Bulk supercell model

Fig. 16.5 (a) Supercell models for bulk anatase (96 atoms) and (b) partial geometry model for three carbon impurities in the anatase supercell. The yellow spheres represent 0 atoms, the small brown spheres represent Ti atoms, and the black spheres represent the carbon impurities. Adapted from Pacchioni etal. [76] with kind permission from the American Chemical Society (2005). 

Figure 49 shows the band structure of the F-center in LiF for the Sig-supercell model obtained at the UlTF level along with the band structure of bulk LiF. Alpha and beta electrons are described by different sets of orbitals. Two band structures are obtained for the a- and p-spin states. The shape of the bands is similar to those of the perfect system, but a new band appears in the... 

The converged FES at T = I m of a (7 x 7 x 12)-supercell model containing 2352 atoms is shown in Fig. 3.4. The two basins associated with the bulk solid and bulk liquid phases have equal minima, as is expected to occur at Tm- In contrast, the... 

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. 

Figure 55 A 64-atom supercell (S32) model of a carbon impurity in bulk silicon.

The defect-free hydroxylated NiO(l 11) structure wtis modeled by aNiO slab covered by an outmost layer of hydroxyl groups similar to that described earlier (Fig. 7.5), except that it contained four NiO bilayers instead of three in order to take into account the H—h- antiferromagnetic state of (lll)-oriented NiO. A (2x2) supercell allowed varying the halide (X) coverage from 25 to 100%. Adsorption and subsurface insertion were modeled using the same protocol as described earlier (Fig. 7.5). The structural optimization was performed at 0 K. The atomic positions of the two upper NiO bilayers were allowed to relax in the X-, y-, and z-directions, and those of the two lower bilayers were frozen to mimic the bulk. The cell parameters were frozen to the bulk values. The energies of substitution/insertion of the halides were calculated using Equation (7.2). 

The theoretical chemistry community developed density functional theory for finite molecular systems which involve molecules and cluster models that describe the catalytic systems. They use the same constructs used in many ab initio wavefunction methods, i.e. Gaussian or Slater basis sets. The solid-state physics community, on the other hand, developed density functional theory to describe bulk solid-state systems and infinite surfaces by using a supercell approach along with periodic basis functions, i.e. plane waves . Nearly all of our discussion has focused on finite molecular systems. In the next section we will describe in more detail infinite periodic systems. 

To eliminate the results dependence on Brillouin-zone samphng the dense Monk-horst-Pack fc-point mesh was used. For primitive unit cells the 12 x 12 x 12 and 12 x 12 special-point sets have been taken for bulk and slab calculations, respectively. In the case of 3D-slabs the number of points in the third fe-direction depends on the chosen value of the c translation vector in direct space. The latter was chosen to provide a similar size in aU directions of the corresponding cycUc model of the crystal (the crystal is composed of equidistant superceUs). The increasing of the 3D unit cell in direct space for producing the 2D supercell was accompanied by the corresponding reduction in the fe-points mesh in the reciprocal space. 

DFT-PW calculations were performed on different H2O Ti02 structures to determine which one corresponds to the most stable arrangement on the rutile (110) surface. To this end, a 3D-supercell consisting of 1 x 1 or 2 x 1 surface unit cells was used to model the (110) surface geometry. The smallest surface unit cell was chosen for hare-surface calculations, having dimensions of c (2.959 A) and /2 a (6.497 A) in the (001) and (-110) directions, respectively, where a and c are translation vectors for the bulk rutile unit cell. This surface unit ceU is doubled in the (001) direction for the hydroxylated or hydrated surfaces. The calculations were performed for 3-layer slabs with a total cell thickness 19 A, i.e. slab thickness + vacuum gap 10 A. A model with the fixed atomic positions of the central layer is assumed to be more appropriate to the real 2D surface relaxation because real crystals are not thin films and the bulk crystal structure probably exists a few atomic layers beneath the mineral... 

---