transition, the nuclei’s coordinate and momentum are unchanged and a sudden electronic configuration change occurs [5]. Once the electronic state has changed, the nuclei relax toward the minimum energy of vibration compatible with the system’s temperature. At 0 K, this transition is toward the minimum vibrational energy, that is the minimum of the parabolas describing the energy of nuclei (dash‐dotted arrows in Figure 1.5). The Stokes shift marks the presence of a non‐null electron–phonon coupling. If the associated energy difference is zero, the two electronic states have the minima of the potential curves for the same value of Q, the two parabolas are vertically aligned, and transition between the same vibrational levels occurs without energy difference between absorption and emission. To go deeper into this aspect, the Born–Oppenheimer approximation is considered again. The wavefunction ψ that solves the Schrodinger equation can be factorized, separating the nuclei’s and the electrons’ contributions
(1.89)
where φ, ϑ refer to the electronic and nuclear wavefunctions, respectively, the first being parametrically dependent on Q, the nuclear coordinate, and the latter being independent on the electronic coordinate r. Furthermore, the electrons’ wavefunction, on the basis of the Condon approximation, depends on the average value of the nuclear coordinate [5, 15]
To evaluate the probability of transition between the two states reported in Figure 1.5, the dipole matrix element introduced in (1.41) should be considered. In particular, the value
(1.91)
has to be determined between the ground state designated by the energy E 1 and the excited state E 2. By considering (1.90) and the separation of nuclear and electronic coordinates, it is found that [5, 15]:
where the indices 1 and 2 refer to the lower and higher electronic energy levels and the indices n and m refer to the vibrational levels of the nuclei. It is worth underlining that the overall energy of the considered molecular system is the combination of the electrons’ and nuclei’s interactions. The latter is determined by the vibrational state marked by the quantum numbers reported in Figure 1.5. Overall, the transition involves electronic states and nuclear vibrational quantum states; so, the transition is called vibronic transition [5]. Equation (1.92) shows that the first factor gives the amplitude of the probability, being linked to the oscillator strength, and the second factor, |M nm |2, is responsible for the shape of the band for the given transition of absorption or emission. This is known as the Franck–Condon factor [5, 15]. In particular, since it is related to the harmonic oscillator solutions of the Schrodinger equation, this factor is null if n ≠ m for a given oscillator [9]. But because the solutions considered pertain to different equilibrium configuration of oscillators, with the same frequency, the orthonormal wavefunctions of the harmonic oscillators
(1.93)
the summation being equal to 1 due to the orthonormality condition of used wavefunctions [5, 9, 15]. This finding explains that the transition probability from a given vibrational state is dependent on the electronic part of (1.92) whereas the nuclear part is responsible for the shape. This result gives origin to the lineshape of the absorption or emission band, since the same considerations can be done inverting the initial state and because |M nm |2 = |M mn |2. In particular, the homogeneous lineshape for a molecular species is dependent on the electron–phonon coupling and on its strength through |M nm |2. It is also found that absorption and emission lineshapes are symmetric with respect to the transition energy individuated by the M 00 element, known as the zero‐phonon line (ZPL, reported symbolically in Figure 1.5) [5, 15]. This transition is from the electronic ground state without vibration excitation to the excited electronic state without vibration excitation and vice versa, and it coincides for absorption and emission. In general, the vibration quantum ℏω is much larger than the thermal energy kT at room temperature; so, only the ground vibrational level, n = 0, is populated in the ground electronic state. In this case, the most relevant terms for the evaluation of the transition are |M 0m |2 and for the absorption process at energy E = E 00 + m ℏω, one can write
(1.94)
giving the amplitude of absorption for the transition from the ground vibrational level of the electronic ground state to the mth vibrational level of the excited electronic state, where E 00 is the zero‐phonon line energy. Analogously, for the emission process, one can write
(1.95)
considering that at room temperature, after the absorption process, the system relaxes to the lowest vibrational level in the excited state, as shown by the dash‐dotted arrow in Figure 1.5, and then it goes back to the electronic ground state, occupying one of the many vibrational levels depending on the |M n0|2 factor. On the basis of the wavefunctions of the harmonic oscillator, it can be demonstrated that [5, 15, 18, 19]
where the Huang–Rhys factor S has been inserted [19]
