VB method

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

In the VB method, a wave equation is written for each of various possible electronic structures that a molecule may have (each of these is called a canonical form), and the total )/ is obtained by summation of as many of these as seem plausible, each with its weighting factor ... 

The two chief general methods of approximately solving the wave equation, discussed in Chapter 1, are also used for compounds containing delocalized bonds. In the VB method, several possible Lewis structures (called canonical forms) are drawn and the molecule is taken to be a weighted average of them. Each in Eq. (1.3),... 

The MO and VB methods provide altema ive but equivalent descriptions of the bonding in a molecule. A set of molecular orbitals can always be transformed into a corresponding set of more localized orbitals, and vice versa. For example, according to the MO de-... 

We have made one rather obvious omission from our descriptions of molecule electronic structure - the structure of transition-metal ions. This is deliberate since, in spite of the well-developed theories of the electronic spectra (U.V., photo-electron) of these compounds, it is still true to say that there is no theory of the bonding in this important class of molecules. The question of the localised or de-localised nature of the electronic structure of the bonds in these systems has not really been solved historically, there has been some skirmishing about the superiority of the MO or VB methods but the nature of the valence in these molecules has received a disproportionately small amount of attention. Thus any attempt to develop a GHO basis for transition-metal compounds is perhaps premature until more experience has been gained with typical element chemistry. 

Simonetta and Heilbronner (1964) recently carried out calculations by the valence bond (VB) method for some simple cations, and compared the results obtained by this method, inter alia, with the results of Colpa and collaborators (1963) and of Koutecky and Paldus (1963). In the case of the proton addition complexes of mesitylene and cyclohepta-triene, the electron excitation energies calculated by the VB method agree very well with experiments, and also agree to a good approximation with the results of Cl calculations. The calculations also successfully reproduce the electron density of the cycloheptatriene cation. In this, a perturbation calculation allowed for the AO s adjoining the —CHg—CH2-lihkage. 

Under the Born-Oppenheimer approximation, two major methods exist to determine the electronic structure of molecules The valence bond (VB) and the molecular orbital (MO) methods (Atkins, 1986). In the valence bond method, the chemical bond is assumed to be an electron pair at the onset. Thus, bonds are viewed to be distinct atom-atom interactions, and upon dissociation molecules always lead to neutral species. In contrast, in the MO method the individual electrons are assumed to occupy an orbital that spreads the entire nuclear framework, and upon dissociation, neutral and ionic species form with equal probabilities. Consequently, the charge correlation, or the avoidance of one electron by others based on electrostatic repulsion, is overestimated by the VB method and is underestimated by the MO method (Atkins, 1986). The MO method turned out to be easier to apply to complex systems, and with the advent of computers it became a powerful computational tool in chemistry. Consequently, we shall concentrate on the MO method for the remainder of this section. 

The approach used first, historically, and the one this book is about, is called the valence bond (VB) method today. Heitler and London[8], in their treatment of the H2 molecule, used a trial wave function that was appropriate for two H atoms at long distances and proceeded to use it for all distances. The ideal here is called the separated atom limit . The results were qualitatively correct, but did not give a particularly accurate value for the dissociation energy of the H—H bond. After the initial work, others made adjustments and corrections that improved the accuracy. This is discussed folly in Chapter 2. A cmcial characteristic of the VB method is that the orbitals of different atoms must be considered as nonorthogonal. 

In 1949 Coulson and Fisher[15] introduced the idea of nonlocalized orbitals to the VB world. Since that time, suggested schemes have proliferated, all with some connection to the original VB idea. As these ideas developed, the importance of the spin degeneracy problem emerged, and VB methods frequently were described and implemented in this context. We discuss this more fully later. 

In all of the various VB methods that have been suggested involving nonlocal orbitals it is obvious that the orbitals must be written as linear combinations of AOs at many centers. Thus one is always faced with some sort of nonlinear minimization of the Rayleigh quotient. 

A goal of this chapter is to show, for the diatomic molecules under discussion, both the capability of the VB method in providing quantitative estimates of molecular properties and its capability of giving qualitative pictures of the bonding. The quantitative results are illustrated in Table 11.1, where we give values for Rg, the equilibrium bond distance, and determined theoretically with ST03G, 6-3IG,... 

VB method has been reported.290 The VB method has also been used to study the spectrum of diphenylsulfide.159... 

Several valence-bond (VB) treatments of heterocyclic compounds were reported in the thirties and forties.1, 2 The known difficulty in applying the VB method to complicated molecules has made an overwhelming majority of authors use the molecular orbital (MO) method. In most cases its simplest version, the naive MO LCAO method, has been used. This approximation differs from the well-known Hiickel... 

An alternative to the SCF MO approach is the valence-bond (VB) method, which is discussed in most quantum-chemistry texts. 

There are many compounds with structures in which electrons are delocalized over more than two atoms. Such molecules should be more stable than would be expected for molecules with the same geometry but with electron pairs constrained to be associated with just one or two atoms. We will shortly discuss some specific examples, but because most of these examples involve the delocalization of it electrons, it is expedient to first discuss ethene as a prototype, using both the MO and VB methods. 

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