International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 2.2, pp. 310312
Section 2.2.16. Examples^{a}Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165TC, A1060 Vienna, Austria 
The general concepts described above are used in many bandstructure applications and thus can be found in the corresponding literature. Here only a few examples are given in order to illustrate certain aspects.
For the simple case of an element, namely copper in the f.c.c. structure, the band structure is shown in Fig. 2.2.16.1 along the symmetry direction from to X. The character of the bands can be illustrated by showing for each band state the crucial information that is contained in the wavefunctions. In the LAPW method (Section 2.2.12), the wavefunction is expanded in atomic like functions inside the atomic spheres (partial waves), and thus a spatial decomposition of the associated charge and its portion of like charge (s, p, dlike) inside the Cu sphere, , provides such a quantity. Fig. 2.2.16.1 shows for each state a circle the radius of which is proportional to the like charge of that state. The band originating from the Cu 4s and 4p orbitals shows an approximately freeelectron behaviour and thus a energy dependence, but it hybridizes with one of the d bands in the middle of the direction and thus the like character changes along the direction.

Character of energy bands of f.c.c. copper in the direction. The radius of each circle is proportional to the respective partial charge of the given state. 
This can easily be understood from a grouptheoretical point of view. Since the d states in an octahedral environment split into the and manifold, the d bands can be further partitioned into the two subsets as illustrated in Fig. 2.2.16.2. The s band ranges from about −9.5 eV below to about 2 eV above. In the direction, the s band has symmetry, the same as one of the d bands from the manifold, which consists of and . As a consequence of the `noncrossing rule', the two states, both with symmetry, must split due to the quantummechanical interaction between states with the same symmetry. This leads to the avoided crossing seen in the middle of the direction (Fig. 2.2.16.1). Therefore the lowest band starts out as an `s band' but ends near X as a `d band'. This also shows that bands belonging to different irreducible representations (small representations) may cross. The fact that splits into the subgroups and is an example of the compatibility relations. In addition, grouptheoretical arguments can be used (Altmann, 1994) to show that in certain symmetry directions the bands must enter the face of the BZ with zero slope.

Decomposition of the Cu d bands into the and manifold. The radius of each circle is proportional to the corresponding partial charge. 
Note that in a sitecentred description of the wavefunctions a similar like decomposition of the charge can be defined as (without the term), but here the partial charges have a different meaning than in the spatial decomposition. In one case (e.g. LAPW), refers to the partial charge of like character inside sphere t, while in the other case (LCAO), it means like charge coming from orbitals centred at site t. For the main components (for example Cu d) these two procedures will give roughly similar results, but the small components have quite a different interpretation. For this purpose consider an orbital that is centred on the neighbouring site j, but whose tail enters the atomic sphere i. In the spatial representation this tail coming from the j site must be represented by the (s, p, d etc.) partial waves inside sphere i and consequently will be associated with site i, leading to a small partial charge component. This situation is sometimes called the offsite component, in contrast to the onsite component, which will appear at its own site or in its own sphere, depending on the representation, sitecentred or spatially confined.
The well known rutile structure (e.g. TiO_{2}) is tetragonal (see Fig. 2.2.16.3) with the basis consisting of the metal atoms at the Wyckoff positions, () and (), and anions at the position, located at () and () with a typical value of about 0.3 for the internal coordinate u. Rutile belongs to the nonsymmorphic space group () in which the metal positions are transformed into each other by a rotation by 90° around the crystal c axis followed by a nonprimitive translation of (). The two metal positions at the centre and at the corner of the unit cell are equivalent when the surrounding octahedra are properly rotated. The metal atoms are octahedrally coordinated by anions which, however, do not form an ideal octahedron. The distortion depends on the structure parameters a, and u, and results in two different metal–anion distances, namely the apical distance and the equatorial distance , the height (z axis) and the basal spacing of the octahedron. For a certain value the two distances and become equal:For this special value and an ideal ratio, the basal plane of the octahedron is quadratic and the two distances are equal. An ideal octahedral coordination is thus obtained with Although the actual coordination of the metal atoms deviates from the ideal octahedron (as in all other systems that crystallize in the rutile structure), we still use this concept for symmetry arguments and call it octahedral coordination.
The concept of a local coordinate system is illustrated for rutile (TiO_{2}) from two different aspects, namely the crystal harmonics expansion (see Section 2.2.13) and the interpretation of chemical bonding (for further details see Sorantin & Schwarz, 1992).
In excitations involving core electrons, simplifications are possible that allow an easier interpretation. As one example, (soft) Xray emission (XES) or absorption (XAS) spectra are briefly discussed. In the oneelectron picture, the XES process can be described as sketched in Fig. 2.2.16.4. First a core electron of atom A in state is knocked out (by electrons or photons), and then a transition occurs between the occupied valence states at energy and the core hole (the transitions between inner core levels are ignored).
According to Fermi's golden rule, the intensity of such a transition can be described bywhere comes from the integral over the angular components (Table 2.2.16.1) and contains the selection rule, is the local (within atomic sphere A) partial (like) DOS, is the radial transition probability [see (2.2.16.6) below], and the last term takes the energy conservation into account.

The are defined as the dipole transition (with the dipole operator r) probability between the valence state at and the core state characterized by quantum numbers ,In this derivation one makes use of the fact that core states are completely confined inside the atomic sphere. Therefore the integral, which should be taken over the entire space, can be restricted to one atomic sphere (namely A), since the core wavefunction and thus the integrand vanishes outside this sphere. This is also the reason why XES (or XAS) are related to , the local DOS weighted with the like charge within the atomic sphere A.
The interpretation of XES intensities is as follows. Besides the factor from Fermi's golden rule, the intensity is governed by the selection rule and the energy conservation. In addition, it depends on the number of available states at which reside inside sphere A and have an like contribution, times the probability for the transition to take place from the valence and to the core hole under energy conservation. For an application, see for example the comparison between theory and experiment for the compounds NbC and NbN (Schwarz, 1977).
Note again that the present description is based on an atomic sphere representation with partial waves inside the spheres, in contrast to an LCAOlike treatment with sitecentred basis functions. In the latter, an equivalent formalism can be defined which differs in details, especially for the small components (offsite contributions). If the tails of an orbital enter a neighbouring sphere and are crucial for the interpretation of XES, there is a semantic difference between the two schemes as discussed above in connection with f.c.c. Cu in Section 2.2.16.1. In the present framework, all contributions come exclusively from the sphere where the core hole resides, whereas in an LCAO representation `cross transitions' from the valence states on one atom to the core hole of a neighbouring atom may be important. The latter contributions must be (and are) included in the partial waves within the sphere in schemes such as LAPW. There is no physical difference between the two descriptions.
In XES, spectra are interpreted on the basis of results from groundstate calculations, although there could be relaxations due to the presence of a core hole. As early as 1979, von Barth and Grossmann formulated a `final state rule' for XES in metallic systems (von Barth & Grossmann, 1979). In this case, the initial state is one with a missing core electron (core hole), whereas the final state is close to the ground state, since the hole in the valence bands (after a valence electron has filled the core hole) has a very short lifetime and is very quickly filled by other valence electrons. They applied timedependent perturbation theory and could show by model calculations that the main XES spectrum can be explained by groundstate results, whereas the satellite spectrum (starting with two core holes and ending with one) requires a treatment of the corehole relaxation. This example illustrates the importance of the relevant physical process in experiments related to the energyband structure: it may not always be the just the ground states that are involved and sometimes excited states must be considered.
References
Altmann, S. L. (1994). Band theory of solids: An introduction from the view of symmetry. Oxford: Clarendon Press.Barth, U. von & Grossmann, G. (1979). The effect of the core hole on Xray emission spectra in simple metals. Solid State Commun. 32, 645–649.
Schwarz, K. (1977). The electronic structure of NbC and NbN. J. Phys. C Solid State Phys. 10, 195–210.
Sorantin, P. & Schwarz, K. (1992). Chemical bonding in rutiletype compounds. Inorg. Chem. 31, 567–576.