Basic VB Theory

The main features of the chemical bonding formed by electron pairs were captured in the early days of quantum mechanics by Heitler and London. Their model, which came to be known, as the valence bond (VB) model in its later versions, will serve as our basic tool for developing potential surfaces for molecules undergoing chemical reactions. Here we will review the basic concepts of VB theory and give examples of potential surfaces for bond-breaking processes. 

The MO theory differs greatly from the VB approach and the basic MO theory is an extension of the atomic structure theory to molecular regime. MOs are delocalized over the nuclear framework and have led to equations, which are computationally tractable. At the heart of the MO approach lies the linear combination of atomic orbitals (LCAO) formahsm... 

After reviewing basic elements of VB theory, we would like to create bridges between the popular and widely used MO theory and the less familiar VB... 

A difference between the qualitative VB theory, discussed in Chapter 3, and the spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO-based determinant, without any a priori bias for a given electronic coupling into bond pairs like those used in the Rumer basis set of VB structures. The bond coupling results from the diagonalization of the Hamiltonian matrix in the space of the determinant basis set. The theory is restricted to determinants having one electron per AO. This restriction does not mean, however, that the ionic structures are neglected since their effect is effectively included in the parameters of the theory. Nevertheless, since ionicity is introduced only in an effective manner, the treatment does not yield electronic states that are ionic in nature, and excludes molecules bearing lone pairs. Another simplification is the zero-differential overlap approximation, between the AOs. 

This text is aimed at a nonexpert audience and designed as a tutorial material for teachers and students who would like to teach and use VB theory, but who otherwise have basic knowledge of quantum chemistry. As such, the primary focus of this textbook is a qualitative insight of the theory and ways to apply this theory to the problems of bonding and reactivity in the ground and excited states of molecules. Almost every chapter contains problem sets followed by answers. These problems provide the teachers, students, and interested readers with an opportunity to practice the art of VB theory. We will be indebted to readers—teachers—students for comments and more suggestions, which can be incorporated into subsequent editions of this book that we hope, will follow. 

In practice, the valence bond picture has probably exerted more influence on how chemists actually think than the HMO picture. However most early applications were primarily qualitative in nature. This qualitative VB picture can be summarized under die name of resonance theory [10]. The basic concept is that in general the more ways one has of arranging the spin pairing in the VB wave function, the more stable the molecule is likely to be. Thus, VB theory predicts that phenanthrene with 14 carbon atoms and 5 Kekule structures should be more stable than anthracene with 14 carbon atoms but just 4 Kekule structures, in complete accord with the experimental evidence. It also predicts that benzenoid hydrocarbons with no Kekule structures should be unstable and highly reactive, and in fact no such compounds are knowa Extensions of this qualitative picture appear, for example, in Clar s ideas of resonant sextets [11], which seem to be very powerful in rationalizing much of the chemistry of benzenoid aromatic hydrocarbons. The early ascendancy of HMO theory was thus largely based on the ease with which it could be used for quantitative computations rather than on any inherent superiority of its fundamental assumptions. 

In Section 3, the Reporter has attempted to cast VB theory into as compact and unified a form as possible by making considerable use of group theoretical techniques. This is followed by a discussion of the various improvements and extensions that have been made over the past few years. The basic difficulty in VB theory is the calculation of the matrix elements of the hamiltonian when there is no orthogonality between the orbitals involved. This problem is also discussed at some length in this section, together with a survey of the various approaches that have been tried or proposed for its solution. 

These expressions illustrate at once the basic difficulty of any general A-elec-tron theory such as VB theory, for unless some assumptions are made about the form of the function 0, each expression consists of Nl terms. This is a matter to which considerable attention will be devoted in this article. However, there is to date no satisfactory general solution to this problem. 

Relationship between Valence Bond and Spin Valence Theories.—We consider first for simplicity a diatomic molecule AB. The basic physical idea behind all the variants of VB theory is that the wavefunction for the molecule, Pab, should in some way be written as a product of the wavefunctions Pa, Pb for particular states of the participating atoms. Thus... 

Describe the basic ideas of the valence bond (VB) theory... 

The basic idea of valence bond (VB) theory is very simple the wavefunctions for the electrons in a molecule are constructed directly from the wavefunctions of the constituent atoms. This implements in a very clear cut way a large part of the experience of chemistry. (For a review of classical valence bond theory, the reader should consult Ref. 1, for example.)... 

In the following section we present a general framework in which non-orthogonal orbitals are used to expand the exact wavefunction. This serves to explain the spin-coupled VB theory which is the basic motif of this chapter, and also to show how this reduces to classical VB theory on the one hand, and to the Cl expansion on the other. 

The basic principle of VB theory is that a covalent bond forms when orbitals of two atoms overlap and the overlap region, which is between the nuclei, is occupied by a pair of electrons. ( Orbital overlap is another way of saying that the two wave functions are in phase, so the amplitude increases between the nuclei.) The central themes of VB theory derive from this principle ... 

Following chemical intuition and basic ideas of VB theory, two additional assumptions are made ... 

The molecular orbital (MO) is the basic concept in contemporary quantum chemistry. " It is used to describe the electronic structure of molecular systems in almost all models, ranging from simple Hiickel theory to the most advanced multiconfigurational treatments. Only in valence bond (VB) theory is it not used. Here, polarized atomic orbitals are instead the basic feature. One might ask why MOs have become the key concept in molecular electronic structure theory. There are several reasons, but the most important is most likely the computational advantages of MO theory compared to the alternative VB approach. The first quantum mechanical calculation on a molecule was the Heitler-London study of H2 and this was the start of VB theory. It was found, however, that this approach led to complex structures of the wave funetion when applied to many-electron systems and the mainstream of quantum ehemistry was to take another route, based on the success of the central-field model for atoms introduced by by Hartree in 1928 and developed into what we today know as the Hartree-Foek (HF) method, by Fock, Slater, and co-workers (see Ref. 5 for a review of the HF method for atoms). It was found in these calculations of atomic orbitals that a surprisingly accurate description of the electronic structure could be achieved by assuming that the electrons move independently of each other in the mean field created by the electron cloud. Some correlation was introduced between electrons with... 

Predict whether a molecule is polar or nonpolar Describe the basic ideas of the valence bond (VB) theory Describe the hybrid orbitals used in bonding in polyatomic molecules and ions... 

In Chapter 7 we described covalent bonding as electron pair sharing that results from the overlap of orbitals from two atoms. This is the basic idea of the valence bond (VB) theory—it describes how bonding occurs. In many examples throughout this chapter, we first use the VSEPR theory to describe the orientations of the electron... 

In VB theory overlap repulsion is clearly associated with the Pauli exclusion principle and occurs in three distinct situations which are shown in drawings 6-8, along with the corresponding monoelectronic expression of the overlap repulsion [8]. As may be seen, the overlap repulsion is given in — 2hs units for each pair of electrons which possess identical spins. In turn, the basic unit of overlap repulsion is identical in absolute magnitude to the HL stabilization in Eq. (3) (disregarding the effect of the normalization constants). 

This chapter is aimed at the nonexpert and designed as a tutorial for faculty and students who would like to teach and use VB theory, but possess only a basic knowledge of quantum chemistry. As such, an important focus of the chapter will be the qualitative wisdom of the theory and the way it applies to problems of bonding and reactivity. This part will draw on material discussed in previous works by the authors. Another focus of the chapter will be on the main methods available today for ab initio VB calculations. However, much important work of a technical nature will, by necessity, be left out. Some of this work (but certainly not all) is covered in a recent monograph on VB theory. ... 

Apart from these simplifying assumptions, a fundamental difference between qualitative VB theory and spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO determinant, without any a priori bias for a given electronic coupling into bond pairs. Instead of an interplay between VB structures, a molecule is viewed then as a collective spinordering The electrons tend to occupy the molecular space (i.e., the various atomic centers) in such a way that an electron of a spin will be surrounded by as many p spin electrons as possible, and vice versa. Determinants having this property, called the most spin-alternated determinants (MSAD) have the lowest energies (by virtue of the VB rules, in Qualitative VB Theory) and play the major role in electronic structure. As a reminder, the reader should recall from our discussion above that the unique spin-alternant determinant, which we called the quasiclassical state, is used as a reference for the interaction energy. 

There are two major approaches to the calculation of molecular structure, valence bond theory (VB theory) and molecular orbital theory (MO theory). Almost all modern computational work makes use of MO theory, and we concentrate on that theory in this chapter. Valence bond theory, however, has left its imprint on the language of chemistry, and it is important to know the significance of terms that chemists use every day. The structure of this chapter is therefore as follows. First, we present VB theory and the terms it introduces. Next, we present in more detail the basic ideas of MO theory. Finally, we see how computational techniques based on MO theory pervade all current discussions of molecular structure, including the prediction of the physiological properties of therapeutic agents. 

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