The results obtained for the regular(H0.5+)nmodel system may be generalised for any 1D conductor. We will now show that any tetramerisation of the regular
(H0.5+)nsystem,4whatever its nature, must be accompanied by a modulation
4In fact this section is completely gen- eral and could be directly applied to any (Hx+)n system and generalised to any 1D system.
of the electronic density of the system. In order to do so, we will analyse the electronic density associated with the CO1(k1)and CO2(k1)crystal orbitals (see Fig. 4.7) of the distorted system. [2]
To begin with let us decompose the Hamiltonian of the system, Hˆ, into a zero-order Hamiltonian describing the regular system, Hˆ0, and a coupling term,Vˆ, describing the coupling between the electronic motion and a vibration allowing the appearance of a tetramerisation. This coupling term allows the interaction between crystal orbitals of the regular system associated with a givenk1 vector of the band structure in Fig. 4.6. We will only consider the interaction between crystal orbitals CO◦1(k1)and CO◦2(k1)(i.e. points A and B).
The mathematical expressions for these orbitals can be easily obtained by exploiting the equivalence between the band structures in Figs 4.6a and 4.6b.
In fact, we are looking for the expressions of the crystal orbitals labelled A and B in Figs 4.6a and b. The CO◦1(k1)orbital, denoted A in Fig. 4.6a, is given by eqn (3.4), giving the crystal orbitals of the regular Hn system generated by a single cell:
CO◦1(k1)=
n
m=−n+1
exp(ik1.rm)1sm
=
n
m=−n+1
exp(i2πka1(m−0.5))1sm (4.1)
The last simplification results from the fact that the rm vector characterises the position of the (1sm)orbital with respect to the origin O located, in the present case (see Fig. 4.5a), in between cells M0and M1(rm =(m−0.5)a).
Likewise, the CO◦2(k1)orbital (point B) may be expressed as the crystal orbital of the regular Hnsystem generated by a single cell (eqn (3.4)) associated with pointka2 = −2kaf +ka1 (see Fig. 4.6a).
CO◦2(k1)=
n
m=−n+1
exp(i2π(ka1−2kaf)(m−0.5))1sm (4.2)
The coupling termVˆ, which is responsible for the tetramerisation, allows the interaction of the two orbitals. After the interaction, the crystal orbitals of the regular form, CO1(k1)and CO2(k1), resulting from the interaction of CO◦1(k1) and CO◦2(k1), may be written in the following non-normalised form:
CO1(k1)=CO◦1(k1)+γexp(−iφγ)CO◦2(k1)
CO2(k1)=γexp(iφγ)CO◦1(k1)−CO◦2(k1) (4.3) whereγexp(iφγ)is a non-nil complex constant depending onka1.
Let us consider the electronic density associated with these two crystal orbitals:
| ( )| =γ | 1( )| + | 2( )|
−2γRe
exp(−iφγ)
CO◦1(k1)∗
CO◦2(k1)
(4.4) If we develop the term|CO◦1(k1)|2by means of eqn (4.1):
|CO◦1(k1)|2=
n
m=−n+1
⎛
⎝ n
m=−n+1
exp i2πka1(m−m) 1sm1sm
⎞
⎠ (4.5)
we can associate with each hydrogen atom the same electronic density,dm1(k1), given by:
dm1(k1)=
n
m=−n+1
exp i2πka1(m−m)
1sm1sm (4.6) Now if we only keep the leading diagonal terms, in other words, if we assume that(1s)m(1s)m =(1s)2mδm,m, we reach the following expression:
dm1(k1)=(1sm)2 (4.7) A similar result is obtained for |CO◦2(k1)|2. Consequently, the electronic density due to the first two terms in eqn (4.4) is uniformly distributed over the different atoms. Let us now turn our attention to the third term of eqn (4.4):
2γRe
exp(−iφγ)
CO◦1(k1)∗
CO◦2(k1)
=2γRe
⎡
⎣exp(−iφγ)
n
m,m=−n+1
exp i2πka1(m−m)
−4iπkaf(m−0.5) 1sm1sm
(4.8)
If we only keep the leading terms we obtain the following expression:
2 γRe
exp(−iφγ)(CO◦1(k1))∗CO◦2(k1)
=2γRe
⎡
⎣ n
m=−n+1
exp(−i4πkaf(m−0.5)−iφγ)(1sm)2
⎤
⎦
=2γ
n
m=−n+1
cos 4πkaf(m−0.5)+φγ
(1sm)2 (4.9)
Fig. 4.11
Schematic representation of the electronic distribution among the difference centres of the crystal orbitals of the distorted system. This
representation applies to the case where φγ is nil andγis positive.
Denoting the contribution of the third term of eqn (4.4) to the electronic density associated with the hydrogen atom of cell Mm asdm3(k1),we obtain:
dm3(k1)=2γcos(4πkaf(m−0.5)+φγ)(1sm)2 (4.10) This contribution is not the same for every atom and is at the origin of a modu- lation of the electron density with periodicitya/2kaf. According to eqn (4.4), a maximum in the electronic density associated with CO1(k1)corresponds to a minimum in the electronic density associated with CO2(k1)and vice versa. In Fig. 4.11 we show the electronic density associated with the crystal orbitals of the regular system, CO◦1(k1)and CO◦2(k1), as well as with those of the distorted system, CO1(k1)and CO2(k1).
Since only one of these two orbitals is occupied in the distorted system (see Fig. 4.7), the electronic density will exhibit a modulation along the chain (see Fig. 4.12 for a different representation).
We have just shown that, for the (H0.5+)n system, any tetramerisation, whatever its nature, stabilises the crystal orbitals lying at or just below the Fermi level because it allows the interaction between filled orbitals near
Fig. 4.12
A different representation of the distribution of the electronic density among different centres before and after a tetramerisation.
that the regular system is unstable with respect to a charge density wave of a/2kaf periodicity. The kind of tetramerisation that is observed experimen- tally optimises electronic stabilisation without increasing the energy terms associated with the nucleus-nucleus and/or electron-electron repulsions too much.