Spectroscopy for Materials Characterization. Группа авторов
the potential energy changes and also the nuclear configuration adapts itself to a novel equilibrium position. By finding the minima of (1.83) and (1.85), it is demonstrated that the difference in the abscissa of minima is
(1.86)
In the linear electron–phonon approximation, the oscillation frequency ω of nuclei is the same in the ground and in the excited state. This is described by parabolic potentials with the same concavity in both states. As reported in Figure 1.5, this also implies that moving by ΔQ far from the minima, the same energy change ΔU is found in both states. By the scheme reported in the above figure, it is found that
where E abs and E em are the absorption and emission energies and the energy difference (E abs − E em) between the maximum of the absorption profile and that of the emission is called Stokes shift. Usually, it is considered that the electronic transition occurs in a much shorter time than the nuclear motion, so the nuclear configuration coordinate is unchanged during both the absorption and emission processes [5]. This statement is known as Franck–Condon principle [5]. From (1.83), it is found that for Q = Q e the configuration energy change of the ground state is
(1.88)
The same absolute value is found for the excited state. This value corresponds to the relaxation energy after the absorption or the emission processes. In particular, based on the Franck–Condon principle, in any 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