Two-electron wave

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

Thus a single two-electron wave is observed and only one product, the alcohol, can be isolated. Finally, at high pH neither the ketone nor the radical anion are protonated by this basic medium and it is not until the dianion, formed by successive electron transfers, that protonation occurs. 

After addition of an excess of a proton donor, a single two-electron wave is observed. This is because the radical anion protonates to give a radical which is more readily reduced than the parent hydrocarbon... 

The Hartree-Fock description of the hydrogen molecule requires two spinorbitals, which are used to build the single-determinant two-electron wave function. In the Restricted Hartree-Fock method (RHF) these two spinorbitals are created from the same spatial... 

Two well-separated Nernstian waves are observed for large values of AE°. As AE° decreases, the two waves come closer to each other and tend to merge. Upon moving to negative values, the response tends asymptotically toward a limiting single two-electron wave, the equation of which is obtained from equation (1.57) by making AF° = —oo ... 

Within the range of small values of AE° that give rise to a single two-electron wave, it is interesting to note that a wave twice the one-electron... 

Within the range of small values of Athat gives rise to a single two-electron wave, it is interesting to note that a wave twice the one-electron Nernstian wave is not obtained for A= 0 but rather for A ° = ( TZT/F)ln4 (see Section 5.1.3). As shown in the next section, this is precisely the value of A ° that corresponds to a molecule containing two identical and independent reducible or oxidizable groups. 

Figure 3.28. Cyclovoltammogram for the hole transporting material Spiro-TAD (36a). The oxidation proceeds in two one-electron and one formal two-electron wave. Solvent Dichloromethane/TBAHFP 0.1 M, Scan rate lOOmV/s. 

Monoactivated and diactivated substituted 6//-l,3-thiazines can be reduced in a sbghtly acidic hydroorganic medium [255-258], In the case of thi-azines monoactivated at carbon 5, an electrolysis performed at the plateau of the two-electron wave gives the 2,3-dihydro compounds (Scheme 141). 

Reduction of alkyl and benzyl halides proceeds in two one-electron addition steps. The first detectable product is the alkyl or benzyl radical and this is reduced further to the carbanion. Some alkyl iodides show two polarographic waves corresponding to the two steps. Alkyl bromides show only one two-electron wave and alkyl chlorides are not reducible in the available potential window. Benzyl halides also show only one wave and benzyl chlorides are reducible in the available potential range. Reduction potentials measured in dimethylformamide are collected in... 

Since their discovery in 1866, it has been known that sulphoxides are reducible by zinc and acid to the conesponding sulphide [63], fhe equivalent electrochemical process cannot be characterised because sulphoxides also decrease the hydrogen overpotential [64], Dialkyl sulphoxides are not reduced in absence of protons and dimethyl sulphoxide is used as a solvent for electrochemical reduction. Phenyl methyl sulphoxide gives a single two-electron wave on polarography in both ethanol (E./, = -2.17 V vs. see) and dimethylformamide (E./, = -2.32 V vs. see), forming phenyl methyl sulphide [65],... 

Schiff bases derived from o-phenylenediamine and from 2-hydroxyaniline show a two-electron wave on cyclic voltammetry in acetonitrile and further waves at more positive potentials due to oxidation of the product from the first wave. Preparative scale reaction at the potential of the first wave leads to formation of a heterocyclic ring product [45]. 

At higher pH values the two waves merge to a single two-electron wave which divides into separate single-electron waves at pH 13. The first wave of these waves... 

Secondary nitramines and nitrosamines both show a two-electron wave on pola-rography in aprotic solvents. Addition of the first electron is synchronous with cleavage of the nitrogen-nitrogen bond in both classes of compounds. Nitramines [114] afford nitrite ion and a nitrogen radical which accepts the second electron ... 

In acidic aqueous buffers, secondary nitramines [116] show a six-electron po-larographic wave which is converted into two two-electron waves above pH 5 (Fig. 11.4). Nitrosamines [117,118] show a four-electron wave in acid solution and this becomes a single two-electron wave above pH 5 (Fig, 11,5). Above pH 5 the halfwave potentials are independent of pH. Primary nitramines are relatively acidic. They show a six-electron wave in acidic aqueous buffers, the height of which falls to zero around the pKg value for the nitramine because the anion is not reducible [119],... 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

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