3.4 Comparison of the regular H n and
3.4.1 Comparison of the band structures of the regular H n
It is now appropriate to look in more details at the relationship between the band structures of the Hnsystem in its regular Hn(see Fig. 3.5) or dimerised (H2)n (see Fig. 3.16) forms. For this purpose we will start by studying the regular system generated in two different ways: first, using a simple repeat unit made of one hydrogen atom and a translation vectora(see representation 1 in Fig. 3.26); second, using a double repeat unit made of two hydrogen atoms and a translation vectora=2a. In the latter case we will arbitrarily distinguish the two equivalent hydrogen atoms of the double cell with indices andr, for ‘left’ and ‘right’, respectively (see representation 2 in Fig. 3.26).
These two approaches will yield two different band structures describing the same system. We will then show that the two schemes are of course equivalent.
Following our treatment in Section 3.2 for the regular Hn chain (see eqn (3.15)), the crystal orbitals CO(1)(k) of the regular chain generated by the simple repeat unit are given by:
CO(1)(k) = 1
√n
n
m=−n+1
exp(i2πmka)1sm
with ka∈
−1 2,1
2
(3.39) The index (1) in crystal orbital CO(1)(k)is a reminder of the fact that this crystal orbital is obtained in the first representation. A representation of this crystal orbital is given in Fig. 3.27. When the chain is generated by the double repeat unit, we will label the crystal orbitals with the index (2). In this second representation, the crystal orbitals result from the interaction of the Bloch orbitals BO(2)(k) and BO(r2)(k) generated by the 1sorbitals centred on either the Hor Hr atoms (see Fig. 3.26b wheren= n4). These crystal orbitals will be denoted CO(±2)(k).
Fig. 3.26
Regular Hnchain generated by a repeat unit of one hydrogen atom
(representation 1) or by a double repeat unit of two hydrogen atoms denoted H and Hr(representation 2).n=n/2 and n=n/2.
Fig. 3.27
Crystal orbital for the hydrogen chain generated in the first representation.
Fig. 3.28
Bloch orbitals for the hydrogen chain obtained in the second representation.
BO(2)(k) = 2
n
n
m=−n+1
exp(i2πmka)(1s)m ka ∈] −1 2,1
2]
BO(r2)(k) = 2
n
n
m=−n+1
exp(i2πmka)(1sr)m ka ∈] −1 2,1
2] (3.40) A representation of the two Bloch orbitals is shown in Fig. 3.28.
The crystal orbitals may be obtained using results derived in Section 3.3.1 (i.e. eqns (3.34) and (3.37)) for the case in whichβi =βe=β, since the chain is regular. In this case, the ratio of coefficients is:
c±(k)
cr±(k) = β+βexp(−i2πka)
±β√
2(1+cos(2πka)) = ±exp(−iπka) (3.41) so that the normalised expression (see eqn (3.36)) for the crystal orbitals as a function of the Bloch orbitals is:
CO(±2)(k) = 1
2
BO(k) ±exp(iπka)BOr(k)
(3.42) Since the a vector is 2a, the a∗ vector is a∗/2 (eqns (3.5) and (3.30)).
A vector kis characterised in the first representation by the real numberka
and in the second representation by the real number ka, which is given by 2ka (k=kaa∗=2kaa∗=kaa∗).
N.B.In order to make the comparison of the crystal orbitals obtained in the two representations(eqns (3.39)–(3.42))easier, the crystal orbitals of the second representation, shown inFig. 3.29, are given in terms of the real number ka
instead of ka.
Since the two representations describe exactly the same system, there must be a clear link between the two series ofncrystal orbitals. For every value of ka in the interval ]−14,14] it is easy to see that the crystal orbitals CO(1)(ka) (see Fig. 3.27) may be identified as the CO+(2)(ka)ones (see Fig. 3.29). Thus the following relation holds:
CO(1)(ka)=CO(+2)(ka) with ka∈
−1 4,1
4
(3.43)
Fig. 3.29
Crystal orbitals obtained in the second representation in terms ofka, which is equal toka/2.
In order to find the relationship between the CO(−2)(ka)crystal orbitals and the orbitals of the first representation, we will integrate the negative signs of the coefficients (see Fig. 3.29) within the exponentials by using the following equivalences:
−exp(i2πka)=exp
i2π(ka+1 2)
=exp
i2π(ka−1 2)
(3.44) or more generally
(−1)mexp(i2πmka)=exp
i2π(ka+1 2)m
=exp
i2π(ka−1 2)m
(3.45) Now we can rewrite the CO(−2)(k) crystal orbitals in two different ways, as shown in Fig. 3.30. Taking into account that in the first representationkavaries in the interval ]−12, 12], it is possible to identify the CO(−2)(k) crystal orbitals with those of the first representation in the following way:
if ka∈ ]0,1
4] then CO−(2)(ka)=CO(1)(ka) with ka =ka−1
2
ka ∈ ] − 1 2,−1
4]
if ka∈ ] −1
4,0] then CO−(2)(ka)=CO(1)(ka) with ka =ka+1
2
ka ∈ ]1 4,1
2]
(3.46)
Fig. 3.30
Representation of the CO(2)−(k)crystal orbitals.
structures is made clear by means of the different numbers in parentheses. In the two cases we have represented the crystal orbital energies as a function of the real numberkain such a way thatk=kaa∗, i.e. by means of a vector of the reciprocal space associated with the Hnsystem in representation 1 (simple repeat unit).
This is a general result, which does not depend on the computational method used to estimate the energies. It is a consequence of the periodic behaviour of the crystal orbitals. We can now compare the band structures obtained for the two representations (see Fig. 3.31).
According to Fig. 3.31 and eqns (3.43) and (3.46), there is a geometric correspondence between the different lines of these two band structures. For instance, the band denoted (2) in the second band structure (ka∈ ]–14,0]), may be found in the first band structure as the part of the band denoted (2) in the interval (ka∈ ]14,12]).
When the periodicity of the chain is artificially doubled it is said that the band structure must be ‘folded’. This is often taken as if the corresponding bands were folded back around the dotted lines centred at thekavalues of14 and−14. However, this is not conceptually correct as illustrated in Fig. 3.31.
The new periodicity of the system in representation 2 is associated with a new Brillouin zone, which is half that of representation 1. Consequently, the single band of representation 1 must become two different bands in representation 2. The folded upper band originates from a translation by
±a∗/2 of the original band (Fig. 3.31a), which in that way is projected into the new, smaller, Brillouin zone (Fig. 3.31b).
The approach followed in this section may at first sight look artificial and useless. Why should we use a double unit cell if we can use a smaller, and thus, simpler one? As we will repeatedly see in this book, it is often very useful to determine the band structure of a system assuming a higher symmetry in order to understand the subtleties leading to the apparently more complex shape of the band structure for a real system with lower symmetry. For the time being, this will lead us to understand the origin of the stabilisation provided by a structural dimerisation for the Hnchain.