Single-particle band-structure calculations

According to Grunes (107) and others (181,303), single-particle band-structure calculations (206) using the final-state potential with neglect of the core hole do well in predicting the observed X-ray absorption spectral features of metals past threshold (200 eV), which reflect the unfilled p density of states. The present insight is that the final-state rule (181,303) for the calculation of X-ray absorption spectra of metals is satisfactory (72). 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

Within a single-particle treatment of the final electron state, XANES can be considered as due to long-range order (5), and consequently band-structure calculations may be applied. The band-structure approach... 

The calculation of realistic quasiparticle bands proceeds in several steps as schematically summarized in fig. 4. The first step is a standard LDA band structure calculation by means of which the effective single-particle potentials are self-consistently generated. The calculation starts, like any other ab-initio calculation, from atomic potentials and structure information. In this step, no adjustable parameters are introduced. The effective potentials and hence the phase shifts of the conduction states are determined from first principles to the same level as in the case of ordinary metals. The f-phase shifts at the lanthanide and actinide sites, on the other hand, are described by a resonance type expression... 

We are interested in a situation where the extra particles in the lattice are described by a single band Hubbard Hamiltonian coupled to the acoustic phonons of the lattice as given in Equation 12.12 [ 128]. In the latter equation, the first and second terms describe the nearest-neighbor hopping of the extra-particles with hopping amplitudes J, and interactions V, computed for each microscopic model by band-structure calculations for Uj = 0, respectively. The third term is the phonon Hamiltonian. The fourth term is the phonon coupling obtained in lowest order in the displacement... 

The single particle band gap of PPV corresponds to a n-n transition [62]. Our calculated value of band gap is 2.0 eV, which is slightly smaller than the experimental value of 2.4 eV [60,63]. Despite this fundamental problem, variations in the band gap produced either by structural or charge distortions, are often more accurately described with density functional methods, and therefore the trends described below should be quite precise... 

Table I summarizes the main results. Figure 19 shows the calculated Pd K-edge absorption coefficient in comparison to the experimental results (208), and Fig. 20 shows other band-structure results that reproduce all features of the single-particle spectrum (204).

A single-particle effect that adds features in the X-ray absorption spectrum of molecules not present in that of atoms is the shape resonance (74, 75). (In the case of solids this effect, caused by a modification of the density of states due to the presence of the other atoms in the molecule, is automatically accounted for in band calculations.) Localization of the excited electron inside the molecule in states resulting from an effective potential barrier located near the electronegative atoms in the molecule causes strong absorption bands in free molecules and near the inner-shell ionization limits of positive ions in ionic crystals (74). Consequently, molecular inner-shell spectra depart markedly from the corresponding atomic spectra. The type of structure of an inner-shell photoabsorption spectrum depends on the geometry of the molecule, the nature of its ligands, etc., and can sometimes be used to determine the structure of the molecule. 

We are now ready to use the Xki(x) functions as the basis to construct crystal wavefianctions and with these calculate the single-particle energy eigenvalues, that is, the band structure of the model. The crystal wavefunctions are obtained from the general expression Eq. (4.4) ... 

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