Construction of Molecular Orbitals

The symbols E and T represent double and triply degenerate species, respectively. 

If a molecule possesses a center of symmetry, the subscript g indicates symmetry with respect to that center, and the subscript u indicates antisymmetry with respect to that center. 

For a molecule that has a rotation axis other than the principal one, symmetry or antisymmetry with respect to that axis is indicated by subscripts 1 or 2, respectively. When no rotation axis other than the principal one is present, these subscripts are sometimes used to indicate symmetry or antisymmetry with respect to a vertical plane, a . 

The marks and are sometimes used to indicate symmetry or antisymmetry with respect to a horizontal plane, ah. 

It should now be apparent how the species A1( A2, Bv and B2 arise. Character tables have been worked out and are tabulated for all the common point groups. Presenting all the tables here would go beyond the scope of the discussion of symmetry and group theory as used in this book. However, tables for some common point groups are shown in Appendix B. 

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Let us first briefly review the construction of molecular orbitals in simple diatomic molecules, AB, using the linear combination of atomic orbitals (LCAO) scheme. The end product for the first long row of the periodic table is the well-known diagram in Fig. 6-1. We focus on two broad principles that are exploited in the construction of this diagram one has to do with symmetry and overlap, the other concerns energies. 

We shall shortly draw on both of these symmetry and energy aspects of Fig. 6-1 in the construction of molecular orbitals for the octahedron. First, however, let us extend the picture to molecules with more than two atoms. 

We are now ready to apply the ideas in the preceding three sections to the construction of molecular orbitals in octahedral complexes. 

In this chapter, the descriptions of molecular structure will be primarily in terms of a valence bond approach, but the molecular orbital method will be discussed in Chapter 5. As we shall see, construction of molecular orbital diagrams for polyatomic species is simplified by making use of symmetry, which will also be discussed in Chapter 5. 

Despite the obvious limitation of the LCAO procedure as revealed by the Hj and H2 problems it still is the most popular scheme used in the theoretical study of polyatomic molecules. There is a bewildering number of approximate methods, commonly distinguished in terms of cryptic acronyms, designated as either ab initio or semi-empirical, but all of them based on the LCAO construction of molecular orbitals. The precise details can be found in many books and reviews. The present summary uses the discussion of Richards and Cooper [92] as a guide. 

Fig. 1. Construction of molecular orbitals for octahedral coordination complexes of non w-bonding ligands utilizing metal 3

This chapter consists of the application of the symmetry concepts of Chapter 2 to the construction of molecular orbitals for a range of diatomic molecules. The principles of molecular orbital theory are developed in the discussion of the bonding of the simplest molecular species, the one-electron dihydrogen molecule-ion, H2+, and the simplest molecule, the two-electron dihydrogen molecule. Valence bond theory is introduced and compared with molecular orbital theory. The photo-electron spectrum of the dihydrogen molecule is described and interpreted. 

The symmetry concepts of Chapter 2 and those of molecular orbital theory were applied to the construction of molecular orbitals for a range of diatomic molecules. 

Let us consider the construction of molecular orbitals from linear combinations of atomic orbitals (LCAO MOs) for an octahedral molecule... 

The concepts of hybridisation and resonance are the cornerstones of VB theory. Unfortunately, they are often misunderstood and have consequently suffered from much unjust criticism. Hybridisation is not a phenomenon, nor a physical process. It is essentially a mathematical manipulation of atomic wave functions which is often necessary if we are to describe electron-pair bonds in terms of orbital overlap. This manipulation is justified by a theorem of quantum mechanics which states that, given a set of n respectable wave functions for a chemical system which turn out to be inconvenient or unsuitable, it is permissible to transform these into a new set of n functions which are linear combinations of the old ones, subject to the constraint that the functions are all mutually orthogonal, i.e. the overlap integral J p/ip dT between any pair of functions ip, and op, (i = j) is always zero. This theorem is exploited in a great many theoretical arguments it forms the basis for the construction of molecular orbitals as linear combinations of atomic orbitals (see below and Section 7.1). 

The real breakthrough in recognizing the role that symmetry plays in determining the course of chemical reactions has occurred only recently, mainly through the activities of Woodward and Hoffmann [5, 6], Fukui [7, 8], Bader [9, 10], Pearson [11], Halevi [12, 13], and others. The main idea in their work is that symmetry phenomena may play as important a role in chemical reactions as they do in the construction of molecular orbitals or in molecular spectroscopy. It is even possible to make certain symmetry based selection rules for the allowedness and forbiddenness of a chemical reaction, just as is done for spectroscopic transitions. 

Figure 7-11. Molecular orbitals of the ethylene-ethylene system and the construction of molecular orbitals of cyclobutane. (The energy scale refers to the reactant orbitals only.)...

Although the state correlation diagram is physically more meaningful than the orbital correlation diagram, usually the latter is used because of its simplicity. This is similar to the kind of approximation made when the electronic wave function is replaced by the products of one-electron wave functions in MO theory. The physical basis for the rule that only orbitals of the same symmetry can correlate is that only in this case can constructive overlap occur. This again has its analogy in the construction of molecular orbitals. The physical basis for the noncrossing rule is electron repulsion. It is important that this applies to orbitals—or states—of the same symmetry only. Orbitals of different symmetry cannot interact anyway, so their correlation lines are allowed to cross. 

Octahedral The construction of molecular orbitals for an octahedral complex involves the same... 

In later chapters, symmetry will be a valuable tool in the construction of molecular orbitals (Chapters 5 and 10) and in the interpretation of electronic spectra of coordination compounds (Chapter 11) and vibrational spectra of organometallic compounds (Chapter 13). 

Ligand field theory - A description of the structure of crystals containing a transition metal ion surrounded by nonmetallic ions (ligands). It is based on construction of molecular orbitals involving the r/-orbitals of the central metal ion and combinations of atomic orbitals of the ligands. 

Construction of molecular orbitals Construction of hybrid orbitals Predicting the decrease of degeneracies of d orbitals under a ligand field Predicting the allowedness of chemical reactions... 

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