Reciprocal Space and k Points

Our first foray into the realm of numerical convergence takes us away from the comfortable three-dimensional physical space where atom positions are defined and into what is known as reciprocal space. The concepts associated with reciprocal space are fundamental to much of solid-state physics that there are many physicists who can barely fathom the possibility that anyone might find them slightly mysterious. It is not our aim here to give a complete description of these concepts. Several standard solid-state physics texts that cover these topics in great detail are listed at the end of the chapter. Here, we aim to cover what we think are the most critical ideas related to how reciprocal space comes into practical DFT calculations, with particular emphasis on the relationship between these ideas and numerical convergence. 

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Therefore k is generally restricted (hence referred to as the reduced wave vector) to a region of k space such that no two points in this region are separated by any vector K. This is a unit cell in reciprocal space, and is referred to as the first Brillouin zone. 

In the theoretical description of regular polymers, the monoelectronic levels (orbital energies in the molecular description) are represented as a multivalued function of a reciprocal wave number defined in the inverse space dimension. The set of all those branches (energy bands) plotted versus the reciprocal wave number (k-point) in a well defined region of the reciprocal space (first Brillouin zone) is the band structure of the polymers. In the usual terminology, we note the analogy between the occupied levels and the valence bands, the unoccupied levels and the conduction band. 

Virtually everything that exists or happens in real space has a corresponding property or effect in diffraction space, and vice versa. The correspondences are established through the Fourier transform, which, as we have seen, operates symmetrically in both directions, getting us from real space into reciprocal space and back again. It may occasionally appear that this rule is violated, but in fact it is not. For example, the chirality of molecules and the handedness of their arrangement in real space would seem to be lost in reciprocal space as a consequence of Friedel s law and the addition of a center of symmetry to reciprocal space. If, however, we could record phases of reflections in reciprocal space, we would see that in fact chirality is preserved in phase differences between otherwise equivalent reflections. The phases of Fhu, for example, are 0, but the phase of F-h-k-i are —0. Fortunately the apparent loss of chiral information is usually not a serious problem in the X-ray analysis of proteins, as it can usually be recovered at some point by consideration of real space stereochemistry. 

No data are reported for s < 4 in Table 1 for a reason. This reason is connected with the tight correlation of the sets of g direct lattice vectors and k reciprocal space points selected in the calculation, when using a local basis set. Iterative Fourier transforms of matrices from direct to reciprocal space, like in Eq. [36], and vice versa (Eq. [38]), are the price to be paid for the already mentioned advantage of determining the extent of the interparticle interactions to be evaluated in direct space on the basis of simple criteria of distance. Consequently, the sets of the selected g vectors and k points must be well balanced. The energy values reported in Table 1 were all obtained for a particular set of g vectors, corresponding to the selection of those AOs in the lattice with an overlap of at least 10 with the AOs in the 0-cell. This process determines the g vectors for which F , S , and the I matrices (Eqs. [34] and [38]) need to be calculated, and if the number of k points included in the calculation is too small compared with the number of the direct lattice vectors, the determination of the matrix elements is poor and numerical instabilities occur. 

If we now move from real space into reciprocal space, the Brillouin zone associated with the crystal lattice is also hexagonal and it shows characteristic high-symmetry points the centre is called F point, while two consecutive corners are denoted as K and K points. Fig. 2b. 

We have omitted the question of boundary conditions. The use of periodic boundary conditions on a finite solid composed of N atoms, which restricts k to a fine grid of points in reciprocal space, and the taking of the infinite limit N - co, is discussed in Refs. 15 and 16. Whenever N is used in the text this limit is implied, t Here and elsewhere u is normalized to unity in the Wigner-Seitz cell. 

Section 2.3 The hep Cu calculations in Fig. 2.4 used a primitive supercell with 2 Cu atoms, a cutoff energy of 292 eV, and 12x12x8 k points placed in reciprocal space using a f-centered grid. 

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. 

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. 

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. 

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