of (H20)3 is 2 x 3 x 6 =8640, but the number of versions in each set that can be interconverted without breaking covalent bonds is only 2 x 3 x (2 ) = 96. The single-flip mechanism links any particular version to two distinct versions, because it may be applied to either majority monomer. The permutation-inversion operation corresponding to the particular labeling scheme in Figure 1 is... 

Table 2 Representative Permutation-inversion Operations for Each Class of Group C(48) in the Same Order as for Table 1...

We now introduce the reciprocal or inverse operator P to the permutation operator P (see Section 3.1) such that... 

INVERSE are the inverse operator list, and the Gensym symbol for it, respectively. The CLASS property value is another Gensym atom which has as its value a list of all of the operators in that class. (In this simple case, the value of // CLS-1 is the list (// GRP-1), etc.) The remaining pairs in each property list represent the group multiplication table. For any particular group multiplication, an element of the group list at the top of Table I pertains to the right operator, the property indicator pertains to the left operator, and the property value pertains to the product. For example, for the product of the permutation operator (123) with itself,... 

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. 

Table 1. Transformation of d, , x. p and of the stretch and bend coordinates of NH3 by the symmetry operations of the permutation-inversion group 03ii-...

Multiplication of the operators P and P means that P and P are to be applied successively. The set of all the permutations of N symbols has the property that the product PP of any two of them is equal to some other permutation of the set. A set of operators with this property is called a group, if in addition the set possesses an identity operation and if every operation P possesses an inverse operation P l such that PP-1 is equivalent to the identity operation. There are N permutations of N different symbols. 

As an example, in Fig. 5.1 we return to our favored ammonia molecule and list all nuclear permutations, with and without the all-particle inversion operator, that leave the full Hamiltonian invariant. Nuclear permutations are defined here in the same way as in Sect. 3.3. A permutation such as (ABC) means that the letters A, B, and C are replaced by B, C, and A, respectively. The inversion operator, E, inverts the positions of all particles through a common inversion center, which can be conveniently chosen in the mass origin. In total, 12 combinations of such operations are found, which together form a group that is isomorphic to Ds. How is this related to our previous point group At this point it is very important to recall that the state of a molecule is not only determined by its Hamiltonian but also, and to an equal extent, by the boundary conditions. The eigenvalue equation is a differential equation that has a very extensive set of mathematical solutions, but not all these solutions are also acceptable states of the physical system. The role of the boundary conditions is to define constraints that Alter out physically unacceptable states of the system. In most cases these constraints also lead to the quantization of the energies. 

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