Quantum Franck-Condon state

The CT complexes are characterized by a new absorption band which is usually red-shifted as compared to local excitation bands [47-49], According to the Mulliken formulation the CT-exdtation corresponds to an electronic transition from the HOMO of the donor to the LUMO of the acceptor, i.e. it accomplishes full electron transfer [47], The transition is instantaneous, producing two intermediates (ions) in a direct contact but in a non-equilibrium, Franck-Condon state. The relaxation of the pair competes with BET, diminishing the quantum yield for ion generation [49], This process is believed to take... 

The assignment of the observed fast kinetic rate to trans-cis isomerization is strongly supported by our experimental data, which show that the time constant is related to the viscosity by approximately The Forster and Hoffman modeP was developed originally to explain the Q= relationship where Q is the fluorescence quantum yield and 17 is the viscosity of the medium for triphenylmethane dyes. In addition, it was predicted that the fluorescence lifetime, r, should follow a similar relationship t = c if According to this model, absorption of light produces a vertically excited Franck-Condon state with the phenyl rings still at a ground state equilibrium... 

Next, there are several indications that the photoreactive species is not in a Franck-Condon state, but rather it is in a thermally equilibrated excited state. One indication is that quantum yields as well as the nature of the photoreaction do not vary appreciably as the irradiating wavelength traverses the width of a ligand field band (although variations may occur on going from one band to another) (13, 14, 15). It appears that a common reactive state is reached, regardless of the degree of vibrational excitation of the initially produced Franck-Condon state. The simplest explanation is that this common state is a thexi state. 

A scheme for the treatment of the solvent effects on the electronic absorption spectra in solution had been proposed in the framework of the electrostatic SCRF model and quantum chemical configuration interaction (Cl) method. Within this approach, the absorption of the light by chromophoric molecules was considered as an instantaneous process. Tliere-fore, during the photon absorption no change in the solvent orientational polarization was expected. Only the electronic polarization of solvent would respond to the changed electron density of the solute molecule in its excited (Franck-Condon) state. Consequently, the solvent orientation for the excited state remains the same as it was for the ground state, the solvent electronic polarization, however, must reflect the excited state dipole and other electric moments of the molecule. Considering the SCRF Hamiltonian... 

In terms of classical mechanics, the Franck-Condon principle is the approximation that an electronic transition is most likely to occur without changes in the positions of the nuclei in the molecular entity and its environment. The resulting state is called a Franck-Condon state, and the transition involved, a vertical transition. The quantum mechanical formulation of this principle is that the intensity of a vibronic transition is proportional to the square of the overlap integral between the vibrational wavefunctions of the two states that are involved in the transition [55]. This principle was readily accepted and from the beginning quoted with the names of the two scientists [56-58]. 

The emission spectrum observed by high resolution spectroscopy for the A - X vibrational bands [4] has been very well reproduced theoretically for several low-lying vibrational quantum numbers and the spectrum for the A - A n vibrational bands has been theoretically derived for low vibrational quantum numbers to be subjected to further experimental analysis [8]. Related Franck-Condon factors for the latter and former transition bands [8] have also been derived and compared favourably with semi-empirical calculations [25] performed for the former transition bands. Pure rotational, vibrationm and rovibrational transitions appear to be the largest for the X ground state followed by those... 

To simulate the vibrational progression, we obtain the Franck—Condon factors using the two-dimensional array method in ref 64. We consider 1 vibrational quantum v = 0) from the EC stationary point and 21 vibrational quanta (z/ = 0, 1,. .., 20) from the GC stationary point. The Franck— Condon factors are then calculated for every permutation up to 21 quanta over the vibrational modes. It is necessary in order to get all Franck—Condon factors of the EC stationary point with respect to each three alg vibrational state (Figure 6) of the GC to sum to one. One obtains a qualitative agreement between the calculated and the experimental emission profiles (Figure... 

Flowever, there is a trade-off in using near-IR emissive lanthanides, in that luminescence lifetimes are shorter, and quantum yields lower, compared to complexes of Tb and Eu. This arises because the near-IR emissive lanthanides are quenched by lower harmonics of the O-H oscillator, increasing the Franck-Condon overlap with the metal excited state. For neodymium, matters are further complicated by the manifold of available metal-centered excited states, which leads to particularly effective quenching by C-H oscillators. Thus, complexes in which there are few C-H oscillators close to the metal are desirable if the luminescence lifetime is to be optimized (e.g. 44).76 97-101... 

FIGURE 4.1 Illustration of adiabatic and vertical ionization potentials. Adiabatic I.P. refers to the energy difference between the lowest quantum states of the molecule and its positive ion. Often, Franck-Condon (vertical) transitions lead to a higher value, the vertical ionization potential. 

In principle, refined and relatively reliable quantum-theoretical methods are available for the calculation of the energy change associated with the process of equation 2. They take into account the changes in geometry, in electron distribution and in electron correlation which accompany the transition M(1 fio) — M+ (2 P/-), and also vibronic interactions between the radical cation states. Such sophisticated treatments yield not only reliable predictions for the different ionization energies 7 , 77 or 7 , but also rather precise Franck-Condon envelopes for the individual bands in the PE spectrum. However, the computational expenditure of these methods still limits their application to smaller molecules. We shall mention them later in connection with examples where such treatments are required. 

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. 

In the quantum mechanical description (in continuation of Box 2.2), the wavefunction can be described by the product of an electronic wavefunction VP and a vibrational wavefunction / (the rotational contribution can be neglected), so that the probability of transition between an initial state defined by ViXa and a final state defined by TQ/b is proportional to electron coordinates, this expression can be rewritten as the product of two terms < f i M vP2> 2 Franck-Condon factor. Qualitatively, the transition occurs from the lowest vibrational state of the ground state to the vibrational state of the excited state that it most resembles in terms of vibrational wavefunction. 

The basic theory of the kinetics of charge-transfer reactions is that the electron transfer is most probable when the energy levels of the initial and final states of the system coincide [5] following the Franck-Condon principle. Thus, the efficiency of the redox reaction processes is primarily controlled by the energy overlap between the quantum states in the energy bands of the semiconductor and the donor and acceptor levels of the reactants in the electrolyte (Fig. 1). In the ideal case, the anodic current density is given by the... 

The ( )j(q)Qm (Q) is a set of linearly independent functions the Qm(Q) functions are not orthogonal in Q-space for arbitrary electronic states the overlap integrals Jd Q Qm(Q) Q m (Q) are the well known Franck-Condon factors. The hypothesis is that an arbitrary quantum molecular state is given by the linear superposition jus as in the general case ... 

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