Symmetry operations, group energies under

As already evident from the previous section, symmetry properties of a molecule are of utmost importance in understanding its chemical and physical behaviour in general, and spectroscopy and photochemistry in particular. The selection rules which govern the transition between the energy states of atoms and molecules can be established from considerations of the behaviour of atoms or molecules under certain symmetry operations. For each type of symmetry, there is a group of operations and, therefore, they can be treated by group theory, a branch of mathematics. 

If two or more atomic orbitals are interrelated under a symmetry operation of the point group and, accordingly, they together belong to an irreducible representation, their energies will also be the same. In other words, these orbitals are degenerate. Such atomic orbitals are parenthesized in the character tables. 

At this point we consult Table 2.2 and see that is the only irrep in which —1 is the character of each of the four symmetry operations in the eecond, excluded set. Conversely, only Bzu along with the totally symmetric representation - has 1 as the character of each of the sym-ops retained in Cf. This latter fact is expressed by the statement that the subgroup is the ker nel of the irrep B u of the parent group D2h- We note further that B u is the representation of the coordinate x, and realize that, after the system has been perturbed by a dipolar field along the x axis, the energy of an electron in a px orbital can remain unaffected only under sym-ops that do not convert x to —x. The perturbation, which has the representation Bs has evidently reduced the symmetry of the system to that of its kernel, i.e. from D2h fo its subgroup... 

Since the related Hamiltonian needs to remain invariant under all the symmetry operations of the molecular symmetry (point) group, the potential energy expansion, see equation (5), may contain only those terms which are totally symmetric under all symmetry operations. Consequently, a simple group theoretical approach, based principally on properties of the permutation groups can be devised, " which yields the number and symmetry classification of anharmonic force constants. The burgeoning number of force constants at higher orders can be appreciated from the entries given in Table 4. 

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