Using the resonance theory

It appears now that, whatever its usefulness, the resonance theory is somewhat inadequate in explaining and predicting either chemical or physical characteristics of dyes compared to more or less sophisticated molecular orbital calculations. 

Semiclassical transition state theory based on second-order perturbation theory (89) provides another way to assign quantized energy levels of the transition state, and an application (90) to the H + H2 reaction yielded encouraging results in comparison to the full quantum (8) calculations. One difference in assignments (8,90) was later explained (88), using the resonance theory reformulation of variational transition state theory, as a consequence of the inadequacy of second-order perturbation theory. 

More recently, Mies and Kraus have presented a quantum mechanical theory of the unimolecular decay of activated molecules.13 Because of the similarity between this process and autoionization they used the Fano theory of resonant scattering.2 Their theory provides a detailed description of the relationships between level widths, matrix elements coupling discrete levels to the translational continuum, and the rate of fragmentation of the molecule. 

The resonance theory is very useful in accounting for, and in many cases predicting, the behavior of substances with tt bonds. However, it is not omnipotent. One example where it fails is cyclobutadiene, for which we can write two equivalent valence-bond structures corresponding to the Kekule structures for benzene ... 

The organic chemist made an important step in the understanding of chemical reactivity when he realized the importance of electronic stabilization caused by the delocalization of electron pairs (bonded and non-bonded) in organic molecules. Indeed, this concept led to the development of the resonance theory for conjugated molecules and has provided a rational for the understanding of chemical reactivity (1, 2, 3). The use of "curved arrows" developed 50 years ago is still a very convenient way to express either the electronic delocalization in resonance structures or the electronic "displacement" occurring in a particular reaction mechanism. This is shown by the following examples. 

Before discussing the computational results, we will attempt to understand the bonding involved in the synthesized anions, using the elementary theories introduced in Chapter 3. First of all, it should be noted that the aromatic system referred to above is not the MA1J anion as a whole. Instead, structurally, the MAl anion should be considered as consisting of an M+ cation coordinated to a square-planar Al - unit and it is the dianion that has aromatic character. For Al -, there are 14 valence electrons and the resonance structures can be easily written ... 

Resonance was introduced when it was found that there are many molecules whose properties cannot be accounted for by means of a single electronic structure of the VB type, but rather by a combination of several structures [1], Although there is an element of arbitrariness in the resonance theory, in the sense of choosing VB structures, Wheland [50] systemized the basic principles to select the important resonance structures as well as to estimate their relative contribution to the ground state of a molecule. In fact, the qualitative resonance theory enjoyed such a great success due to its convenience and usefulness that resonance has become one of the most fundamental concepts in chemical theory. 

With the advent of the computer era, it is now possible to reexamine and rethink the resonance theory at the ab initio level. For example, throughout Pauling and Wheland s books, benzene is supposed to be a hybrid of two Kekule structures, by noting that Dewar and other ionic structures make little contribution to the resonance in benzene. However, classical ab initio VB calculations with all possible 175 resonance structures by Norbeck et al. [51] and Tantardini et al. [3], where strictly atomic orbitals are used to construct VB functions, manifested that the five covalent Kekule and Dewar structures make even less contribution to the ground state of benzene than the other 170 ionic structures. This prompts us to reconsider the mathematical formulations for resonance structures [52]. 

The resonance theory says that if two compounds have the same set of atoms and bonds, then that compound for which more resonance structures can be drawn will be the more stable. Using this rule, which would be more stable, anthracene, phenanthrene, or benzazulene ... 

Figure 14.9 Resonant Cu KL23L23 Auger spectra excited from Cu metal foil using near-threshold- and sub-threshold-energy X-ray photons [25]. The energies of the photons related to the K-absorption threshold are indicated. Dots experimental data corrected for inelastic electron scattering solid line calculated spectra using the resonant X-ray scattering theory and the DV-Xa cluster MO model.

The great usefulness, and hence the great value, of the resonance theory lies in the fact that it retains the simple though crude type of structural representation which we have used so far in this book. Particularly helpful is the fact that the stability of a structure can often be roughly estimated from its reasonableness. If only one reasonable structure can be drawn for a molecule, the chances are good that this one structure adequately describes the molecule. 

Discuss the nature of the bonding in the nitrite ion (NO2). Draw the possible Lewis resonance diagrams for this ion. Use the VSEPR theory to determine the steric numbeg the hybridization of the central nitrogen atom, and the geometry of the ion. Show how the use of resonance structures can be avoided by introducing a de-localized 77 MO. What bond order does the MO model predict for the N—O bonds in the nitrite ion ... 

Note that the microscopic theory of Fermi resonance with polaritons, developed above, cannot be directly applied to cubic crystals, because triply degenerate states correspond to dipole-active transitions in such crystals (for the corresponding generalization of the theory, see (41)). However, as was mentioned previously, the polariton spectrum can also be found within the framework of macroscopic electrodynamics, which requires that the dielectric tensor of the crystal be known. The results of a proper analysis, as could be expected, are equivalent to those obtained in microscopic theory. We shall use the macroscopic theory in the following in application to cubic crystals. Using this approach we shall show additionally how the longitudinal and surface biphonons can also be found (see also (15)). 

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