Exchange matrix element

Terms containing the W intermediates no longer contain a factor of The energy-independent, third-order term, Epp (oo), is a Coulomb-exchange matrix element determined by second-order corrections to the density matrix, where... 

The dependence of the exchange matrix element on the distance i between the donor and the acceptor is discussed in Sect. 3. 

The probability of an electron tunneling reaction is directly proportional to the square of the exchange matrix element (see the following section). Therefore, the dependence of the rate constant of an electron tunneling reaction on the distance between the donor and the acceptor and on their mutual orientation is also determined, to a considerable degree, by the dependence of the exchange matrix element on these parameters. 

In the three-dimensional case, the computation of the exchange matrix element can be carried out by the asymptotic method [15]. Without dwelling upon the details of the calculation, we quote the final result... 

Along with the harmonic approximation, the Franck-Condon approximation is usually also used in calculating the probability of electron tunneling. In accordance with the latter approximation, the exchange matrix element V(q) in eqn. (20) is factorized outside the integral sign... 

It is also interesting to estimate the maximum value of the frequency factor in the case of purely quantum nuclear motion. This can be done with the help of the formula W 2nV2Sp, where V2 exp(—2yR) is the exchange matrix element, S is the Franck-Condon factor, p 1 jco is the density of the vibrational levels, and co 1000 cm-1 is the characteristic vibrational frequency of the nuclei. In the atomic unit system, the multiplier 2np has the order 103 and the atomic unit of frequency is 4.13 x 1016s-1 consequently, in the usual unit system, the frequency factor is of the order 4 x 1019Ss-1. The frequency factor reaches its maximum value when S 1. Thus, in the case of purely quantum nuclear motion, the maximum value of the frequency factor is also 1019-102°s-1. 

Direct and superexchange electron tunneling between adjacent and remote sites of plastocyanins were considered in Ref. [326], Exchange matrix elements for direct... 

The coordinate representation of the two-electron potential in the exchange matrix elements of (7.35) is... 

For a closed-shell target the LS-coupling case again simplifies. For the orbital the spin integrations give factors (vo vo)(v V <) for the space-direct matrix element and —(vo v )(v vo) for the space-exchange matrix element. Since is equal to v(, for only half the orbitals we have for space-orbitals a... 

Electron-Exchange Matrix Element VfRK The electron-exchange matrix element depends on the center-to-center distance R of electron donor and acceptor and is usually given by (8,43) ... 

Maximal Rate Constant. Ic ay, and Maximal Electron-Exchange Matrix Element. Vjpax Non-Bridaed ir-Svstems. The maximal rate constant of nonadiabatic electron transfer in the medium temperature regime is given by (43)... 

In the following we would like to speculate on the maximal rate constant and maximal electron-exchange matrix element possible in pure through-vacuum electron transfer of organic molecules in the face-to-face geometry at van-der-Waals contact, and on a potential adiediatic-to-nonadied)atic transition. The Landau-Zenner parameter used to characterize the... 

Exchange interaction results from the overlap of the wave-functions of S and A, and is consequently a very short interaction falling off exponentially with the distance between the ions. If the ions S and A have only a single active electron each, the initial and final states S A> and ISA > must be antisymmetrized in the coordinates of the two electrons, and the exchange matrix elements will have the expression... 

For the general case where S, S, A, A are many electron states the exchange matrix element is... 

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