The Cyclic Three-Orbital Mixing Problem

The mixing of three s orbital in a cyclic array. A. The mixing pattern and orbital energies. B. Coefficients for the degenerate pair of orbitals. 

We can derive the orbital coefficients from the rules outlined in Section 14.3.2. In the lowest MO all coefficients are 1 / 3, by symmetry. For the degenerate pair, the MO with a node through one atom has coefficients of 1 /, 2. For the other MO there will be two different coefficients. The two AOs corresponding to the AOs involved in the other member of the degenerate pair will have one coefficient, a, while the third AO will have a different coefficient, 6. It is a simple matter to set up two equations using the rules of Section 14.4.2. 

Solving gives r = 1 / 6 and b = 2/ %. As before, the AO that lies on the node in one member of a degenerate pair has a larger coefficient in the other member. 

The results of the three-orbital mixing problem can be summarized as follows  

These rules are general for the mixing of three degenerate orbitals in a cyclic arrangement at the level of Hiickel theory. 

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Use the methods of secular determinants to derive the MOs and energies for the cyclic three-orbital mixing problem. Let the three orbitals be simple s orbitals. As with Hiickel theory, let the overlap integrals equal zero. The answer should be the... 

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