Supercell transformation

Matrices of the Symmetrical Supercell Transformations of 14 Three-dimensional Bravais Lattices... 

Reciprocal Matrices of the Symmetric Supercell Transformations of the Three Cubic Bravais Lattices... 

When a defect is introduced into a crystalline environment, crystal translational symmetry can no longer be invoked to transform the problem into tractable form as is done in band-structure calculations. Most of the computational treatments of defects in semiconductors rely on approximations to the defect environment that fall into one of three categories cluster, supercell (or cyclic cluster), and Green s function. 

Figure 1 The topological transformation of ice XI (at 50 kbar) to ice II with 02, easy directions displayed as arrows for hydrogen and O—O BCPs, (a) ice XI at 50 kbar, (b) supercell of ice II (most of O—O BCPs removedfor clarity).

Figure 4 The topological transformation of ice II to ice IX. (a), supercell of ice II highlighted ice IX-like structure (b) Ice II unit cell rotated to emphasize ice IX-like structure.

We use a periodic supercell model based on the large unit cell (LUC) method [22] which is free from the limitations of different cluster models applicable mainly to ionic solids, e.g., alkali halides. The main computational equations for calculating the total energy of the crystal within the framework of the LUC have been given in Refs. [22-24]. Here, we shall outline some key elements of the method. The basic idea of the LUC is in computing the electronic structure of the unit cell extended in a special manner at k = 0 in the reduced Brillouin zone (BZ), which is equivalent to a band structure calculation at those BZ k points, which transform to the reduced BZ center on extending the unit cell [22]. The total energy of the crystal is... 

Much more flexible is to describe the solute-solvent system within the supercell technique employing a set of 3D plane waves. The convolution in the OZ equation can be then approximated by means of the 3D fast Fourier transform (3D-FFT) procedure. For the first time, this approach was elaborated by Beglov and Roux [16] for numerical solution of the 3D-OZ integral equation with the hypernetted chain (HNC) closure for the 3D distribution of a simple LJ solvent around non-polar solutes... 

Much as for the ID solvent-solvent correlations, the renormalization of the 3D-RISM/HNC equations is not necessary in respect to convergence. Nor is it required for the 3D fast Fourier transform (3D-FFT) employed to evaluate the convolution in Eq. (4.A.41). For a periodic solute neutralized by a compensating background charge, the Coulomb potential of the solute charge is screened at a supercell length. Therefore the 3D site direct, and hence total correlation functions, are free from the Coulomb singularity at fc = 0 and can be transformed directly by the 3D-FFT. 

In our work, we first constructed a large supercell from the primitive cell, then perturbed each atom in the original I = (0, 0, 0) cell by a small amount along the coordinate axes and calculated the forces on the atoms in the perturbed supercell. We obtained an approximate force constant matrix by the numerical differentiation of the forces, which we Fourier-transform to obtain the dynamical matrix. This procedure can be performed using any interatomic potential model, although using... 

The character of the locahzation of Wannier functions depends on the analytical properties of Bloch states (as a function of the wavevector) that are essentially determined by the nature of the system under consideration. One can arbitrary change only the form of an unitary transformation of Bloch functions. It is just this arbitrariness that is used in the variational approach [42] to assure the best localization of Wannier functions. The accuracy of the Wannier functions obtained by the proposed method is determined solely by the accuracy of the Bloch functions and the size of the supercell used. As the calculations have shown, the proposed method is reUable and useful in the problem of generation of the locaUzed Wannier functions. In the two examples... 

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