Electron coupling matrix element

For the intersystem crossing T Si, the electronic coupling matrix element (T il so S ) often vanishes. In this case, we have to take into account the higher-order terms in Eq. (3.69), that is,... 

The probability P in equations (61) and (62) may be related to the electronic coupling matrix element through equation (63) by application of the Landau-Zener model ... 

Thus, from equation (63), the magnitude of the electronic coupling matrix element may finally be estimated, leading to values of 21 and 24 meV for EDA and perylene, respectively. That these values are quite reasonable derives from the observation that they correspond to moderately non-adiabatic electron transfer at the ground state (with electronic factors of 2 /(1 + P) - 0.5 and 0.6 with EDA and perylene, respectively). 

Explicit calculation of the electronic coupling matrix element, Hah, is performed by modeling the transition state (Fig. 3) as a supermolecule, [M(H20)6]2+, and optimizing its geometry under the constraint of having an inversion center of symmetry The numerical value of Hab is then obtained from the energy gap between the appropriate molecular orbitals of the supermolecule. 

Figure 1. Potential energy plot of the reactants (precursor complex) and products (successor complex) as a function of nuclear configuration Eth is the barrier for the thermal electron transfer, Eop is the energy for the light-induced electron transfer, and 2HAB is equal to the splitting at the intersection of the surfaces, where HAB is the electronic coupling matrix element. Note that HAB << Eth in the classical model. The circles indicate the relative nuclear configurations of the two reactants of charges +2 and +5 in the precursor complex, optically excited precursor complex, activated complex, and successor complex.

Table III. Estimates of Electronic Coupling Matrix Elements and Adiabatic ty Factors3...

In a semiclassical picture, the rate kda of nonadiabatic charge transfer between a donor d and an acceptor a is determined by the electronic coupling matrix element Vda and the thermally weighted Franck-Condon factor (f C) [25, 26] ... 

Several such approaches for calculating electronic coupling matrix elements Vda have been proven to be useful. Most of them employ a two-state approximation [27, 28] where one assumes that donor and acceptor elec-... 

Comparing matrix elements of simple models of two purine nucleobases with those calculated for WCP dimers (Table 3), one wonders about the effect of pyrimidine nucleobases on the electronic coupling matrix elements of hole transfer in DNA [14, 73]. 

Electronic coupHng matrix elements for DNA-related systems are much more difficult to calculate than the coupHng of hole transfer. First, in the case of an excess electron, the one-electron approximation likely is insufficient and electron correlation is expected to play a crucial role. Second, preliminary results revealed a considerable influence of the basis set on the calculated coupling. Third, an excess electron is expected to be delocalized over several pyrimidine bases this will render the evaluation of V a even more difficult. Thus far, no rehable estimates of electronic coupling matrix elements seem to be available. 

The theory of electron transfer in chemical and biological systems has been discussed by Marcus and many other workers 74 84). Recently, Larson 8l) has discussed the theory of electron transfer in protein and polymer-metal complex structures on the basis of a model first proposed by Marcus. In biological systems, electrons are mediated between redox centers over large distances (1.5 to 3.0 nm). Under non-adiabatic conditions, as the two energy surfaces have little interaction (Fig. 5), the electron transfer reaction does not occur. If there is weak interaction between the two surfaces, a, and a2, the system tends to split into two continuous energy surfaces, A3 and A2, with a small gap A which corresponds to the electronic coupling matrix element. Under such conditions, electron transfer from reductant to oxidant may occur, with the probability (x) given by Eq. (10),... 

The pre-exponential factor A in Eq. 1 is a weak function of the temperature and the reorganization energy, and strongly dependent upon the electronic coupling matrix element V. In the simplest case, V may be assumed to be exponentially dependent upon the through-space donor-acceptor separation r. This yields a distance dependence for electron transfer of ... 

Apart from the selection rules for the electronic coupling matrix element, spin-forbidden and spin-allowed nonradiative transitions are treated completely analogously. Nonradiative transitions caused by spin-orbit interaction are mostly calculated in the basis of pure spin Born-Oppenheimer states. With respect to spin-orbit coupling, this implies a diabatic behavior, meaning that curve crossings may occur in this approach. The nuclear Schrodinger equation is first solved separately for each electronic state, and the rovibronic states are spin-orbit coupled then in a second step. 

From Eq. [239], it is apparent that the size of a particular is not only determined by the magnitude of the electronic coupling matrix element but also by the overlap of the vibrational wave functions v,- and i/. Squared overlap integrals of the type (Xi/, (Q) IXt/ (Q))q 2 are frequently called Franck-Con-don (FC) factors. In contrast to radiative processes, FC factors for nonradiative transitions become particularly unfavorable if two states differing considerably in their electronic energies exhibit similar shapes and equilibrium coordinates of their potential curves. Due to the near-degeneracy requirement, an upper state vibrational wave function, with just a few nodes... 

The following variables are used in Eqs. (1) and (2) HAB is the electronic coupling matrix element that permits ET to occur h is Planck s constant k is Boltzmann s constant T is absolute temperature s is the reorganization energy of the solvent vibrations associated with ET wy is the angular frequency of the quantized, high-energy molecular vibration associated with ET, such that ooy/27r = m in the... 

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