Electrons single-particle picture

In a series of calculations on ethylene, butadiene and hexatriene, Deleuze and co-workers [105] showed that the ADC(3) method can provide a very accurate picture of the electronic processes associated with ionisation in the valence region. Poly(acetylene) has a large feature above 21 eV, which was previously assigned to shake up. The theoretical work showed conclusively that in fact even the band at around 17eV, which had previously been assigned to a C 2s excitation could not be explained by a single particle picture but was due to satellite excitations. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

One basic assumption almost universally made in interpreting XANES is the validity of the single-particle picture, which assumes that a single-particle density of states can explain the various spectral features. This assumption has been strengthened by recent calculations (83, 126, 155, 205, 206). The appealing result of 15 years of both theoretical and experimental work is that a one-electron approach is usually really relevant. However, in some instances this simple single-particle picture for XANES should be used with caution (13, 273). 

The spectra exhibits deviations from the single-particle picture of the potential for the ejected electron, which can be divided into two categories i) broadening of the spectra due to lifetime effects, and ii) energy shifts due to exchange correlation effects]... 

In chapter 2 we provided the justification for the single-particle picture of electrons in solids. We saw that the proper interpretation of single particles involves the notion of quasiparticles these are fermions which resemble real electrons, but are not identical to them since they also embody the effects of the presence of all other electrons, as in the exchange-correlation hole. Here we begin to develop the quantitative description of the properties of solids in terms of quasiparticles and collective excitations for the case of a perfectly periodic solid, i.e., an ideal crystal. 

The first term describes the electron interactions in the single-particle picture with the ions frozen, and the second term describes the electron interactions mediated by the exchange of phonons, when the ionic motion is taken into account. In principle, we can write the first term as a sum over single-particle terms and the second as a sum over pair-wise interactions ... 

As already mentioned, the treatment is mostly restricted to the single-particle picture. The meaning of this is clarified and its advantages and limitations are described in great detail in the second chapter. Briefly, the electrons responsible for the cohesion of a solid interact through long-range Coulomb forces both with the... 

Usually the photoionization cross section of 5s-electrons in xenon serves as an example of the very significant role of cprrela--tions. As is known, 5s-electrons are strongly effected by their neighbours 5p and 4d subshells and are completely collectivised, their photoionization cross section being even qualitatively different from the single-particle picture - Fig. 4. 

Furthermore, the coupling of the quasiparticle surface phonon with other quasiparticles such as Bloch electrons, magnons, plasmons, or polaritions are important ongoing research topics that lead in many cases beyond the well-established single-particle picture. Here, Section 8.2.5.2 could only give a small impression on phonon coupling issues. 

While the detailed calculations gave energies and widths in reasonable agreement with the observed experimental results, they did not provide a simple physical picture of why the He spectrum was the way it was. To address this issue Macek described the He atom using hyperspherical coordinates. In this approach, the two electrons, each described by three coordinates, are replaced by an equivalent single particle in six dimensions.8,9 In Fig. 23.3 we show the He atom in which rt and r2 are the vectors from He2+ to each of the two electrons. The hyperradius Rh is defined by... 

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