Electron matrix elements

There is a clear one-to-one correspondence between the theoretical expressions and the computational implementation in terms of one- and two-electron matrix elements. Implementations of the expressions are therefore facilitated. 

Following Eq. (75), the rate constant for spin conversion may be expressed as a product of the electronic matrix element V and the nuclear Franck-Condon... 

The elements of the matrix H involve the electron matrix elements of the Hamiltonian of the system. The general expression... 

For singlet-singlet transfer, the electronic matrix element is given by... 

In the lowest order (quadratic) approximation for n electronic states of asymmetrical (ABCD) tetraatomics, the electronic matrix elements (60) have the forms [18,152,153] ... 

The integral in large parentheses is over the electronic coordinates r only, and still depends on the nuclear coordinates R. At this stage we invoke the Condon approximation, which is familial from the theory of electronic spectroscopy. Because of the large nuclear mass the wave-functions Xi and Xf are much more strongly localized than the electronic wavefunctions 4>i and some value R, and it suffices to replace the electronic matrix element by its value M at R. So we write ... 

Due to the integral approximations used in the MNDO model, closed-shell Pauli exchange repulsions are not represented in the Hamiltonian, but are only included indirectly, e.g., through the effective atom-pair correction terms to the core-core repulsions [12], To account for Pauli repulsions more properly, the NDDO-based OM1 and OM2 methods [23-25] incorporate orthogonalization terms into the one-center or the one- and two-center one-electron matrix elements, respectively. Similar correction terms have also been used at the INDO level [27-31] and probably contribute to the success of methods such as MSINDO [29-31],... 

Hinze, J. (ed.) (1981), The Unitary Group for the Evaluation of Electronic Matrix Elements, Springer-Verlag, Berlin. 

One expects the impact of the electronic matrix element, eqs 1 and 2, on electron-transfer reactions to be manifested in a variation in the reaction rate constant with (1) donor-acceptor separation (2) changes in spin multiplicity between reactants and products (3) differences in donor and acceptor orbital symmetry etc. However, simple electron-transfer reactions tend to be dominated by Franck-Condon factors over most of the normally accessible temperature range. Even for outer-... 

Actually, the separation of donor and acceptor contributions to the electronic matrix element is not likely to be simple. One might define the resonance exchange reactions,... 

Specific donor-acceptor charge transfer interactions can lead to a relatively large numerical value of the electronic matrix element, possibly attributable to an increase in V, and, thus, to larger rate constants than those predicted by distance variations alone. 

The general approach is illustrated in detail for the case of aqueous ferrous and ferric ions, and the calculated rate constant and activation parameters are found to be in good agreement with the available experimental data. The formalisms we have employed in studying such complicated condensed phase processes necessarily rely on numerous approximations. Furthermore, some empirical data have been used in characterizing the solvated ions. We emphasize, nevertheless, that (1) none of the parameters were obtained from kinetic data, and (2) this is, as far as we are aware, the first such theoretical determination to be based on fully Ab initio electronic matrix elements, obtained from large scale molecular orbital (MO) calculations. A molecular orbital study of the analogous hexaaquo chromium system has been carried out by Hush, but the calculations were of an approximate, semi-empirical nature, based in part on experi-... 

Charge Localized vs. Delocalized Wavefunctions. In the spirit of the Condon approximation discussed above, we do not include the full dependence of the purely electronic matrix elements on q n, but rather evaluate them where the diabatic curves... 

Definition (5) shows that TJb which is sometimes called the electronic matrix element , represents the residual interaction resulting from the overlap of the wavefunctions v /j and These functions, which describe the initial and final electronic states of the whole system, respectively, depend closely on the nature of the redox centers and of the medium, so that reliable values of T are very difiicult to obtain from ab initio calculations in complex systems. For that reason, some authors have proposed determining T b semi-empirically by using the results of spectroscopic measurements. We begin by a brief presentation of... 

Here the Vkk represent the electronic matrix elements of the diabatic potential matrix and T denotes the kinetic-energy operator of the nuclei. 

Electronic contributions to gr(7 ) and gy(R) relate to nonadiabatic rotational and vibrational effects, respectively, as electronic matrix elements [9], whereas a nuclear contribution, the same in each case. 

Table 15.4. Comparison of some one- and two-electron matrix elements for pure and SCVB 2pz orbitals. All energies are in hartrees.

The parameter A, tensors A and G, molecular dipole moment D and the constants Wi are expressed in terms of one-electron matrix elements concrete expressions for the above parameters can be found in [90], and for Wd and Ay they are also given in the next sections. The results of the calculations are presented in Table 2. 

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