Estimation of Orbital Energies

The numerical evaluation of the energies of orbitals and states is fundamentally a matter of making quantum mechanical computations. As indicated in Chapter 1, quantum mechanics per se is not the subject of this book, and indeed we have tried in general to avoid any detailed treatment of methods for solving the wave equation, emphasis being placed on the properties that the wave functions must have purely for reasons of symmetry and irrespective of their explicit analytical form. However, this discussion of the symmetry aspects of ligand field theory would be artificial and unsatisfying without some 

Our discussion here of computational procedures will be very superficial and aimed at bringing out the physical features of the models. For a full treatment of this subject with references to the original literature, and for further discussion of the interpretation of the chemical behavior of transition metal compounds in terms of ligand field theory, the reader is referred to the publications cited in Appendix IX. 

As noted in Section 9.1, there are three closely related theories of the electronic structures of transition metal complexes, all making quite explicit use of the symmetry aspects of the problem but employing different physical models of the interaction of the ion with its surroundings as a basis for computations. These three theories, it will be recalled, are the crystal field, ligand field, and MO theories. There is also the valence bond theory, which makes less explicit use of symmetry but is nevertheless in accord with the essential symmetry requirements of the problem. We shall now briefly outline the crystal field and ligand field treatments and comment on their relationship to the MO theory. 

This model of a complex or of a crystalline salt of a metal ion in a compound such as a halide or oxide is of an electrostatic, point charge, or point dipole type. The ligands or neighbors of the metal ion are treated as structureless, orbital-less point charges, which set up an electrostatic field. The effect of this field on electrons in the d orbitals of the metal ion is then investigated. 

It should be noted that the splitting A0 is generally of the order of 1-3 eV, whereas the elevation of the set of d levels as a whole is of the order of 20-40 eV. Thus it should always be borne in mind that the crystal and ligand field theories focus attention on only one relatively small aspect of the overall energy of formation of a complex. 

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We describe two procedures of how to construct the OS global hybrid functionals using Eq. (14.61) the determination of OS parameters and estimation of orbital energies. 

FIGURE 2.7. (a) Three active pz orbitals that are used in the quantum treatment of the X + CH3-Y— X-CH3 + Y Sw2 reaction, (b) Valence-bond diagrams for the six possible valence-bond states for four electrons in three active orbitals, (c) Relative approximate energy levels of the valence-bond states in the gas phase (see Table 2.4 for the estimation of these energies). 

The different struetures and transitions states of interest in the neutral and negative ion reaetions are represented in Fig. 2. A first approach was done at the SCF level, using the split-valence 4-3IG basis set. In order to provide a better estimation of the energy differences implied in this reaction schemes, extensive calculations have been performed at the MP2 level of theory using the 6-311++G basis set which contains the diffuse orbitals necessary to quantitatively describe the negative ions. 

It must be admitted that 0-0 and a-n switch processes are of limited stereochemical interest. Given the prototypes (92) or (97), the reactions must be syw-stereospecific any other stereochemical result would in fact be used as a criterion of a multi-step mechanism. This is also true of the mixed types, (101)-(104). But these reactions are exceedingly useful as models for orbital-based calculations or estimates of free energies of activation, with the use of extended HMO theory. The four- or six-center process (92)-(104) does not appear to be more complex than the Diels-Alder reaction, which has been investigated theoretically (Herndon and Hall, 1967). 

A similar idea to that involved in EPCE-F26" was used for the estimates of correlation energy in different electronic states of diatomic molecules. The essence of that approach is the conversion of the MO representation of the electronic structure of a particular electronic state to the atomic representation by means of the population analysis and the assumption that the predominant correlation effects are due to electron pairs in the same orbital. In this way, the correlation energy is again given by the products of AO populations and AO correlation contributions (available in the literature). Although the approach is rough, it may be useful for spectroscopic purposes. Table 4.2 presents an example of the treatment of this type. 

In the following pages, the valence bond theory and the crystal field theory are described very briefly to set more recent developments in their historical context. The rest of the chapter describes the ligand field theory and the method of angular overlap, which can be used to estimate the orbital energy levels. These two supply the basic approach to bonding in coordination compounds for the remainder of the book. 

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