Orbital energies degeneracy

We have already seen (Chapters 8 and 13) that the orbital energies, degeneracies, and coefficients in benzene are to a great extent determined by symmetry. But these symmetry constraints can also be viewed as resulting from the cyclic periodicity of benzene, and this finite cyclic periodicity is, in essential ways, like the extended infinite periodicity of, say, regular polyacetylene or of graphite. It is reasonable, therefore, that some of the aspects of the energies and orbitals in these extended periodic stmctures are closely related to those in cyclic periodic systems. 

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.

In the cyclophane 1, although the overlap between the n-system (2p) and the bridging cr-bonds (2s2p) is most effective, these orbital energy levels match worst, the first ionization potentials being 9.25 eV for benzene and 12.1 eV for ethane. As a result, the HOMOs are the almost pure it MOs with the b2g and b3g combinations. Both the PE spectrum and theoretical calculation demonstrate the degeneracy of the two HOMO levels. The absorption bands are attributed to the 17-17 transitions associated with the HOMOs. 

Within each (n +1) manifold, the dependence on n (or l) can also be readily understood. The hydrogen-like tendency toward near-degeneracy of l values, but strong dependence on n, tends to persist even in many-electron atoms. As a result, the orbital energies continue to depend much more strongly on n than on l. The lowest-energy orbital of the (n + l) manifold is therefore that of lowest n (or, equivalently, highest i), e.g.,... 

The t degeneracy is cancelled, to first order, hy the perturbation and the orbital energy changes are determined by the eigenvalues of the matrix above. They are expressed as... 

Figure 6. Energy splitting of the d orbitals in octahedral and tetrahedral coordination. The numbers at each level indicate the energy degeneracy that still remains after ligand-field splitting. Note that the energy barycenter (i.e., center of energy ) need not be the same in octahedral and tetrahedral coordination as pictured.

Orbital energy Designation Degeneracy Ground-state occupancy ... 

In a hydrogen atom, the orbital energy is determined exclusively by the principal quantum number n—all the different values of / and mi are degenerate. In a multielectron atom, however, this degeneracy is partially broken the energy increases as / increases for the same value of n. 

The spin-orbit coupling therefore removes the degeneracy of the so-called fine-structure states listed above. For example, simply by substituting the appropriate values of J, L and S, the three fine-structure components arising from the L = 1, S= 1 configuration are calculated to have the following first-order spin orbit energies ... 

Secondly it was assumed that Koopmans theorem holds i. e. that the orbital energy computed in an ab initio SCF calculation is approximately equal to the ionisation potential, I, of an electron from that orbital. For closed shell systems each orbital energy level is then represented by a band in the photoelectron spectrum, the integrated intensity of which depends on the occupancy of each level, the degeneracy of the ionic state produced by ionisation, and matrix elements. The approximations introduced by the assumption of Koopmans theorem is discussed fully in several papers e.g. Refs. (2) and (J)]. A comprehensive review of gas phase UPS is to be found in the recent book by Eland (4). 

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