6.4 Band structure methods and basis functions 133Γ X Γ X
–6–4–20246–6–4–20246Figure 6.6. Energy bands for a(12, 0)and a(13, 0)tube. In reality, both have an
energy gap. That of the even tube is very small. It is absent in the present analysis
owing to the neglect of tube curvature.6.4 Band structure methods and basis functions
Many band structure methods exist and they all have their own particular features.
A distinction can be made betweenab initiomethods, which use no experimental
input, andsemi-empiricalmethods, which do. The latter should not be considered
as mere fitting procedures: by fitting a few numbers to a few experimental data,
many new results may be predicted, such as full band structures. Moreover, the
power ofab initiomethods should not be exaggerated: there are always approx-
imations inherent to them, in particular the reliance on the Born–Oppenheimer
approximation separating the electronic and nuclear motion, and often the local
density approximation for exchange and correlation.
Inab initiomethods, the potential is usually determined self-consistently with
the electron density according to the DFT scheme. Some methods, however, solve
the Schrödinger equation for agiven, cleverly determined potential designed to give
reliable results. The latter approach is particularly useful for nonperiodic systems
where many atoms must be treated in the calculation.
In a general electronic structure calculation scheme we must give the basis func-
tions a good deal of attention since we know that by cleverly choosing the basis
states we can reduce their number, which has a huge impact on the computer time
needed as the latter is dominated by theO(N^3 )matrix diagonalisation.
Two remarks concerning the potential in a periodic solid are important in this
respect. First, the potential grows very large near the nuclei whereas it is (relatively)
small in the interstitial region. There is no sharp boundary between the two regions,
but it is related to the distance from the nucleus where the atomic wave function