Bond Energy Formulas

Now, the problems associated with the search for meaningful atomic charges, such as those required in applications of our bond energy formulas, are manifold. One... 

This concludes the presentation of bond energy formulas. 

Calculation of Reference Bond Energies [Eq. (10.39)]. The parameters indicated in Table 10.4 are ready for use in the bond energy formula, Eq. (10.39). The following examples, in part based on detailed results given in Chapters 15 and 16 for nitrogen- and oxygen-containing molecules, illustrate the procedure and report the input data. 

This concludes the derivation of our bond energy formula and the presentation of simple examples pertaining to saturated systems, namely, the alkanes, including their numerical parameterization. 

The flexibility and internal consistency of the present theory are well illustrated by the transformations that generate the sets of parameters required for the unsamrated hydrocarbons from those of their saturated models. But most importantly they preserve the original form and great simphcity of the basic bond energy formula, Ski = Ski + akAqk + aik qu as well as its accuracy. 

Example 12.4 Influence of the Environment on D i. Nitromethane is interesting to some people because it explodes. The reason is, of course, in the cleavage of the carbon-nitrogen bond. The monomer, compared to its trimer (taken as a model for the crystal), reveals that the C and N net charges change by A c — 8.7 and A n—1-1 me. respectively, on crystallization. Our bond energy formula and the appropriate parameters thus indicate that the crystalline environment reinforces the CN bond by 4.7 kcal/mol, which is significant at the local point of rupture, responsible for the reaction [251]. 

The first thing to do is to learn how to write the bond energy formula [Eq. (10.37)] for carbon-nitrogen bonds... 

Any bond energy formula can be expressed either i) by reference to a selected bond with reference net atomic charges q and q°i at the bond-forming atoms k and I, or ii) by reference to hypothetical k-l bonds constructed with the assumption q = q° = 0. The former reflects a physical situation, but requires additional work in order to sadly charge normalization constraints it is most useful in the constmction of general energy formulas for molecules that use chemical shifts espressed with respect to the appropriate references. The latter method simplifies bond-by-bond calculations. The two forms are... 

The following bonds occur in benzenoid hydrocarbons. Consider first the endocyclic carbon-carbon bonds, namely, those found in benzene, with = 115.39 (or Sqq = 124.84) kcal/mol, which—in a sketchy way—are some kind of averages between a single and a double sp -sp bond. (Their number is double that of the number of double bonds that can be written in classical Kekule structures, e.g., 2 X 5 in naphthalene, 2 x 7 in anthracene.) But in polynuclear benzenoid structures there are not twice as many averages as there are Kekule double bonds. Hence, consider the extra single C sp )—C sp ) bonds like the one found in naphthalene, or the two extra single bonds found in anthracene. The appropriate bond energy formulas are... 

Values of the heat of adsorption of hydrogen on various metals calculated by Eley (110) using Pauling s covalent bond energy formula, i.e.. 

Formula for the chemical potentials have been derived in terms of the formation energy of the four point defects. In the process the conceptual basis for calculating point defect energies in ordered alloys and the dependence of point defect concentrations on them has been clarified. The statistical physics of point defects in ordered alloys has been well described before [13], but the present work represents a generalisation in the sense that it is not dependent on any particular model, such as the Bragg-Williams approach with nearest neighbour bond energies. It is hoped that the results will be of use to theoreticians as well as... 

C06-0094. Use average bond energies (Table 6-2) to compare the stabilities of ethanol, C2 H5 OH, and dimethyl ether, (CH3)2 O, which have the same empirical formula, C2 Hg O (all the bonds are single bonds). 

Let us summarize briefly at this stage. We have seen that the point of degeneracy forms an extended hyperline which we have illnstrated in detail for a four electrons in four Is orbitals model. The geometries that lie on the hyperline are predictable for the 4 orbital 4 electron case using the VB bond energy (Eq. 9.1) and the London formula (Eq. 9.2). This concept can be nsed to provide nseful qualitative information in other problems. Thns we were able to rationalize the conical intersection geometry for a [2+2] photochemical cycloaddition and the di-Jt-methane rearrangement. 

A knowledge of these enthalpic terms, and therefore of the relative bond energies, would be expected to considerably clarify many of these fundamental aspects. The data in Table 4 show that, with the main exception of rhenium and osmium, the metal-metal distances in the tetranuclear clusters and in the pure metals are quite similar this relationship is generally valid for all the polynuclear carbonyls60. The metal-metal bond energies in clusters are therefore expected to be of the same order as those in the metallic state for a close-packed arrangement, these are given by the formula Z m-m = A//f M(g)/6. 

Equation (50) uses the fact that the a-bond energies and the entropy components may be assumed constant. However, this formula does not allow for the fact that several isomeric proton addition complexes may be present in the solution. In that case one obtains the more general relation ... 

One of the early efforts to evaluate quantitatively the bond dissociation energy of particular bonds in a compound was the work initiated by Mulliken (-3) in his so-called Magic Formula. Although this formula contains five terms, the two most important for the evaluation of a bond dissociation energy, Dq (uncorrected for zero-point vibrational energy), between two atoms i and j, are the covalent bond energy, Xjj, and the ionic resonance energy, IRE. The evaluation of Ay takes the form ... 

In this formula, m and n are the number of ligand and receptor atoms, respectively r is the interatomic distance between atoms i and j the q s are the point charges on the atom, and A and B are adjustable van der Waals repulsion and attraction parameters, and D is the dielectric function. They assumed that this scoring function could account for hydrogen bond energies in the electrostatic term. 

Formula HBr MW 80.912. H— Br bond energy 88.0 kcal/mol internuclear distance 1.41 A An aqueous solution of hydrogen bromide gas is hydrobromic acid. 

Formula HCl MW 36.461 a polar molecule, dipole moment 1.12D H—Cl bond energy 105.5 kcal/mol internuclear distance 1.28A. Hydrochloric acid is an aqueous solution of hydrogen chloride. 

Formula SiCh MW 169.90 bond energy 91.06 kcal/mol Synonym tetrachlorosilane... 

The perturbation A(T f + 2T ) describes the replacement of model densities and inter-nuclear distances by the values that are appropriate for the molecule under scrutiny. Similarly, appropriate reference atomic energies must be used in the atomic-like formula (4.15) to get A °. Ingeniously selected references require small corrections. Nature helps a lot in that matter by keeping the changes of p(r) as small as possible. The bond energy theory is rooted in Eq. (4.47). 

Direct applications of Eq. (10.12) are generally difficult to handle—this is why the more efficient charge-dependent energy formulas were developed in the first place. Most thorough tests were made for selected carbon-carbon bonds [13,14,44,108] (Table 1.1). 

This is one of our working formulas. It is an approximation, of course, but we are presently unable to do better evidently, by implementing this approximation, we possibly transfer additional contingent variations of the sh/ bond energy to other bonds formed by atom 1 in the molecule. With this reservation in mind, we shall illustrate the use of Eq. (10.15) in Chapter 15, thus revealing instructive bond properties. 

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