Representations and bases of the space group

Schwarz, K.

doi:10.1107/97809553602060000639

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

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International Tables for Crystallography (2006). Vol. D. ch. 2.2, pp. 297-298

Section 2.2.6.1. Representations and bases of the space group

K. Schwarz^a ^*

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

2.2.6.1. Representations and bases of the space group

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The effect of a space-group operation $[\{p|{\bf w}\}]$ on a Bloch function, labelled by $[{\bf k}]$ , is to transform it into a Bloch function that corresponds to a vector $[p{\bf k}]$ , $[\{p|{\bf w}\}\psi_{{\bf k}}=\psi_{p{\bf k}},\eqno(2.2.6.1)]$ which can be proven by using the multiplication rule of Seitz operators (2.2.3.12) and the definition of a Bloch state (2.2.4.17).

A special case is the inversion operator, which leads to $[\{i|{\bf E}\}\psi_{{\bf k}}=\psi_{-{\bf k}}.\eqno(2.2.6.2)]$ The Bloch functions $[\psi_{{\bf k}}]$ and $[\psi_{p{\bf k}}]$ , where p is any operation of the point group P, belong to the same basis for a representation of the space group G. $[ \langle \psi_{{\bf k}}|= \langle \psi_{p{\bf k}}|\hbox{ for all }p\in P\hbox{ for all }p{\bf k}\in {\rm BZ}.\eqno(2.2.6.3)]$ The same $[p{\bf k}]$ cannot appear in two different bases, thus the two bases $[\psi_{{\bf k}}]$ and $[\psi_{{\bf k}^{\prime}}]$ are either identical or have no $[{\bf k}]$ in common.

Irreducible representations of T are labelled by the N distinct $[{\bf k}]$ vectors in the BZ, which separate in disjoint bases of G (with no $[{\bf k}]$ vector in common). If a $[{\bf k}]$ vector falls on the BZ edge, application of the point-group operation p can lead to an equivalent $[{\bf k}^{\prime}]$ vector that differs from the original by $[{\bf K}]$ (a vector of the reciprocal lattice). The set of all mutually inequivalent $[{\bf k}]$ vectors of $[p{\bf k}]$ ( $[p\in P]$ ) define the star of the k vector ( $[S_{{\bf k}}]$ ) (see also Section 1.2.3.3 of the present volume).

The set of all operations that leave a $[{\bf k}]$ vector invariant (or transform it into an equivalent $[{\bf k+K}]$ ) forms the group $[G_{{\bf k}}]$ of the $[{\bf k}]$ vector. Application of q, an element of $[G_{{\bf k}}]$ , to a Bloch function (Section 2.2.8) gives $[q\psi_{{\bf k}}^{j}({\bf r})=\psi_{{\bf k}}^{j^{\prime}} ({\bf r})\hbox{ for }q\in G_{{\bf k}},\eqno(2.2.6.4)]$ where the band index j (described below) may change to $[j^{\prime}]$ . The Bloch factor stays constant under the operation of q and thus the periodic cell function $[u_{{\bf k}}^{j}({\bf r})]$ must show this symmetry, namely $[qu_{{\bf k}}^{j}({\bf r})=u_{{\bf k}}^{j^{\prime}}({\bf r})\hbox{ for }q\in G_{{\bf k}}.\eqno(2.2.6.5)]$ For example, a $[p_{x}]$ -like orbital may be transformed into a $[p_{y}]$ -like orbital if the two are degenerate, as in a tetragonal lattice.

A star of $[{\bf k}]$ determines an irreducible basis, provided that the functions of the star are symmetrized with respect to the irreducible representation of the group of $[{\bf k}]$ vectors, which are called small representations. The basis functions for the irreducible representations are given according to Seitz (1937) by $[ \langle s\psi_{{\bf k}}^{j}|,\hbox{ where }s\in S_{{\bf k}},]$ written as a row vector $[ \langle |]$ with $[j=1,\ldots,n]$ , where n is the dimension of the irreducible representation of $[S_{{\bf k}}]$ with the order $[\left| S_{{\bf k}}\right| ]$ . Such a basis consists of $[n\left| S_{{\bf k} }\right| ]$ functions and forms an $[n\left| S_{{\bf k}}\right|]$ -dimensional irreducible representation of the space group. The degeneracies of these representations come from the star of $[{\bf k}]$ (not crucial for band calculations except for determining the weight of the $[{\bf k}]$ vector) and the degeneracy from $[G_{{\bf k}}]$ . The latter is essential for characterizing the energy bands and using the compatibility relations (Bouckaert et al., 1930; Bradley & Cracknell, 1972).

References

Bouckaert, L. P., Smoluchowski, R. & Wigner, E. (1930). Theory of Brillouin zones and symmetry properties of wavefunctions in crystals. Phys. Rev. 50, 58–67.Google Scholar

Bradley, C. J. & Cracknell, A. P. (1972). The mathematical theory of symmetry in solids. Oxford: Clarendon Press.Google Scholar

Seitz, F. (1937). On the reduction of space groups. Ann. Math. 37, 17–28.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 2.2, pp. 297-298