Molecular orbitals, degenerate singly occupied

Cyclopropane (D3 , symmetry) has a degenerate pair of in-plane e orbitals (S, A). Vertical ionization leads to a doubly degenerate E state, and JT distortion results in two nondegenerate electronic states, A and 82 (Cav symmetry), corresponding to two different molecular structures. The Ai state (orbital S singly occupied) corresponds to a structure with one lengthened C—C bond it is lowest in energy for many cyclopropane radical cations (Fig. 6.10). 

Note that in fact cyclobutadiene does not have degenerate, singly-occupied molecular orbitals, as a Jahn-Teller type (actually a pseudo-Jahn-Teller) distortion lowers its symmetry from square to rectangular and leads to a closed-shell paired-electron molecule [4]. 

FIGURE 13.27 In the tropyhum ion, the lowest single orbital and the bonding degenerate molecular orbitals are fully occupied. 

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.

Cycl[2,2,2]azinium salts7 (5) have not been reported so far. Hiickel molecular orbital (HMO)-calculations of the isoelectronic acepentylene (6) lead to two singly occupied degenerate HOMO s.8,9 Since there is no contribution in both the HOMO s and the LUMO from atom 6b this situation does not change in going from 6 to the 5-ion,10 which can be represented as an annulenone derivative of the (4n + 1) rc-type. 

Exact solutions such as those given above have not yet been obtained for the usual many-electron molecules encountered by chemists. The approximate method which retains tile idea of orbitals for individual electrons is called molecular-orbital theory (M. O. theory). Its approach to the problem is similar to that used to describe atomic orbitals in the many-electron atom. Electrons are assumed to occupy the lowest energy orbitals with a maximum population of two electrons per orbital (to satisfy the Pauli exclusion principle). Furthermore, just as in the case of atoms, electron-electron repulsion is considered to cause degenerate (of equal energy) orbitals to be singly occupied before pairing occurs. 

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. 

At this point it is worth rephrasing some of the issues of the above discussions. The UPS spectra are a measure of the single-particle excitation spectrum of the molecule, in so far as removal of an electron is concerned, while UAS data are a measure of the particle-hole excitation spectrum. In other terms, UPS measures the molecular-ion states while UAS measures excited states of the neutral molecule. For a molecule in isolation, in a one-electron picture the valence electron molecular cation states are comprised of the set of one-electron molecular orbitals (mo s) containing one half-filled (usually non-degenerate) molecular orbital and the totality of other fully occupied orbitals, distorted from their situation in the neutral molecule due to the removal of an electron from the molecule in a photoelectron... 

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