Vibrational overlap integrals

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

The last factor, the square of the overlap integral between the initial and final vibrational wavefunctions, is called the Franck-Condon factor for this transition. 

The first term in this expansion, when substituted into the integral over the vibrational eoordinates, gives ifj(Re) , whieh has the form of the eleetronie transition dipole multiplied by the "overlap integral" between the initial and final vibrational wavefunetions. The if i(Rg) faetor was diseussed above it is the eleetronie El transition integral evaluated at the equilibrium geometry of the absorbing state. Symmetry ean often be used to determine whether this integral vanishes, as a result of whieh the El transition will be "forbidden". 

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. 

Figure 7.21 illustrates a particular case where the maximum of the v = 4 wave function near to the classical turning point is vertically above that of the v" = 0 wave function. The maximum contribution to the vibrational overlap integral is indicated by the solid line, but appreciable contributions extend to values of r within the dashed lines. Clearly, overlap integrals for A close to four are also appreciable and give an intensity distribution in the v" = 0 progression like that in Figure 7.22(b). 

This specfmm is dominated by ftmdamenfals, combinations and overtones of fofally symmefric vibrations. The intensify disfribufions among fhese bands are determined by fhe Franck-Condon factors (vibrational overlap integrals) between the state of the molecule and the ground state, Dq, of the ion. (The ground state of the ion has one unpaired electron spin and is, therefore, a doublet state, D, and the lowest doublet state is labelled Dq.) The... 

The intensity of a vibronic transition depends upon the square of the overlap integral of the vibrational wave functions,... 

As discussed in Chapter 1, the probability of a nonradiative transition is proportional to the square of the vibrational overlap integral J xiXa drv ... 

In the limit that Huty >> kgT, the rate constant for nonradiative decay is simply the product of the square of the vibrational overlap integral and an electronic term for the... 

In these terms, the electronic integrals such as (Mge)° and (Mge) a are constrained by the symmetry of the electronic states. While term I involves Frank-Condon overlap integrals, terms II and HI involve integrals of the form i Qa v) in the harmonic approximation, the integrals of this type obey the selection rule v = i + 1. Keeping these considerations in mind, we will next discuss how terms I, II and in contribute to distinct vibrational transitions. 

The probability of a transition v" v is determined by the Franck-Condon factor, which is proportional to the squared overlap integral of both vibrational eigenfunctions in the upper and lower state. 

Every fifth vibrational level is marked with a dashed line. The a) state is the v = 20 vibrational level of the second potential. The i>) state is the v = 30 vibrational level of the third potential. Both are marked by solid lines. The overlap integral between these two states is 0.118. Reprinted figure by permission from Ref. [38]. Copyright 2003 by the American Physical Society. 

Adiabatic and Condon." Since the Condon approximation separates the electronic and the vibrational parts of the problem (see Section 10c), most of the papers using this approximation emphasize the latter aspect. Specifically, they analyze the so-called Franck-Condon overlap integrals [the (Xm X ) of Eq. (42)] and the occupancy factors of these vibrational levels. Such analyses include (1) the influence of mode type and/or number as well as... 

A possible explanation for this increase in lifetime is a reduction of the nonradiative processes. As pointed out by Robinson (95), these radiationless rates must depend upon the magnitudes of the product of the vibrational overlap integrals between the initial and final states. The substitution of deuterium for hydrogen results in lower vibronic amplitudes, yielding a smaller overlap product. 

Electronic transitions between two energy states are governed bytheFranck-Condon principle. In quantum mechanical terminology, the Franck-Condon overlap integral f xf x, dxv is important. xf and x" are, respectively, vibration wave functions for v in the final electronic state, and v" in the initial electronic state. 

To determine the vibrational structure of electronic transitions of polyatomics, we can make the same approximation [Equation (7.21)] as for diatomics of replacing Ptl by some sort of average Pel, which is independent of the Qi s. The intensities of vibrational bands in an electronic transition then depend on the vibrational overlap integral, which is like... 

Condon principle, and the square of the vibrational overlap integral (7.23) is the Franck-Condon factor for the transition. 

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