10.6: k-point sampling

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. $b > k$).