The Franck-Condon principle

There is no reason why the equilibrium bond length in an excited state should be the same as that in the ground state. Indeed, in the case of a diatomic molecule, it should be greater because one of the bonding electrons has moved into an antibonding MO. Excitation will then involve a transition between states with different geometries (Fig. 6,5). Now molecules at 

FIGURE 6.4. Plot of energy vs. bond length for a diatomic molecule XY, So being its residual energy of vibration and Vq the equilibrium bond length. 

While vibrational excitation is important in spectroscopy, it is usually unimportant in photochemistry because the vibrationally excited excited state is really just a hot excited state containing a lot of excess thermal energy. This is lost extremely rapidly by collisions with surrounding molecules, so that chemical reactions in solution involve only the vibrational ground state. 

One can also see from Fig. 6.5 that the absorption band will lie to the high-frequency or short-wavelength side of the zero-zero transition, i.e., that between the two vibrational ground states, whereas the corresponding emission will lie to the low-frequency or long-wavelength side. The two 

According to the Born-Oppenheimer approximation, the motions of electrons are much more rapid than those of the nuclei (i.e. the molecular vibrations). Promotion of an electron to an antibonding molecular orbital upon excitation takes about 10-15 s, which is very quick compared to the characteristic time for molecular vi- 

5) Spin-orbit coupling can be understood in a primitive way by considering the motion of an electron in a Bohr-lilce orbit. The rotation around the nucleus generates a magnetic moment moreover, the electron spins about 

3 Classical and quantum mechanical description of the Franck-Condon principle31 

In the quantum mechanical description (in continuation of Box 2.2), the wavefunction can be described by the product of an electronic wavefunction VP and a vibrational wavefunction / (the rotational contribution can be neglected), so that the probability of transition between an initial state defined by ViXa and a final state defined by TQ/b is proportional to vPi/a M f,2/b) 2- Because M only depends on the electron coordinates, this expression can be rewritten as the product of two terms f i M vP2 2 /a /b) 2 where the second term is called the Franck-Condon factor. Qualitatively, the transition occurs from the lowest vibrational state of the ground state to the vibrational state of the excited state that it most resembles in terms of vibrational wavefunction. 

Molecular Quantum Mechanics, Oxford University Press, Oxford. 

In our treatment of IR selection rules (Appendix 6) we have written wavefunctions for the harmonic oscillator without reference to the electronic state of the molecule. In fact, all the detail of the electronic states is assumed to be contained within the spring constant for the bond. To characterize the molecule fully we would need to take into account the nuclear and electronic coordinates when defining the potential energy. Rotational and translational degrees of freedom could also be included, adding more coordinates to describe the molecular motion of the system. However, we will only consider the internal structure of molecules, and so these additional factors will be left to one side. 

The wavefunction with the required information for the electronic and vibrational aspects of a molecule would depend on the coordinate set of the electrons r to account for the multi-electronic state and those of the nuclei R to account for the vibrational state. As a shorthand, the combined picture is referred to as a vibronic state. We use the symbol to refer to states that contain information about multiple particles and write F( r,R) to mean that the state is a function of both electron and nuclear coordinates. 

The mass of a proton is 1836 times that of an electron, and so the electron has a much smaller mass than even the H atom nucleus. This difference allows the functional form of the total wavefunction to be simplified by treating the electronic and vibrational states separately. The separation of the nuclear and electronic degrees of freedom in this way is 

Molecular Symmetry David J. Willock 2009 John Wiley Sons, Ltd. ISBN 978-0 70-85347  

The semicolon here is used to separate the coordinates which enter directly into the functional form of the wavefunction from other factors on which the wavefunction depends. So, under the Born-Oppenheimer approximation  

To understand how Nature makes use of colored materials in such important processes as photosynthesis and vision, we need to know the factors that control the intensity of electronic transitions and the shapes of absorption bands. 

The vibrational structure of a band is explained by the Franck-Condon principle  

Because nuclei are so much more massive than electrons, an electronic transition takes place faster than the nuclei can respond. 

In an electronic transition, electron density is lost rapidly from some regions of the molecule and is built up rapidly in others. As a result, the initially stationary nuclei suddenly experience a new force field. They respond by beginning to vibrate, and (in classical terms) swing backwards and forwards from their original separation, which they maintained during the rapid electronic excitation. The equilibrium separation of the nuclei in the initial electronic state therefore becomes a turning point, one of the end points of a nuclear swing, in the final electronic state (Fig. 12.33). 

When discussing symmetry selection rules it was mentioned that vibrational motion can influence both the shape and the intensity of electronic absorption bands. In the usual Born-Oppenheimer approximation with molecular wave functions written as products as in Equation (1.12) this can be understood as follows. 

Electronic motion with a typical frequency of 3 x 10 s (i = 10 cm ) is much faster than vibrational motion with a typical frequency of 3 x 10 s (v = 10 cm ). As a result of this, the electric vector of light of frequencies appropriate for electronic excitation oscillates far too fast for the nuclei to follow it faithfully, so the wave function for the nuclear motion is still nearly the same immediately after the transition as before. The vibrational level of the excited state whose vibrational wave function is the most similar to this one has the largest transition moment and yields the most intense transition (is the easiest to reach). As the overlap of the vibrational wave function of a selected vibrational level of the excited state with the vibrational wave function of the initial state decreases, the transition moment into it decreases cf. Equation (1.36). Absorption intensity is proportional to the square of the overlap of the two nuclear wave functions, and drops to zero if they are orthogonal. This statement is known as the Franck-Condon principle (Franck, 1926 Condon, 1928 cf. also Schwartz, 1973)  

The squared overlap integrals of the vibrational wave functions are referred to as the Franck-Condon factors. 

If the potential governing the nuclear motion is accidentally similar in the initial and final electronic states described by % and respectively, with a minimum at the same equilibrium geometry, the two operators vibO . Q) for these two states as well as their vibrational wave functions are identical. The vibrational wave functions and then are orthonormal. The nonvanishing factors will be and only the 0- 0, 1- 1,. . .,  

Figmc 1.12. Illustration of the Franck-Condon principle in the case of a diatomic molecule the absolute value of the integral Xt - largest for vertical tran- 

Evidently, the evaluation of optically important parameters, such as nd Tj, depends on the evaluation of the transition dipole moments, (0 /( J). In calculating the dipole moments a considerable simplification arises if we adopt the Franck-Condon principle. This is discussed in the next section. 

A general state, J), of the polymer is a function of many degrees of freedom, corresponding to the electron and nuclear coordinates. As usual, it is convenient to represent the nuclear degrees of freedom as normal modes, with each normal mode being associated with a normal coordinate, Qq, and a characteristic frequency, Wa. To simplify the discussion of the Franck-Condon principle we will make the reasonable assumption that only one normal mode is strongly coupled to the electronic degrees of freedom. 

The Franck-Condon principle is essentially a restatement of the Born-Oppenheimer approximation (introduced in Chapter 2), as it assmnes that the electronic transition occurs so quickly that the nuclear coordinates remain stationary. Phonon 

For small displacements of Q from equilibrium the adiabatic energy profiles are quadratic, and thus fluctuations in Q may be quantized as linear harmonic oscillators. The energy of the oscillators is represented by the horizontal lines in Fig. 8.2. Thus, quantum mechanically, there is a progression of linear harmonic oscillator states for each electronic state. Vibronic transitions can occur between pairs of vibronic states. 

To calculate the amplitude of these transitions we adopt the Born-Oppenheimer approximation and factorize J) as a single, direct product of the electronic and nuclear degrees of freedom 

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The Franck-Condon principle says that the intensities of die various vibrational bands of an electronic transition are proportional to these Franck-Condon factors. (Of course, the frequency factor must be included for accurate treatments.) The idea was first derived qualitatively by Franck through the picture that the rearrangement of the light electrons in die electronic transition would occur quickly relative to the period of motion of the heavy nuclei, so die position and iiioiiientiim of the nuclei would not change much during the transition [9]. The quaiitum mechanical picture was given shortly afterwards by Condon, more or less as outlined above [10]. 

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). 

Condon E U 1947 The Franck-Condon principle and related topics Am. J. Phys. 15 365-79... 

Duschinsky F 1937 On the interpretation of electronic spectra of polyatomic molecules. I. Concerning the Franck-Condon Principle Acta Physicochimica URSS 7 551... 

The Franck-Condon principle reflected in tire connection between optical and tliennal ET also relates to tire participation of high-frequency vibrational degrees of freedom. Charge transfer and resonance Raman intensity bandshape analysis has been used to detennine effective vibrational and solvation parameters [42,43]. 

In electronic spectra there is no restriction on the values that Au can take but, as we shall see in Section 1.2.53, the Franck-Condon principle imposes limitations on the intensities of the transitions. 

Section 6.13.2 and illustrated in Figure 6.5. The possible inaccuracies of the method were made clear and it was stressed that these are reduced by obtaining term values near to the dissociation limit. Whether this can be done depends very much on the relative dispositions of the various potential curves in a particular molecule and whether electronic transitions between them are allowed. How many ground state vibrational term values can be obtained from an emission spectrum is determined by the Franck-Condon principle. If r c r" then progressions in emission are very short and few term values result but if r is very different from r", as in the A U — system of carbon monoxide discussed in Section 7.2.5.4, long progressions are observed in emission and a more accurate value of Dq can be obtained. 

Figure 8.8 The Franck-Condon principle applied to the ionization of Fl2...

Using the Franck-Condon principle in this way we can see that the band system associated with the second lowest ionization energy, showing a long progression, is consistent with the removal of an electron from a bonding n 2p MO. The progressions... 

The state may decay by radiative (r) or non-radiative (nr) processes, labelled 5 and 7, respectively, in Figure 9.18. Process 5 is the fluorescence, which forms the laser radiation and the figure shows it terminating in a vibrationally excited level of Sq. The fact that it does so is vital to the dye being usable as an active medium and is a consequence of the Franck-Condon principle (see Section 7.2.5.3). 

Excited-State Relaxation. A further photophysical topic of intense interest is pathways for thermal relaxation of excited states in condensed phases. According to the Franck-Condon principle, photoexcitation occurs with no concurrent relaxation of atomic positions in space, either of the photoexcited chromophore or of the solvating medium. Subsequent to excitation, but typically on the picosecond time scale, atomic positions change to a new equihbrium position, sometimes termed the (28)- Relaxation of the solvating medium is often more dramatic than that of the chromophore... 

Solvatochromic shifts are rationalized with the aid of the Franck-Condon principle, which states that during the electronic transition the nuclei are essentially immobile because of their relatively great masses. The solvation shell about the solute molecule minimizes the total energy of the ground state by means of dipole-dipole, dipole-induced dipole, and dispersion forces. Upon transition to the excited state, the solute has a different electronic configuration, yet it is still surrounded by a solvation shell optimized for the ground state. There are two possibilities to consider ... 

Even where the promotion is to a lower vibrational level, one that lies wholly within the 2 curve (such as Vi or V2), the molecule may still cleave. As Figure 7.2 shows, equilibrium distances are greater in excited states than in the ground state. The Franck-Condon principle states that promotion of an electron takes place much faster than a single vibration (the promotion takes... 

The elementary act of an electrochemical redox reaction is the transition of an electron from the electrode to the electrolyte or conversely. Snch transitions obey the Franck-Condon principle, which says that the electron transition probability is highest when the energies of the electron in the initial and final states are identical. 

It follows from the Franck-Condon principle that in electrochemical redox reactions at metal electrodes, practically only the electrons residing at the highest occupied level of the metal s valence band are involved (i.e., the electrons at the Fermi level). At semiconductor electrodes, the electrons from the bottom of the condnc-tion band or holes from the top of the valence band are involved in the reactions. Under equilibrium conditions, the electrochemical potential of these carriers is eqnal to the electrochemical potential of the electrons in the solution. Hence, mntnal exchange of electrons (an exchange cnrrent) is realized between levels having the same energies. 

We have seen in Chapter 1 that absorption and emission spectra are controlled at least in part by the Franck-Condon principle. However, this is only one of three major factors that must be considered. 

Effect of diagonal dynamic disorder (DDD). Fluctuations of the polarization and the local vibrations produce the variation of the positions of the electron energy levels eA(Q) and eB(C ) to meet the requirements of the Franck-Condon principle. 

These selection rules are affected by molecular vibrations, since vibrations distort the symmetry of a molecule in both electronic states. Therefore, an otherwise forbidden transition may be (weakly) allowed. An example is found in the lowest singlet-singlet absorption in benzene at 260 nm. Finally, the Franck-Condon principle restricts the nature of allowed transitions. A large number of calculated Franck-Condon factors are now available for diatomic molecules. 

Certain features of light emission processes have been alluded to in Sect. 4.4.1. Fluorescence is light emission between states of the same multiplicity, whereas phosphorescence refers to emission between states of different multiplicities. The Franck-Condon principle governs the emission processes, as it does the absorption process. Vibrational overlap determines the relative intensities of different subbands. In the upper electronic state, one expects a quick relaxation and, therefore, a thermal population distribution, in the liquid phase and in gases at not too low a pressure. Because of the combination of the Franck-Condon principle and fast vibrational relaxation, the emission spectrum is always red-shifted. Therefore, oscillator strengths obtained from absorption are not too useful in determining the emission intensity. The theoretical radiative lifetime in terms of the Einstein coefficient, r = A-1, or (EA,)-1 if several lower states are involved,... 

The optical absorption of the solvated electron, in the continuum and semicontinuum models, is interpreted as a Is—-2p transition. Because of the Franck-Condon principle, the orientational polarization in the 2p state is given... 

The preceding calculations can also be performed for finite cavity sizes. For this case, there are some additional sources of small amounts of energy associated with cavity formation arising from surface tension, pressure-volume work, and electrostriction. Because of the Franck-Condon principle these do not affect the transition energy, but they have some influence on the heat of solvation. Jortner s (1964) results are summarized as follows ... 

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