Benzene, degenerate orbitals

Cs subgroup which was used above in the allyl ease) has no degenerate representations. Moleeules with higher symmetry sueh as NH3, CH4, and benzene have energetieally degenerate orbitals beeause their moleeular point groups have degenerate representations. 

Consider now the rr-system in benzene. The MO approach will generate linear combinations of the atomic p-orbitals, producing six rr-orbitals delocalized over the whole molecule with four different orbital energies (two sets of degenerate orbitals). Figure 7.3. The stability of benzene can be attributed to the large gap between the HOMO and LUMO orbitals. 

Benzene is aromatic because it has a filled shell of equal-energy orbitals. The degenerate orbitals and 773 are filled, and all the electrons are paired. Cyclobutadiene, by contrast, has an open shell of electrons. There are two half-filled orbitals easily capable of donating or accepting electrons. To derive Hiickel s rule, we must show under what general conditions there is a filled shell of orbitals. 

Obviously, only molecules with partially filled orbitals display Jahn-Teller distortion. As was shown in Section 6.3.2, the electronic ground state of molecules with completely filled orbitals is always totally symmetric, and thus cannot be degenerate. In comparison with the above-mentioned unstable H3 molecule, Hj" has only two electrons in an a symmetry orbital therefore, its electronic ground state is totally symmetric, and the D3/,-symmetry triangular structure of this ion is stable (see, e.g., Reference [62]). On the other hand, take the benzene molecule, e.g., whose ground electronic state is of Alg symmetry and the molecule is stable and its structure is well understood. At the same time, in its cation, C6Hg, it loses one electron from an c -symmetry doubly-degenerate orbital, so that orbital is left with only one electron. The electronic state of the cation has E g symmetry and thus, it is subject to Jahn-Teller effect. Indeed, its vibrational spectrum is extremely complicated and can only be satisfactory explained if the Jahn-Teller distortion is taken into consideration (see, e.g., Reference [63]). 

The highest energy orbital that contains electrons is called the highest occupied molecular orbital (HOMO). For benzene, the degenerate orbitals V2 j/3 are the HOMOs. 

In symmetrical cyclic conjugated systems, such as benzene, there are sets of molecular orbitals with the same energy (degenerate orbitals). (The pattern of energy levels for these aromatic systems can be determined by Frost... 

Another uniformity concerns the special stability of aromatic hydrocarbons having 4n + 2 tt electrons n = 1 for benzene, n = 2 for naphthalene, etc. The reeison lies in the observation that, in such cases, all bonding m.o.s are filled whereas the anti-bonding ones are vacant. In these hydrocarbons, except for the lowest energy tt m.o. (two electrons), the bonding tt m.o.s invariably occur in pairs of degenerate orbitals - 4n electrons hence a total number of 4 -I- 2 bonding electrons. 

Fig ure 15.10 Energy levels of the six benzene - molecular orbitals. There is a single, lowest-energy orbital, above which the orbitals come in degenerate pairs. 

The pattern for planar conjugated systems established for cyclobutadiene, benzene, and cyclooctatetraene persists for larger rings. All 4 - -2 systems are predicted to have all electrons paired in bonding MOs with net stabilization relative to isolated double bonds. In contrast, planar systems containing 4n tt electrons are predicted to have two degenerate orbitals, each with one unpaired electron. This pattern is the theoretical basis of the Hiickel rule. 

The electronic structure is computed using the CASSCF method. Using the standard 6-3IG basis set, we choose the 6 7t orbitals as active. The degenerate HOMO, HOMO-1 and matching degenerate LUMO, LUMO-l-1 are needed to recover the non-dynamic electron correlation. The remaining pair of benzene n orbitals contributes to dynamic correlation and has to be included for stability (because of a large dynamic correlation effect). 

For benzene (N = 6), there are two degeneracies. The degenerate orbitals may be added and subtracted (in general, transformed unitarily). Figure 3.6 is obtained. 

Unpaired electron population Panel (e) of Fig. 7.26 suggests that the initial tt - tt state contains about three radicals. It is obvious that there should exist at least two radicals, and it is likely that the additional one radical center appears due to the strong electron correlation arising from the nearly degenerate orbitals in the benzene ring (near degeneracy in each manifold of tt and tt orbital). Beyond the barrier, one radical remains localized in PhO site and another is found in AMC (mainly on AMI), which is really the biradical state. [71]... 

Figure 15.3 The six benzene tt molecular orbitals. The bonding orbitals >p2 and t 3 have the same energy and are said to be degenerate, as are the antibonding orbitals tf/4 and 5. The orbitals and 4 have no tt electron density on two carbons because of a node passing through these atoms.

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