Frank-Condon

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

The absorption and fluorescence spectra of a neat film made of RdB-den-drimer are shown in Fig. 2. The absorption spectrum in visible-wavelength region was similar to that obtained from a solution of RdB with a concentration less than 0.1 mmol/1. Interpretation of the fluorescence in terms of the Frank-Condon mechanism indicated that the core RdB chromophore behaved with a site-isolation effect and had little interaction with the neighboring chro-... 

The rate constants for these relatively short range hole transfer reactions generally decrease exponentially with distance. Yet, characterizing these DNA-mediated reactions with the parameter (3 is a simplification and is certainly inappropriate in cases where the Frank-Condon factor varies with distance (such as has been observed for the acridine photooxidant). Keeping these limitations in mind, however, /i-values for DNA-mediated hole transfer of -0.6-0.7 A-1 have been suggested using several different oxidant-DNA assemblies (Ap, St, Ap radical cation). 

In the quantum mechanical formulation of electron transfer (Atkins, 1984 Closs et al, 1986) as a radiationless transition, the rate of ET is described as the product of the electronic coupling term J2 and the Frank-Condon factor FC, which is weighted with the Boltzmann population of the vibrational energy levels. But Marcus and Sutin (1985) have pointed out that, in the high-temperature limit, this treatment yields the semiclassical expression (9). 

In these terms, the electronic integrals such as (Mge)° and (Mge) a are constrained by the symmetry of the electronic states. While term I involves Frank-Condon overlap integrals, terms II and HI involve integrals of the form i Qa v) in the harmonic approximation, the integrals of this type obey the selection rule v = i + 1. Keeping these considerations in mind, we will next discuss how terms I, II and in contribute to distinct vibrational transitions. 

A natural question is In which temporal order do the reorganization processes and the proper electron transfer take place The answer is given by the Frank-Condon principle, which in this context states First the heavy particles of the inner and outer sphere must assume a suitable intermediate configuration, then the electron is exchanged isoenergetically, and finally the system relaxes to its new equilibrium... 

This equation is in accord with the Frank-Condon principle The nuclei stand still during an electronic transition, so that a good overlap between the nuclear wavefunctions is required for the transition. 

In Sect. 2.1, the electron transfer rate was defined as the Boltzmann average of transition probabilities, which were calculated through time-dependent perturbation theory by using the Born-Oppenheimer and Frank-Condon approx-... 

In the spirit of the adiabatic approximation, the transitions between two vibrational states (belonging to initial and final electronic states) must occur so rapidly that there is no change in the configurational coordinate Q. This is known as the Frank Condon principle and it implies that the transitions between i and / states can be represented by vertical arrows, as shown in Figure 5.12. Let us now assume our system to be at absolute zero temperature (0 K), so that only the phonon level = 0 is populated and all the absorption transitions depart from this phonon ground level to different phonon levels m = 0, 1, 2,... of the excited state. Taking into account Equation (5.25), the absorption probability from the = 0 state to an m state varies as follows ... 

SF-OD level with the basis set composed of a cc-pVTZ basis on carbons and a cc-pVDZ basis on hydrogens). These energies are very close to the MRPT values (26) of 0.72 and 0.83 eV (for the 1 fi and 1 Ai states, respectively). With regard to experiment, the lowest adiabatic state, 1 B, has not been observed in the photoelectron spectrum (40) because of unfavorable Frank-Condon factors. The experimental adiabatic energy gap (including ZPE) between the ground triplet state and the VA state is 0.70 eV. The estimated experimental >s 0.79 eV, which is 0.15 eV lower than the SF-OD estimate. 

Two main liminations have, however, become evident the hrst is that molecules whose gas-phase ionization energy exceeds > 9.5 eV cannot be oxidized by ionized solid Ar. The reason for this limitation is unclear, because the process is exothermic (the ionization energy of solid Ar is 13.9 eV, that of organic molecules in Ar is typically lowered by 1 eV in solid Ar relative to the gas phase ). Perhaps the localization of the spin and charge onto the substrate entails a Frank-Condon barrier that cannot easily be surmounted at 12 K. 

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