Molecular-orbital approach

In the simplest case there are two localised magnetically active orbitals pA and tpg that can be combined into two molecular orbitals (MO) 

A comparison with the VB method shows that the MO method includes both the covalent structures ( l5 P2) and the ionic structures (4 3, P4), but with the same weights. This leads to a drawback which will be discussed later. 

The full molecular wave function is given by the antisymmetrised product (a Slater determinant), for instance 

The evaluation of this expression leads to the final formula 

The exchange coupling constant evaluated at this level of approximation reads 

The basic starting point is the consecrated time-independent Schrbdinger equation 

Of course, as it is, Eq. (4.249) cannot be solved exactly, in its most general way. The approximations have to be implemented in such a way as to include the specific reality of the dynamic electronic-nuclear system. In this respect, considering an approximation is not viewed as a limitation here, but rather as a sort of rescaling of the concerned issue. Epistemologically, it is equivalent with Descartes scholastic methodology of reducing a problem to smaller problems through the method of analysis. 

Such a procedure has been long verified in mathematical-physics with impressive praetical applieations, for example, the integral-differential recipes, and with be thus safely implemented also here without loss in generality of the basie problem. 

In quantum chemistry the speeifie method was consecrated as Bom-Oppenheimer approximation that separate the electronic-nuclear system and problem in two smaller parametrieally linked subsystems associated with an electronic motion, defined by equations 

It is worth noting that this phenomenological separation of electronic and nuclear problems may be possible due to the impressive difference in their mass that practically fixes the nuclei as the reference system in which the electronic system evolves. This is, nevertheless, only the first and most straight (however appropriate) approximation considered upon a many-body (electrons and nuclei) problem. 

The effect of the environment (which is denoted by s) surrounding the reacting system (r, referred to as solute) can be included in MO calculations in a fairly straightforward way if the overlap integrals between the solute and solvent are neglected [7]. The effect of the solvent polarisation on the solute electronic states can be estimated within the zero differential overlap approximation by rewriting the diagonal matrix elements of r as 

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The molecular orbital approach to chemical bonding rests on the notion that as elec trons m atoms occupy atomic orbitals electrons m molecules occupy molecular orbitals Just as our first task m writing the electron configuration of an atom is to identify the atomic orbitals that are available to it so too must we first describe the orbitals avail able to a molecule In the molecular orbital method this is done by representing molec ular orbitals as combinations of atomic orbitals the linear combination of atomic orbitals molecular orbital (LCAO MO) method... 

The mechanism of the Diels-Alder reaction is best understood on the basis of a molecular orbital approach To understand this approach we need to take a more detailed look at the rr orbitals of alkenes and dienes... 

A common example of the Peieds distortion is the linear polyene, polyacetylene. A simple molecular orbital approach would predict S hybddization at each carbon and metallic behavior as a result of a half-filled delocalized TT-orbital along the chain. Uniform bond lengths would be expected (as in benzene) as a result of the delocalization. However, a Peieds distortion leads to alternating single and double bonds (Fig. 3) and the opening up of a band gap. As a result, undoped polyacetylene is a semiconductor. 

It should be noted that a comprehensive ELNES study is possible only by comparing experimentally observed structures with those calculated [2.210-2.212]. This is an extra field of investigation and different procedures based on molecular orbital approaches [2.214—2.216], multiple-scattering theory [2.217, 2.218], or band structure calculations [2.219, 2.220] can be used to compute the densities of electronic states in the valence and conduction bands. 

The frontier molecular orbital approach provides a description of the bonding interactions that occur in the 8 2 process. The orbitals involved are depicted in Fig. 

This chapter will try to cover some developments in the theoretical understanding of metal-catalyzed cycloaddition reactions. The reactions to be discussed below are related to the other chapters in this book in an attempt to obtain a coherent picture of the metal-catalyzed reactions discussed. The intention with this chapter is not to go into details of the theoretical methods used for the calculations - the reader must go to the original literature to obtain this information. The examples chosen are related to the different chapters, i.e. this chapter will cover carbo-Diels-Alder, hetero-Diels-Alder and 1,3-dipolar cycloaddition reactions. Each section will start with a description of the reactions considered, based on the frontier molecular orbital approach, in an attempt for the reader to understand the basis molecular orbital concepts for the reaction. 

We said in Section 1.5 that chemists use two models for describing covalent bonds valence bond theory and molecular orbital theory. Having now seen the valence bond approach, which uses hybrid atomic orbitals to account for geometry and assumes the overlap of atomic orbitals to account for electron sharing, let s look briefly at the molecular orbital approach to bonding. We ll return to the topic in Chapters 14 and 15 for a more in-depth discussion. 

In Chapter 9, we considered a simple picture of metallic bonding, the electron-sea model The molecular orbital approach leads to a refinement of this model known as band theory. Here, a crystal of a metal is considered to be one huge molecule. Valence electrons of the metal are fed into delocalized molecular orbitals, formed in the usual way from atomic... 

Burdett JK (1976) The Shapes of Main-Group Molecules A Simple Semi-Quantitative Molecular Orbital Approach. 31 67-105... 

Lewis DFV, loannides C, Parke DV. Validation of a novel molecular-orbital approach (COMPACT) for the prospective safety evaluation of chemicals, by comparison with rodent carcinogenicity and Salmonella mutagenicity data evaluated by the United States NCI NTP. Mut Res 1993 291 61-77. 

Molecular orbital theory is more complex than the hybrid orbital approach, but the foundations of the model are readily accessible. Though complex, molecular orbital theory opens the door to many fascinating aspects of modem chemistry. In this section, we introduce the molecular orbital approach through diatomic molecules. 

Our treatment of O2 shows that the extra complexity of the molecular orbital approach explains features that a simpler description of bonding cannot explain. The Lewis structure of O2 does not reveal its two unpaired electrons, but an MO approach does. The simple (t-tt description of the double bond in O2 does not predict that the bond in 2 is stronger than that in O2, but an MO approach does. As we show in the following sections, the molecular orbital model has even greater advantages in explaining bonding when Lewis structures show the presence of resonance. 

Noteworthy also is the extensive compilation of early data on layered MX2 given by Wilson and Yoffe [37], who worked out a group-by-group correlation of transmission spectra of the compounds to available electrical and structural data and produced band models in accord with a molecular orbital approach. 

Table 2.4 shows a comparison of the experimental and PPP-MO calculated electronic spectral data for azobenzene and the three isomeric monoamino derivatives. It is noteworthy that the ortho isomer is observed to be most bathochromic, while the para isomer is least bathoch-romic. From a consideration of the principles of the application of the valence-bond approach to colour described in the previous section, it might have been expected that the ortho and para isomers would be most bathochromic with the meta isomer least bathochromic. In contrast, the data contained in Table 2.4 demonstrate that the PPP-MO method is capable of correctly accounting for the relative bathochromicities of the amino isomers. It is clear, at least in this case, that the valence-bond method is inferior to the molecular orbital approach. An explanation for the failure of the valence-bond method to predict the order of bathochromicities of the o-, m- and p-aminoazobenzenes emerges from a consideration of the changes in 7r-electron charge densities on excitation calculated by the PPP-MO method, as illustrated in Figure 2.14. 

The considerable number of molecular orbital calculations which have recently been made for sandwich compounds are however considered in some detail in Section 6. This has been done in order to make clear the relationship between the ligand field and molecular orbital approaches, and also to indicate the need for the use of a more sophisticated molecular orbital scheme than that adopted in this Introduction, i.e. one in which the a-framework of the rings is specifically included in the basis set as well as the rr-type orbitals. 

As indicated in Section 1. (i) our approach to the interpretation of the electronic spectra of the hexafluorometallates of the Ad and 5d series closely follows that employed previously (7), and need not be recapitulated in detail. The ligand field model is adopted for the interpretation of the d—d transitions, and the qualitative molecular orbital approach for the treatment of both these and, more particularly, the charge-... 

The preceding discussion is presented in order to show how the basic ideas of the molecular orbital approach are employed. It is also intended to show how to approach getting improved results after the basic ideas are used to generate molecular wave functions. For the purposes here, it is sufficient to indicate the nature of the changes rather than presenting quantitative results of the calculations. 

In this chapter, the basic ideas related to the molecular orbital approach to covalent bonds have been presented. Other applications of the molecular orbital method will be discussed in Chapters 5 and 17. 

When the bonding is considered in the molecular orbital approach, it can be seen that each oxygen atom contributes one p orbital to yield three molecular orbitals. The orbital overlap combinations lead to the wave functions... 

However, a different view of the bonding in XeF2 is provided by a molecular orbital approach in which a p orbital on Xe combines with a p orbital from each F to form a three-center, four-electron linear bond. Actually, the three atomic orbitals form three molecular orbitals, but only the bonding and nonbonding orbitals are populated. This population of the orbitals places nonbonding electron density on the F atoms to give polar Xe-F bonds. 

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