Molecular orbitals diatomic molecules localized

As their names suggest, molecular orbitals can span an entire molecule, while localized bonds cover just two nuclei. Because diatomic molecules contain just two nuclei, the localized view gives the same general result as molecular orbital theoiy. The importance of molecular orbitals and delocalized electrons becomes apparent as we move beyond diatomic molecules in the follow-ing sections of this chapter. Meanwhile, diatomic molecules offer the simplest way to develop the ideas of molecular orbital theory. 

The following presentation is limited to closed-shell molecular orbital wave-functions. The first section discusses the unique ability of molecular orbital theory to make chemical comparisons. The second section contains a discussion of the underlying basic concepts. The next two sections describe characteristics of canonical and localized orbitals. The fifth section examines illustrative examples from the field of diatomic molecules, and the last section demonstrates how the approach can be valuable even for the delocalized electrons in aromatic ir-systems. All localized orbitals considered here are based on the self-energy criterion, since only for these do the authors possess detailed information of the type illustrated. We plan to give elsewhere a survey of work involving other types of localization criteria. 

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

In Chapters 4 and 6 we have distinguished between a molecular orbitals and 7T molecular orbitals in diatomic molecules on the basis of the symmetry of the m.o.s with respect to rotation around the intemuclear axis whereas a a orbital has cylindrical symmetry, a tt orbital changes sign upon a rotation of 180°. In Chapters 7 and 8 we have extended the notion of a orbitals to polyatomic molecules by referring to the local symmetry with respect to each X-Y intemuclear axis. We will now study systems of tt m.o.s in polyatomic species. 

The success of the preceding scheme for diatomic molecules I7,i8,i , 20,21) ied Hund 22> and Mulliken 23> to apply the same theory to polyatomic molecules. In the beginning, there seemed to be no direct relation between molecular orbitals (MO s) and the bonds in a chemical formula, because MO s normally extend over the whole molecule and are not restricted to the region between two atoms. The difficulty was overcome by using equivalent localized MO s instead of the delocalized ones 24>25>. The mathematical definition of equivalent MO s was given only in 1949 by Lennard-Jones and his coworkers 26.27), but the concept of localization... 

A molecular orbital is most likely to be composed of atomic orbitals of similar energy levels. In diatomic molecules, or more generally in localized two-centered bonds, orbitals of the two atoms that may be combined to form a molecular orbital are those that have the same symmetry about the internuclear axis. (In constructing molecular orbitals that extend over three or more atoms, more complicated rules of symmetry must be considered, involving the symmetry of the extended nuclear framework.)... 

In thissection we cons de [ homonuclear diatomic molecules (those composed of two identical atoms) of elements in Period 2 of the periodic table. Since the lithium atom has a 15 25 electron configuration, it would seem that we should use the Li I5 and 2s orbitals to form the molecular orbitals of the Ll2 molecule. However, the I5 orbitals on the lithium atoms are much smaller than the 2s orbitals and therefore do not overlap in space to any appreciable extent (see Fig. 9.31). Thus the two electrons in each I5 orbital can be assumed to be localized and not to participate in the bonding. To participate in molecular orbitals, atomic orbitals must overlap in space. This means that only the valence orbitals of the atoms contribute significantly to the molecular orbitals of a particular molecule. 

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