Selection rules vibronic

When a normal mode is distorted but is not displaced, then, according to the selection rules, vibronic transitions with odd n, aO —> bn , are forbidden. For even n, Franck-Condon factors are expressed as follows ... 

Often it is possible to resolve vibrational structure of electronic transitions. In this section we will briefly review the symmetry selection rules and other factors controlling the intensity of individual vibronic bands. 

The selection rule for vibronic states is then straightforward. It is obtained by exactly the same procedure as described above for the electronic selection rules. In particular, the lowest vibrational level of the ground electronic state of most stable polyatomic molecules will be totally synnnetric. Transitions originating in that vibronic level must go to an excited state vibronic level whose synnnetry is the same as one of the coordinates, v, y, or z. 

The synnnetry selection rules discussed above tell us whether a particular vibronic transition is allowed or forbidden, but they give no mfonnation about the intensity of allowed bands. That is detennined by equation (Bl.1.9) for absorption or (Bl.1.13) for emission. That usually means by the Franck-Condon principle if only the zero-order tenn in equation (B 1.1.7) is needed. So we take note of some general principles for Franck-Condon factors (FCFs). 

A very weak peak at 348 mn is the 4 origin. Since the upper state here has two quanta of v, its vibrational syimnetry is A and the vibronic syimnetry is so it is forbidden by electric dipole selection rules. It is actually observed here due to a magnetic dipole transition [21]. By magnetic dipole selection rules the A2- A, electronic transition is allowed for light with its magnetic field polarized in the z direction. It is seen here as having about 1 % of the intensity of the syimnetry-forbidden electric dipole transition made allowed by... 

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

The general vibronic selection rule replacing that in Equation (9.20) is... 

The ro-vibronic spectrum of molecules and the electronic transitions in atoms are only part of the whole story of transitions used in astronomy. Whenever there is a separation between energy levels within a particular target atom or molecule there is always a photon energy that corresponds to this energy separation and hence a probability of a transition. Astronomy has an additional advantage in that selection rules never completely forbid a transition, they just make it very unlikely. In the laboratory the transition has to occur during the timescale of the experiment, whereas in space the transition has to have occurred within the last 15 Gyr and as such can be almost forbidden. Astronomers have identified exotic transitions deep within molecules or atoms to assist in their identification and we are going to look at some of the important ones, the first of which is the maser. 

In the lowest optically excited state of the molecule, we have one electron (t u) and one hole (/i ), each with spin 1/2 which couple through the Coulomb interaction and can either form a singlet 5 state (5 = 0), or a triplet T state (5 = 1). Since the electric dipole matrix element for optical transitions H em = (ep A)/(me) does not depend on spin, there is a strong spin selection rule (A5 = 0) for optical electric dipole transitions. This strong spin selection rule arises from the very weak spin-orbit interaction for carbon. Thus, to turn on electric dipole transitions, appropriate odd-parity vibrational modes must be admixed with the initial and (or) final electronic states, so that the weak absorption below 2.5 eV involves optical transitions between appropriate vibronic levels. These vibronic levels are energetically favored by virtue... 

As the isoquinoline molecule reorients in the order listed above, the absorption of infrared radiation by the in-plane vibrational modes would be expected to increase, while that of the out-of-plane modes would be predicted to decrease (in accordance with the surface selection rule as described above). In the flat orientation there is no component of the dipole moment perpendicular to the surface for the in-plane modes, and under the surface selection rule these modes will not be able to absorb any of the incident radiation. However, as mentioned above, infrared active modes (and in some cases infrared forbidden transitions) can still be observed due to field-induced vibronic coupled infrared absorption (16-20). We have determined that this type of interaction is present in this particular system. 

Hence, according to the symmetry selection rule, n —> n transitions are allowed but n —> ti transitions are forbidden. However, in practice the n —> it transition is weakly allowed due to coupling of vibrational and electronic motions in the molecule (vibronic coupling). Vibronic coupling is a result of the breakdown of the Born-Oppenheimer approximation. 

The general form of the energy of the harmonic oscillator indicates that the vibrational energy levels are equally spaced. Due to the vector character of the dipole transition operator, the transition between vibronic energy levels is allowed only if the following selection rule is satisfied ... 

It is seen from Equation 19 that the electronic transitions take place without changing the equilibrium positions of the nuclei, and the electronic component of the dipole transition moment is non-zero only if there is no change of the vibronic state during this transition. Dg is non-zero only if the transitions occur between the vibronic states within one electronic state, and the selection rules of Equation 16 are derived from the conditions for a non-vanishing matrix element in Dg. ... 

In (a) the ion is so situated as to be in a noncentrosymmetric field, even when it is not vibrating. In this case electric-dipole emission is allowed. In (b) there is inversion symmetry when the ion is not vibrating, but vibration carried it to some other point Py at which the center of symmetry is lost. It should be self-evident that, even when the ion is in a noncentrosymmetric environment, vibrations may be important. That is, changes in the crystal-field symmetry induced by the vibronic motion will lead to violations of the crystal-field-selection rules. 

For C22H2, the nuclear spin of C12 is zero and contributes a factor of 1 to the nuclear statistical weights. The statistical weights are therefore the same as in H2. For the ground vibronic state, the even J levels are s and have nuclear statistical weight 1, corresponding to the one possible ns the odd J levels are a and have nuclear statistical weight 3. The usual selection rule (4.138) holds for collisions as well as radiative transitions, and we have ortho and para acetylene. The two forms have not been separated. 

Just as with vibronically allowed transitions, in symmetry groups in which all Cartesian axes are not equivalent (noncubic groups), it is found that, in general, transitions will be allowed only for certain orientations of the electric vector of the incident light. One class of compounds in which this phenomenon has been studied both theoretically and experimentally consists of trischelate compounds such as tris(acetylacetonato)M(III) and tris(oxalato)M(III) complexes. In these complexes the six ligand atoms form an approximately octahedral array but the true molecular symmetry is only Dy Since there is no center of symmetry in these molecules, the pure electronic selection rules might be expected to be dominant. 

It can be seen that these are very powerful selection rules indeed. On the other hand, we might have assumed that the symmetry of the environment of the metal ion could have been adequately approximated by considering only the six coordinated oxygen atoms. In this case, the symmetry would be Dm, in which there is a center of inversion and the transitions would be governed by vibronic selection rules. When these are worked out, it is found that all of the transitions are vibronically permitted. Thus, experimental study of the polarizations should provide clear-cut evidence as to the correct effective symmetry and selection rules. Such a study has been reported and shows conclusively that the selection rules followed are those given above for pure electronic transitions in Dy symmetry. 

The well-known selection rules proposed by Woodward and Hoffman25 to predict the stereochemical course of electrocyclic reactions can be viewed as emphasizing the symmetry requirements for electronic coupling of final and initial states. The rules are expressed in terms of rotatory motions required to convert one electronic state into another, so the matrix element is really vibronic rather than pure electronic. In terms of this paper, it appears that Woodward and Hoffman have identified necessary rotation properties of the perturbation operator. 

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