In the previous section we found it useful to use the cyclic representation of the periodic system because it allowed us to reason on the basis of the properties of point groups. In the following, it will be easier to reason directly using
the real geometry of the systems under consideration, i.e. linear geometry.
In this section we will show that it is possible to transpose the results of the previous section to linear periodic systems since they fulfil the Born–von Karman periodic boundary conditions.
Taking into account the translational properties
Essentially, the Born–von Karman periodic boundary conditions impose the equivalence of aCnrotation on the cyclic representation and a translation by a vector a, ta, on the linear representation, as well as the equivalence of a tna translation and the identity. Consequently, the group of rotationsCn of the cyclic system is equivalent to the group of translations Tn of the linear system. This group contains all translations by vectors that are multiples of theavector,tma(m=0,±1,±2, . . . , (n−1),n). The product tables of the different symmetry operations of theCnandTngroups are identical provided that the roles played by theCnmrotation and thetmatranslation are considered identical. These two groups are said to be isomorphic. This means that the rotations in the cyclic system play the same role as the translations in the linear system. These two groups possess exactly the same properties, namely the same character table. The Bloch orbitals of the linear system obtained in Chapter 3 (see Fig. 3.3) are consequently bases for the different irreducible representations of the translation group.
Notion of space group
By definition, we will denote as the space group G of a periodic system the group containing the symmetry operations leaving the system invariant.
Besides the group of translations,Tn, this group also contains the point sym- metry operations as well as the combined operations resulting from a product between a point symmetry operation and a translation.
Here we will restrict ourselves to the space symmetry groups G that may be written as the direct product of the translation group Tnand a point group P (G=Tn⊗P). These groups are known as symmorphic.We will consider the existence of non-symmorphic groups in Section 6.4 of this chapter.1
1Working with space groups which may be written as a direct product Tn⊗P may involve neglecting some symme- try operations, i.e. those written as the product of a point symmetry operationR and a translationtbsuch that the opera- tions R andtbdo not leave the system invariant.
Under such conditions the combined symmetry operations resulting from the product between a point symmetry operation and a translation necessarily involve a symmetry operation of thePgroup and a translation of theTngroup.
In addition, since any point group leaves one point invariant, we choose the P group in such a way that it contains the largest possible number of symmetry operations and leaves invariant at least the point O of the reference cell.
Space group of the (Pt(CN)4)2−chain
We will now illustrate the notion of space group by looking at the case of a (Pt(CN)4)2− chain in an eclipsed configuration, as shown in Fig. 6.4.
What space group P leaves invariant this chain as well as a point O of the reference cell? In view of the symmetry of the system it seems natural to place point O either on a platinum atom of the reference cell Pt0 or at the middle of the Pt0–Pt1 segment. Let us place it at the Pt0 site. It would be
Fig. 6.4
Eclipsed structure of the (Pt(CN)4)2−
chain.
completely equivalent to place it in the middle of the Pt0–Pt1 segment (see Exercise (6.2)). The symmetry operations of the D4hgroup, which leave this O point invariant, also leave the whole chain invariant. We will label this group(D4h)0because it leaves atom Pt0invariant. Consequently, the(D4h)0
group is the P group that we are looking for.2 Now let us determine the 2
One must be very careful in defining the O zaxis in the cyclic and linear represen- tations. In the cyclic system theO zaxis is the highest order rotation axis, i.e. the Cnrotation axis. In contrast, theO zaxis in the linear system is the axis of theP group which is associated with the high- est order rotation axis, i.e. in the present case, the horizontal axis (see Fig. 6.4).
different symmetry operations contained in the space groupG,which is equal toTn⊗(D4h)0. It is possible to associate the different symmetry operations of (D4h)0in two subgroups E1and(E2)0.E1contains the symmetry operations leaving invariant every point of theO zaxis, i.e.E1= {E,2C4,C2,2σv,2σd}. (E2)0 contains the symmetry operations that only leave invariant one point of theO zaxis, Pt0, so that(E2)0= {(i)0,2(S4)0, (σh)0,2(C2)0,2C2)0}. The index 0 is a reminder of the fact that the symmetry operation leaves the Pt0
point invariant. The product of ata translation and a symmetry operation of E1 generates a combined symmetry operation (point symmetry operation–
translation). In contrast, the product of a ta translation and a symmetry operation of (E2)0 generates the whole set of point symmetry operations of(E2)0/1, i.e.{(i)0/1,2(S4)0/1, (σh)0/1,2(C2)0/1,2C2)0/1}, which leave the middle point of the Pt0–Pt1segment invariant.3 3
The symmetry operations of(E2)0do not commute with thetatranslation. The product oftaand a symmetry operationg of(E2)0, i.e.ta.g, generates a symmetry operation of(E2)0/1leaving the middle point of the Pt0–Pt1 segment invariant.
However, the product of a symmetry operation of(E2)0andta, i.e.g.ta, gen- erates a symmetry operation of(E2)−1/0 leaving invariant the middle point of the Pt−1–Pt0segment.
Taking into account the different symmetry operations ofG
We can now generalise the results discussed in Section 6.1.1 for the cyclic system to 1D compounds.
When taking into account the symmetry properties of a cyclic periodic system we must successively consider the following.
• The translational symmetry operations {tma,m=0,±1, . . . ,±(n− 1),n} of the group Tn. These allow the construction of the Bloch orbitals, BOj(k). Only two Bloch orbitals associated with the same k point may interact.
• The symmetry operations of P leaving every cell invariant. These sym- metry operations may be rotations around the 1D axis and reflections related to the symmetry planes containing this 1D axis. Only the Bloch orbitals possessing similar properties with respect to such operations may interact.
• The symmetry operations of P which do not leave invariant every cell.
These symmetry operations may be the inversion with respect to the O point, the reflection with respect to a plane perpendicular to the 1D axis, and theC2rotations with respect to an axis perpendicular to the 1D axis.
Atand X these symmetry operations may be responsible for the non- interaction between two Bloch orbitals, whereas at any otherkpoint it is possible to neglect them.