10.6: k-point sampling

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DFT reduces the problem of \(10^{26}\) interacting electrons to \(10^{26}\) noninteracting quasiparticles. To reduce this to a manageable number, we recall that electrons in solids can be labelled by a wavevector \({\bf k}\), and that they form bands in which electrons with similar \({\bf k}\) have similar energy. The energy is the integral of these, thus we can obtain a good estimate by sampling states from an evenly-spaced grid of “\({\bf k}\)-points”. As this grid becomes finer, so the accuracy of the integral improves. For most systems a surprisingly small number suffices: tens for insulators and hundreds for metals.

According to the Bloch theorem, any wavefunction must be written:

\[\Phi_{\bf k} = u({\bf r}) \text{ exp } −i{\bf k.r} \nonumber\]

If the wavefunction is expanded in plane waves, then

\[\Phi_{\bf k} = \sum_{\bf b} \text{ exp } −i({\bf k} + {\bf b}).{\bf r} \nonumber\]

where \(k\) correspond to Bloch waves longer than the unit cell, and \(b\) to basis function plane waves shorter than the cell (i.e. \(b > k\)).