The Tetrahedron Method

The density N(E) of electronic states is defined such that N(E)dE is the number of one-electron states per unit cell and per spin, with energy between E and E + dE. It may be calculated from 

1 Note that for historical reasons the programme evaluates 2N(E) where N(E) is defined in (9.25), cf. also (1.13). 

In the tetrahedron method, one divides the irreducible wedge of the Bril-louin zone into tetrahedra, and the integral (9.24) is then evaluated as the sum 

When the eigenvalues in the j th band at the four corners of the i th tetrahedron are ordered such that 

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With the advent of the computationally much faster linearized methods, the computation of reliable densities of states has become much easier. These techniques allow the calculation of energies for a great number of k points in an energy band. The DOS is then obtained by means of the tetrahedron method (67). 

Calculations were done with a full-potential version of the LMTO method with nonoverlapping spheres. The contributions from the interstitial region were accounted for by expanding the products of Hankel functions in a series of atom-ce- -ered Hankels of three different kinetic energies. The corrected tetrahedron method was used for Brillouin zone integration. Electronic exchange and correlation contributions to the total energy were obtained from the local-density functional calculated by Ceperley and Alder " and parametrized by Vosko, Wilk, and Nusair. ... 

Seiber P. A new acid-labile anchor groups for the solid phase synthesis of C-terminal peptide amides by the Fmoc method. Tetrahedron Lett 1987 28 2107-2110. 

Ede NJ, Bray AM. A simple linker for the attachment of aldehydes to the solid phase. Application to solid phase synthesis by the Multipin method. Tetrahedron Lett 1997 38 7119-7122. 

Figure 7.8. (a) enlarged region for 0 < i < 0.5 calculated according to CVM in the tetrahedron approximation (T), the tetrahedron-octahedron approximation (T-O) and using a Monte Carlo method (MC) (Sanchez et al. 1982). (b) results obtained from using the quasi-chemical or CSA approximation (Kikuchi 1977). 

Figure 7.9. Variation of (a) the Gibbs energy, (b) ttie enthalpy and (c) the entropy with temperature (scaled to the nearest-nei bour interaction energy J ) for the complex structure A15. Comparison between BWG CVM in the tetrahedron approximation ( ) and the Monte Carlo method (—) (Turchi and Finel 1992).

Ede, N. J. Bray, A. M. A Simple Linker for the Attachment of Aldehydes to the Solid Phase. Application to Solid Phase Synthesis by the Multipin Method, Tetrahedron Lett. 1997, 38, 7119. 

The methods discussed in this section can be extended to systems that have more than three components. The problem is to convert each system to a pseudobinary system. For a quaternary system, the properties of an equilateral tetrahedron may be used to depict the composition of the system. The composition axes would be four lines drawn from the four apexes perpendicular to the opposite faces. Planes cutting the tetrahedron parallel to the bases would represent pseudoternary systems for which one composition variable would be constant. Pseudobinary systems would be depicted by the intersections of two of the pseudoternary planes. Indeed, the experimental measurements and calculations would be extensive. 

Polycrystalline [Mn(taa)] samples were prepared according to the literature method [11,12] and tetrahedron-shaped single crystals were obtained by recrystallization from CH2Cl2/hexane solutions. 

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