The One-Electron Model

Before getting started with second quantization it appears to be useful to summarize some important features of the usual first quantized theory. As it was already noted, the following treatment of second quantization is intended 

The basic concept of standard many-electron theory is that of the Slater determinant  

Owing to the mathematical properties of determinants, D is antisymmetric in the coordinates of electrons. This fulfills the Pauli principle for electrons. Note that in general the Pauli principle requires the wave function to be symmetric in the variables of particles of integer spin (called bosons) and antis)mimetric for particles of half-integer spin (called fermions) such as electrons, for example. 

Any wave function of a many-electron system which is composed of one-electron functions, should be a linear combination of Slater determinants to fulfill the Pauli principle. 

Matrix elements of physical observables are to be calculated over such determinantal wave functions. The rules for the values of matrix elements of different operators between two determinants, and D2  

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Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

For approximate wavefunctions, however, the various formulations give rise to different theoretical predictions. This has been demonstrated in detail, for example, by Hush and Williams (31) for large aromatic systems. Thus, when we wish to obtain exact values of J, we must be very careful in deciding which formalism to use. A final point here is that the one-electron model does not take into account configuration interaction. Calculations for relatively simple systems would be useful here. 

The vibronic Hamiltonian in the one-electron model is H = Hq + V. The kernels of these operators are... 

One-electron picture of molecular electronic structure provides electronic wavefunction, electronic levels, and ionization potentials. The one-electron model gives a concept of chemical bonding and stimulates experimental tests and predictions. In this picture, orbital energies are equal to ionization potentials and electron affinities. The most systematic approach to calculate these quantities is based on the Hartree-Fock molecular orbital theory that includes many of necessary criteria but very often fails in qualitative and quantitative descriptions of experimental observations. 

In the one-electron transition model it is assumed that only one core electron is excited to an unfilled state present in the initial, unperturbed solid. The remaining electrons are assumed to be unaffected, remaining frozen in their original states. In essence the one-electron model describes the potential seen by the final-state electrons as nonoverlapping spherically symmetric spin-independent potentials (muffin-tin scatterers) centered around an atom from which the X-ray cross section from a deep core level of an atom to final states above the Fermi level can in principle be calculated for any energy above threshold. 

Absorption of the X-ray makes two particles in the solid the hole in the core level and the extra electron in the conduction band. After they are created, the hole and the electron can interact with each other, which is an exciton process. Many-body corrections to the one-electron picture, including relaxation of the valence electrons in response to the core-hole and excited-electron-core-hole interaction, alter the one-electron picture and play a role in some parts of the absorption spectrum. Mahan (179-181) has predicted enhanced absorption to occur over and above that of the one-electron theory near an edge on the basis of core-hole-electron interaction. Contributions of many-body effects are usually invoked in case unambiguous discrepancies between experiment and the one-electron model theory cannot be explained otherwise. Final-state effects may considerably alter the position and strength of features associated with the band structure. 

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. ... 

For example, the one-electron models incorrectly predict (even at a qualitative level) the Knight shifts in and NMR spectra of ammoniated electron, and solvated electrons in amines (Sec. 4.1). The same problem arises in the explanation of magnetic (hyperfine) parameters obtained from ESEEM spectra of trapped (hydrated) electrons in low-temperature alkaline ices. The recent resonance Raman spectra of also appear to be incompatible with the one-... 

Electron momentum spectroscopy can therefore be considered in terms of (q/ 0). For the one-electron model of the target... 

If the perimeter is uncharged, n = 4N + 2 and the configurations d>j and I>4 belong to the same irreducible representation b = e ,2 — e ,2 of the point group C . They differ in two spin orbitals and therefore interact only through the electron repulsion part iU,j) of the Hamiltonian. Consequently, the one-electron model fails and first-order configuration interaction has to be taken into account. f Q = , f d>4> is the interaction matrix element between configurations ] and d>4, the Cl matrix reads... 

There are following steps to get a correct estimation of the spectral peak energy within the one-electron model,... 

The importance of the one-electron model relies upon two facts (i) the approximation represented by Eq. (1) is usually a good one, and (ii) assuming a complete set of one-electron orbitals q>i one may develop the exact wave function in terms of (an infinite number) of determinants of the form of Eq. (1). 

It has not proven possible to find suffieiently accurate approximations for E [/-], so that a single Euler-Langrage equation for the density would yield an accurate density and energy. In particular, it is very difficult to find a good density functional for the kinetic energy. Therefore applications of DF theory in chemistry invariably use the one-electron model of Kohn and Sham (KS). They... 

Experience with the one-electron case the one-electron model potentials work. 

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