Number of bonds

Flence, on summing over the graphs, the only non-zero tenns are closed polygons with an even number of bonds at each site, i.e. s. must appear an even number of times at a lattice site in a graph that does not add up to zero on suimning over the spins on the sites. 

Unlike the forces between ions which are electrostatic and without direction, covalent bonds are directed in space. For a simple molecule or covalently bonded ion made up of typical elements the shape is nearly always decided by the number of bonding electron pairs and the number of lone pairs (pairs of electrons not involved in bonding) around the central metal atom, which arrange themselves so as to be as far apart as possible because of electrostatic repulsion between the electron pairs. Table 2.8 shows the essential shape assumed by simple molecules or ions with one central atom X. Carbon is able to form a great many covalently bonded compounds in which there are chains of carbon atoms linked by single covalent bonds. In each case where the carbon atoms are joined to four other atoms the essential orientation around each carbon atom is tetrahedral. 

With such a matrix representation, the storage space is dependent only on the number of nodc.s (atoms) and independent of the number of bonds. As Figure 2-14 dcmon.stratcs, all the e.sscntial information in an adjacency matrix can also be lound in the much smaller non-rediindant matrix. But the adjacency matrix is unsuitable for reconstructing the constitution of a molecule, because it does not provide any information about the bond orders. 

Figure 2-15, Distance matrices of ethanal with a) geometric distances in A and b) topological distances. The matrix elements of b) result from counting the number of bonds along the shortest walk between the chosen atoms,...

The elements of a distance matrix contain values which specify the shortest distance between the atoms involved. Distances can be expressed either as geometric distances (in A) or as topological distances (in number of bonds) (Figure 2-15a,b). 

Figure 3-16. A reaction scheme that changes the number of bonds at one atom, and some specific examples.

The two reaction schemes of Figures 3-13 and 3-15 encompass a large proportion of all organic reactions. However, these reactions do not involve a change in the number of bonds at the atoms participating in them. Therefore, when oxidation and reduction reactions that also change the valency of an atom ate to be considered, an additional reaction scheme must be introduced in which free electron pairs are involved. Figure 3-16 shows such a scheme and some specific reaction types. 

The optimization of the backtracking algorithm usually consists of an application of several heuristics which reduce the number of candidate atoms for mapping from Gq to Gj. These heuristics are based on local properties of the atoms such as atom types, number of bonds, bond orders, and ring membership. According to these properties the atoms in Gq and Gj are separated into different classes. This step is known in the literature as partitioning [13]. Table 6.1 illustrates the process of partitioning. 

The contribution of an atomj to the polarizability effect is attenuated by the number of bonds, H , between this atom and the site of protonation, i. 

We denote the topological distance between atoms i and j (i.e., the number of bonds for the shortest path in the structure diagram) dy, and the properties for atoms i and j are referred to as pi and pj, respectively. The value of the autocorrelation function a d) for a certain topological distance d results from summation over all products of a property p of atoms i and j having the required distance d. 

The neighborhoods of the atoms directly bonded to tbe chiral center must be defined. The neighborhood of an atom A. dircetly bonded to the ehiral eenter, is dc-fned as the set of atoms whose distance (in number of bonds) to A is less than their distance to any of the other three atoms bonded to the chiral center (Figure 8-9. In cyclic structures different neighborhoods can overlap. 

In order to consider the 3D structure but make the chirality code independent of a specific conformer, r- is taken as the sum of the bond lengths between atoms i and j on the path with a minimum number of bond counts. 

Each combination of four atoms (A, B. C. and D) is characterized by two parameters, e and e.. As for the CICC, is a parameter that depends on atomic properties and on distances, and is calculated by Eq. (27), with r, again being the sum of bond lengths between atoms on the path with the minimum number of bond counts. However c is now a geometric parameter (dependent on the conformation)... 

The similarity of the retrieved protons to those of the query structure, and the distribution of chemical shifts among protons with the same HOSE codes, can be used as measures of prediction reliability. When common substructures cannot be found for a given proton (within a predefined number of bond spheres) interpolations are applied to obtain a prediction proprietary methods are often used in commercial programs. 

Figure 10.3-42. Deriving the reaction substructure, A diFferent number of bond spheres around the strategic bond can be Included in the reaction substructure, thereby influencing the specificity of a search in a reaction database.

The position of the ehosen strategic bond locates the reaction center. To derive the reaction siibstrncture, the user can select the number of bond, spheres around the strategic bond which should be included. The reaction substructure obtained is then n.scd as the query for a reaction substructure search in the database. Figure 10,3-42 illustrates the first and second bond spheres around a selected strategic bond of a retrosynthetic step. 

The number of bond spheres chosen influences the specificity of tlie reaction substructure query and the result of the search. In Figure 10.3-43 the reaction substructure queiy including the first bond sphere of the retrosynthetic step of Figure 10,3-42 is shown. 

The computational effort is significantly increased if three-body terms are included in the model. Even with a simple pairwise model, the non-bonded interactions usually require by far the greatest amount of computational effort. The number of bond, angle and torsional terms increases approximately with the number of atoms (N) in the system, but the number of non-bonded interactions increases with N. There are N(N —l)/2 distinct pairs of... 

The key term is which is the bond order between the atoms i and j. This parameter depends upon the number of bonds to the atom i the strength of the bond between i and j decreases as the number of bonds fo fhe atom i increases. The original bond-order potential [Abell 1985] is mathematically equivalent to the Finnis-Sinclair model if the bond order by is given by ... 

It can be readily confirmed thaf by decreases as the number of bonds N increases and/or llieir length (r ) decreases. This relationship between the bond strength and the number of neighbours provides a useful way to rationalise the structure of solids. Thus the high coordination of metals suggests that it is more effective for them to form more bonds, even though each individual bond is weakened as a consequence. Materials such as silicon achieve the balance for an infermediate number of neighbours and molecular solids have the smallest atomic coordination numbers. 

Derivation of bond enthalpies from themioehemieal data involves a system of simultaneous equations in which the sum of unknown bond enthalpies, each multiplied by the number of times the bond appears in a given moleeule, is set equal to the enthalpy of atomization of that moleeule (Atkins, 1998). Taking a number of moleeules equal to the number of bond enthalpies to be determined, one ean generate an n x n set of equations in whieh the matrix of eoeffieients is populated by the (integral) number of bonds in the moleeule and the set of n atomization enthalpies in the b veetor. (Obviously, eaeh bond must appear at least onee in the set.)... 

Number of atoms of various elements Number of bonds of various orders Number of rings... 

Count the number of electrons in shared electron pair bonds (twice the number of bonds) and sub tract this from the total number of electrons to give the number of electrons to be added to com plete the structure... 

Inductive effects depend on the electronegativity of the substituent and the num ber of bonds between it and the affected site As the number of bonds increases the inductive effect decreases... 

Oxidation of carbon corresponds to an increase in the number of bonds between carbon and oxygen or to a decrease in the number of carbon-hydrogen bonds Conversely reduction corresponds to an increase in the number of carbon-hydrogen bonds or to a decrease in the number of carbon-oxygen bonds From Table 2 4 it can be seen that each successive increase m oxidation state increases the number of bonds between carbon and oxygen and decreases the number of carbon-hydrogen bonds Methane has four C—H bonds and no C—O bonds car bon dioxide has four C—O bonds and no C—H bonds... 

The decreased shielding caused by electronegative substituents is primarily an inductive effect and like other inductive effects falls off rapidly as the number of bonds between the substituent and the proton increases Compare the chemical shifts of the pro tons m propane and 1 mtropropane... 

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