Supercell size

Use of the plane wave based electronic structure methods introduces two basic parameters the kinetic energy cutoff value, controlling the basis set quality, and the periodic unit-cell (supercell) size, present due to periodic nature of these approaches. Both of these parameters should be large enough to guarantee the convergence in the total energy and in all the physical quantities that are supposed to be determined from the simulation. 

Table 8.4 Defect formation energy when Ca is substituted for Mg in MgO as a function of supercell size. The number of shells of atoms neighbouring the defect is shown in the second column.

Obviously, computed properties are required to converge with the supercell size. This internal consistency check is important for estimating the interaction between defects in neighboring cells. In fact, two kinds of limitations in the model exist, which correspond to two different levels of complexity ... 

The supercell size must be such as to contain the defect zone, which includes all atoms involved in the structural and electronic relaxation. 

Within the supercell approach, AE" should tend to a well-defined limit with increasing the supercell size ... 

Table 21 Effect of the Supercell Size on the Defect Formation Energy, A (in kj/mol) ...

The convergence of the defect data with the supercell size must be checked again. Table 25 reports the defect formation energies of the Sg, Sig, S32, and Sg4 supercells, with respect to the atomic energies of the species involved in the substitution (Li —7.429609 hartree Mg —199.602732... 

Table 25 Dependence of the MgO [Li]° Defect Formation Energy (in kj/mol) with Respect to the Supercell Size and Geometry Relaxation...

The formation energy obtained with three supercells of increasing size is reported in Table 29, which shows that the convergence of the formation energy with the supercell size is much slower than the one reported for MgO[Li]° (Table 25), where the formation energy is stable already for small supercells. 

The present example shows that it is not difficult to find situations where the supercell size must be large, and the accuracy of the calculations, referring to different supercells, must be high. 

We consider first the convergence of the results with respect to the supercell size. Four supercells with 8 (S4), 16 (Sg), 32 (Sie), and 64 (S32) atoms (the last is shown in Figure 55) are considered, all with the cubic symmetry. As in previous examples, to investigate how far the perturbation propagates, an increasing number of defect neighbors has subsequently been allowed to relax in each supercell. 

These examples show that the supercell approach is an accurate and, in many cases, relatively cheap tool for the study of neutral defects in crystalline systems, once properly gauged with respect to supercell size. 

The first term remains constant with the sample size because it only depends on the concentration and supercell size (of course, the correction defined by Equation 11.11 can also be applied here), and only the second term should be converged with respect to sample size. Even after this correction, free energies and entropies tend to show slower convergence with sample size than the energy. Therefore it is always important to check convergence with respect to supercell size for the thermodynamic potential of interest. We note that the differences in free energies of two macroscopic states, however, converge much faster. 

The one-to-one correspondence was demonstrated between a fixed fc-mesh and the supercell in a real space, see Ekpiations (4.77), (4.80) and (4.84). Due to the one-to-one correspondence between fc-point sampling and the supercell size in a real space it is possible to find such a fc-mesh that ensures a compromise between its size and a reasonable reproduction of the total- and one-electron energies, as well as the electron-density distribution in the host crystal. At this stage, the fc-point sets satisfying (4.84) are used. 

HF and CNDO calculations were done to investigate convergence with respect to supercell size. All basis atoms, except those involved in a defect, occupy lattice sites. The CNDO results show that the defect-formation energy changes by only 0.3%... 

Cl, Br impurities in Ti02 [720,721) without preliminary symmetry analysis of the cyclic clusters chosen. Such an analysis was made in [683] and allowed the convergence of the supercell UHF LCAO calculations of V-doped rutile with the supercell size to be investigated. This first nonempirical superceU LCAO calculation of the defective rutile crystal is discussed here in more detail. 

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