Localized molecular orbitals , definition

By virtue of the definition of Eq. (16) it is recognized that this expression represents the sum of the self-energies of all occupied molecular orbitals. The higher the self-energy of a particular orbital, the smaller the space to which this orbital is confined. The localization sum of Eq. (28) represents, therefore, an average measure for the degree of concentration of all orbitals in the set. 

In this chapter, we focus on the class of reactive intermediates that bear at least two unpaired electrons diradicals and carbenes. The exact definition of a diradical is somewhat in the eye of the beholder. Salem and Rowland provided perhaps the most general, yet effective, definition—a diradical is a molecule that has two degenerate or nearly degenerate orbitals occupied by two electrons. With this definition, carbenes can be considered as a subcategory of diradicals. In a carbene, the two degenerate molecular orbitals are localized about a single carbon atom. 

F 2p character than F 2s character and is also bonding with respect to the FI orbital. This set of orbitals (2cr, 3a) illustrates a central feature of the MO approach. Whereas a simple Lewis structure or valence picture would draw a localized electron pair interaction between two orbitals, the MO picture attributes some bonding character to two separate molecular orbitals. This simple MO diagram illustrates the difficulty of determining a meaningful definition for bond order in a polyatomic molecule. No single MO completely represents the bonding between two atoms. 

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... 

Theoreticians often resort to an analysis of the molecular orbitals. However, orbitals are not observable entities, they have only a mathematical meaning. Furthermore, they can be transformed by any unitary transformation into a new set of orbitals without changing energy or other properties of the molecule. For example, they can be presented as delocalized or localized MOs. A unique definition of chemical bonding, however, should be independent of the form of the MOs. The chemical bond should preferentially be described with the help of a molecular quantity that is observable. 

On the other hand, if a(r) = f(r) the local hardness becomes proportional to the electrostatic potential generated by the molecular Fukui distribution. Since far away from the nuclei the electronic density will be practically equal to the density of the highest occupied molecular orbital pHOMo(r)> and since the Fukui function is closely related to pHOMo(r), one should expect that in this limit both definitions will lead to very similar results. However, at intermediate distances from, and close to, the nuclei they may provide different information on the reactive sites of a chemical species. 

Another molecular orbital approach is to calculate the energy of a model in which the tt bonds are constrained to be localized double bonds by the definition of the wave function. The calculated energy of this model can then be compared with the computed energy of the molecule in which delocalization is permitted. By this definition, butadiene has a resonance stabilization of about 9.3 kcal/mol, while benzene has a resonance energy of about 56 kcal/mol. To compare this value with that obtained with the polyene reference, one must subtract a correction for the butadiene resonance energy (3 x 9.3), which gives a value of about 28 kcal/mol as the resonance stabilization of benzene. 

These changes are defined by the nature of molecular orbital (MO) that is filled in the Si state. In the case of photoactive azides, this is an orbital of definite structure, namely, <7nn -MO, which is localized on the azido group and is... 

Since rigorous theoretical treatments of molecular structure have become more and more common in recent years, there exists a definite need for simple connections between such treatments and traditional chemical concepts. One approach to this problem which has proved useful is the method of localized orbitals. It yields a clear picture of a molecule in terms of bonds and lone pairs and is particularly well suited for comparing the electronic structures of different molecules. So far, it has been applied mainly within the closed-shell Hartree-Fock approximation, but it is our feeling that, in the future, localized representations will find more and more widespread use, including applications to wavefunctions other than the closed-shell Hartree-Fock functions. 

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