Non-independent bonds

Structured formulae and non-independent bonds non-bonding electrons (at higher energy) ... 

As we have already stated, if it is true that some properties (in particular those related to the electron energies) can only be understood by considering non-independent bonds, in other cases (e.g. bond energies) such consideration is not important. For example, the C-C bond energies do not vary significantly from molecule to molecule, in agreement with a description in terms of the overlap of sp atomic orbitals of adjacent C atoms, as illustrated next for ethane in a direct extension of the case of CH4 ... 

Fig. 4-2. Optimized STO-3G structure for the benzenediazonium ion. I Bond lengths (pm) and angles. Non-independent angles are shown in parentheses.

It is only when the interactions expressed by Eq. (8.27) are neglected that orbital hybridization seems to have an effect and, thus, seems to explain some molecular properties. This is so, both in m.o. theory and in v.b. theory. Ironically, it is more difficult to take these interactions into account in v.b. theory where the concept of hybrid orbital was first introduced. For example, it is more difficult to do a complete v.b. treatment of CH4, with inclusion of the less traditional formulae in the resonance scheme (and which accounts for the non-independence of bonds), than to do a fully delocalized treatment by m.o. theory, at the same degree of approximation. But if the complete treatment is carried out, the v.b. result is the same irrespective of the hybridization considered, just as any m.o. result is the same whether pure, hybrid orbitals or a mixture of both are taken as the basis for the linear combinations. 

The continuum model has been applied to an experimental study of the solvent effect on the 6-chloro-2-hydroxypyridine/6-chloro-2-pyridone equilibrium in a variety of essentially non-hydrogen-bonding solvents (Beak et al., 1980). In this study, a plot of log A nh/oh) versus (e - 1)/ (2e + 1), the solvent dielectric term, yielded a linear least-squares fit with a slope of 2.5 0.2, an intercept of -1.71, and a correlation coefficient of 0.9944. This result was used to estimate the gas phase free-energy difference of 9.2 kJ mole-1, which compares favorably with the observed value of 8.8 kJ mole-1 for this system. The authors also reported that alcohol solvents are correlated fairly well in this study but that other solvents seem to be divided into two classes, those that are electron-pair donors and those that are electron-pair acceptors in a hydrogen bond. The hydrogen bonding effect is assumed to be independent from the reaction field effect and is included in the continuum model by means of the Kamlet and Taft (1976) empirical parameters. The interested reader is referred to the original paper for a detailed discussion of the method and its application. 

Similarly, each non-terminal bond between atoms of ligancy p and q can be associated with (p- l)(Qf-1) distinct torsion angles, of which only one is independent once the bond angles have been specified. Additional torsional degrees of freedom may be introduced, but only at the cost of reducing the number of independent bond angles. 

To derive the MM picture we must consider the question of transfer-ability of the ESPs. Above we defined two groups of parameters entering the expression for the energy Eq. (17). Let us consider the first group of parameters. In the case of lone pairs they are perfectly transferable. The SCF approximation for non-polar bonds gives the geometry independent density matrix elements ... 

We have introduced a coefficient of proportionality, because the A-B bonds are not totally independent of one another. Figure 3.1 shows an example of this non-independence. Indeed, suppose that on the lattice (which we shall take to be flat for the purpose of simplicity), we have placed the molecules Aa, Bb, and B respectively on sites a, b and c. If we place a molecule B on site d, it is clear that the bond between sites a and d is necessarily a mixed bond AaBd. In other words, the pairs are not distributed entirely randomly. The coefficient k depends on the lattice chosen to represent the cell. 

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. 

Independent molecules and atoms interact through non-bonded forces, which also play an important role in determining the structure of individual molecular species. The non-bonded interactions do not depend upon a specific bonding relationship between atoms, they are through-space interactions and are usually modelled as a function of some inverse power of the distance. The non-bonded terms in a force field are usually considered in two groups, one comprising electrostatic interactions and the other van der Waals interactions. 

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