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. 304305
Section 2.2.13. The local coordinate system^{a}Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165TC, A1060 Vienna, Austria 
The partition of a crystal into atoms (or molecules) is ambiguous and thus the atomic contribution cannot be defined uniquely. However, whatever the definition, it must follow the relevant site symmetry for each atom. There are at least two reasons why one would want to use a local coordinate system at each atomic site: the concept of crystal harmonics and the interpretation of bonding features.
All spatial observables of the bound atom (e.g. the potential or the charge density) must have the crystal symmetry, i.e. the pointgroup symmetry around an atom. Therefore they must be representable as an expansion in terms of sitesymmetrized spherical harmonics. Any pointsymmetry operation transforms a spherical harmonic into another of the same . We start with the usual complex spherical harmonics, which satisfy Laplacian's differential equation. The are the associated Legendre polynomials and the normalization is according to the convention of Condon & Shortley (1953). For the dependent part one can use the real and imaginary part and thus use and instead of the functions, but we must introduce a parity p to distinguish the functions with the same . For convenience we take real spherical harmonics, since physical observables are real. The even and odd polynomials are given by the combination of the complex spherical harmonics with the parity p either or − by
The expansion of – for example – the charge density around an atomic site can be written using the LAPW method [see the analogous equation (2.2.12.5) for the potential] in the form where we use capital letters for the indices (i) to distinguish this expansion from that of the wavefunctions in which complex spherical harmonics are used [see (2.2.12.1)] and (ii) to include the parity p in the index M (which represents the combined index ). With these conventions, can be written as a linear combination of real spherical harmonics which are symmetryadapted to the site symmetry,i.e. they are either [(2.2.13.2)] in the noncubic cases (Table 2.2.13.1) or are well defined combinations of 's in the five cubic cases (Table 2.2.13.2), where the coefficients depend on the normalization of the spherical harmonics and can be found in KurkiSuonio (1977).


According to KurkiSuonio, the number of (nonvanishing) terms [e.g. in (2.2.13.3)] is minimized by choosing for each atom a local Cartesian coordinate system adapted to its site symmetry. In this case, other terms would vanish, so using only these terms corresponds to the application of a projection operator, i.e. equivalent to averaging the quantity of interest [e.g. ] over the star of . Note that in another coordinate system (for the L values listed) additional M terms could appear. The grouptheoretical derivation led to rules as to how the local coordinate system must be chosen. For example, the z axis is taken along the highest symmetry axis, or the x and y axes are chosen in or perpendicular to mirror planes. Since these coordinate systems are specific for each atom and may differ from the (global) crystal axes, we call them `local' coordinate systems, which can be related by a transformation matrix to the global coordinate system of the crystal.
The symmetry constraints according to (2.2.13.4) are summarized by KurkiSuonio, who has defined picking rules to choose the local coordinate system for any of the 27 noncubic site symmetries (Table 2.2.13.1) and has listed the combinations, which are defined by (a linear combination of) functions [see (2.2.13.2)]. If the parity appears, both the and the − combination must be taken. An application of a local coordinate system to rutile TiO_{2} is described in Section 2.2.16.2.
In the case of the five cubic site symmetries, which all have a threefold axis in (111), a well defined linear combination of functions (given in Table 2.2.13.2) leads to the cubic harmonics.
Chemical bonding is often described by considering orbitals (e.g. a or a atomic orbital) which are defined in polar coordinates, where the z axis is special, in contrast to Cartesian coordinates, where x, y and z are equivalent. Consider for example an atom coordinated by ligands (e.g. forming an octahedron). Then the application of group theory, ligandfield theory etc. requires a certain coordinate system provided one wishes to keep the standard notation of the corresponding spherical harmonics. If this octahedron is rotated or tilted with respect to the global (unitcell) coordinate system, a local coordinate system is needed to allow an easy orbital interpretation of the interactions between the central atom and its ligands. This applies also to spectroscopy or electric field gradients.
The two types of reasons mentioned above may or may not lead to the same choice of a local coordinate system, as is illustrated for the example of rutile in Section 2.2.16.2.
References
Condon, E. U. & Shortley, G. H. (1953). The mathematical theory of symmetry in crystals. Cambridge University Press.KurkiSuonio, K. (1977). Symmetry and its implications. Isr. J. Chem. 16, 115–123.