Scalars lattice structure

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

It is easily seen by inspection that the biorthogonal basis set definition (3.55) cmnddes with the definifion (3.18) ven in the discussion of the Lanczos method. We recall that the dynamics of operators (liouville equations) or probabilities (Fokker-Planck equations) have a mathematical structure similar to Eq. (3.29) and can thus be treated with the same techniques (see, e.g., Chapter 1) once an appropriate generalization of a scalar product is performed. For instance, this same formalism has been successfully adopted to model phonon thermal baths and to include, in principle, anharmonicity effects in the interesting aspects of lattice dynamics and atom-solid collisions. ... 

Table 12. Charges inside the atomic spheres Qy and the ratios Qy VOy of the 1 partial charges for LiTl and NaTl. For both compounds the self consistent scalar relativistic APW calculations have been performed for both structures B2 and B32. The lattice constants are chosen aB32 = 2 agi where aB32(NaTl) = 7.47 A and aB2(Lin) = 3.42 A are the lattice constants found experimentally. The volumes of the muffin-tin sphere are Wu(LiTl) = w-nCLiTI) = 12.47 A and ti)nj(NaTl) = o)Ti(NaTl) = 16.84 A ...

At least in principle, 2D NMR techniques can be used to establish connectivities in the solid-state, and for crystalline three-dimensional framework structures (in contrast to the case of molecular crystals), these connectivities could be used to define the three-dimensional lattice itself. In the present section, we examine the potential of 2D 39Si MAS NMR measurements involving scalar coupling interactions to establish three-dimensional Si-O-Si lattice connectivities in zeolite framework structures. 

This is also an important feature to avoid the variational collapse. The proposed approach was apphed to An and InSb crystals. The basis functions were chosen so that they have enongh variational flexibility not only were used AOs of neutral atoms but also those of positive ions Au+, Au +, Au +, In +, Sb +. In the comparison for Au the results of scalar- and two-component relativistic calculations (lattice constant and bulk modulus) indicates that the spin-orbit coupling plays a minor role in the structural properties of Au. The comparison of results of the relativistic calculations with those of the nonrelativistic calculations shows that the lattice constant is overestimated by 5% and the bnlk modulus is underestimated by 35% in nonrelativistic calculations. This strongly shows the importance of the inclusion of the relativistic effects in the stndy of the structural properties of Au. In contrast with the case of An, the resnlts of nomelativistic calculations of InSb are not so poor the error in the lattice constant is 1% and the error in the bulk modulus is 10%. This should be due to the fact that In and Sb are not so heavy that the relativistic effects do not play an important role in stndying the structural properties of InSb. 

Comparison of the nonrelativistic and scalar-relativistic results for fee Au reveals the large impact that relativity has on the lattice constant (6%) and bulk modulus (57%) [542]. The most important quaUtative change in the band structure of fee Au is the more than 2-eV lowering of the s-band relative to the bottom of d-bands. In addition, the overall width of the d-bands is increased by more than 15% due to a relativistic delocaUzation of the d- states. The spin-orbit coupling included LGGTO DFT-GGA calculations were made for fluorite structure actinide oxides MO2 (M=Th,U,Pu) and their clean and hydroxylated surfaces, [556], magnetic ordering in fee Pn [557] and bulk properties of fee Pb [558]. 

The electronic structure is determined using the ab initio all-electron scalar-relativistic tight-binding linecir muffin-tin orbital (TB-LMTO) method in the atomic-sphere approximation (ASA). The nnderlying lattice, zincblende structure, refers to an fee Bravais lattice with a basis which contains a cation site (at a(0,0,0)), an anion site (at o(j,, )), and two interstitial sites occupied by empty spheres (at a(, 5, h) and a(, , )) which in turn are necessary for a correct description of open lattices . ... 

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