Monkhorst-Pack scheme

Brillouin zone integrations, pervasive in any solid-state calculation, are best performed with the Monkhorst-Pack scheme [6]. These integrations are essentially equally demanding in any representation and with any basis set. This satisfies criterion 5. 

All calculations in this chapter used the PBE GGA functional. For calculations related to Cu surfaces, a cutoff energy of 380 eV and the Methfessel-Paxton scheme was used with a smearing width of 0.1 eV. For calculations related to Si surfaces, the cutoff energy was 380 eV and Gaussian smearing with a width of 0.1 eV was used. The k points were placed in reciprocal space using the Monkhorst-Pack scheme. For all surface calculations, the supercell dimensions in the plane of the surface were defined using the DFT-optimized bulk lattice parameter. 

As is seen, the faster convergence takes place for the modified Monkhorst-Pack schemes. For comparison, the experimental 13.0 eV and HartreeMA>ck 9.0 eV may be found in [94]. 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

Secondly, there exists several techniques for integrating approximately over the k-points of the first Brillouin zone (BZ). For materials with ully occupied bands (e.g., semiconductors) the special points method is by far the most efficient (Chadi and Cohen, 1974 Monkhorst and Pack, 1976). The method appeals to the tight-binding picture of atomic interactions, integrating a definite number of interactions exactly with a suitably chosen set of k-points. For metallic systems it is necessary to exhaust the irreducible BZ with a fine mesh, and to choose a method of assigning occupation numbers to the electron states. Several methods prevail, and we refer to Fu and Ho (1983) for a detailed comparison of two schemes. 

---