The Franck-Condon principle is a fundamental concept in spectroscopy, particularly in electronic spectroscopy, that describes the transition between electronic states of molecules. Here's a breakdown of what it entails:

Electronic Transitions: When a molecule absorbs light, it can undergo a transition from one electronic state to another. Each electronic state corresponds to a different distribution of electrons around the nuclei of the atoms in the molecule.
Vibrational States: Within each electronic state, the molecule can also exist in different vibrational states, where the nuclei vibrate with different amplitudes and frequencies.
Franck-Condon Principle: This principle states that during an electronic transition, the nuclei of the molecule move much slower than the electrons. Therefore, the nuclear positions essentially remain unchanged before and after the electronic transition occurs.
Implications:
- Intensity of Spectral Lines: The intensity of the spectral lines in electronic spectra depends on the overlap (or lack thereof) of the vibrational wavefunctions of the initial and final electronic states.
- Spectroscopic Features: The principle helps explain the shapes and intensities of electronic absorption or emission spectra observed experimentally.
Applications: It is crucial in understanding and interpreting spectroscopic data, such as absorption spectroscopy and fluorescence spectroscopy, which are widely used in chemistry and physics to study molecular properties, reaction dynamics, and more.

In essence, the Franck-Condon principle simplifies the analysis of electronic transitions by assuming that the nuclei are stationary during the transition, focusing attention on the electronic changes that occur.

Here solid horizontal lines in each curve for an electronic state refer to vibrational energy level ( eg LN line). The probability of transition between two given vibrational levels of two electronic states A and B is determined by Franck Condon principle. The principle is stated below.

1. The observed transition between two states should start from extreme position of vibrational levels.

2. They should be represented by vertical lines

That probability of transition from one state to another is maximum when the nuclei are in the mean position. For eg L and M for the level LN. This is because nuclei spend longest time at these positions due to vibrational K.E.

Quantum mechanically, when the square of vibrational eigen function is greatest, the probability of finding the nuclei at the extreme position is greatest. Quantum mechanics also predict that the nuclei's most likely position is equilibrium for lower vibrational position level (γ = 0), the most probable position of the nuclei is the equilibrium. Thus, the most probable internuclear distance for vibrational level other than 𝛾 = 0 corresponds to the extreme position. That means transition should start from extreme position for the levels other than 𝛾 = 0 which is due to the fact that e moves and rearrange themselves much faster than nuclei of the molecular eg the time period of e is 10^-16 s which is thousand times shorter than period of molecule (10^-13). It means electronic configuration will change in a time so short that nuclei will not change their position in that period. As the internuclear distance of a vibrating molecule in the course of transition between two electronic states doesn't change appreciably such a transition should then be represented by a vertical line.

Fortrate Diagram: The frequency in wave number unit for P, R and Q branches in terms of variable parameters p and q are

𝛾(PR) = 𝛾o + (B' + B'')p + (B'- B'')p² ....1 for p = ± 1, ±2, ...

𝛾(Q) = 𝛾o + (B' - B'')q + (B'- B'')q² ....1 for q = ± 1, ±2, ..

Each of equations represents a parabola, p takes both positive and negative values while q takes only positive values, we sketch these parabola called fortrate parabola taking 10% different between upper and lower B values (B'and B'') and choosing B'< B''. The regions of positive p are labelled as 𝛾(R) and regions of negative p are labelled as 𝛾(P) p and q may takes only integral values ( but not zero) showing by drawing order around the allowed points on the parabola.

The diagram enable us to read off the value of r for the spectral lines clearly A usual property of fortrate diagram is that the band head is at the vertex of PR parabola. The position of the vertex is given by differentiating equation 1 and equating to zero.

i.e d𝛾(PR)/ dp = 0

(B'+ B'') + 2p(B'- B'') = 0

p= - (B'+ B'') / 2(B'- B'') ....3 for band head

Thus if B'< B'' the band head occurs at positive p values i.e in R branch.

If B'> B'' the band head occurs at negative p values i.e in P branch.

𝛾(PR) = 𝛾o + (B' + B''){- (B'+ B'')/ 2(B'- B'')} + (B'- B'') (B'+ B'')²/ 4(B'- B'')²

𝛾(PR) = 𝛾o - (B' + B'')²/ 2(B'- B'') + (B'+ B'')²/ 4(B'- B'')

𝛾(PR) - 𝛾o = - (B' + B'')²/ 2(B'- B'')

which is wave number separation between head and origin.

𝛾(PR) - 𝛾o is positive for band degraded to red (B' < B'') and 𝛾(PR) - 𝛾o is negative for bond degraded to violet (B' > B'')

The vertex of Q branch parabola can be obtained by differentiating equation 2 with respect to q and equating to 0.

i.e d𝛾 / dq = 0

or, (B'- B'') + 2q(B'- B'') = 0

∴ q = -1/2

whatever the values of B' and B'', the band head appears on q = J' = J'' = -1/2. If (B'- B'') diff is large, the spacing between Q lines increases rapidly and Q head is not pronounced.

This note is a part of the Physics Repository

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