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To describe the optical transition between two electronic states, Figure 2.1 shows a configuration coordinate (q _s ) diagram where the potential energy curves of the ground, W _{I, s} (q _s )), and excited, W _II,s (q _s ), states are represented together with the vibrational levels, and ε ₀ is the energy difference between them. Since the electronic state changes in a time (∼10⁻¹⁵ s) much shorter than the nuclear vibration (∼10⁻¹² s), it can be assumed that the nuclei do not move nor change their momenta during the electronic transition (Franck–Condon Principle) [8, 9]. Therefore, both the absorption (from the ground to the excited state) and the luminescence (from the excited to the ground state) are represented by vertical arrows.

According to quantum mechanics, optical absorption and luminescence processes are quite well described by the first‐order time‐dependent perturbation theory. Let us consider the electronic state I, ψ _{I, n} (r, q) = ϕ _I (r, q)φ _{I, n} (q), with the set of vibrational levels n ≡ {n ₁, n ₂, …, n _f } thermally populated in accordance with the Boltzmann distribution; the absorption transition probability W(I, n → II, m) to the set of vibrational levels m ≡ {m ₁, m ₂, …, m _f } of the electronic state II, ψ _{II, m} (r, q) = ϕ _II(r, q)φ _{II, m} (q), is proportional to the absolute square of the matrix element of the perturbation operator. Since the light wavelength is much greater than the size of the defect (electric dipole approximation), the perturbation operator will be the dipole moment due to the electronic and nuclear charges:

(2.10)

It is worth noting that in the description of electronic transitions, the contribution D _nucl, involving purely vibrational transitions within a single electronic state, can be neglected. The matrix element of P is therefore:

Figure 2.1 Configuration coordinate diagram. The potential energy of the ground W _{I, s} and the excited W _{II, s} electronic states are depicted together with the associated vibrational levels. For simplicity sake, the electronic transitions (vertical arrows) are supposed to take place from the lower vibrational level.

(2.11)

Hereafter, we will use the notation:

(2.12)

to indicate the electronic matrix element. D _{I → II}(q) can be argued to be only weakly dependent on the nuclear coordinates, and in agreement with the Condon Approximation, it can be replaced by its value at the nuclear equilibrium position, .

Thus, the absorption transition probability is given by:

(2.13)

The first factor is the electronic part and is proportional to the overall intensity of the absorption band. The second factor, called Franck–Condon integral, is the nuclear part; it measures the overlap between the vibrational functions of the ground and excited state and determines the band shape [10, 11].

After absorption, the nuclei relax toward the minimum energy configuration in a much shorter time (10⁻¹²–10⁻¹¹ s) than the luminescence lifetime (≥10⁻⁹ s). This implies that at the time when light emission occurs, the vibrational levels of the electronic excited state are populated in accordance with a thermal distribution. The rate of light emission is given by the relationship between the Einstein coefficients of stimulated absorption (b) and spontaneous emission (a):

(2.14)

where n _ref is the refraction index of the medium. The emission transition probability is obtained by combining Eqs. (2.13) and (2.14). The indexes of the nuclear functions are different from those of Eq. (2.13) as a consequence of the displacement of the equilibrium position:

(2.15)

A case of particular interest is that in which the vibration frequency does not change during the electronic transition (ℏω _{I, s} = ℏω _{II, s} ), and, consequently, W _{I, s} (q _s ) and W _{II, s} (q _s ) have the same curvature; this is referred to as linear electron–phonon coupling. Consequently, the vibrational wave functions in the ground and excited electronic states are equal in pairs, thus leading to a symmetrical relationship:

(2.16)

Since the absorption and luminescence transitions occur under temperature equilibrium, the vibrational nth level, both in the ground and in the excited states, are populated following the same distribution. Therefore, the symmetry property of the Franck–Condon integral (Eq. 2.16) applies to the absorption and luminescence lineshapes; as a consequence they are characterized by a mirror symmetry with symmetry plane at the energy of the transition between the lowest vibrational levels.