Molecular orbitals degeneracy

Figure 3. Molecular-orbital diagrams as obtained by the ROHF method. Dashed lines indicate MOs dominated by the metal d-orbitals, the solid lines stand for doubly occupied or virtual ligand orbitals. Orbitals which are close in energy are presented as degenerate the average deviation from degeneracy is approximately 0.01 a.u. In the case of a septet state (S=3), the singly occupied open-shell orbitals come from a separate Fock operator and their orbital energies do not relate to ionization potentials as do the doubly occupied MOs (i.e. Koopmann s approximation). For these reasons, the open-shell orbitals appear well below the doubly occupied metal orbitals. Doubly occupying these gives rise to excited states, see text.

While the canonical orbitals of a system are unique, aside from degeneracies due to multidimensional representations, this is not always the case for localized orbitals, and there may be several sets of localized orbitals in a particular molecule. This situation is related to the fact that the localization sum of Eq. (28) may have several relative maxima under suitable conditions. If one of these maxima is considerably higher than the others, then the corresponding set of molecular orbitals would have to be considered as the localized set. In some cases, however, the two maxima are equal in value, so that there exist two sets of localized orbitals with equal degree of localization. 23) In such a case there... 

Every occupied molecular orbital (MO) is taken with and spin and if there is symmetry degeneracy the whole degenerate set is included. 

Fio. 8. Relative energy levels and degeneracies of the 13 bonding and 23 antibonding skeletal molecular orbitals (MO s) of the icosahedral anion BiaHia (148). (AO, atomic orbital.)... 

The loss of degeneracy of atomic or molecular levels in a molecular entity with a given symmetry by the attachment or removal of ligands to produce reduced symmetries. Ligand field theory treats metal ligand complexation as a consequence of molecular orbital formation, whereas crystal field splitting considers ligands as point... 

Herzberg (Nobel prize for Chemistry, 1971) commented on the two distinct photoionizations from methane that this observation illustrates the rather drastic nature of the approximation made in the valence bond treatment of CH4, in which the 2s and 2p electrons of the carbon atom are considered as degenerate and where this degeneracy is used to form tetrahedral orbitals representing mixtures of 2s and 2p atomic orbitals. The molecular orbital treatment does not have this difficulty". 

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