Bond matrix

The bond matrix is related to the adjacency matrix but gives information also on the bond order of the connected atoms. Elements of the matrix obtain the value of 2 if there is a double bond between the atoms, c.g, between atoms 2 and 3 

In essence, a BE-matrix li.sts all the valence clcctron.s of the atoms in a molecule, both the ones involved in bonds and those associated as free electrons with an atom. A BE-matrix has a scries of interesting mathematical properties that directly 

The sum, S over all entries of a row or column indicates the number of valence electrons of atom i (Eq, (1)). 

The sum over all entries of the BE-matrix (S ) gives the total number of valence electrons in the molecule (Eq. (2)). 

If the nttmber of valence electrons thus calculated does not agree with the standard number of valence electrons in an atom, this atom carries a charge, in this case, the diagonal element h, has more or fewer valence electrons than the nominal value of the respective atom i. The charge value, Ah, can be determined by subtracting the sum of the row values from the nominal value (Eq, (3)). 

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The bond-clcctron matrix (BE-matrix) was introduced in the Dugundji-Ugi model [39], It can be considered as an extension of the bond matrix or as a mod-ific atinn of Spialter s atom connectivity matrix [38], The BE-inatrix gives, in addition to the entries of bond values in the off-diagonal elements, the number of free valence electrons on the corresponding atom in the diagonal elements (e.g., 03 = 4 in Figure 2-18). 

Bond matrix describes connections and bond orders of atoms no number of free electrons cannot be rcpicscntcd by bits... 

The task is now to take one of the numberings as the standard one and to derive a unique code from it, which is called canonicalization. This can be accomplished by numbering the atoms of a molecule so that it is represented later by only one connection table or bond matrix. Such a unique and reproducible numbering or labeling of the atoms is obtained by a set of rules. 

Carrying out this proeedure for propane and butane, CH3—CH2—CH3 and CH3—CH2—CH2—CH3, yields the bond matrix and enthalpies of atomization ... 

The bond matrix expresses 2 C—C bonds plus 8 C—H bonds for propane and 3 C—C bonds plus 10 C—H bonds for n-butane. Eaeh enthalpy of atomization is obtained by subtraeting the enthalpy of formation of the alkane from the sum of atomie atomization enthalpies (C 716 H 218 kJ mol ) for that moleeule. For example, the moleeular atomization enthalpy of propane is 3(716) +8(218) — (—104) = 3996 kJ mol . Enthalpies of formation are available from Pedley et al. (1986) or on-line at www.webbook.nist.gov. 

Solve the same problem for propane and isobutane (2-methylpropane). The bond matrix is the same as it is for n-butane, but the enthalpy of formation is somewhat different (n-butane) = —127.1 kJ mol vs. (isobutane) = —134.2... 

Refractories. Calcined alumina is used in the bond matrix to improve the refractoriness, high temperature strength/creep resistance, and abrasion/corrosion resistance of refractories (1,2,4,7). The normal, coarse (2 to 5 )J.m median) crystalline, nominally 100% a-Al202, calcined aluminas ground to 95% —325 mesh mesh are used to extend the particle size distribution of refractory mixes, for alumina enrichment, and for reaction with... 

Other effects (solvent, hydrogen bonding, matrix)... 

Another illuminating implication of the proposed tangled network model was that it explained an old riddle, namely how could mucin networks, which are normally released from the granules in small discrete packages, interpenetrate each other, annealing into a continuum gel mass While the annealing property of mucus remained unexplained by the notion of an interchain S S bonded matrix, it was readily explained by the idea of a tangled mucin polymer network. 

Hemicelluloses are constituted of different hexoses and pentoses glucose, mannose, xylose, etc. Since these heteropolysaccharides are often branched polymers, they cannot constitute crystalline structures. However, their function in the constitution of natural fibres is crucial. Together with lignin, they constitute the bonding matrix of the cellulose microfibres. 

Carbon-carbon (C-C) composites with a variety of unique properties can be fabricated by altering the combinations of the type and distribution of filaments and the bonding matrix used. Many engineering applications can be satisfied with a composite material whose density is just 70% that of aluminum and 25% that of steel, but for which the specific strength and stiffness values are four or five times those of steel. To attain the desired properties for such applications requires an understanding of the interrelationship of the fibers and the matrix that holds them. 

The newest advances in C-C composite materials have resulted from fiber improvements, such as modulus increases of two to three times and diameter decreases of more than 50% in the last ten years. Weaving techniques have also been improved, so that increased fiber content and reproducible distribution of filaments and yarns (down to 0.75 mm center-to-center spacing between yarns in 3D composites) are possible. However, because the interrelationship of the filaments and yarns with the bonding matrix is not well understood, the improved fiber properties have not yet been fully translated into a corresponding magnitude of improvement in C-C composites. 

The behavior of cavities during deformation also depends on the refractoriness of the bonding matrix. In a recent study,33,34 stable cavities were observed to form at the grain boundaries of a grade of silicon nitride containing 4wt.% yttria, even though there was very tittle glass at these boundaries Fig. 4.14a. The cavities observed were reminiscent of Hull-... 

Another form of alkali metal attack on the hot faces of refractory linings involves their high temperature reaction with various components of the brick to form expansive crystalline phases which cause brick to bloat on their hot faces and, subsequently, erode or spall. An example Is the case of alumina brick exposed to sodium at temperatures from about 1700°F to 3000°F. Although sodium does not form a low temperature melt with alumina, it reacts with the alpha phase of alumina, corundum, to form beta alumina, sodium aluminate. Beta alumina has a much greater volume than the very dense corundum and, therefore, disrupts the brick bonding matrix, causing eventual bond failure. 

The authors described the reaction mechanism as occurring by impregnation of the sample, bond reaction, and bond depletion. Immediately the alkali consumes the cristobalite and attacks the glass of the matrix bond. Next the fine crystalline mullite bridges associated with the bonding matrix react with the soda. The mullite x-ray peak intensity is quickly reduced. The extent of attack on the coarser crystalline mullite is intensified at higher temperatures or as the reaction proceeds. When equilibrium... 

Derived from the —> molecular graph Q, the edge adjacency matrix, denoted by E, or more formally as A, also called bond matrix, encodes information about the coimectivity between... 

At the University of Santiago de Compostela, recent work has developed a family of 2D molecular descriptors based on the local spectral moments of a bond adjacency matrix [25b,26], Particular attention has been paid to using the bond spectral moments of the bond matrix that correspond to the central bond of a dihedral angle as descriptors for the dihedral angle. 

Magnetic characteristics of bonded (matrix) magnets made with R—Co alloy powders. Ranges are for products commercially available or near-production maturity... 

For convenience, we shall classify the molecular models according to their topological dimensionality, p. A molecular conformation defined by the set of nuclear position vectors is a zero-dimensional (OD) model. A one-dimensional (ID) model corresponds to a molecular skeleton, defined by the set of nuclear positions and their connectivity (bond) matrix. Contour surfaces of one-particle molecular properties such as electron density or electrostatic potential are topologically two-dimensional (2D) models embedded in three dimensions. Finally, we find a true three-dimensional (3D) model whenever an entire one-electron property over all space is involved. This model can be regarded as the continuum of all 2D isoproperty surfaces. The difference among the models is summarized in Figure 1. We shall deal with pD models in this work (p = 0,1, 2, 3). Each of them requires a different type of shape descriptor. 

This inclusion compound has a van der Waals bonded matrix and therefore is a true clathrate (cf. Chapter 1 of this volume) the hydrogen bonding is internal to the molecular pairs, and these interact with one another and with the guest by van der Waals forces only. A similar situation is found in the orthorhombic inclusion compounds of deoxychoUc acid (see Fig. 5 in Chapter 1). 

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