Franck-Condon principle initial excitation

We return to the self-trappping (ST) of Frenkel excitons in 3D organic structures. As a result of the Franck-Condon principle, photo-exciting a crystal from its ground state, with a regular lattice, leads to the initial creation of a coherent... 

The multiplicity of excitations possible are shown more clearly in Figure 9.16, in which the Morse curves have been omitted for clarity. Initially, the electron resides in a (quantized) vibrational energy level on the ground-state Morse curve. This is the case for electrons on the far left of Figure 9.16, where the initial vibrational level is v" = 0. When the electron is photo-excited, it is excited vertically (because of the Franck-Condon principle) and enters one of the vibrational levels in the first excited state. The only vibrational level it cannot enter is the one with the same vibrational quantum number, so the electron cannot photo-excite from v" = 0 to v = 0, but must go to v = 1 or, if the energy of the photon is sufficient, to v = 1, v = 2, or an even higher vibrational state. 

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) ... 

We start by tracing the time-development of the vibrational excitation following a broad-band optical or UV absorption. The initially excited state will be a wave-packet residing essentially at the point B in Fig. 10 (the Franck-Condon principle). 

D14.2 The Franck-Condon principle states that because electrons are so much lighter than nuclei, an electronic transition occurs so rapidly compared to vibrational motions that the internuclear distance is relatively unchanged as a resu It of the transition. This implies that the most probable transitions vf <— vj are vertical. This vertical line will, however, intersect any number of vibrational levels Vf in the upper electronic state. Hence transitions to many vibrational states of the excited state will occur with transition probabilities proportional to the Frank-Condon factors which are in turn proportional to the overlap integral of the wavefunctions of the initial and final vibrational states. A vibrational progression is observed, the shape of which is determined by the relative horizontal positions of the two electronic potential energy curves. The most probable transitions are those to excited vibrational states with wavefunctions having a large amplitude at the internuclear position Re. 

Due to their direct relation to the spectral overlap integral, see Eq. (9), the emission and absorption spectra of the dye molecules are of interest in the context of EET processes. The simplest way to model excitation spectra employs the calculation of vertical energy separations, i.e., the separation of the Bom-Oppenheimer potential energy surfaces of the initial state and the final state at the equilibrium structure of the initial state. This energy separation is expected to coincide with the absorption maximum, as rationalized by the Franck-Condon principle (see for example [135]). This assumption is not always appropriate, rylene dyes being a prominent example. These dyes feature a strong 0-0 transition and a pronounced vibronic progression that is even visible in solution at room temperature (see for example [137]). A detailed simulation of the vibrational substructure of the absorption and emission bands is necessary to understand the details of the spectram. 

The calculation of UV/vis spectra, or any other form of electronic spectra, requires the robust calculation of electronic excited states. The absorption process is a vertical transition, i.e. the electronic transition happens on a much faster timescale than that of nuclear motion (i.e. Bom-Oppenheimer dynamics, more correctly referred to as the Franck-Condon principle in the context of electronic spectroscopy). The excited state, therefore, maintains the initial ground-state geometry, with a modified electron density corresponding to the excited state. To model the corresponding emission processes, i.e. fluorescence or phosphorescence, it is necessary to re-optimize the excited-state nuclear geometry, as emission in condensed phases generally happens from the lowest vibrational level of the emitting excited state. This is Kasha s Rule. 

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