Vibronic equations

Wilkowski, A. and Moffitt, W. (1960) Electronic spectra of dimers derivation of the fundamental vibronic equation. 

This last transition moment integral, if plugged into equation (B 1.1.2). will give the integrated intensity of a vibronic band, i.e. of a transition starting from vibrational state a of electronic state 1 and ending on vibrational level b of electronic state u. 

Equation (B 1.1.8) gives the intensity of one vibronic band in an absorption spectrum. It is also of interest to consider... 

For molecules we can use Bom-Oppenlieimer wavefimctions and talk about emission from one vibronic level to another. Equation (B1.1.5T equation tb 1.1.6) and equation tb 1.1.7) can be used just as they were for absorption. If we have an emission from vibronic state iih to the lower state a, the rate constant for emission would be given by... 

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). 

If the solution of the zero-order Schiodinger equation [i.e., all teiins in (17) except V(r,Ro) are neglected] yields an/-fold degenerate electronic term, the degeneracy may be removed by the vibronic coupling tenns. If F) and T ) are the two degenerate wave functions, then the vibronic coupling constant... 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

The most consequent and the most straightforwaid realization of such a concept has been carried out by Handy, Carter, and Rosmus (HCR) and their coworkers. The final form of the vibration-rotation Hamiltonian and the handling of the corresponding Schrddinger equation in the absence of the vibronic... 

In this approximation, the angle r does not appear explicitly in the vibronic secular equation. From Eq. (76) it follows that... 

For the variational calculations of the vibronic spectrum and the spin-orbit fine structure in the X H state of HCCS the basis sets involving the bending functions up to 0i = 02 = 11 with all possible and I2 values are used. This leads to the vibronic secular equations with dimensions 600 for each of the vibronic species considered. The bases of such dimensions ensure full... 

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 intensity distribution among the rotational transitions is governed by the population distribution among the rotational levels of the initial electronic or vibronic state of the transition. For absorption, the relative populations at a temperature T are given by the Boltzmann distribution law (Equation 5.15) and intensities show a characteristic rise and fall, along each branch, as J increases. 

If vibrations are excited in either the lower or the upper electronic state, or both, the vibronic transition moment corresponding to the electronic transition moment Rg in Equation (7.115), is given by... 

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

In addition, it can be shown that second-order vibronic perturbation will make possible some intersystem crossing to the 3B3u(n, tt ) state. However, this second-order perturbation should be much less important than the first-order spin-orbit perturbation.(19) This will produce the unequal population of the spin states shown in Figure 6.1. In the absence of sir the ratios of population densities n are given by the following equations ... 

Here for simplicity it is assumed that there is only one promoting mode. Equation (3.85) should be compared with Eq. (3.73). In Eq. (3.85) both vibronic and spin-orbit coupling are involved in W-,. /. Due to the rapid progress in ab initio calculations, it has now become possible to evaluate Wi >/ by using potential energy surfaces information obtained from ab initio calculations. 

Hamiltonian equations, 627-628 perturbative handling, 641-646 II electronic states, 631-633 vibronic coupling, 630-631 ABC bond angle, Renner-Teller effect, triatomic molecules, 611-615 ABCD bond angle, Renner-Teller effect, tetraatomic molecules, 626-628 perturbative handling, 641-646 II electronic states, 634-640 vibronic coupling, 630-631 Abelian theory, molecular systems, Yang-Mills fields ... 

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