International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 2.2, pp. 304-305

Section 2.2.13. The local coordinate system

K. Schwarza*

aInstitut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.13. The local coordinate system

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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.

2.2.13.1. Crystal harmonics

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All spatial observables of the bound atom (e.g. the potential or the charge density) must have the crystal symmetry, i.e. the point-group symmetry around an atom. Therefore they must be representable as an expansion in terms of site-symmetrized spherical harmonics. Any point-symmetry operation transforms a spherical harmonic into another of the same [\ell]. We start with the usual complex spherical harmonics, [Y_{\ell m}(\vartheta,\varphi)=N_{\ell m}P_{\ell}^{m}(\cos\vartheta)\exp({im\varphi}),\eqno(2.2.13.1)]which satisfy Laplacian's differential equation. The [P_{\ell}^{m} (\cos\vartheta)] are the associated Legendre polynomials and the normalization [N_{\ell m}] is according to the convention of Condon & Shortley (1953[link]). For the [\varphi]-dependent part one can use the real and imaginary part and thus use [\cos(m\varphi)] and [\sin(m\varphi)] instead of the [\exp({im\varphi})] functions, but we must introduce a parity p to distinguish the functions with the same [\left| m\right| ]. 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 [\eqalignno{y_{\ell mp} &=\left\{ \matrix{ y_{\ell m+}=(1/\sqrt{2})(Y_{\ell m}+Y_{\ell\bar m}) \hfill&+\, \,{\rm parity}\cr y_{\ell m-}=-(i/\sqrt{2})(Y_{\ell m}-Y_{\bar m}) \hfill &-\, {\rm parity}},\right. \,\, m=2n &\cr y_{\ell mp} &=\left\{\matrix{ y_{\ell m+}=-(1/\sqrt{2})(Y_{\ell m}-Y_{\ell\bar m})\hfill&+\,\, {\rm parity}\cr y_{\ell m-}=(i/\sqrt{2})(Y_{\ell m}+Y_{\ell\bar m}) \hfill& -\, {\rm parity}},\right. \,\, m=2n+1.&\cr &&(2.2.13.2)}]

The expansion of – for example – the charge density [\rho({\bf r})] around an atomic site can be written using the LAPW method [see the analogous equation (2.2.12.5)[link] for the potential] in the form [\rho({\bf r})=\textstyle\sum\limits_{LM}\rho_{LM}(r)K_{LM}(\hat{r})\hbox{ inside an atomic sphere},\eqno(2.2.13.3)]where we use capital letters [LM] 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)[link]] and (ii) to include the parity p in the index M (which represents the combined index [mp]). With these conventions, [K_{LM}] can be written as a linear combination of real spherical harmonics [y_{\ell mp}] which are symmetry-adapted to the site symmetry,[K_{LM}(\hat{r})=\left\{\matrix{ y_{\ell mp}\hfill & \hbox{non-cubic}\hfill\cr \textstyle\sum_{j}c_{Lj}y_{\ell jp}\hfill & \hbox{cubic}\hfill}\right. \eqno(2.2.13.4)]i.e. they are either [y_{\ell mp}] [(2.2.13.2)[link]] in the non-cubic cases (Table 2.2.13.1[link]) or are well defined combinations of [y_{\ell mp}]'s in the five cubic cases (Table 2.2.13.2[link]), where the coefficients [c_{Lj}] depend on the normalization of the spherical harmonics and can be found in Kurki-Suonio (1977[link]).

Table 2.2.13.1| top | pdf |
Picking rules for the local coordinate axes and the corresponding [LM] combinations ([\ell mp]) of non-cubic groups taken from Kurki-Suonio (1977[link])

Symmetry Coordinate axes [\ell, m, p] of [y_{\ell mp}] Crystal system
1 Any All [(\ell,m,\pm) ] Triclinic
[\overline{1}] Any [(2\ell,m,\pm)]
2 [2\parallel z] [(\ell,2m,\pm) ] Monoclinic
[m ] [m\perp z ] [ (\ell,\ell-2m,\pm) ]
[2/m ] [ 2\parallel z, m\perp z ] [(2\ell,2m,\pm) ]
222 [ 2\parallel z, 2\parallel y\,\, (2\parallel x) ] [(2\ell,2m,+), (2\ell+1,2m,-) ] Orthorhombic
[mm2] [ 2\parallel z, m\perp y\,\, (2\perp x) ] [ (\ell,2m,+) ]
[mmm] [2\perp z, m\perp y, 2\perp x ] [(2\ell,2m,+) ]
4 [4\parallel z ] [ (\ell,4m,\pm) ] Tetragonal
[\overline{4}] [-4\parallel z ] [ (2\ell,4m,\pm), (2\ell+1,4m+2,\pm) ]
[4/m ] [4\parallel z, m\perp z ] [ (2\ell,4m,\pm) ]
422 [4\parallel z, 2\parallel y\,\, (2\parallel x) ] [ (2\ell,4m,+), (2\ell+1,4m,-) ]
[4mm] [ 4\parallel z, m\perp y\,\, (2\perp x) ] [ (\ell,4m,+) ]
[\overline{4}2m ] [-4\parallel z, 2\parallel x \,\,(m=xy\rightarrow yx) ] [ (2\ell,4m,+), (2\ell+1,4m+2,-) ]
[4mmm ] [ 4\parallel z, m\perp z, m\perp x ] [ (2\ell,4m,+) ]
3 [3\parallel z] [ (\ell,3m,\pm) ] Rhombohedral
[\overline{3}] [-3\parallel z] [ (2\ell,3m,\pm) ]
32 [ 3\parallel z, 2\parallel y ] [ (2\ell,3m,+), (2\ell+1,3m,-) ]
[3m ] [ 3\parallel z, m\perp y ] [ (\ell,3m,+) ]
[\overline{3}m ] [ -3\parallel z, m\perp y ] [ (2\ell,3m,+) ]
6 [6\parallel z ] [ (\ell,6m,\pm) ] Hexagonal
[\overline{6}] [-6\parallel z ] [ (2\ell,6m,+), (2\ell+1,6m+3,\pm)]
[6/m ] [ 6\parallel z, m\perp z ] [ (2\ell,6m,\pm) ]
622 [ 6\parallel z, 2\parallel y \,\,(2\parallel x) ] [ (2\ell,6m,+), (2\ell+1,6m,-)]
[6mm ] [6\parallel z, m\parallel y\,\, (m\perp x) ] [(\ell,6m,+) ]
[\overline{6}2m ] [-6\parallel z, m\perp y\,\, (2\parallel x) ] [ (2\ell,6m,+), (2l+1,6m+3,+) ]
[6mmm ] [ 6\parallel z, m\perp z, m\perp y\,\, (m\perp x) ] [ (2\ell,6m,+) ]

Table 2.2.13.2| top | pdf |
LM combinations of cubic groups as linear cominations of [y_{\ell mp}]'s (given in parentheses)

The linear-combination coefficients can be found in Kurki-Suonio (1977[link]).

SymmetryLM combinations
23 (0 0), (3 2−), (4 0, 4 4[+]), (6 0, 6 4[+]), (6 2[+], 6 6[+])
[m3 ] (0 0), (4 0, 4 4[+]), (6 0, 6 4[+]) (6 2[+], 6 6[+])
432 (0 0), (4 0, 4 4[+]), (6 0, 6 4[+])
[\overline{4}3m ] (0 0), (3 2−), (4 0, 4 4[+]), (6 0, 6 4[+]),
[m3m ] (0 0), (4 0, 4 4[+]), (6 0, 6 4[+])

According to Kurki-Suonio, the number of (non-vanishing) [LM] terms [e.g. in (2.2.13.3)[link]] is minimized by choosing for each atom a local Cartesian coordinate system adapted to its site symmetry. In this case, other [LM] 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. [\rho({\bf r})]] over the star of [{\bf k}]. Note that in another coordinate system (for the L values listed) additional M terms could appear. The group-theoretical 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)[link] are summarized by Kurki-Suonio, who has defined picking rules to choose the local coordinate system for any of the 27 non-cubic site symmetries (Table 2.2.13.1[link]) and has listed the [LM] combinations, which are defined by (a linear combination of) functions [y_{\ell mp}] [see (2.2.13.2)[link]]. If the [\pm] parity appears, both the [+] and the − combination must be taken. An application of a local coordinate system to rutile TiO2 is described in Section 2.2.16.2[link].

In the case of the five cubic site symmetries, which all have a threefold axis in (111), a well defined linear combination of [y_{\ell mp}] functions (given in Table 2.2.13.2[link]) leads to the cubic harmonics.

2.2.13.2. Interpretation for bonding

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Chemical bonding is often described by considering orbitals (e.g. a [p_{z}] or a [d_{z^{2}}] 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, ligand-field 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 (unit-cell) 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.[link]

References

Condon, E. U. & Shortley, G. H. (1953). The mathematical theory of symmetry in crystals. Cambridge University Press.
Kurki-Suonio, K. (1977). Symmetry and its implications. Isr. J. Chem. 16, 115–123.








































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