Non-rigid molecules, group theory

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. 

Group Theory for non-rigid molecules considers only large amplitude mou-vements ignoring the small amplitude motions, such as vibrations. In the following, the Non-Rigid Molecule Group (NRG) will be strictly defined as the complete set of the molecular conversion operations, which commute... 

The Group Theory for Non-Rigid Molecules considers isoenergetic isomers, and the interconversion motions between them. Because of the discernability between the identical nuclei, each isomer possesses a different electronic Hamiltonian operator in (3), with different eigenfunctions, but the same eigenvalue. In contrast, a non-rigid molecule has an unique effective nuclear Hamiltonian operator (7). 

Introduction to Group Theory for Non-Rigid Molecules version, phenomenon which has not been observed experimentally. 

In order to go farther into the Non-Rigid Molecule Theory, let us analyze expression (20) and compare the full and restrictred non-rigid groups. For this purpose, let us consider two border-line cases ... 

To solve the Schrodinger equation of a non-rigid system, the potenti2d energy function has to be known. This may be expanded in multidimensional Fourier series [30]. Group Theory for Non-Rigid Molecule permits this type of expansions to be simplified and gives rise to approximated analytical forms [30-37] and [47-52]. 

In order to illustrate the power of the Group Theory for Non-Rigid Molecules, let us consider the double internal rotation problem in acetone, solved in [34]. This motion is described by a restricted Hamiltonian operator such as that of pyrocatechin (32) in which only the threefold periodicity of the potential for acetone (47) has been introduced ... 

In the semi-classic approach. Group Theory for Non-rigid molecules allows to select the isoenergetic conformations, and to save calculation time. 

Group Theory for Non-Rigid Molecule (NRG), permits us to classify the torsional wave-functions according to the irreducible representations of the symmetry group of the molecule. As it is well known, the scalar product of (119) does not vanish when the direct products of the irreducible representations, under which 4, and / transform, contain at least one of the components of the dipole moments variation. Thus, when symmetry properties of these components are known, Selection Rules may be established. 

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