Special Points of Brillouin Zone

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. 

Table 4.1. Special points of the Brillouin zone for the simple cubic lattice generated by the symmetrical transformation Bi = - (1,0,0), B2 = (0,1,0), B3 = (0,0,1) Kg = ctiBi -f OC2B2 + OC3B3 = (oi, 02,03)...

Table 4.2. Special points of the Brillouin zone for the face-centered cubic lattice generated by the symmetrical transformation...

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

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. 

All parameters have now been specified in detail energy bands can be obtained as a function of k by diagonalizing the matrix in Table 3-1 for each k. For arbitrary wave numbers this would need to be accomplished numerically, but at special wave numbers or for wave numbers along symmetry lines in the Brillouin Zone it can be accomplished analytically. Let us diagonalize the matrix analytically for the point F at the center of the Brillouin Zone, k = 0. At k = 0, 9i= 9z - 03 = Qq = 4. Thus all off-diagonal matrix elements in Table 3-1 vanish except those coupling. s with s , those coupling and p", and so on. The... 

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.

For this PP calculation the exact Ex has been combined with the LDA for Ec. In spite of the use of the exact exchange functional, FeO is predicted to be a metal, in contrast to experiment. At present, it is not clear whether this failure to reproduce the insulating ground state of FeO originates from the use of the LDA for Ec or from the technical limitations of the PP calculation (KLI approximation for vx, only three special it-points for the integration over the Brillouin zone, 3s elections in the core). It must be emphasized, however, that the band structure shown in Figure 4.6 is rather different from its LDA counterpart (Dufek et al. 1994), which emphasizes the importance of the exact vx for this system. 

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. 

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