Ten-fold speed up of DFT: Improving k-point integration

The amount of recent cpu time ($>100$ mega cpu hours) spent in our group on high-throughput materials prediction led us to re-examine convergence issues in standard DFT calculations. For total energy calculations, k-point convergence can be increased two-fold and ten-fold for semiconductor and metals, respectively. For semiconductors, the popular ``rectangle approximation method'' using Monkhorst-Pack grids convergences much faster than expected (for reasons that will be explained), which explains why it gained popularity in the early development of DFT codes. (Its simplicity is also a likely factor.) However, it is not possible to adapt the method to the case of partially-occupied bands in metals while preserving the rapid convergence of semiconductors. Using a rectangle rule for metals, irrespective of any smearing method, leads to the well-known problem that convergence rates are 100s times worse than for semiconductors. Revisiting the k-point integration issue in light of modern DFT practice, we demonstrate that this ``metal deficit'' can be reduced to only a factor of 5--10 worse than semiconductors. The further complication of integrating inside the Fermi surface for metals is solved with our approach without the need for smearing and its associated ad hoc parameters.