7.3 General remarks concerning Peierls distortions
8.1.1 Band structure of the eclipsed chain [Pt(CN) 4 ] (2−δ)−
Symmetry
As shown in Section 6.1.2, the space group of this chain may be written as a direct product Tn⊗D4h, the D4h group leaving the platinum atom of the reference cell invariant. Consequently, the appropriate group for theand X points is D4h, while it isC4vfor the otherkpoints.
Choice of fragment orbitals generating the Bloch orbitals
Let us start our analysis by generating the Bloch orbitals of the system from the Pt(CN)(42−δ)−fragment orbitals adapted to theC4vsymmetry, which must hold for anykpoint. For instance, we can build a basis of Bloch orbitals from the orbitals of the metal complex Pt(CN)24−, which is of D4h symmetry and thus well adapted to the C4v symmetry. The mainly metal-based orbitals of
Fig. 8.2
Molecular orbitals with strong metal component (a) of the octahedral Pt(CN)4−6 ; (b) of the square-planar Pt(CN)2−4 compound. For clarity, the ligand contributions have not been shown. The symmetry labels
corresponding to theC4vgroup, which is the appropriate symmetry group to label the Bloch orbitals generated from these fragment orbitals, are also given. The labelsb,nb,andabdenote the bonding, non-bonding, or antibonding character of the associated orbitals.
this complex may be deduced from those for the Pt(CN)46−octahedral complex shown at the left side of Fig. 8.2.
In the octahedral compound, the orbitals with strong metal character impli- cate either the (5dx2−y2,5dz2) orbitals ofegsymmetry, the (5dx z, 5dyz, 5dx y) orbitals of t2g symmetry, the (6px, 6py, 6pz) orbitals of t1u symmetry, or the 6s orbital of a1 symmetry. If we now remove the two ligands in the O z axis, the square-planar complex is generated. The departure of the two ligands will cause the stabilisation of the molecular orbitals of the octahedral complex that are antibonding along theO zdirection, and a destabilisation of the molecular orbitals that are bonding along this direction. The molecular orbitals of the square-planar complex with symmetry labels corresponding to theC4vgroup are shown at the right side of Fig. 8.2. It is clear that the departure of two axial ligands is associated with a stabilisation of the orbital 5dz2, which becomes almost non-bonding in the square-planar complex because of the loss of metal–ligand antibonding character along the axial direction and the mixing of the 6sorbital which in this geometry is allowed by symmetry. Also worthy of a mention is a strong stabilisation of the molecular orbital denoted 6pz, which in the square-planar complex does not exhibit anyσ metal–ligand antibonding interaction. Finally let us note that in Pt(CN)24− all bonding or non-bonding orbitals of thed-block are filled whereas the strongly antibonding
5dx2−y2 orbital is empty.2In our analysis of the band structure of the chain we
2In this complex, the platinum oxidation state is+II, i.e. its electronic configura- tion isd8.
will consider all orbitals of the 5d block and the almost non-bonding orbital 6pz involving the metal. Why do we need to retain all these orbitals of the square-planar fragment? To begin with we do not wish to skip any step of the reasoning in this case, the first example in which we consider a real chain containing transition metal atoms. In addition, all fragment orbitals that may play a role in the electronic structure must of course be retained. Since we wish to comment on the conductivity of the system, it is mandatory to retain all orbitals of the Pt(CN)(42−δ)−unit that may be involved in strong interactions along the chain direction, i.e. the O z axis. This is why in addition to the d-block orbitals we have retained the empty orbital mostly based on the metal 6pz orbital. Even if this orbital lies relatively high in energy for the isolated fragment, the bottom part of the energy band based on this orbital may occur at energies of the same order, or even lower, than the top of some bands based on the 5dblock orbitals.
Analysis of the Bloch orbitals for theand X points
We will start by looking at the Bloch orbitals for theand X points generated by the fragment orbitals retained. We will characterise the symmetry of these orbitals with respect to the D4h group that leaves the platinum atom of the reference cell invariant, i.e. the symmetry group of the and X points (see Fig. 8.3). Since all these Bloch orbitals possess different symmetry properties,
Fig. 8.3
Bloch orbitals at theand X points. The symmetry labels according to theD4h group are given.
The opposite case will be discussed in Section 8.1.2.
Band structure of the eclipsed chain
Since the symmetry group fork points other thanand X is the C4vgroup leaving every platinum atom invariant, the symmetry label for the different Bloch orbitals is the same as the symmetry label of the fragment orbital from which it originates (Fig. 8.2). Consequently, only the bands generated from the 6pz and 5dz2 fragment orbitals ofa1 symmetry may interact for anyk point other thanand X. All other orbital mixings are precluded by symmetry, so that the Bloch orbitals of symmetry other thana1may be identified with the crystal orbitals of the system.
How can the different bands be positioned on an energy scale? Let us consider first thea1-type bands resulting from the interaction of the BOd
z2(k) and BOpz(k)Bloch orbitals. The energies of the BOdz2(k)and BOpz(k)Bloch orbitals as a function ofka are shown as dotted lines in Fig. 8.4. The energy of the BOdz2(k)Bloch orbital increases whenka goes from 0 to 1/2, while the energy of the BOpz(k) Bloch orbital decreases along the same interval.
Since these orbitals interact at any point other thanand X, the two dotted lines repel each other fromto X giving rise to the continuous lines, which represent the energy of the crystal orbitals resulting from the mixing of the two Bloch orbitals.
Since the crystal orbitals ofb1,b2,andesymmetries coincide with the Bloch orbitals, the shape of the curves representing their energies may be obtained by connecting the energies of the corresponding crystal orbitals at and X through a cosine-type line (see eqn (3.20)). Thus, knowing the energies of the crystal orbitals atand X (Fig. 8.3) as well as the shape and relative energy of thea1-type bands (Fig. 8.4), we may arrive at the band structure of Fig. 8.5 for the eclipsed [Pt(CN)4]nchain. According to this diagram the 1a1and 2a1
bands are considerably wider than the 1ebands which, in turn, are wider than
Fig. 8.4
Energy bands ofa1-type symmetry: the dotted lines are the Bloch orbital energies while the continuous lines are the energies of the crystal orbitals resulting from their interaction.
Fig. 8.5
Schematic band structure for the eclipsed [Pt(CN)4]nchain.
the 1b1and 1b2 bands. These differences originate from the different nature of the overlaps associated with each type of band: very strong (σ) for the 1a1
and 2a1bands, moderate (π) for the 1ebands, and finally very weak (δ) for the 1b1and 1b2bands. In general, a flat band suggests very weak interactions along the chain. However, even if flat, this band contains as many states as any other band, for instance the strongly dispersive ones. The band structure of the system can also be represented in a simplified way as shown in Fig. 8.6.
Fig. 8.6
General band structure for the eclipsed [Pt(CN)4]nchain.
This qualitative approach does not allow us to be completely sure about the relative positioning of all bands. It is important to determine which features of the band structure can be derived from the simple qualitative approach and which aspects can only be firmly established by computation. It follows from the analysis in this section that the band structure must necessarily exhibit the following features:
(i) non-existence of a crossing between the 1a1and 2a1bands (ii) relative band widths in agreement with the overlap involved.
Among the uncertainties we have at this point, the positioning of the bottom part of the 2a1band (essentially 6pzin character) with respect to the very flat 1b1band (essentiallydx2−y2 in character) is not clear from purely qualitative reasoning. This feature depends on the Pt–Pt bond length, which will vary as a function of the degree of doping or as a function of pressure. The reader can verify that the properties of these chains do not depend on the relative position of the 2a1and 1b1bands. If this were not the case, a computation using the actual geometry of the chain would be necessary in order to establish the band structure of the chain for certain.
Electronic structure of the eclipsed chain
At this point we must look at the filling of the different bands for the non- doped K2Pt(CN)4.3H2O compound at T = 0 K. Every unit with formula Pt(CN)24− contributed eight electrons, i.e. its electronic configuration is for- mally(1a1)2(1b2)2(1e)4. This being the case, the four lower bands of Fig. 8.5 must be filled and thus the Fermi level occurs at the top of the 1a1band, which
electrons, since 0.3 electrons have, on average, been transferred to the bromine atoms, i.e. the formal electronic configuration of every square-planar unit is (1a1)1.7(1b2)2(1e)4. The 1a1band is thus only 85% filled, and the Fermi level occurs atkaf = ±0.425= ±0.5∓0.075. Since the 1a1 band is 15% empty and given that a 15% empty band behaves like a 15% filled band as far as a potential Peierls distortion is concerned (see Chapter 4), we can conclude that this system must be unstable with respect to a 6.666-merisation (see Fig. 4.14). Consequently, it is expected that when the temperature is lowered this compound exhibits a Peierls distortion and loses its metallic properties and exhibits a modulation of its structure.
In view of the previous analysis, the hypothetical eclipsed compound, doped as in KCP, must be metallic at high temperature but should undergo a 6.666- merisation at lower temperature, which makes the system semiconducting.