Two-electron potential

Radical cations can be generated by many chemical oxidizing reagents, including Brpnsted and Lewis acids, the halogens, peroxide anions or radical anions, metal ions or oxides, nitrosonium and dioxygenyl ions, stable aminium radical cations, semiconductor surfaces, and suitable zeolites. In principle, it is possible to choose a reagent with a one-electron redox potential sufficient for oxidation-reduction, and a two-electron potential insufficient for oxidation-reduction of the radical ion. 

For the system under consideration, with only two electronic potential-energy surfaces, the conservation of population implies that... 

The diagram in Fig. 1 is drawn in such a manner that the lines leaving the vertex (which is marked by a dot) correspond to creation operators, and the entering lines to annihilation operators. The dashed line connecting the vertices shall be called the interaction line. This line stands for the two-electron potential l/r,y. Then we can schematically rewrite Eq. (74) in the form... 

One-electron potential (NHE) in water with oxo and aquo ligands, unless indicated otherwise. Two-electron potential. MeCN. 

The direct reduced potential matrix element in j j coupling is calculated by substituting the full potential matrix elements into (7.37) with the continuum orbitals given by (7.46), the bound orbitals by (7.49), the two-electron potential by (7.60) and the reduced density-matrix element by (7.59). 

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

The factorisation is characteristic also of the plane-wave Born approximation, which is (10.30) with distorted waves replaced by plane waves. Here the two-electron T-matrix element is replaced by the two-electron potential matrix element (3.41). 

In the previous sections, we demonstrated how a space-dependent coupling between two electronic potentials influences the transition probabilities between molecular energy levels. In the following, we ask whether the space-dependence of the coupling can be used to steer molecular transitions. There-... 

The zero-order Hamiltonian H0 corresponds to the Fock operator, whereas the fluctuation potential V represents the difference between the full, instantaneous two-electron potential and the averaged SCF potential of the Hartree-Fock model ... 

The correction B(i,j) to the Coulomb potential is treated as a perturbation of the zero-order Hamiltonian, and may include relaxation effects, correlations, quantum electrodynamic corrections and the relativistic retardation of the two-electron potential. 

D14.2 The Franck-Condon principle states that because electrons are so much lighter than nuclei, an electronic transition occurs so rapidly compared to vibrational motions that the internuclear distance is relatively unchanged as a resu It of the transition. This implies that the most probable transitions vf <— vj are vertical. This vertical line will, however, intersect any number of vibrational levels Vf in the upper electronic state. Hence transitions to many vibrational states of the excited state will occur with transition probabilities proportional to the Frank-Condon factors which are in turn proportional to the overlap integral of the wavefunctions of the initial and final vibrational states. A vibrational progression is observed, the shape of which is determined by the relative horizontal positions of the two electronic potential energy curves. The most probable transitions are those to excited vibrational states with wavefunctions having a large amplitude at the internuclear position Re. 

The calculation of the two-electron potential in equation 5.39 involves the calculation of the charge density, p(r), from the Is orbital approximated to in equation 5.40. Since the two Is spin orbitals are occupied in the helium atom this charge density is... 

Then the components of the two-electron potential are constructed in cells H 10 to H 15 following exactly the prescription in equation 5.33, we have... 

Figure 5.9 Detail from the spreadsheet, fig5-9.xls, showing the renormalization worksheet, individual components of the two-electron potential for the basis set written in the form of equation 1.26 and, in the third diagram, the remodelling of the worksheet oneel from fig4-18.xls.

Use the ij kl) as a template to generate the two-electron potential worksheets over the Gaussian primitives for the Is and 2s linear combinations used in the calculation. 

Link the individual contributions to the two-electron potential to the appropriate cells in hfs l... 

The calculation for the hydrogen molecule ion makes plain the basic problem with attempting to extend the numerical approach to molecular calculations. Even, for the simplest molecular species, we have to build a spreadsheet with multiple worksheets devoted to the calculation of the components of the potential and kinetic energies and, this, without any consideration of the extra need in neutral species to evaluate the two-electron potential term. It is appropriate, therefore, to extend the direct approach of Chapter 5, for the calculation of the two-electron integral components for helium, to the calculation of all the integrals in any calculation. 

Figure 6.8a The two-electron potential term worksheets for the dihydrogen calculation note how the colour coding of the entries for the exponents and coefficients of the Gaussian primitives identify individual ij kl) integrals.

Let us consider this potential. Suppose electron 2 occupies Xb- two-electron potential rii felt by electron 1 and associated with the instantaneous... 

To derive the transformed differential equation for the two-electron potential... 

State in the reverse direction. These results are the first dynamical experiments to extract the symmetries of the two electronic potentials involved in the curve crossing, and they demonstrate a powerful new way to investigate electronic curve crossing phenomena in general. 

Loss of a proton from the semiquinone radical HaQ+ takes place after the ratedetermining step, kx. Consideration of the rate constants (Table 3) for these reactions in terms of Marcus theory suggests that the one-electron redox potential of HaQ lies around 0.38 V above the value of the two-electron potential regardless of substituents on the hydroquinone. A similar mechanism was noted in the oxidation of a series of catechols by [Fe(phen)3] and its derivatives. 

As an example, values of the one- and two-electron reduction potential of pheophytin-a were calculated in N-N-dimethylformamide by a combination of quantum mechanical, statistical mechanical, and polarizable continuum methods (Mehta and Datta 2008). Two different computational methods gave -0.92 and -1.03 V for the one-electron, while -1.34 and -1.30 V for the two-electron reduction potential values, respectively. The observed one-and two-electron potentials are -0.90 and -1.25 V, respectively. Solution of the Poisson-Boltzmann equation gave for the reduction potential of pheophytin-a within the thylakoid membrane -0.58 V, which is in good agreement with the reported value of -0.61 V. 

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