• However, as was known at the time, molecules will only absorb energy corresponding to allowed quantum transitions, and there are no vibrational levels above the dissociation
  energy level of the potential well.

• Note that while the electronic transitions are quantized, the chromophore-solvent interaction energy is treated as a classical continuum due to the large number of molecules
  involved.

• The physical intuition of this principle is anchored by the idea that the nuclear coordinates of the atoms constituting the molecule do not have time to change during the
  very brief amount of time involved in an electronic transition.

• Since the dependence is usually rather smooth it is neglected (i.e., the assumption that the transition dipole surface is independent of nuclear coordinates, called the Condon
  approximation is often allowed).The first integral after the plus sign is equal to zero because electronic wavefunctions of different states are orthogonal.

• Figure 1 illustrates the Franck–Condon principle for vibronic transitions in a molecule with Morse-like potential energy functions in both the ground and excited electronic
  states.

• Classically, the Franck–Condon principle is the approximation that an electronic transition is most likely to occur without changes in the positions of the nuclei in the molecular
  entity and its environment.

• The remaining two integrals contributing to the probability amplitude determine the electronic spatial and spin selection rules.The Franck–Condon principle is a statement
  on allowed vibrational transitions between two different electronic states; other quantum mechanical selection rules may lower the probability of a transition or prohibit it altogether.

• The principle states that during an electronic transition, a change from one vibrational energy level to another will be more likely to happen if the two vibrational wave
  functions overlap more significantly.

• Combining these equations leads to an expression for the probability amplitude in terms of separate electronic space, spin and vibrational contributions: The spin-independent
  part of the initial integral is here approximated as a product of two integrals:This factorization would be exact if the integral over the spatial coordinates of the electrons would not depend on the nuclear coordinates.

• Just like in the Franck–Condon principle, the probability of transitions involving phonons is determined by the overlap of the phonon wavefunctions at the initial and final
  energy levels.

• In this case we have a very great change in the oscillation energy on excitation by light… — James Franck, 1926 James Franck recognized that changes in vibrational levels
  could be a consequence of the instantaneous nature of excitation to higher electronic energy levels and a new equilibrium position for the nuclear interaction potential.

• [1] Electronic transitions are relatively instantaneous compared with the time scale of nuclear motions, therefore if the molecule is to move to a new vibrational level during
  the electronic transition, this new vibrational level must be instantaneously compatible with the nuclear positions and momenta of the vibrational level of the molecule in the originating electronic state.

• In the low temperature approximation, the molecule starts out in the vibrational level of the ground electronic state and upon absorbing a photon of the necessary energy,
  makes a transition to the excited electronic state.

• In this situation, transitions to higher electronic levels can take place when the energy of the photon corresponds to the purely electronic transition energy or to the purely
  electronic transition energy plus the energy of one or more lattice phonons.

• The rearrangement of the solvent molecules according to the new potential energy curve is represented by the curved arrows in Figure 7.

• The electron configuration of the new state may result in a shift of the equilibrium position of the nuclei constituting the molecule.

• Since the electronic transition is essentially instantaneous on the time scale of solvent motion (vertical arrow), the collection of excited state chromophores is immediately
  far from equilibrium.

• At the same time the equilibrium position of the nuclei moves with the excitation to greater values of r. If we go from the equilibrium position (the minimum of potential
  energy) of the n curve vertically [emphasis added] upwards to the a curves in Diagram I. the particles will have a potential energy greater than D’ and will fly apart.

• This effect is analogous to the original Franck–Condon principle: the electronic transition is very fast compared with the motion of nuclei—the rearrangement of solvent molecules
  in the case of solvation.

• In examining how much vibrational energy a molecule could acquire when it is excited to a higher electronic level, and whether this vibrational energy could be enough to immediately
  break apart the molecule, he drew three diagrams representing the possible changes in binding energy between the lowest electronic state and higher electronic states.

• The probability that the molecule can end up in any particular vibrational level is proportional to the square of the (vertical) overlap of the vibrational wavefunctions of
  the original and final state (see Quantum mechanical formulation section below).

• Franck–Condon metaphors in spectroscopy The Franck–Condon principle, in its canonical form, applies only to changes in the vibrational levels of a molecule in the course of
  a change in electronic levels by either absorption or emission of a photon.

• Although emission is depicted as taking place from the minimum of the excited state chromophore-solvent interaction potential, significant emission can take place before equilibrium
  is reached when the viscosity of the solvent is high or the lifetime of the excited state is short.

• The Franck–Condon principle (named for James Franck and Edward Condon) is a rule in spectroscopy and quantum chemistry that explains the intensity of vibronic transitions
  (the simultaneous changes in electronic and vibrational energy levels of a molecule due to the absorption or emission of a photon of the appropriate energy).

• Franck–Condon principles can be applied when the interactions between the chromophore and the surrounding solvent molecules are different in the ground and in the excited
  electronic state.

• The vibrational structure of molecules in a cold, sparse gas is most clearly visible due to the absence of inhomogeneous broadening of the individual transitions.

• Weaker magnetic dipole and electric quadrupole electronic transitions along with the incomplete validity of the factorization of the total wavefunction into nuclear, electronic
  spatial and spin wavefunctions means that the selection rules, including the Franck–Condon factor, are not strictly observed.

• The quantum mechanical formulation of this principle is that the intensity of a vibronic transition is proportional to the square of the overlap integral between the vibrational
  wavefunctions of the two states that are involved in the transition.

• When the solution is illuminated by light corresponding to the electronic transition energy, some of the chromophores will move to the excited state.

• In this use of the Franck–Condon metaphor, the vibrational levels of the chromophores, as well as interactions of the chromophores with phonons in the liquid, continue to
  contribute to the structure of the absorption and emission spectra, but these effects are considered separately and independently.

• This change in interaction can originate, for example, due to different dipole moments in these two states.

• In the original Franck–Condon principle, after the electronic transition, the molecules which end up in higher vibrational states immediately begin to relax to the lowest
  vibrational state.

• Rotational contributions can be observed in the spectra of gases but are strongly suppressed in liquids and solids.It should be clear that the quantum mechanical formulation
  of the Franck–Condon principle is the result of a series of approximations, principally the electrical dipole transition assumption and the Born–Oppenheimer approximation.

• Immediately after the transition to the ground electronic state, the solvent molecules must also rearrange themselves to accommodate the new electronic configuration of the
  chromophore.

• Electronic transitions to and from the lowest vibrational states are often referred to as 0–0 (zero zero) transitions and have the same energy in both absorption and fluorescence.

• High-energy photon absorption leads to a transition to a higher electronic state instead of dissociation.

• The overall wavefunctions are the product of the individual vibrational (depending on spatial coordinates of the nuclei) and electronic space and spin wavefunctions: This
  separation of the electronic and vibrational wavefunctions is an expression of the Born–Oppenheimer approximation and is the fundamental assumption of the Franck–Condon principle.

• Equal spacing between vibrational levels is only the case for the parabolic potential of simple harmonic oscillators, in more realistic potentials, such as those shown in
  Figure 1, energy spacing decreases with increasing vibrational energy.

 

Works Cited

[‘Franck, J. (1926). “Elementary processes of photochemical reactions”. Transactions of the Faraday Society. 21: 536–542. doi:10.1039/tf9262100536.
2. ^ Condon, Edward (1926-12-01). “A Theory of Intensity Distribution in Band Systems”. Physical Review.
28 (6): 1182–1201. Bibcode:1926PhRv…28.1182C. doi:10.1103/PhysRev.28.1182.
Photo credit: https://www.flickr.com/photos/swallowtailgardenseeds/16110795216/’]