Resonating VB theory

Further in 1986 P. W. Anderson suggested [76] that resonating VB theory would be crucial to the understanding of high-temperature superconductivity. This... 

Surely the continuing VB-theoretic successes which have been achieved in the last few decades indicate much promise for future extensions. Resonating VB theory MO theory may be seen as complementary descriptions, each with greater utility in different circumstances. 

And finally resonating VB theory was proposed [18] as crucial in understanding high-temperature superconductivity, whereafter the ideas and solution techniques were vastly developed, in a many-body context. 

The positive charge in 28 is stabilized by /j-cr-C-C-hyperconj ligation with the C-C-ring bonds of the two cyclopropyl moieties. In the parlance of VB theory this is described by resonance of 28 with non-bonding resonance limiting structures, the homoallenyl cation type structure 28a, the homopropargyl cation type structure 28b and the Dewar-type limiting resonance structure 28c. 

The overlap integrals between the 3d(M ) and 4d(M) or 5rf(M) functions show that at an M —M separation of ca. 2.8 A direct metal-metal interactions should also be taken into consideration. The four-membered ring systems M (Sbr)2M can be described (in terms of VB theory) with the resonance structures (23) and (24). 

Another feature of the resonance mixing, already alluded to, is the sign inversion which is caused by the different nature of the matrix elements that mix the Kekule structures for aromatics and antiaromatics. Thus, in the case of benzene (part a), the ground state is the positive combination of the two Kekule structures, while in cyclobutadiene (part b), the ground state is the negative combination.15116158210 214 Consequently, the twin excited states are the negative and positive linear combin-tions, respectively, for aromatics and antiaromatics.13-15-115-209 212 This relationship of the ground and excited states to the fundamental Kekule structures has been derived early on by the pioneers of VB theory.211-214... 

The molecule can now be fitted together. If more than one Lewis structural formula can be written down and can be plausibly described in terms of VB theory, resonance is invoked. 

The concepts of hybridisation and resonance are the cornerstones of VB theory. Unfortunately, they are often misunderstood and have consequently suffered from much unjust criticism. Hybridisation is not a phenomenon, nor a physical process. It is essentially a mathematical manipulation of atomic wave functions which is often necessary if we are to describe electron-pair bonds in terms of orbital overlap. This manipulation is justified by a theorem of quantum mechanics which states that, given a set of n respectable wave functions for a chemical system which turn out to be inconvenient or unsuitable, it is permissible to transform these into a new set of n functions which are linear combinations of the old ones, subject to the constraint that the functions are all mutually orthogonal, i.e. the overlap integral J p/ip dT between any pair of functions ip, and op, (i = j) is always zero. This theorem is exploited in a great many theoretical arguments it forms the basis for the construction of molecular orbitals as linear combinations of atomic orbitals (see below and Section 7.1). 

In the 1950s and 1960s, inorganic chemists became increasingly interested in phenomena such as paramagnetism, electron spin resonance, electronic spectroscopy and photoelectron spectroscopy. VB theory is ill-adapted to the interpretation of these properties, and inorganic chemists were forced to look elsewhere (see Sections 2.6-2.8). 

Most organic chemists are familiar with two very different and conflicting descriptions of the 7r-electron system in benzene molecular orbital (MO) theory with delocalized orthogonal orbitals and valence bond (VB) theory with resonance between various canonical structures. An attitude fostered by many text books, especially at the undergraduate level, is that the VB description is much easier to understand and simpler to use, but that MO theory is in some sense more fundamental . 

In the preceding section, we discussed the electron pair (2c-2e) bond and how it can be influenced by Pauli repulsion of the SOMOs with other electrons. In the three-electron (2c-3e) bond, Pauli repulsion plays an even more fundamental role, as we will see.72 The idea of the three-electron bond was introduced in the early 1930s by Pauling in the context of the valence bond (VB) model of the chemical bond.70 71 Since then, it has been further developed both in VB and in MO theory and has become a standard concept in chemistry.118-129 In VB theory,7°>71 118 123 the two-center, three-electron (2c-3e) bond between two fragments A and B is viewed as arising from a stabilizing resonance between two valence bond structures in which an electron pair is on fragment A and an unpaired electron on B (13a), or the other way around (13b) ... 

This idea is a failure of simple resonance theory, not of VB theory. Taking into account the sign of the matrix element (overlap) between the five VB structures shows that singlet CsH5+ is Jahn—Teller unstable, and the ground state is in fact the triplet state. As shown later in Chapter 5, this is generally the case for all of the antiaromatic ionic species having 4n electrons over 4n + 1 or 4n - 1 centers (61). 

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