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. 294-295

Section 2.2.3. Symmetry operators

K. Schwarza*

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

2.2.3. Symmetry operators

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The concepts of symmetry operations in connection with a quantum-mechanical treatment of the electronic states are essential for an understanding of the electronic structure. In this context the reader is referred, for example, to the book by Altmann (1994[link]).

For the definition of symmetry operators we use in the whole of this chapter the active picture, which has become the standard in solid-state physics. This means that the whole configuration space is rotated, reflected or translated, while the coordinate axes are kept fixed.

A translation is given by[\eqalignno{{\bf r}^\prime &={\bf r}+{\bf T}&(2.2.3.1)\cr t{\bf r} &={\bf r}+{\bf T},&(2.2.3.2)}%fd2.2.3.2]where t on the left-hand side corresponds to a symmetry (configuration-space) operator.

2.2.3.1. Transformation of functions

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Often we are interested in a function (e.g. a wavefunction) [f({\bf r})] and wish to know how it transforms under the configuration operator g which acts on [{\bf r}]. For this purpose it is useful to introduce a function-space operator [\widetilde{g}] which defines how to modify the function in the transformed configuration space so that it agrees with the original function [f({\bf r)}] at the original coordinate [{\bf r}]:[\widetilde{g}f(g{\bf r})=f({\bf r}).\eqno(2.2.3.3)]This must be valid for all points [{\bf r}] and thus also for [g^{-1}{\bf r}], leading to the alternative formulation [\widetilde{g}f({\bf r})=f(g^{-1}{\bf r}).\eqno(2.2.3.4)]The symmetry operations form a group G of configuration-space operations [g_{i}] with the related group [\widetilde{G}] of the function-shape operators [\widetilde{g}_{i}]. Since the multiplication rules [g_{i}g_{j}=g_{k}\rightarrow\widetilde{g}_{i}\widetilde{g}_{j}=\widetilde {g}_{k}\eqno(2.2.3.5)]are preserved, these two groups are isomorphic.

2.2.3.2. Transformation of operators

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In a quantum-mechanical treatment of the electronic states in a solid we have the following different entities: points in configuration space, functions defined at these points and (quantum-mechanical) operators acting on these functions. A symmetry operation transforms the points, the functions and the operators in a clearly defined way.

Consider an eigenvalue equation of operator [{\bb A}] (e.g. the Hamiltonian):[{\bb A}\varphi=a\varphi,\eqno(2.2.3.6)]where [\varphi({\bf r})] is a function of [{\bf r}]. When g acts on [{\bf r}], the function-space operator [\widetilde{g}] acts [according to (2.2.3.4)[link]] on [\varphi] yielding [\psi]: [\psi=\widetilde{g}\varphi\rightarrow\varphi=\widetilde{g}^{-1}\psi.\eqno(2.2.3.7)]By putting [\varphi] from (2.2.3.7)[link] into (2.2.3.6)[link], we obtain [{\bb A}\widetilde{g}^{-1}\psi=a\widetilde{g}^{-1}\psi. \eqno(2.2.3.8)]Multiplication from the left by [\widetilde{g}] yields [\widetilde{g}{\bb A}\widetilde{g}^{-1}\psi=a\widetilde{g}\widetilde{g}^{-1}\psi=a\psi.\eqno(2.2.3.9)]This defines the transformed operator [\widetilde{g}{\bb A}\widetilde {g}^{-1}] which acts on the transformed function [\psi] that is given by the original function [\varphi] but at position [g^{-1}{\bf r}].

2.2.3.3. The Seitz operators

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The most general space-group operation is of the form [wp] with the point-group operation p (a rotation, reflection or inversion) followed by a translation w: [wp=\{p|{\bf w}\}.\eqno(2.2.3.10)]With the definition [\{p|{\bf w}\}{\bf r}=wp{\bf r}=w(p{\bf r})=p{\bf r} +{\bf w}\eqno(2.2.3.11)]it is easy to prove the multiplication rule [\{p|{\bf w}\}\{p^{\prime}|{\bf w}^{\prime}\}=\{pp^{\prime} |p{\bf w}^{\prime}+{\bf w}\}\eqno(2.2.3.12)]and define the inverse of a Seitz operator as [\{p|{\bf w}\}^{-1}=\{p^{-1}|-p^{-1}{\bf w}\},\eqno(2.2.3.13)]which satisfies [\{p|{\bf w}\}\{p|{\bf w}\}^{-1}=\{E|{\bf 0}\},\eqno(2.2.3.14)]where [\{E|{\bf 0}\}] does not change anything and thus is the identity of the space group G.

2.2.3.4. The important groups and their first classification

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Using the Seitz operators, we can classify the most important groups as we need them at the beginning of this chapter:

  • (i) the space group, which consists of all elements [G=\{\{p|{\bf w}\}\}];

  • (ii) the point group (without any translations) [\ P=\{\{p|{\bf 0}\}\}]; and

  • (iii) the lattice translation subgroup [T=\{\{E|{\bf T}\}\}], which is an invariant subgroup of G, i.e. [T\triangleleft G]. Furthermore T is an Abelian group, i.e. the operation of two translations commute ([t_{1}t_{2}=t_{2}t_{1}]) (see also Section 1.2.3.1[link] of the present volume). A useful consequence of the commutation property is that T can be written as a direct product of the corresponding one-dimensional translations, [T=T_{x}\otimes T_{y}\otimes T_{z}.\eqno(2.2.3.15)]

  • (iv) A symmorphic space group contains no fractional translation vectors and thus P is a subgroup of G, i.e. [P\triangleleft G].

  • (v) In a non-symmorphic space group, however, some p are associated with fractional translation vectors [{\bf v}]. These [{\bf v}] do not belong to the translation lattice but when they are repeated a specific integer number of times they give a vector of the lattice. In this case, [\{p|{\bf 0}\}] can not belong to G for all p.

  • (vi) The Schrödinger group is the group S of all operations [\widetilde{g}] that leave the Hamiltonian invariant, i.e. [\widetilde {g}{\bb H}\widetilde{g}^{-1}={\bb H}] for all [\widetilde{g}\in S]. This is equivalent to the statement that [\widetilde{g}] and [{\bb H}] commute: [\widetilde{g}{\bb H}={\bb H}\widetilde{g}]. From this commutator relation we find the degenerate states in the Schrödinger equation, namely that [\widetilde{g}\varphi] and [\varphi] are degenerate with the eigenvalue E whenever [\widetilde{g}\in S], as follows from the three equations [\eqalignno{{\bb H}\varphi &=E\varphi &(2.2.3.16)\cr \widetilde{g}{\bb H}\varphi &=E\widetilde{g}\varphi &(2.2.3.17)\cr {\bb H}\widetilde{g}\varphi &=E\widetilde{g}\varphi .&(2.2.3.18)}%fd2.2.3.18]

References

Altmann, S. L. (1994). Band theory of solids: An introduction from the view of symmetry. Oxford: Clarendon Press.








































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