Applying VB theory

We illustrate the applications and limitations of VB theory by considering octahedral complexes of Cr(III) d ) and Fe(III) d ) and octahedral, tetrahedral and square planar complexes of Ni(II) d ). The atomic orbitals required for 

Vacemt oitntals available to accept ligand electrons 

Coordination number Arrangement of donor atoms Orbitals hybridized Hybrid orbital description Example 

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When we try to apply VB theory to methane we run into difficulties. A carbon atom has the configuration [HeJ2s22pvl2p l,1 with four valence electrons (34). However, two valence electrons are already paired and only the two half-filled 2/ -orbitals appear to be available for bonding. It looks as though a carbon atom should have a valence of 2 and form two perpendicular bonds, but in fact it almost always has a valence of 4 (it is commonly tetravalent ) and in CH4 has a tetrahedral arrangement of bonds. 

In Chapters 6 and 7, it was observed that VB theory works well in Main Group chemistry provided that we are dealing with closed-shell systems. It runs into difficulties with open-shell systems, especially those where - in an MO treatment - we have partial occupancy of an antibonding level. The same constraints are valid when we come to apply VB theory in d block chemistry they are necessarily more severe. 

These molecules, propenal, methoxyethene, and etheneamine, show how we can apply VB theory and resonance to questions of reactivity. We looked at how structure and conjugation affect electron density and bond formation in both the reactant and the intermediate. When VB theory indicates that the particular disposition of function groups will change the electron distribution relative to an unsubstituted molecule, we can expect to see those differences reflected in altered reactivity. For propenal, the electron withdrawal by the formyl group causes decreased reactivity toward electrophiles and increased reactivity toward nucleophiles. For methoxyethene and ethenamine, the electron release of the substituents is reflected by increased reactivity toward electrophiles with strong selectivity for the P-carbon. 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

In this contribution, we describe and illustrate the latest generalizations and developments[1]-[3] of a theory of recent formulation[4]-[6] for the study of chemical reactions in solution. This theory combines the powerful interpretive framework of Valence Bond (VB) theory [7] — so well known to chemists — with a dielectric continuum description of the solvent. The latter includes the quantization of the solvent electronic polarization[5, 6] and also accounts for nonequilibrium solvation effects. Compared to earlier, related efforts[4]-[6], [8]-[10], the theory [l]-[3] includes the boundary conditions on the solute cavity in a fashion related to that of Tomasi[ll] for equilibrium problems, and can be applied to reaction systems which require more than two VB states for their description, namely bimolecular Sjy2 reactions ],[8](b),[12],[13] X + RY XR + Y, acid ionizations[8](a),[14] HA +B —> A + HB+, and Menschutkin reactions[7](b), among other reactions. Compared to the various reaction field theories in use[ll],[15]-[21] (some of which are discussed in the present volume), the theory is distinguished by its quantization of the solvent electronic polarization (which in general leads to deviations from a Self-consistent limiting behavior), the inclusion of nonequilibrium solvation — so important for chemical reactions, and the VB perspective. Further historical perspective and discussion of connections to other work may be found in Ref.[l],... 

VB and MO theories can be applied to simple molecular systems as follows. According to VB theory, if (p.A and

wave functions of independent systems a and b then the total wave function iff and total energy E are written as follows ... 

Ik this chapter we explore how symmetry considerations can be applied to one of the most pervasive concepts in all of chemistry bonding between atoms by the sharing of pairs of electrons. Though the idea of an electron-pair bond was first introduced in 1916 by G. N. Lewis, it was only after the advent of quantum mechanics that it could be given a proper theoretical basis. This came about through the development of two theories valence bond (VB) theory and localized MO theory both of which describe the electron pair in terms of orbitals of the component atoms of the bond. 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

This chapter was dedicated to demonstrations that all the so-called failures of VB theory are in fact not real. It was shown that in each such failure , one could use a simple VB theory, based on the principles outlined in Chapter 3, and arrive at the correct predictions—results. In so doing, this chapter also provided the reader with an opportunity to apply qualitative VB theory to some classical problems in bonding. Having done so, the reader is now more prepared for the material in Chapter 6, where VB theory is applied to chemical reactivity. 

One of the major problems in applying quantum chemical calculations to excited states is the restricted ability to interpret the calculations in large Cl expansions, such as CIS and CASPT2. This limitation often does not exist in VB theory, which in many cases can assign a few chemical structures to describe a given excited state. As such, the major goal of this chapter is to teach a conceptual VB approach to excited states, based on the qualitative VB theory discussed throughout Chapters 1—6. 

A simple principle of the spin-Hamiltonian VB theory, first formulated by Ovchinnikov (13), applies to alternant conjugated molecules, that is, those molecules that possess fully spin-alternant determinants. The rule is stated as follows ... 

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. 

The matrix element rules (40) and (41), which apply to the superposition pattern for any two structures, 4>a for a singlet state, were first given by Pauling [14] in 1933. They were to form the basis of nearly all semi-empirical applications of VB theory to polyatomic molecules during the next few decades. 

Applying perturbation theory, the value of the ground state energy, relative to the energy of the VB state, is found to be ... 

The failure of crystal field theory and VB theory to explain the spectrochemical series stimulated the development of ligand field theory, which applies qualitative methods of molecular orbital theory to describe the bonding and structure of coordination complexes. The terms ligand field theory and molecular orbital theory are often used interchangeably in inorganic chemistry today. 

To explain bonding in methane, VB theory uses hypothetical hybrid orbitals, which are atomic orbitals obtained when two or more nonequivalent orbitals of the same atom combine in preparation for covalent bond formation. Hybridization is the term applied to the mixing of atomic orbitals in an atom (usually a central atom) to generate a set of hybrid orbitals. We can generate four equivalent hybrid orbitals for carbon by mixing the 2s orbital and the three 2p orbitals ... 

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