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28 CHAPTER2. COMPUTATIONALMETHODS where VBZ is the volume of the first BZ and fik is the number of electrons in state ik (the occupation number, 0 ≤ fik ≤ 1). The sum is evaluated over i bands, and the integral over the first BZ is an integral over all k-points. Another example is the kinetic energy, which in a periodic KS scheme can be evaluated as[135] ˆˆ1 E [{ψ}] = ψ∗ (r) − ∇2 ψ (r)drdk (2.26) kin,s ∑ikrik 2ik The range of k-points in the first BZ is of course continuous, but an integral over an infinite number of k-points (not to mention the additional sum over all i bands) is not practical to evaluate. However, one can make use of the fact that wave functions ψik at k-points spaced only a short distance apart in the first BZ are very similar. Then, the integral over all k-points can be approximated as a weighted sum over a number of k-points. Using more k-points gives an improved approximation of the integral. There are several methods that attempt to find ways of selecting (or sampling) the k-points efficiently, to reduce the number of k-points needed. One method is that of Monkhorst and Pack [172]. The number of k-points needed also depends on the volume of the first BZ. As the BZ spans reciprocal space, an increased real space volume of a unit cell will yield a smaller BZ. Therefore, fewer k-points will be needed to yield the same spacing between points in recip- rocal space. A doubling of the real unit cell size will thus mean that half as many k-points need to be sampled for maintained accuracy of description. However, the number of k-points that needs to be sampled can also be reduced by using symme- tries in the BZ. From a computational cost point of view, it is more advantageous to use more k-points rather than a larger unit cell, since the cost scales linearly with k-points but (usually) more than linearly with increasing the number of atoms. Surfaces are commonly treated in a periodic picture. The surface is then modeled as a slab of e.g. four layers of atoms which is separated by vacuum from its own image in the neighboring cell. Usually e.g. the bottom layers of the slab are fixed to the bulk geometry, and the number of layers of atoms in the slab is high enough to reach convergence in the property of interest. The slabs must be separated by a large enough vacuum distance so that they do not interact. The smallest distance that can be used is found by performing several calculations with larger and larger slab distances and seeing at which distance the property of study (e.g. an adsorption energy) no longer changes. The distance between the slabs is then said to be converged. Treatments based on the Bloch theorem can also be applied to study defects (such as vacancies or interstitial atoms) in a material or on a surface. In a similar way as when using periodic slabs to model surfaces, it is important that the unit cell is chosen to be large enough that the defect in a unit cell is located far away from its own repeated image so that the defects do not interact, if that is the situation to be modeled.PDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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