Point supercell

Such an approximation of periodicity was made for the calculations discussed in the next section (section 4). The supercells for these calculations were composed of either 12 or 32 primitive Li M02 unit cells (M = 3d TM ion 0 < X < 1) that contained various M defects. The lattice parameters of the supercells were kept constant at the parameters for the undetected structure, while the ionic coordinates were allowed to relax. A 2 x 2 x 2 Tr-point mesh was used for the calculations on the 12-unit supercells and a 1x1x1 Tr-point mesh for the 32-unit supercells. The primitive LiJV[02 unit cells used to construct the super cells had previously been calculated with full relaxation of lattice parameters as well as ionic coordinates. 

We now need to define a collection of atoms that can be used in a DFT calculation to represent a simple cubic material. Said more precisely, we need to specify a set of atoms so that when this set is repeated in every direction, it creates the full three-dimensional crystal stmcture. Although it is not really necessary for our initial example, it is useful to split this task into two parts. First, we define a volume that fills space when repeated in all directions. For the simple cubic metal, the obvious choice for this volume is a cube of side length a with a corner at (0,0,0) and edges pointing along the x, y, and z coordinates in three-dimensional space. Second, we define the position(s) of the atom(s) that are included in this volume. With the cubic volume we just chose, the volume will contain just one atom and we could locate it at (0,0,0). Together, these two choices have completely defined the crystal structure of an element with the simple cubic structure. The vectors that define the cell volume and the atom positions within the cell are collectively referred to as the supercell, and the definition of a supercell is the most basic input into a DFT calculation. 

Section 2.1 The cubic Cu calculations in Fig. 2.1 used a cubic supercell with 1 Cu atom, a cutoff energy of 292 eV, and 12x12x12/ points. 

Increasing the volume of a supercell reduces the number of k points needed to achieve convergence because volume increases in real space correspond to volume decreases in reciprocal space. 

If calculations involving supercells with different volumes are to be compared, choosing k points so that the density of k points in reciprocal space is comparable for the different supercells is a useful way to have comparable levels of convergence in k space. 

At this point it may seem as though we can conclude our discussion of optimization methods since we have defined an approach (Newton s method) that will rapidly converge to optimal solutions of multidimensional problems. Unfortunately, Newton s method simply cannot be applied to the DFT problem we set ourselves at the beginning of this section To apply Newton s method to minimize the total energy of a set of atoms in a supercell, E(x), requires calculating the matrix of second derivatives of the form SP E/dxi dxj. Unfortunately, it is very difficult to directly evaluate second derivatives of energy within plane-wave DFT, and most codes do not attempt to perform these calculations. The problem here is not just that Newton s method is numerically inefficient—it just is not practically feasible to evaluate the functions we need to use this method. As a result, we have to look for other approaches to minimize E(x). We will briefly discuss the two numerical methods that are most commonly used for this problem quasi-Newton and... 

For more information on Pulay stress and related complications associated with finite sets of plane waves and k points when calculating forces in supercells with varying volumes, see G. P. Francis and M. C. Payne, /. Phys. Condens. Matter 2 (1990), 4395. 

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. 

Section 4.5 Surface relaxations were examined using asymmetric slab models of five, six, seven, or eight layers with the atoms in the two bottom layers fixed at bulk positions and all remaining atoms allowed to relax. For Cu(100), the supercell had c(2 x 2) surface symmetry, containing 2 atoms per layer. For Cu(l 11), (y/3 X /3)R30 surface unit cell with 3 atoms per layer was used. All slab models included a minimum of 23 A of vacuum along the direction of the surface normal. A 6x6x1 /c-point mesh was used for all calculations. 

Hydron atoms readily dissolve into bulk Pd, where they can reside in either the sixfold octahedral or fourfold tetrahedral interstitial sites. Determine the classical and zero-point corrected activation energies for H hopping between octahedral and tetrahedral sites in bulk Pd. In calculating the activation energy, you should allow all atoms in the supercell to relax but, to estimate vibrational frequencies, you can constrain all the metal atoms. Estimate the temperature below which tunneling contributions become important in the hopping of H atoms between these two interstitial sites. 

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. 

The calculations in Section 7.1.1 examining vacancies in Cu20 used a cubic supercell containing 16 formula units of the compound and 5x5x5 k points. All other details of these calculations were the same as those for the bulk materials in Section 7.1. 

Figure 8.1 Electronic DOS for bulk Ag, calculated using a supercell containing two atoms and sampling k space with 24 x 24 x 24 k points.

Figure 8.4 Electronic DOS of bulk Si (black line) and bulk Si53Au (grey line). Both DFT calculations were performed with a 54 atom supercell and sampled k space with 12 x 12 x 12 k points.

Figure 8.5 Electronic DOS of bulk quartz from DFT calculations with nine atom supercell that sampled k space with 9x9x9 k points. Energy is defined so the top of the valence band is at 0 eV.

Figure 8.6 The DOS of Ag20 calculated with DFT using a six atom supercell and 20 x 20 x 20 k points.

Section 8.3 All calculations for bcc Fe were performed with a cubic supercell containing two atoms, 5x5x5 1 points, and an energy cutoff of 270 eV. 

Specify the numerical details used in performing the calculations. A nonexhaustive list of these details include the size and shape of the supercell, the number and location of the k points, the method and... 

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