The One-Electron Approximation

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. 

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...

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 ... 

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. 

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 ... 

Finally, the SOC term is a true two-electron operator, which can, however, be approximated by a one-electron operator involving adjusted effective nuclear charges. Several studies have shown that this model operator works fairly well in the case of light atoms, providing results close to those obtained using more refined expressions for the SOC operator [23], The one-electron approximate SOC operator reads ... 

This result is actually valid far beyond the one-electron approximation because in the presence of interaction between the core hole and the surrounding electron cloud the spectral function Ai(e-co) will describe the full core level spectrum with shifted and broadened main lines, satellite lines and continua. 

For a deep core hole in the XPS regime it is quite reasonable to explain the breakdown of the one-electron approximation in terms of interaction between the core hole and the average electronic medium. However, for core holes in the outer shells, Fermi sea correlations can give significant contributions to satellite spectra and for low photoelectron energies interactions between the photoelectron and the residual ion can become important. In the rest of this section several of these aspects will be illustrated. 

For our formal treatment of the fermionic 2-level system we assume that we may describe the behaviour of electrons in the one-electron approximation. Then each electron is represented by a wave function that is independent of the wave functions of other electrons, and the individual wave functions may be linearly superimposed. This picture often proves useful in the context of inorganic semiconductors [2,3]. However, it may be highly questionable in organic and molecular matter, where excitonic [4] and polaronic effects are often predominant [5]. 

In Fig. 10 the calculated band gap energies EG are compared with the experimental emission and reflectivity energies of the E c-polarized transitions (see Sect. E.I.) which are correlated in the limit of the one-electron approximation. Apparently the one-electron model calculations can be used to fit the experimentally observed trends. A similar fit has been obtained in Ref. 83. Further investigations concerning the one-electron band structure calculations are found in Ref. 84 and in a very recent paper85. ... 

Eq. (10) is obviously erroneous because in the one-electron approximation the total energy does not reduce, in fact, to the sum of the orbital energies, i.e. 

Energy bands can be calculated from first principles, without any experimental input. The main approximation required is the one-electron approximation (see Appendix A), which we use throughout this text. Then the two remaining questions are what does one use for the potential and what representation does one use to describe the wave function At present the same essential view of the potential is taken by almost all workers, based upon free-electron exchange and little, if any, modification for correlation. (This is discussed in Appendixes A and... 

C.) The principal differences in different calculations are in the accuracy with which the potentials are determined self-consislently with the charge densities from the states that are being calculated. It may well be that the principal remaining inaccuracies are in use of the one-electron approximation itself, and that little is to be gained from further improvements within that context. 

The remarkable agreement indicates that the one-electron approximation is capable of a very complete and adequate description of these systems if carefully carried through. Even the irregularities in the cohesion near the center of the two scries, which arise from spin polarizations associated with Hand s rule, are rather well given. The only significant discrepancies are in the bulk modulus of the strongly magnetic metals at the center of the iron series. 

A similar general approach was given by Slater (1951) and made variational by Kohn and Sham (1965). In fact, the same theory had been developed earlier as an extension of Fermi-Thomas theory by Gombas (1949). It is the basis of the one-electron approximation contemplated throughout this text, and the resulting equation,... 

To solve this equation the one electron approximation is used. The differences in the treatments of the various authors lie in the differences in their assumptions about orthogonality in the wave functions used and in the use of either a variational or perturbation method to find the minimum energy. 

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