Electronic model, matrix elements

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 ... 

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.

Both the electronic couphng matrix element and the outer-sphere component of the nuclear reorientation parameter are thought to vary with donor-acceptor separation and orientation [29, 49]. It has been shown in studies of Os and Ru-ammines bridged by polyproline spacers that the distance dependence of X can be greater than that of [50]. Dielectric continuum models of solvent reorganization predict that Xg will increase with... 

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]. 

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),... 

N electron/hole excitations is characterized by the set of excitation energies Cfe P2 where the index k labels the quantum number of excitation p. k labels its quantum number. Assuming a simple model for electron tunneling matrix elements [7]... 

A hybrid approach to the electronic coupling problem in proteins has been introduced by Siddarth and Marcus (10-12). In this method, a pathway-searching algorithm is combined with extended Hiickel (EH) calculations on a subset of amino acids in the protein. As with the TP model, a search of the protein structure for important residues reduces the complexity of the electronic structure problem. These residues are then used as a basis in an EH calculation of the electronic coupling matrix element TDA. The method is not expected to produce highly accurate absolute values of TDA, but it is expected to describe relative values of this matrix element. 

The electronic coupling matrix element (//ab) reflects the strength of the interaction between reactants and products at the nuclear configuration of the transition state. Square-barrier ET tunneling models predict that the coupling will... 

Classical, semi-classical, and quantum mechanical procedures have been developed to rationalize and predict the rates of electron transfer. In summary, the observed rate of a self-exchange reaction can be calculated as a function of interatomic distances, force constants, electronic coupling matrix element, and solvent parameters. These model parameters are either calculated, estimated, or determined by experiment, in each case with a corresponding standard deviation. Error propagation immediately demonstrates that calculated rates have error ranges of roughly two orders of magnitude, independent of the level of sophistication in the numerical procedures. 

The encounter complexes exhibit high degrees of charge-transfer [20, 91], and on the basis of absorption and emission data electronic coupling matrix elements for similar complexes (exciplexes) have been determined [205] which are comparable to those of mixed-valence metal complexes commonly used as prototypical models for the bridged-activated complex in inner-sphere electron transfers [2, 26, 197]. Accordingly, we ascribe the unusually high rate constants, their temperature-independence, and their on-Marcus behavior to an inner-sphere electron transfer process [31]. 

The principal effects of the electron-electron interaction can be embodied in a disordered Hubbard model containing three parameters t, the nearest-neighbor electron transfer matrix element assumed constant U, the Coulomb repulsion between electrons of... 

N. Koga, K. Sameshima, K. Morokuma, Ab initio MO calculations of electronic coupling matrix elements on model systems for intramolecular electron transfer, hole transfer, and triplet energy transfer distance dependence and pathway in electron transfer and relationship of triplet energy transfer with electron and hole transfer, J. Phys. Chem., 1993, 97, 13117-13125. 

These models were developed with respect to the metal-centered breathing modes (analogous to combining Limits 1 and 2 so that = ( ml + "mb))- The electron density delocalized, in an adiabatic description of the reaction coordinate, is a minimum at the PE minima and a maximum at the transition state, and consequently the electronic coupling matrix element in the adiabatic description is a function of the reaction coordinate (/ da = aO- This deseription can be transformed into a diabatic description in which the electronic coupling, /da, is independent of the nuclear coordinates. In the diabatic description, the PE is a linear (i.e., dPEjdr f 0), as well as quadratic function of the reaction coordinate and the diabatic potential can be written. 

Here, emphasis is given to the application of few-state models in the description of the near-resonant vacancy exchange between inner shells. It is well known that the quantities relevant for inner-shell electrons may readily be scaled. Therefore, the attempt is made to apply as much as possible analytic functional forms to describe the characteristic quantities of the collision system. In particular, analytic model matrix elements derived from calculations with screened hydrogenic wave functions are applied. Hydrogenic wave functions are suitable for inner shells, since the electrons feel primarily the nuclear Coulomb field of the collision particles. Input for the analytic expressions is the standard information about atomic ionization potentials available in tabulated form. This procedure avoids a fresh numerical calculation for each new collision system. 

The parameters of the model matrix elements used here are adjusted to fit Hartree-Fock energies. Inner-shell electrons are well described by means of the independent particle model within which the Hartree-Fock method yields accurate one-electron (MO) energies. At low incident velocities, where the MO concept is applicable, the vacancy transfer probabilities are sensitively dependent on the one-electron energies involved. 

---