Valence-Bond Treatment of Polyatomic Molecules

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. 

The oxygen-atom ground-state electron configuration is l 2s 2p with an unpaired electron in each of the AOs 2py and 2p. We thus assume that these AOs along with the hydrogen Is AOs will form electron-pair bonds. The three possible ways of pairing these four AOs to get covalent structures are shown in Hg. 15.18. 

We then take as a trial variation function i[t, a linear combination of the bond eigenfunctions of structures A, B, and C. However, the functions and I c are not lin- 

It is wasted effort to include all three structures we shall drop structure C, taking + Cb B- 

Rumer s procedure is easily justified. Let I ( ), ( H), and I ( x ) be three bond eigenfunctions that involve any number of AOs, but that differ only in the way they pair up a certain subset of four AOs each of these functions corresponds to one of the three different ways of pairing these four AOs (see Fig. 15.18). By a slight extension of (15.151), it follows that 

The weight of each resonance structure in the wave function is sometimes taken as proportional to the square of its coefficient in the wave function. Because the bond eigenfunctions are not mutually orthogonal, the electron probability density is not equal to the weighted sum of the probability densities of the various structures, and the c, p quantities are somewhat lacking in direct physical significance. There are several other ways of 

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Section 15.24 Valence-Bond Treatment of Polyatomic Molecules 605... 

Two basic methods, the valence-bond (VB) and the molecular orbital (MO) method, have been developed for the determination of approximate state functions. In practice, the MO method constitutes the simplest and most efficient approach for the treatment of polyatomic molecules. And, in fact, all the calculations for the systems under consideration have been carried out within the framework of the MO theory. 

In subsequent independent papers, Pauling [4] and Slater [6] generalized the valence-bond treatment made for the H2 molecule to polyatomic systems as H2O, NH3, CH4 etc. .. where an atom of the first period (the second row) is linked to hydrogens by several two-electron bonds they described the valence orbitals coming from the central atom by appropriate s and p combinations known later as hybrid orbitals. At the same time Hund [7] and Mulliken [8] presented another quantum theory of valence, the molecular orbital method in LCAO form, using the spectroscopic concept of molecular configuration built from s, p, d. ..pure atomic orbitals. The actual status of the hybridization process was clarified by Van Vleck [9], who showed that the various approximations... 

The valence bond concepts of atomic orbital overlap discussed in Chapter 3 also apply to polyatomic molecules. A satisfactory bonding scheme, however, must account for molemlar geometry. 1 d S three exmpJes of VB treatment of bonding in polys E( o9lBculel ... 

Classical Dynamics of Nonequilibrium Processes in Fluids Integrating the Classical Equations of Motion Control of Microworld Chemical and Physical Processes Mixed Quantum-Classical Methods Multiphoton Excitation Non-adiabatic Derivative Couplings Photochemistry Rates of Chemical Reactions Reactive Scattering of Polyatomic Molecules Spectroscopy Computational Methods State to State Reactive Scattering Statistical Adiabatic Channel Models Time-dependent Multiconfigurational Hartree Method Trajectory Simulations of Molecular Collisions Classical Treatment Transition State Theory Unimolecular Reaction Dynamics Valence Bond Curve Crossing Models Vibrational Energy Level Calculations Vibronic Dynamics in Polyatomic Molecules Wave Packets. 

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