Some Molecular Orbital Results

In the course of our discussion we shall draw heavily on MO terminology and results but shall avoid MO calculations as such. In the primitive aspect of the theory, n one-electron MO s, are constructed by a linear combination of atomic orbitals (LCAO)  

Compilations of the results of Hiickel MO (HMO) calculations for a variety of polyene systems are available (Coulson and Streitwieser, 1965 Heilbronner and Straub, 1966). These include the coefficients cn in any MO, and the MO energies 

For a three-atom system consisting of three identical j orbitals, which may be s or p, the MO s are given as follows  

The ground state of linear Ajf (four electrons) is indicated as = while the first excited state may be written as W2 = By reference 

With regard to the shape of the MO s themselves, consider a set of four parallel and equally spaced -orbitals of a carbon chain, as a model 

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Some molecular orbital results for first- and second-row diatomic molecules, as well as relevant experimental data, are summarized in Table S.5. 

It has been customary to classify methods by the nature of the approximations made. In this sense CNDO, INDO (or MINDO), and NDDO (Neglect of Diatomic Differential Overlap) form a natural progression in which the neglect of differential overlap is applied less and less fully. It is now clearer that there is a deeper division between methods, related to their objectives. On the one hand are approximate methods which set out to mimic the ab initio molecular orbital results. The objective here is simply to find a more economical method. On the other hand, some workers, recognizing the defects of the MO scheme, aim to produce more accurate results by the extensive use of parameters obtained from experimental data. This latter approach appears to be theoretically unsound since the formalism of the single-determinant wavefunction and the Hartree-Fock equations is retained. It can be argued that the use of the single-determinant wavefunction prevents the consistent achievement of predictions better than those obtained by the ab initio scheme where no further... 

So far, it has been assumed in deriving the g values, etc., that the unpaired electron is localized entirely on the metal atom, i.e., the crystal field approximation. This is not usually the case since the formation of molecular orbitals results in the transfer of some of the unpaired electron density onto the ligands. The effect is to reduce the orbital angular momentum, i.e., further quench the orbital contribution and so make the magnetic parameters have values closer to spin-only, i.e., g = 2. 

The results of some molecular orbital calculations that treat explicitly either all of the electrons or all of the valence-shell electrorrs, support the molecular orbital theory that has been described in this Section. It may also be noted that the through-bond coupling of lone-pair orbitals over three o-bonds is equivalent to lone-pair delocalization into the antibonding a orbital between the central o-bond. 

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

Valence band spectra provide information about the electronic and chemical structure of the system, since many of the valence electrons participate directly in chemical bonding. One way to evaluate experimental UPS spectra is by using a fingerprint method, i.e., a comparison with known standards. Another important approach is to utilize comparison with the results of appropriate model quantum-chemical calculations 4. The combination with quantum-chcmica) calculations allow for an assignment of the different features in the electronic structure in terms of atomic or molecular orbitals or in terms of band structure. The experimental valence band spectra in some of the examples included in this chapter arc inteqneted with the help of quantum-chemical calculations. A brief outline and some basic considerations on theoretical approaches are outlined in the next section. 

We shall illustrate these rules first with H2 and then with other diatomic molecules. The same principles apply to polyatomic molecules, but their molecular orbitals are more complicated and their energies are harder to predict. Mathematical software for calculating the molecular orbitals and their energies is now widely available, and we shall show some of the results that it provides. 

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