Franck- Condon Level

These levels of interfacial redox electrons are connected with the hydrogen and oxygen electrode reactions. As noted in Sec. 5.1.2, the electron level of adsorbate particles is broadened by contact adsorption and undergoes the Franck-Condon level splitting due to a difference in adsorption energy between the oxidized particle and the reduced particle on the interface of semiconductor electrodes as shown in Fig. 5-59. 

Figure 8-1 shows the potential energy barrier for the transfer reaction of redox electrons across the interface of metal electrode. On the side of metal electrode, an allowed electron energy band is occupied by electrons up to the Fermi level and vacant for electrons above the Fermi level. On the side of hydrated redox particles, the reductant particle RED is occupied by electrons in its highest occupied molecular orbital (HOMO) and the oxidant particle OX, is vacant for electrons in its lowest imoccupied molecular orbital (LUMO). As is described in Sec. 2.10, the highest occupied electron level (HOMO) of reductants and the lowest unoccupied electron level (LUMO) of oxidants are formed by the Franck-Condon level sphtting of the same frontier oihital of the redox particles... 

Is the first excited state a franck-condon level ... 

IS THE FIRST EXCITED STATE A FRANCK-CONDON LEVEL ... 

This calculation is, in principle, possible but is quite difficult. Either one has to go to a non-periodic model or one has to use a sufficiently large super-cell to accommodate the displacement field. We are currently looking at this problem. The alternative approach is to examine the experimental data for evidence of a Franck-Condon level. The idea is in fact very attractive as it explains the sharpness of the peak for the first excited state in PdH [18]. The accumulated measurements on this peak show that it gets narrower as the hydrogen concentration approaches stoichiometry and the sample temperature... 

Probed excited state Franck-Condon level of initially excited state Vibrationally relaxed emitting state Lowest-lying excited state (can be a dark state)... 

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

The observation of a bend progression is particularly significant. In photoelectron spectroscopy, just as in electronic absorption or emission spectroscopy, the extent of vibrational progressions is governed by Franck-Condon factors between the initial and final states, i.e. the transition between the anion vibrational level u" and neutral level u is given by... 

This example relates to the well known Franck-Condon principal of spectroscopy in which squares of overlaps between the initial electronic state s vibrational wavefunction and the final electronic state s vibrational wavefunctions allow one to estimate the probabilities ofpopulating various final-state vibrational levels. 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

Quite apart from the necessity for Franck-Condon intensities of vibronic transitions to be appreciable, it is essential for the initial state of a transition to be sufficiently highly populated for a transition to be observed. Under equilibrium conditions the population 1, of any v" level is related to that of the u" = 0 level by... 

The state may decay by radiative (r) or non-radiative (nr) processes, labelled 5 and 7, respectively, in Figure 9.18. Process 5 is the fluorescence, which forms the laser radiation and the figure shows it terminating in a vibrationally excited level of Sq. The fact that it does so is vital to the dye being usable as an active medium and is a consequence of the Franck-Condon principle (see Section 7.2.5.3). 

The ZEKE-PE process shown in Figure 9.50(c) can be modified as shown by changing the wavenumber Vj of the first laser to excite the molecule to an excited vibrational level of M. Then the Franck-Condon factors for the band system are modified. This can allow... 

Even where the promotion is to a lower vibrational level, one that lies wholly within the 2 curve (such as Vi or V2), the molecule may still cleave. As Figure 7.2 shows, equilibrium distances are greater in excited states than in the ground state. The Franck-Condon principle states that promotion of an electron takes place much faster than a single vibration (the promotion takes... 

As we have reviewed here, the linear region is not fully repulsive, and transitions of the ground-state, linear conformer access vibrationally excited intermolecular levels that are delocalized in the angular coordinate. As depicted in Fig. 1, however, the internuclear distance is significantly longer in the excited state at the linear geometry. Consequently, there is favorable Franck-Condon overlap of the linear conformer with the inner-repulsive wall of the excited-state potential. It is therefore possible for the linear Rg XY conformers to be promoted to the continuum of states just above each Rg - - XY B,v ) dissociation limit. 

It follows from the Franck-Condon principle that in electrochemical redox reactions at metal electrodes, practically only the electrons residing at the highest occupied level of the metal s valence band are involved (i.e., the electrons at the Fermi level). At semiconductor electrodes, the electrons from the bottom of the condnc-tion band or holes from the top of the valence band are involved in the reactions. Under equilibrium conditions, the electrochemical potential of these carriers is eqnal to the electrochemical potential of the electrons in the solution. Hence, mntnal exchange of electrons (an exchange cnrrent) is realized between levels having the same energies. 

Effect of diagonal dynamic disorder (DDD). Fluctuations of the polarization and the local vibrations produce the variation of the positions of the electron energy levels eA(Q) and eB(C ) to meet the requirements of the Franck-Condon principle. 

Due to strong interaction of the reactants with the medium, the influence of the latter may not be reduced only to the widening of the vibrational levels of the proton in the molecules AH and BH. The theory takes into account the Franck-Condon factor determined by the reorganization of the medium during the course of the reaction. 

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