Dipole vibrational transition elements

The subscripts I, m and n represent the molecular inertial axes a, b and c) aim and iM represent the Raman and dipole vibrational transition elements, respectively... 

Fig. 4.2. The principal induced dipole components of H2-H2 ground state vibrational averages are shown in the left-hand plot. Overlap (dotted), multipole-induced terms (solid lines) classical multipole approximation (dashed). Right-hand plot ditto, but vibrational transition elements, v = 0 - v = 1. Molecule 2 undergoes the vibrational transition [281].

Fig. 4.1. The four significant induced dipole components of H2-He pairs ground state vibrational average (left panel) and vibrational transition matrix element v = 0 —> v = 1 (center panel) and of the ground state vibrational average of H2-Ar pairs (right panel). Overlap components are dotted multipole-induced components are represented by solid lines dashed lines indicate the associated classical multipole component after [279, 151, 280],...

The most striking fact about the dipole matrix elements for the H2-H2 overtone band is the large number of components, Tables 4.13 and 4.14. Besides the single vibrational transitions (V2 = 0 — 2 while t i = const, Table 4.13, and vice versa), we now have to consider vibrational double transitions ( i = 0 — 1 and V2 —> 1, Table 4.14). The associated spectra appear at nearly the same frequencies. If one adds to these the various rotational bands, Eq. 4.40, a very large number of spectral components arises that must be accounted for in the computations of the overtone... 

Table 4.13. Analytical form (Eqs. 4.39 and 4.40) of the dipole matrix elements for the H2-H2 first overtone band, single vibrational transitions [284].

The pattern of intensities in Fig. 4.9 deserves mention. The intensities of absorption lines are proportional to the population of the lower level, and to the square of the dipole-moment matrix element (4.97). It turns out that for vibration-rotation transitions in the same band, the integral (4.97)... 

There is a formal similarity in the mathematics used to describe vibrational transitions pumped by a resonant radiation field [148] and vibrational transitions pumped by phonons in a crystal lattice. In the lowest-order approximations, the radiation field and the vibrational transition are coupled by a transition dipole matrix element that is a linear function of a coordinate. The transition dipole describes charge displacement that occurs during the transition. Some of the cubic anharmonic coupling terms described by Eq. (10) result in a similar coupling between vibrational transitions and a phonon coordinate. These generally have the form / vibVph, so that the energy of the vibration with normal coordinate /vib is linearly proportional to the phonon coordinate /ph. Thus either an incoherent photon field or an incoherent phonon field can result in incoherent... 

Symmetry rules dictate that the electronic matrix elements are only nonzero for electronic states of definite symmetries. We introduce these rules in the next section, followed by a discussion of the the Pranck-Condon factors, which determine the intensity of the vibrational transitions. Finally, we use the exciton model (described in Chapter 6) to evaluate the electronic dipole moments. 

Due to the electron-phonon interactions, which contribute to the emission process by phonon-assisted transitions, the above expression may be modified to take into account the vibrational states. In that case, the dipole momentum matrix elements include both electronic... 

For vibrational transitions it is not appropriate to use the Born-Oppenheimer (BO) approximation because the ground and excited states in the context of vibrational transitions have the same electronic wavefunction and differ only in the nuclear wavefunctions, a consequence of which is that the electronic contribution to the magnetic dipole transition matrix element vanishes in the BO approximation. In order to include the important electronic contribution to magnetic dipole transition moments, one has to choose either to make further approximations to the magnetic dipole operator yielding effective non-vanishing magnetic transition moments, or to go beyond the BO approximation. Various approximate models and exact a priori methods have resulted in the last 25 years. 

The intensity of a vibronic band is proportioned to the square of the electronic dipole moment matrix element between the initial and final states. These states are well described by Bom-Oppenheimer products of electronic and nuclear functions. For a symmetry allowed transition the nuclei coordinate dependence of the electronic function can be ignored so that the intensity of a vibronic band can be factored into a purely electronic factor that is independent of nuclear coordinate times the square of the overlap of the vibrational functions (Franck-Condon factor). 

To ewiliiate the dipole matrix element for a transition between two vibrational states v" -> Y it is necessary to sum expression (1.16) over all rotational quantum numbers R and R" associated with the vibrational transition. Considering Eqs. (1.12) and (1.13) and the invariance of molecular dipole moment with respect to orientation in space and, therefore, to rotational coordinates, die following expression for the transition matrix element between vibrational-rotational states n and n" is obtained (4)... 

The harmonic oscillator selection rules for vibrational transition can be evaluated using the expression of the dipole moment as a power series with respect to normal coordinates. On the basis of expressions (1.25) - (1.27) tiie dipole moment matrix element for a transition between vibrational states V and V" may be written as... 

The solution of the inverse intensity problem, which in this formulation implies die evaluation of bond polar parameters from experimental dipole moment derivatives with respect to normal coordinates, can only be performed fm a molecule possessing higfrer symmetry. The situation in this respect is the same as in the alternative theoretical formulations. The requirement is that the direction of the vibrational transition dipole is fixed by symmetry. In odier words, there should be only one non>zero element in each colunm of the Pq matrix [Eq. (3.1)]. Again, all calculations are considerably simplified if the dp/dQi derivatives are first transformed into dipole moment derivatives with respect to internal symmetry coordinates. The determination of the elements of P(, can then be realized using die following general expression, in matrix notation... 

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... 

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