Franck integral

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]).

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

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. 

There are cases where the variation of the electtonic ttansition moment with nuclear configuration caimot be neglected. Then it is necessary to work with equation (B 1.1.6) keeping the dependence of on Q and integrating it over the vibrational wavefiinctions. In most such cases it is adequate to use only the tenns up to first-order in equation (B 1.1.7). This results in modified Franck-Condon factors for the vibrational intensities [12]. 

If we can use only the zero-order tenn in equation (B 1.1.7) we can remove the transition moment from the integral and recover an equation hrvolving a Franck-Condon factor ... 

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. 

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... 

Both the weak- and strong-coupling results (2.82a) and (2.86) could be formally obtained from multiplying Aq by the overlap integral (square root from the Franck-Condon factor) for the harmonic q oscillator,... 

Palma, A., and Sandoval, L. (1988), The Nonabelian Two-Dimensional Algebra and the Franck-Condon Integral, IntlJ. Quant. Chem. S22, 503. 

The function G in eq 1 is the Franck-Condon factor which accounts for the contribution of nuclear degrees of freedom and represents the thermal average of the overlap integrals between nuclear wavefunctions with respect to conservation of energy, and is given by (2, 3, 8, 9)... 

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. 

The ( )j(q)Qm (Q) is a set of linearly independent functions the Qm(Q) functions are not orthogonal in Q-space for arbitrary electronic states the overlap integrals Jd Q Qm(Q) Q m (Q) are the well known Franck-Condon factors. The hypothesis is that an arbitrary quantum molecular state is given by the linear superposition jus as in the general case ... 

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... 

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. 

Figure 5.2 Potential energy surface for So, S, and S2 and Franck-Condon overlap integral. Inset (A) Overlap integral between AJ 12 and SJ""° and (B) Sj and SJ ° states for horizontal transfer of energy.

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

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