Frank-Condon transitions

The minimal active space needed to describe the electronic structure of the NDI moiety includes the five occupied and five unoccupied 7t-orbitals of the naphthalene core and four lone pair orbitals of the carbonyl groups. The 57t[4n]57t active space contains 14 electrons and electronic transitions arise from seven states. Only two of the seven states, 1 1 B2u and 1 B3U, show transitions in the region of interest between 320 and 420 nm. Other transitions have no effect on the bands in this region and hence were not considered. The main features in the experimental absorbance spectrum were reproduced using the most intense Frank-Condon transitions (Fig. 15). The calculated spectrum (dashed lines) showed a red shift of 9 nm relative to the experiment, which may be due to the representation of each transition by only two charges, and also due to the neglect of other transitions. 

Fig. 1 Qualitative cross section through the potential energy surface, along JT active vibration Qa, Definition of the JT parameters - the JT stabilisation energy, Eji, the warping barrier, A, the JT radius, iJjx, the energy of the vertical Frank-Condon transition, fc...

Frank-Condon transition) define the potential energy surface. The meaning of the parameters is clear - energy stabilization due to the JT effect is given by the value of Ejy (or alternatively by E c = Eji), and direction and magnitude of the distortion by the Rjy. 

The use of FOISTs, even for this more complicated system, leads to requisite alterations in the spatial profiles, so that they peak at the internuclear distances required to facilitate Frank-Condon transitions to the appropriate portion of the excited state potential energy curves, enhancing photodissociation ont of the desired channel. 

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. 

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 ... 

Corresponding to the Frank-Condon principle is an associated concept called the Frank-Condon factors. Thus, when an electronic transition occurs from the vibrational levels of a lower vibrational state to the corresponding vibrational levels of a higher electronic state, there are various intensities of transition, depending on the vibrational states to which a transition is made. 

Development of the Frank-Condon principle in quantum mechanical terms (involving a transition dipole moment14) allows a calculation of the intensities referred to in terms of a series of Frank-Condon factors by which expressions for the transition probabilities are multiplied to obtain a net transition probability from one level to another for an electron-transfer process. 

There are two different temperature regimes of diffusive behavior they are analogous to those described by Holstein [1959] for polaron motion. At the lowest temperatures, coherent motion takes place in which the lattice oscillations are not excited transitions in which the phonon occupation numbers are not changed are dominant. The Frank-Condon factor is described by (2.51), and for the resonant case one has in the Debye model ... 

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