Localized molecular orbitals occupied

The last equation has been used to analyze occupied-unoccupied localized molecular orbital pair contributions for excitations in chiral metal complexes and metallahelicenes [260, 261], as well as in chiral organic acids derived from amino acids by substitution of the amino group with —OH and —F [170]. The analyses in terms of canonical MOs and LMOs may be considered complementary tools, with the canonical MO analysis generally leading to fewer contributions since the canonical MOs are well adapted to describe electronic excitations. The analysis in terms of LMOs allows one to focus on selected chemist s orbitals of interest, such as contributions to excitations from a given lone pair or localized n orbital, or from metal-centered orbitals, which can also be very useful. 

It is interesting to compare the implications expressed by these subrules with the depiction of some localized molecular orbitals in Figure 3-41 [97], The lone pair of electrons occupies more space than do the bonding pairs in the vicinity of the central atom. Also, a bond to a more electronegative ligand such as fluorine occupies less space in... 

Although only one valence electron resides in a 5f-localized orbital on uranium in [U(C7H7)2], a formal oxidation state of -1-3 (5f ) was assigned, based on the fact that the HOMO Se molecular orbitals (occupied by four electrons) are nearly 50% 5f in character, and so two of these electrons were assigned to the metal. EPR and electron-nuclear double resonance (ENDOR) studies of [(C7H7)2U] suggest that the complex could be treated as 5f, with a ground-state molecular orbital comprised of both 5br and 5f(r orbitals. ... 

Among the 12 occupied m.o.s, the three tt m.o.s - one bonding and two non-bonding - are not only non-localized but also non-localizable. That is, it is not possible to make a unitary transformation with them that leads to three localized molecular orbitals . 

The canonical m.o.s of diamond are delocalized over the entire crystal. However, as we have seen in Chapter 8 for other systems, the occupied m.o.s can be the object of a unitary transformation leading to a set of equivalent and quasi-localized molecular orbitals . This is why the structure of diamond can (for some purposes) be described in terms of the overlap of sp hybrid orbitals, four for each C atom. As we have seen in Chapter 8, we must stress that such an alternative description cannot be used to infer information about electron energies. In particular, the localized bond description of the structure of diamond does not imply that all valence electrons have the same energy. This would be the case only if the sp -sp bonds were independent. It is because of residual interactions such as f and (3"... 

If the Coulomb interaction between electrons of different pairs is ignored, each localized bond and lone pair contribute independently to the total energy, which implies a perfect additivity of bond energies. In the independent-particle model, the localized bond function can be visualized as a two-center molecular orbital occupied by two electrons. Nevertheless, it is possible to reproduce deviations from additivity rules within this scheme, for instance, by taking into account overlap (for a review, see e.g. 2>). 

In the localized molecular orbital (LMO) approach [87], 2X2 rotations are applied to annihilate the interactions between occupied and virtual LMOs that are located within a certain cutoff radius, whereas all other interactions are considered to be negligible and therefore not treated. The resulting small numerical errors can be controlled by a renormalization of the LMOs and a suitable choice of the cutoff radius. 

Still another measure of electron delocalization can be seen in the forms of natural localized molecular orbitals (NLMOs).i By construction, each NLMO remains as close as possible to a parent NBO but includes the weak delocalization tail necessary to preserve exact double occupancy. A determinant of doubly occupied NLMOs is therefore unitarily equivalent to the standard MO determinant, and the Lewis-type NLMOs are able to describe all observable properties of the system just as well as canonical MOs. The perturbative expression (Equation 7.6a) approximates the form of the Lewis-type NLMO a B with parent NBO and weak delo-... 

Figure 12.23 shows that, in small molecules, electrons occupy discrete molecular orbitals whereas in macroscale solids the electrons occupy delocalized bands. At what point does a molecule get so large that it starts behaving as though it has delocalized bands rather than localized molecular orbitals For semiconductors, both theory and experiment tell us that the answer is roughly at 1 to 10 nm (about 10—100 atoms across). The exact number depends on the specific semiconductor material. The equations of quantum mechanics that were used for electrons in atoms can be applied to electrons (and holes) in semiconductors to estimate the size where materials undergo a crossover from molecular orbitals to bands. Because these effects become important at 1 to 10 nm, semiconductor particles with diameters in this size range are called quantum dots. 

Let us carry out the Hartree-Fock calculations for the water molecule. We focus on a subsequent calculation of the localized molecular orbitals and get five doubly occupied molecular orbitals, as shown in Fig. 8.30. 

An appealing feature of this partitioning is that the correlation (dispersion) energy computationally can be assigned exactly to contributions from orbital pairs. If localized molecular orbitals (LMOs) are used in the correlation treatment, this allows a local (group- or fragment-wise) description of dispersion effects [52]. The correlation energy ( [,) for an HF reference state can be written as a sum over occupied orbital pairs ij... 

This reaction can be analysed by means of a simple extension of the analysis of the [l,3]-shift just discussed. Pentadiene has two carbon and two hydrogen atoms more than propylene the total number of orbitals is thus raised by ten to twenty-eight, as is the number of valence electrons. However, after the four acc and three fixed ctch bonds are combined to form seven localized molecular orbitals and their antibonding counterparts to seven more, we are left with 14 mobile electrons in 7 doubly occupied MOs. There is no need to draw a new correspondence diagram for the reaction to a slight amplification of Fig. 8.6 is all that is required. 

The localized molecular orbitals (LMO) are extensively used not only for the chemical-bonding analysis in molecules but also in the local correlation methods [41] (we consider the problem of electron correlation in molecules and crystals in Chap. 5). The LMO are generated from the canonical MO occupied by electrons and found in the Hartree-Fock or DFT calculations. This generation is based on one or other localization criteria [42]. 

---