Brillouin special points

Monkhorst H J and J D Pack 1976. Special Points for Brillouin-zone Integration. Physical Review B13 5188-5192. 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

HJ. Monkhorst and J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 5188... 

The Brillouin zone, 104, has some special points labeled in it. There are conventions for this labeling.915 The zone is, of course, three-dimensional. The hand structure (Fig. 37) shows the evolution of the levels along several directions in the zone. Count the levels to confirm the presence of six low-lying bands (which a decomposition of the DOS shows to be mainly S 3p) and 10 V 3d bands. The two S 3s bands are below the energy window of the drawing. At some special points in the Brillouin zone there are degeneracies, so one should pick a general point to count bands. 

There are 24 such points in the Brillouin Zone, but only one need be used since the band energies are the same at the others. The same special point can be used to... 

Chadi and Martin (1976) used essentially the same LCAO parameters that are given in the Solid State Table to obtain the energies at the two special points in the Brillouin Zone. They then redetermined the wave numbers of the special points for the distorted crystal and recalculated the energy. The clastic distortion which they used is a shear strain, in which there is no change in bond length to first order in the strain thus the radial force constant ofEq. (8-1) docs not enter the calculation. That strain can be written as... 

Fig. 2 Brillouin zone for a hexagonal unit cell showing some of the special points. F is the centre point.

Kobayasi, T. and Nara, H. (1993) Properties of nonlocal pseudopotentials of Si and Ge optimized under full interdependence among potential parameters. Bull. Coll. Med. Set Tohoku Univ., 2, 7-16. Chadi, DJ. and Cohen, M.L. (1973) Special points in the Brillouin zone, Phys. Rev., B8, 5747-5753. Clementi. E. (1965) Tables of atomic functions, in Supplement to the paper Ab initio computations in atoms and molecules . IBMJ. Res. Develop., 9, 2-19. 

Fig. 2.16 Sketch of the five crystal-orbital phases at three different special points (a) within the Brillouin zone (b) of a two-dimensional Fe crystal compare with Figure 2.15.

In order to ease interpretation, the corresponding Brillouin zone is also included, and a sketch of the associated crystal orbitals at the special points r and N is presented in Figure 3.19. These crystal orbitals can be roughly grouped into the f2g and Cg sets. The, primarily, Cg bands are mostly flat except from r H and H N because they are involved in a interactions with the second-nearest neighbors. The f2g crystal orbitals, however, experience 7r-like interactions with their nearest neighbors, thus there is reasonable dispersion along every symmetry direction. 

Monkhorst HJ, Pack JD (1976) Special points for Brillouin zone integratioa Phys Rev B 13 5188-5192 Montroll EW (1942) Fieqnency spectrum of crystalline solids. J ChemPhys 10 218-228 Njo SL, Koningsveld H van, Gr B van de (1997) A computational stndy on zeolite MCM-22. Chem Cotmnun 1243-1244... 

In all lattice dynamics treatments for librational degrees of freedom discussed in Section IIC, interactions between these coordinates and translations must be considered. As already pointed out, interaction matrix elements vanish only at the zone center and some very special points on the zone boundary and this only for centro-symmetric solids. The first calculation of the dispersion curves for a molecular solid throughout the Brillouin zone was carried out by Cochran and Pawley (1964) for hexamethylenetetramine (hexamine). Once the librational displacement coordinates have been defined and a potential function chosen, the interaction force constants O, can be calculated. The subscripts refer to the displacement coordinate components, where we use i to designate a translational displacement component and a to designate a librational displacement component m,. The corresponding dynamical matrix elements are Mi, analogous to the M,-, defined in (2,9). In general, the matrix elements are complex, and the matrix is hermitian. 

Secondly, there exists several techniques for integrating approximately over the k-points of the first Brillouin zone (BZ). For materials with ully occupied bands (e.g., semiconductors) the special points method is by far the most efficient (Chadi and Cohen, 1974 Monkhorst and Pack, 1976). The method appeals to the tight-binding picture of atomic interactions, integrating a definite number of interactions exactly with a suitably chosen set of k-points. For metallic systems it is necessary to exhaust the irreducible BZ with a fine mesh, and to choose a method of assigning occupation numbers to the electron states. Several methods prevail, and we refer to Fu and Ho (1983) for a detailed comparison of two schemes. 

The Brillouin zone (BZ) k-integration is performed by the special points method, suitably generalized for distorted lattices The set of undlstorted k-points (Monkhorst and Pack,... 

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