Monkhorst Pack method

Unless otherwise stated, all of our calculations used the generalized gradient approximation as defined by the Perdew-Wang 91 functional. In shorthand, we used PW91-GGA calculations. Unless otherwise stated k points were placed in reciprocal space using the Monkhorst-Pack method. Some specific details for the calculations from each section of the chapter are listed below. 

TABLE 3.2 Results from Computing the Total Energy of fee Cu with M x M x M k Points Generated Using the Monkhorst Pack Method... 

Section 3.5.1 Calculations for molecular N2 and CO2 used a cubic supercell of side length 10 A, with reciprocal space sampled using 3x3x3 k points placed with the Monkhorst-Pack method. The energy cutoff for these calculations was 358 eV. 

In Chapter 3, we discussed choosing k points for integrating over the Brillouin zone. The Monkhorst-Pack method for choosing k points may be used when... 

All calculations in Section 7.1 used the PW91 GGA functional. Bulk Ag and Cu were treated with cubic supercells containing four atoms, while the cubic supercells for bulk Ag20 and Cu20 contained six atoms. For each bulk material, reciprocal space was sampled with 10 x 10 x 10 points placed with the Monkhorst-Pack method. Calculations for 02 used a 10 x 10 x 10 A supercell containing one molecule and 3x3x3 k points. Spin polarization was used in the calculations for 02, but spin has no effect on the bulk materials considered. The energy cutoff for all calculations was 396 eV. 

All results in this chapter were obtained from calculations with the PW91 GGA functional. The Monkhorst-Pack method for sampling k space was used in all cases. 

DFT method SIESTA, GGA-PBE, 2x2 unit cell, four-layer slab, 5x5x1 k-points in the Monkhorst-Pack grid and energy cutoff of200 Ry. Corrected for thermodynamic... 

The electronic structure for the MgO crystal was calculated in [608] both in the LCAO approximation and in the PW basis. In both cases the calculations were done by the density-functional theory (DFT) method in the local density approximation (LD A). The Monkhorst Pack set of special points of BZ, which allows a convergence to be obtained (relating to extended special-points sets) in the calculations of electronic structure, was used in both cases. For the LCAO calculations the Durand Barthelat pseudopotential [484] was used. In the case of the PW calculations the normconserving pseudopotential and a PW kinetic energy cutoff of 300 eV were used. 

Pack and Monkhorst °° have suggested that a commensurate grid of points is a suitable option for this purpose. In their method, the grid size depends on a parameter, the shrinking factor s, that specifies how many equidistant k points must be taken along each direction of bi, hx, and bs inside one reciprocal lattice unit cell so that the total number of points in the grid, s, is equal to s , with n denoting the order of periodicity ( = 3 for three-dimensional crystals). 

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. 

The Brillouin zone (BZ) k-integration is performed by the special points method, suitably generalized for distorted lattices The set of undlstorted k-points (Monkhorst and Pack,... 

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