One Electron Approximation

The reason the Schrodinger equation could be solved exactly for the hydrogen atom was that the potential V was a simple function of r oiUy that describes the attraction between the single electron and the nucleus. In a system with many electrons, the potential function would have to include the repulsive interaction between an electron and all the other electrons. This is an extremely difficult calculation which can only be carried out numerically using an iterative scheme. A guess is made for the wavefunction of each electron from the hydrogen wavefunction. Then the trial wavefunctions are adjusted 

Different quantum chemical approaches can be invoked to calculate the electronic couplings. In many cases one can reliably estimate electron-transfer matrix elements on the basis of a one-electron approximation [27-29]. 

It is also practical to invoke a one-electron approximation in the FCD method [41] when one estimates donor and acceptor charges. Thus, one approximates the fragment charges of the radical cations [(G C),(AT)] and [(GC),(A+T)] via the corresponding Mulliken populations of the HOMO and HOMO-1 of the neutral dimer. Then, the charge on fragment / in [(G+C),(AT)] is 

Sy is the overlap of atomic orbitals i and j i runs over atomic orbitals (AOs) associated with the selected fragment / while j runs over all AOs. The fragment charges of the second adiabatic state are calculated analogously using the coefficients c homo-i of orbital HOMO-1 in place of c homo in Eq. 20. By the same token, the quantity qmnif) can be defined as 

Similarly, to apply the GMH method, Eq. 11, one calculates the difference of the adiabatic dipole moments and the transition moment /Un as follows  

are the matrix elements of the dipole operator defined for AOs i and j. 

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We shall call this basis also linear because in the one-electron approximation at p 0 the functions (26) become... 

In the framework of the one-electron approximation, the second term in braces in Eq. (9) can be written as... 

In electrocatalysis, the reactants are in contact with the electrode, and electronic interactions are strong. Therefore, the one-electron approximation is no longer justified at least two spin states on a valence orbital must be considered. Further, the form of the bond Hamiltonian (2.12) is not satisfactory, since it simply switches between two electronic states. This approach becomes impractical with two spin states in one orbital also, it has an ad hoc nature, which is not satisfactory. 

A quantitative consideration on the origin of the EFG should be based on reliable results from molecular orbital or DPT calculations, as pointed out in detail in Chap. 5. For a qualitative discussion, however, it will suffice to use the easy-to-handle one-electron approximation of the crystal field model. In this framework, it is easy to realize that in nickel(II) complexes of Oh and symmetry and in tetragonally distorted octahedral nickel(II) complexes, no valence electron contribution to the EFG should be expected (cf. Fig. 7.7 and Table 4.2). A temperature-dependent valence electron contribution is to be expected in distorted tetrahedral nickel(n) complexes for tetragonal distortion, e.g., Fzz = (4/7)e(r )3 for com-... 

Fig. 7.7 Schematic diagrams for common electron configurations of Ni " complexes in the one-electron approximation. The resulting valence electron contributions V z are obtained from Table 4.2...

Since the form of the electronic wave functions depends also on the coordinate p (in the usual, parametric way), the matrix elements (21) are functions of it too. Thus it looks at first sight as if a lot of cumbersome computations of derivatives of the electronic wave functions have to be carried out. In this case, however, nature was merciful the matrix elements in (21) enter the Hamiltonian matrix weighted with the rotational constant A, which tends to infinity when the molecule reaches linear geometry. This means that only the form of the wave functions, that is, of the matrix elements in (21), in the p —> 0 limit are really needed. In the above mentioned one-electron approximation... 

If we fix the cores and assume that each electron moves in the average potential generated by the cores and other electrons, this is the so-called Hartree or one-electron approximation. For this model we arrive at a relatively simple expression for the Hamiltonian for determining the electronic band structure and wave functions. The one-electron Hamiltonian is... 

This is the one-electron approximation, also called the independent electron approximation and hence the le superscript, where a Hamiltonian Hq of an A e-electron system can be expressed as the sum of Ne one-electron Hamiltonians and the Schrodinger equation to be solved becomes ... 

Thus, the electronic couplings for hole transfer in cation radical systems can be treated using computational results for the corresponding neutral species. In this way, all parameters needed for the minimum sphtting method as well as the FCD and GMH schemes can be efficiently estimated in a one-electron approximation, as given by the equations of this section. 

To study the effect of the energy gap between donor and acceptor, we carried out HF/6-31G calculations on [(GC),(AT)] [41] and compared various procedures for determining the coupling elements within the one-electron approximation. In the complex [(GC),(AT)], guanine and adenine act as donor and acceptor sites. The donor-acceptor energy gap can be modulated by ap-... 

If one attempts an accurate description of the coupling between neighboring pairs, one needs to answer several general questions. How accurate is the one-electron approximation How essential are the effects of electron corre-... 

Electronic coupHng matrix elements for DNA-related systems are much more difficult to calculate than the coupHng of hole transfer. First, in the case of an excess electron, the one-electron approximation likely is insufficient and electron correlation is expected to play a crucial role. Second, preliminary results revealed a considerable influence of the basis set on the calculated coupling. Third, an excess electron is expected to be delocalized over several pyrimidine bases this will render the evaluation of V a even more difficult. Thus far, no rehable estimates of electronic coupling matrix elements seem to be available. 

Within the adiabatic and one-electron approximations, when electron spin is neglected, electron states in crystals are described by the eigenfunctions and their corresponding eigenvalues, which are the solutions of the Schrodinger equation,... 

In 1954 Sandorfyand Daudel15) published their C" approximation, a one-electron approximation which employs a linear combination of sp3 orbitals and borrows most of its simplifying features from the Htickel theory. The originality of this method lies essentially in the introduction of a resonance integral m/J between sp3 orbitals of the same carbon atom. Sandorfy 16> showed that the inductive effect due to a heteroatom can be reproduced by such a calculation. 

Our conclusion today is that ligand-field theory is essentially the one-electron approximation used for the classification of the energy levels of inorganic chromophores. 

In the second step, the crystal-field potential is introduced to the Hamiltonian. The potential in the one-electron approximation can be written as follows ... 

Improvement of the crystal-field splitting calculation has been achieved by two different approaches. On one hand the basis set of wavefunctions was extended to include also excited configurations. This approach will be dealt with in sect. 4.4.6. On the other hand, the one-electron approximation has been relaxed to take into account electron correlation effects. The original formulation of the correlation crystal-field parameterization has been proposed by Bishton and Newman (1968). Judd (1977) and Reid (1987) redefined the operators to ensure their mutual orthogonality ... 

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