Vibronic transitions intensity

Warshel, A. and Huler, E. (1974) Theoretical Evaluation of Potential Surfaces, Equilibrium Geometries and Vibronic Transition Intensities of Excimers The Pyrene Crystal Excimer, Chem. Phys. 6, 463—468. 

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). 

Quite apart from the necessity for Franck-Condon intensities of vibronic transitions to be appreciable, it is essential for the initial state of a transition to be sufficiently highly populated for a transition to be observed. Under equilibrium conditions the population 1, of any v" level is related to that of the u" = 0 level by... 

In 1925, before the development of the Schrodinger equation, Franck put forward qualitative arguments to explain the various types of intensity distributions found in vibronic transitions. His conclusions were based on an appreciation of the fact that an electronic transition in a molecule takes place much more rapidly than a vibrational transition so that, in a vibronic transition, the nuclei have very nearly the same position and velocity before and after the transition. 

In 1928, Condon treated the intensities of vibronic transitions quantum mechanically. The intensity of a vibronic transition is proportional to the square of the transition moment which is given by (see Equation 2.13)... 

The quantity J dr is called the vibrational overlap integral, as it is a measure of the degree to which the two vibrational wave functions overlap. Its square is known as the Franck-Condon factor to which the intensity of the vibronic transition is proportional. In carrying out the integration the requirement that r remain constant during the transition is necessarily taken into account. 

The answer, very often, is that they do not obtain any intensity. Many such vibronic transitions, involving non-totally symmetric vibrations but which are allowed by symmetry, can be devised in many electronic band systems but, in practice, few have sufficient intensity to be observed. For those that do have sufficient intensity the explanation first put forward as to how it is derived was due to Herzberg and Teller. 

The first term on the right-hand side is the same as in Equation (7.128). Herzberg and Teller suggested that the second term, in particular (dRg/dQj), may be non-zero for certain non-totally symmetric vibrations. As the intensity is proportional to Rgy this term is the source of intensity of such vibronic transitions. 

Although we have considered cases where (9/ g/90,)gq in Equation (7.131) may be quite large for a non-totally symmetric vibration, a few cases are known where (9/ g/90,)gq is appreciable for totally symmetric vibrations. In such cases the second term on the right-hand side of Equation (7.131) provides an additional source of intensity forAj orX vibronic transitions when Vx is totally symmetric. 

The spectrum shows that many vibronic transitions are allowed, and that some are more probable than others that is, the intensities of the... 

At room temperature, most of the molecules are in the lowest vibrational level of the ground state (according to the Boltzmann distribution see Chapter 3, Box 3.1). In addition to the pure electronic transition called the 0-0 transition, there are several vibronic transitions whose intensities depend on the relative position and shape of the potential energy curves (Figure 2.4). 

Another important detail on the Sugano-Tanabe diagram shown in Figure 6.8 is the vertical line at the value DqlB = 2.2, at which the states T2g and E are equal in energy. This value of DqlB is usually referred to as the crossover value. Materials (crystal - - ion) for which Dq/B is less than the crossover value are usually called low crystal field materials. Eor these materials, the lowest energy level is the " T2g and so they present a characteristic broad and intense emission band associated with the spin-allowed T2g A2g transition (which is usually a vibronic transition). On the other hand, the materials on the right-hand side of the crossover line are called high crystal field materials. These materials (such as the ruby crystal) present a narrow-line emission related to the spin-forbidden Eg A2g transition, usually called R-line emission. 

Fig. 29. Group-theoretical predictions of the polarizations of the vibronic transitions, allowed to second order, from the individual zero-field levels of the lowest triplet state of 2,3-dichIoro-quinoxaline to vibrational levels of the ground electronic state. Solid line transitions gain intensity by spin-orbit mixing between states which differ in the electronic type of one electron e.g., S n and T . The dashed line transitions require the mixing to occur between states of the same electronic type (e.g., S and T n ) and is expected to be weaker. The dash-dotted transition could involve the favorable mixing between states that differ in the electronic type of one electron, but a spin-vibronic perturbation is needed. (From Tinti and El-Sayed, Ref. ))...

As detailed in Section 2, we have derived and programmed the expression for line strengths of individual rotation-vibration transitions of XY3 molecules the line strengths depend on the vibronic transition moments entering into equation (70). With the theory of Section 2, we can simulate rotation-vibration absorption spectra of XY3 molecules. In computing the transition wavenumbers, line strengths, and intensities we use rovibronic wavefunctions generated as described in Ref. [1]. 

The most likely electronic transition will occur without changes in the positions of the nuclei (e.g., little change in the distance between atoms) in the molecular entity and its environment. Such a state is known as a Franck-Condon state, and the transition is referred to as a vertical transition. In such transitions, the intensity of the vibronic transition is proportional to the square of the overlap interval between the vibrational wavefunctions of the two states. See Fluorescence Jablonski Diagram Comm, on Photochem. (1988) Pure and Appl. Chem. 60, 1055. 

If we use an ns probe pulse, we can tune its wavelength resonant to one particular vibronic transition. In this case, the LIF signal reflects the population of a single vibrational level involved in the WP. By scanning the wavelength of the probe pulse, we can observe the population distribution of the eigenstates involved in the WP. The peak intensities of the LIF signal are influenced by the Franck-Condon factors and the probe laser intensities, so that the relevant corrections are necessary to obtain the population distribution. 

Here, S is the Huang-Rhys factor [63], which is related to the intensity of the 0-n vibronic transition, Io-n = e sSn/n, and reflects the time-dependent Stokes shift associated with a given type of vibrational mode (e.g., S 0.6 for the high-frequency C=C stretch modes [61,64,65]). For the class of systems studied here, two types of phonon modes are considered per monomer unit, i.e., high-frequency C=C stretch modes and low-frequency ring-torsional modes. 

Figure 5-1. Mass resolved excitation spectrum of bare aniline in a molecular jet. Several of the more intense vibronic transitions are assigned. 

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