Irreducible representations vibrational wave function

We have seen that the molecular electronic and vibrational wave functions el and vib each transform according to the irreducible representations of the molecular point group. We now consider the rotational wave function ptot. 

Let r and be the irreducible representations to which the vibrational wave functions in (9.189) belong. According to the italicized statement following (9.187), the integral (9.189) vanishes unless T+<8>riratls contains T, . This is the basic IR selection rule. (Note that its deduction does not involve the harmonic-oscillator approximation or the approximation of... 

It is further shown in Chapter 10 that, when each of the normal modes is in its ground state, each of the y/j is totally symmetric and hence y/v is totally symmetric. If one of the normal modes is excited by one quantum number, the corresponding it may then belong to one of the irreducible representations other than the totally symmetric one, say T, and thus the entire vibrational wave function f/Y will belong to the representation T,. Simple methods for finding the representations to which the first excited states of the normal modes belong are explained in Chapter 10. In this section we will quote without proof results obtained by these methods. 

As shown in Section 5.1, the wave functions must form bases for irreducible representations of the symmetry group of the molecule, and the same holds, of course, for all kinds of wave functions, vibrational, rotational, electronic, and so on. Let us now see what representations are generated by the vibrational wave functions of the normal modes. Inserting Hn(Va4,) into 10.6-1, we obtain... 

The vibrational wave function, as any wave function, must form a basis for an irreducible representation of the molecular point group [3],... 

The vibrational wave function of the ground state belongs to the totally symmetric irreducible representation of the point group of the molecule. The wave function of the first excited state will belong to the irreducible representation to which the normal mode undergoing the particular transition belongs. 

Group theory can be applied to several different areas of molecular quantum mechanics, including the symmetry of electronic and vibrational wave functions and the study of transitions between energy levels. There is also a theorem which says that there is a correspondence between an energy level and some one of the irreducible representations of the symmetry group of the molecule, and that the degeneracy (number of states in the level) is equal to the dimension of that irreducible representation. 

The seleetion rules derived above pertained to electronic transitions, where the positions of the nuelei were fixed in space. Now a change of the vibrational states of the molecule will be eonsidered, while the electronic state is assumed to be unchanged. The vibrations of a molecule are related to its vibrational levels (each of them corresponding to an irreducible representation) and the eorresponding vibrational wave functions, and the IR spectrum results from transitions between sueh levels. Fig. C.6 shows the energy levels of three normal modes. 

Equation 144 is just the equation for the electronic energy which was discussed in Chapter XI. The electronic wave functions F x, y, z, r) therefore have the symmetry properties of the various irreducible representations of the groups Dooa or Coop according as the nuclei are identical or different. The vibrational wave function R r) depends only on the distance between the two nuclei and therefore belongs to the totally symmetrical representation. The complete wave fimction will thus have the symmetry properties of the product F x, y, z, r)U(6, x)-In order to discuss the nature of the solutions of 14 6 it will be con-... 

In this expression, cr, p = x, y, or z, where a p is one of the polarizability components and (i and f) are the initial and final vibrational wave functions. Each of the factors in Eq. (80) belongs to an irreducible representation T, and according to group theory, in order for the integral in Eq. (80) to be nonvanishing, the direct product TiXT xTf must contain the totally symmetric representation. Thus, for a fundamental vibration to be Raman active, YfYf must transform under symmetry operations in the same manner as one of the a p components. This is the basis of the normal Raman vibrational selection rule. 

In order to give the concept of an irreducible representation a more concrete reality for the reader, the transformations of normal coordinates have been used throughout as examples, but the concept itself is quite independent of the idea of normal coordinates or the problem of molecular vibrations. It arises whenever a set of linear transformations has the properties of a group, no matter what the meaning of the transformation variables. Later, it will be necessary to regard the transformations of a set of wave functions as forming a group, and the present theory will be applied. 

In the many electron case, where the electronic system of the central ion has a d" configuration, the ligand field operator and its derivatives are sums over the number of d-elec-trons and the wave functions W describe many electron states (multiplets) which transform according to irreducible representations Fp of the symmetry point group Oh. Then the perturbation of a Fp state due to coupling to the vibration Oi is given by... 

Tensor representations, synonymous for product representations and their decomposition into irreducible constituents, are useful concepts for the treatment of several problems in spectroscopy. Important examples are the classification of the electronic states in atoms and the derivation of selection rules for infrared absorption or the vibrational Raman or hyper-Raman effect in crystals. In the first case the goal is to reduce tensors which are defined as products of one-particle wave functions, while in the second case tensors for the dipole moment, the electric susceptibility or the susceptibilities of higher orders have to be reduced according to the irreducible representations of the relevant point groups. 

Irreducible representation is the group theory term for a certain combination of symmetry properties of wave functions, normal vibrations, etc., which cannot be simplified (reduced) further by a transformation. The whole of the irreducible representation describes the symmetry or point group. 

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