The Valence-Bond Treatment of

The first quantum-mechanical treatment of the hydrogen molecule was by Heitler and London in 1927. Their ideas have been extended to give a general theory of chemical bonding, known as the valence-bond (VB) theory. The valence-bond method is more closely related to the chemist s idea of molecules as consisting of atoms held together by localized bonds than is the molecular-orbital method. The VB method views molecules as composed of atomic cores (nuclei plus inner-sheU electrons) and bonding valence electrons. For H2, both electrons are valence electrons. 

The first step in the Heitler-London treatment of the H2 ground state is to approximate the molecule as two ground-state hydrogen atoms. The wave function for two such noninteracting atoms is 

This leads to a 2 X 2 secular determinant that is the same as (8.56), except that W is replaced by see Prob. 9.20. 

We now solve the secular equation. The Hamiltonian is Hermitian, all functions are real, and /i and /2 are normalized. Therefore 

Interchange of the coordinate labels 1 and 2 in H22 converts H22 to /fn, since this relabeling leaves H unchanged. Hence Hu = 22- The secular equation det - SijW) = 0 becomes 

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Herzberg (Nobel prize for Chemistry, 1971) commented on the two distinct photoionizations from methane that this observation illustrates the rather drastic nature of the approximation made in the valence bond treatment of CH4, in which the 2s and 2p electrons of the carbon atom are considered as degenerate and where this degeneracy is used to form tetrahedral orbitals representing mixtures of 2s and 2p atomic orbitals. The molecular orbital treatment does not have this difficulty". 

The application of the (8—n) rule to elements preceding group 4 implies the availability of (8—n) electrons per atom for covalent bond formation and is to this extent artificial unless a mechanism for the provision of these electrons can be proposed. A possible mechanism in the case of zinc, based on the valence-bond treatment of metal theory, has already been outlined in 5.28, but it is difficult to feel satisfied that this is more than an ad hoc explanation designed to explain the observed crystal structure of the element if the structure of zinc were unknown there would be few grounds for treating it as other than a simple divalent element. 

The valence-bond treatment of polyatomic molecules is closely tied to chemical ideas of structure. One begins with the atoms that form the molecule and pairs up the unpaired electrons to form chemical bonds. There are usually several ways of pairing up (coupling) the electrons. Each pairing scheme gives a VB structure. A Heitler-London-type function (called a bond eigenfunction) is written for each structure i, and the molecular wave function is taken as a linear combination 2, c,4>, of the bond eigenfunctions. The variation principle is then applied to determine the coefficients c,. The VB wave function is said to be a resonance hybrid of the various structures. 

Sketch the valence bond treatment of BeHj, using boxes to represent the orbitals. Show the process of electron promotion, hybridization, and bond formation. Also sketch a picture of the hybrid orbitals on the Be atom overlapping with the Is AOs on the H ligands. 

Application of valence bond theory to more complex molecules usually proceeds by writing as many plausible Lewis structures as possible which correspond to the correct molecular connectivity. Valence bond theory assumes that the actual molecule is a hybrid of these canonical forms. A mathematical description of the molecule, the molecular wave function, is given by the sum of the products of the individual wave functions and weighting factors proportional to the contribution of the canonical forms to the overall structure. As a simple example, the hydrogen chloride molecule would be considered to be a hybrid of the limiting canonical forms H—Cl, H Cr, and H C1. The mathematical treatment of molecular structure in terms of valence bond theory can be expanded to encompass more complex molecules. However, as the number of atoms and electrons increases, the mathematical expression of the structure, the wave function, rapidly becomes complex. For this reason, qualitative concepts which arise from the valence bond treatment of simple molecules have been applied to larger molecules. The key ideas that are used to adapt the concepts of valence bond theory to complex molecules are hybridization and resonance. In this qualitative form, valence bond theory describes molecules in terms of orbitals which are mainly localized between two atoms. The shapes of these orbitals are assumed to be similar to those of orbitals described by more quantitative treatment of simpler molecules. 

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