Bonds in Two and Three Dimensions

In principle, the calculation of bonding in two or three dimensions follows the same scheme as outlined for the chain extended in one dimension. Instead of one lattice constant a, two or three lattice constants a, b and c have to be considered, and instead of one sequential number k, two or three numbers kx, ky and k- are needed. The triplet of numbers k = (kx, ky, kz) is called wave vector. This term expresses the relation with the momentum of the electron. The momentum has vectorial character, its direction coincides with the direction of k the magnitudes of both are related by the de Broglie relation [equation (10.5)]. In the directions a, b and c the components of k run from 0 to nja, njb and n/c, respectively. As the direction of motion and the momentum of an electron can be reversed, we also allow for negative values of kx, ky and kz, with values that run from 0 to —nja etc. However, for the calculation of the energy states the positive values are sufficient, since according to equation (10.4) the energy of a wave function is E(k) = E(—k). 

The magnitude of k corresponds to a wave number 2n/X and therefore is measured with a unit of reciprocal length. For this reason k is said to be a vector in a reciprocal space or k space . This is a space in a mathematical sense, i.e. it is concerned with vectors in a coordinate system, the axes of which serve to plot kx, ky and kz. The directions of the axes run perpendicular to the delimiting faces of the unit cell of the crystal. 

The region within which k is considered (—n/a kx nja etc.) is the first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

The pictures in Fig. 10.12 give an impression of how s orbitals interact with each other in a square lattice. Depending on the k values, i.e. for different points in the Brillouin zone, different kinds of interactions result. Between adjacent atoms there are only bonding 

10 MOLECULAR ORBITAL THEORY AND CHEMICAL BONDING IN SOLIDS 

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The first Hamiltonian was used in the early simulations on two-dimensional glass-forming lattice polymers [42] the second one is now most frequently used in two and three dimensions [4]. Just to illustrate the effect of such an energy function, which is given by the bond length, Fig. 10 shows two different states of a two-dimensional polymer melt and, in part. 

Polymers are frequently classified in terms of bonding in one dimension versus bonding in two or three dimensions. Bonding in one dimension results in linear polymers with single-strand chains. Bonding in two or three dimensions results in cross-linked polymers having infinite sheets or three-dimensional networks. Linear polymers are produced by addition polymerization if the reactant has only one double bond or by condensation polymerization if the reactant or reactants each have two reactive sites. Such polymers are usually soluble in suitable solvents. Since they also tend to soften when heated, they are called thermoplastic polymers. Cross-linked polymers may be produced by addition polymerization if the reactant has more than one double bond, or by condensation polymerization if the reactant or reactants each have more than two reactive sites. Such network polymers are usually insoluble and Infusible and are called thermosetting polymers. 

Some polymers, such as polyethylene, polystyrene, and polypropylene, have chainlike molecules. These polymers usuaUy soften when heated and are sometimes caUed thermoplastic polymers. Other polymers are made up of networks instead of chains. Some of the network polymers have long chains with short chains (cross links) fastening two or more chains together, and others, such as Bakehte, have networks that are bonded in two or three dimensions. These polymers are sometimes called thermosetting because they are usuahy formed at high temperatures. 

To describe steric hindrance effects in gelation one may study percolation on a lattice in which bonds are restricted in a way that no more than v bonds can enamate from the same site, or no site may have more than v nearest neighbors . Similarly, valence saturation may occur for the monomers in the gelation process. The case v = 2 is similar to self-avoiding walks , while for larger v one expects random percolation exponents, as confirmed by the Monte Carlo methods in two and three dimensions. Then, on a large... 

G. A. Papoian, R. Hoffmann, Hypervalent bonding in one, two and three dimensions extending the Zintl-Klemm concept to nonclassical electron-rich networks. Angew. Chem. Int. Ed. 39 (2000) 2408. 

T. Hughbanks, Bonding in clusters and condesed cluster compounds that extend in one, two and three dimensions. Prog. Solid State Chem. 19 (1990) 329. 

If electron-pair, or covalent, bonding is periodic in two or three dimensions, crystals result. The most important case is the carbon-carbon bond. If it is extended periodically in two-dimensions the result is graphite in three-dimensions it is diamond. Other elements that form electron-pair bonds are Si, Ge, and a-Sn. Some binary compounds are A1P (isoelectronic with Si),... 

Aromaticity is a manifestation of electron delocalization in closed circuits, either in two or three dimensions. This results in energy lowering, often quite substantial, and a variety of unusual chemical and physical properties. These include a tendency toward bond-length equalization, unusual reactivity, and characteristic spectroscopic features. Since aromaticity is related to induced ring currents, magnetic properties are particularly important for its detection and evaluation. 

Most of the simulations reviewed were performed in two dimensions, although some simulations of three-dimensional detonations are discussed in Sec. 3.1.2. Equilibrated molecular crystals for AB Model I in both two and three dimensions are shown in Fig. 2. This model has slightly longer bond lengths than either Model II or III, but otherwise the starting crystal structures are similar for all three models. To model an infinite crystal, almost all detonation simulations were carried out with periodic boundary conditions enforced perpendicular to the direction of the shock propagation. 

The many theories behind the various models developed to calculate the solubility of polymers, and to predict the ability of liquids to dissolve them, are described clearly and in high detail by Burke (Burke, 1984). All define a term known as solubility parameter for liquids and polymers using one or more of the intermolecular force components and represent the parameter in two or three dimensions. Calculating solubility parameters is a mathematically complex process which will not be discussed here. The most widely used method today for predicting whether a polymer is soluble in a liquid was developed by Charles M. Hansen in 1966. Hansen parameters ( ) for solvents and polymers are calculated from the dispersion force component ( d), polar component ((5p) and hydrogen bonding component ( h) for each using the formula ... 

Hughbanks, T. (1989). Bonding in Clusters and Condensed Cluster Compounds that Extend in One, Two and Three Dimensions, Prog. Solid St. Chem. 19, 329-372. 

In the earlier mixture-model theories of liquid water, this principle was introduced implicitly into the theory by choosing two components an open structure, but highly hydrogen-bonded (HBed), and a close-packed form but of weakly interacting molecules. In the modern era, the principle was used either implicitly or explicitly in the construction of many models of water-like particles in one, two, and three dimensions. 

Individual fibres are aligned in nonwoven structures in various directions. These fibre alignments are inherited from fibre alignments in both fibrous web forms and fibre relocation in the nonwoven bonding process. This characteristic can be described by fibre orientation angles in two or three dimensions (Fig. 6.1). 

Proper design of the components is the key to assembly. The unique ends of each component will only bond to its intended counterpart. Designs can create circuits in two or three dimensions and can be independent (free floating) or be joined to a substrate like silicon for connection to more conventional circuitry. With complete circuits thousands of times smaller than their conventional counterparts, an entirely new (previously unthinkable) world of development is created. 

More than 90% of the rocks and minerals found in the earth s crust are silicates, which are essentially ionic Typically the anion has a network covalent structure in which Si044-tetrahedra are bonded to one another in one, two, or three dimensions. The structure shown at the left of Figure 9.15 (p. 243), where the anion is a one-dimensional infinite chain, is typical of fibrous minerals such as diopside, CaSi03 - MgSi03. Asbestos has a related structure in which two chains are linked together to form a double strand. 

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