Core excitation energies discussions

Abstract This chapter discusses descriptions of core-ionized and core-excited states by density functional theory (DFT) and by time-dependent density functional theory (TDDFT). The core orbitals are analyzed by evaluating core-excitation energies computed by DFT and TDDFT their orbital energies are found to contain significantly larger self-interaction errors in comparison with those of valence orbitals. The analysis justifies the inclusion of Hartree-Fock exchange (HFx), capable of reducing self-interactions, and motivates construction of hybrid functional with appropriate HFx portions for core and valence orbitals. The determination of the HFx portions based on a first-principle approach is also explored and numerically assessed. 

A very narrow 0 2p band is observed with earlier measurements (74) on partially oxidized Cs films, and the result has been discussed in terms of isolated ions incorporated into the films under a surface layer of metallic cesium. This interpretation is very near to reality. Unfortunately, these investigations have been performed with 10.2 eV excitation energy. So the core levels of Cs are not recorded but the measurements definitely refer to Cs suboxides. In this respect it is interesting that during the early stages of oxidation of K (75) and Sr (76, 77) very narrow 0 2p bands are observed, although bulk suboxides of these materials are not yet known. 

The spectrum serves to Illuminate several of the characteristics of core-excited resonances. The "doublet" structure, repeated at intervals of approximately 170 meV, is characteristic of the >2 C-C symmetric stretch. The spacing between the first and second features in each pair is 60 meV. We attribute these to two quanta of the CH2 torsional mode, l.e. 2V1,. These characteristic energies in the anion are quite close to those of the Rydberg "parent" state, as expected. The existence of the low frequency modes is a clear indication of the long lifetime of the Feshbach resonance relative to that of the B2g shape resonance discussed previously. 

The quality of this approximation, which was introduced by Heisenberg to describe the deep core-hole excitations of X-ray spectroscopy, varies according to the element and also according to the excitation energy. This aspect will be discussed in chapter 7 (see in particular fig. 7.1 and the related discussion in section 7.2). The fact that holes are stable results in some extremely useful simplifications of atomic spectra for example, if a vacancy is created in the fu subshell, i.e. if we remove one electron to create the /13 hole, then this hole can behave like a single particle, i.e. a closed subshell minus one electron, despite the fact that it is made up from thirteen electrons. 

Of course, it is an approximation to regard quasiparticles as particles, and this approximation can be expected to break down in several ways. First, core holes always have a finite lifetime, i.e. they are broadened, and disperse on a short timescale. The effects of core-hole broadening will be discussed in chapters 8 and 11. Secondly, the very concept of a core hole may become inapplicable, i.e. it may prove impossible to identify a single structure in the spectrum as the result of exciting a quasiparticle. This form of breakdown is discussed in chapter 7. Experience shows that well-characterised holes tend to be the deepest ones, which are fully screened, while vacancies in the subvalence shells cannot always be described in this way. Thus, the concept of core holes is most useful in X-ray spectroscopy, but can sometimes break down quite severely at lower excitation energies. 

What makes the neutral final state or complete screening condition approximate rather than exact is the time required for the system to respond to the core excitation (the various time scales are discussed by Gadzuk (1978)). In this work we are concerned only with the excitation energy, the threshold for absorption. Since the threshold corresponds to a long-time process (in the sense of a Fourier transform of a correlation function), the actual dynamics of complete screening will have negligible effect on the threshold energy but will affect the lineshape. 

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