faster than the nuclei’s motion and that each electron explores a slowly varying configuration of the nuclei. Such “adiabatic” approximation is usually known as Born–Oppenheimer approximation [5, 8, 15, 17]. It assumes that the nuclei are practically fixed in space during the electrons’ motion, influencing the latter's energy statically. On the other side, the nuclei experience the electrons’ motion as an average value since the latter instantaneously adapt themselves to the given nuclear configuration. Such rapid rearrangement imposes that the nuclear energy is essentially related to the instantaneous nuclei positions. This enables to define a nuclear configuration and associate to it the overall electrons–nuclei system energy. Under this approximation, a configurational energy can be introduced
where the first three terms, depending on electrons’ motion, are an averaged (over the electrons’ coordinates and motion) contribution for a given spatial configuration of nuclei. The modified Hamiltonian of the overall system is then
(1.82)
where the configurational potential energy and the kinetic energy of the nuclei are explicitly reported. In the harmonic approximation, U conf is simplified as a quadratic function of the generalized coordinate Q [15, 18]. This is the internuclear distance in the most simple diatomic molecule or a normal coordinate in the case of a polyatomic molecule [5, 7]. It can be written
where the equilibrium position (minimum of the potential energy) Q = 0 has been chosen for simplicity and m and ω are the mass and frequency of the oscillation mode whereas U g is a constant representing the minimum energy of the given electronic configuration (related to spin and orbital motion of electrons). The complete Hamiltonian then becomes
(1.84)
that has the harmonic oscillator form [9]. The solution of the Schrodinger equation using this Hamiltonian gives eigenvalues and eigenfunctions of the nuclear coordinate [9]. The total energy of the system based on (1.81) will include the electrons’ energy. A representation of this energy is shown in Figure 1.5.
Figure 1.5 On the left: Schematic representation of the configuration total energy for the ground (lower) and the excited (upper) electronic states of a molecule. The continuous lines refer to the total energy, the horizontal dashed lines refer to the nuclei vibrational energy levels, marked by their quantum numbers. The equilibrium configuration coordinates are Q = 0 and Q = Q e for the ground and excited states, respectively. The zero‐phonon line is highlighted by the double arrow. On the right: The excitation (E abs) and emission (E em) pathways are reported by continuous arrows; ΔQ and ΔU highlight the change in configuration coordinate and potential energy, respectively; the dash‐dotted arrows mark the nuclear relaxation processes.
Each parabola schematizes the energy of the electronic configuration. In the lower parabola, representing the ground state of the system, the electrons have given spin and orbital angular momenta. In the upper parabola, schematizing the first excited state, the spin and orbital angular momenta have in general changed. The parabolic approximation gives the shape of the total energy, and the solutions of the harmonic oscillator define the possible quantized energy for the nuclear motion (represented by dashed lines in the figure). This means that not all the energy values are possible but just those marked by the dashed lines. Anyway, the energy of vibration is much lower than the energy of electrons’ interaction, and almost a continuous change can be considered within the given parabola, schematizing the electron configuration.
Considering the excited energy level, it is characterized, in general, by a configuration coordinate of equilibrium Q = Qe different with respect to the electronic ground state. Assuming again a harmonic approximation, the total energy of the excited state can be written as
where E is the absorption energy and the linear electron–phonon coupling approximation has been applied, represented by the term with F [15, 18]. The parameter F shows that the configuration coordinate Q of the minimum potential energy in the ground state and that in the excited state are different due to the connection between this energy and the electronic configuration (analytically, (1.85) is the formula of a parabola whose vertex position in the energy‐Q space is different from that of the parabola in (1.83) but they have the same concavity). By changing the electronic configuration, 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,
