Perturbation Theory for a Degenerate State

SOLUTION The HOMO-LUMO contribution to the charge shift favors atom 4, since its coefficients in these MOs are larger than those for atoms 2 or 3. However, the acmal data (not given in Appendix 6) are tti 2 = 0.21342, tti 3 = —0.01771, = 0.13937. So atom 2 experiences 

In general, nik canhave either sign. This means that making atom k more attractive for electrons will, to first order, cause the n electron density to decrease at some carbons and increase at others. (This is necessary since the total 7r-charge density must he conserved.) 

Equations (12-19), (12-21), and (12-24) have denominators containing — Ej. If jfi and jfj are degenerate, this leads to difficulty, alerting us to the fact that our earlier derivations do not take proper account of states of interest that are degenerate. The problem results from our initial expansion of / / as a power series in k. The leading term in the expansion, is the wavefimction that / , becomes in the limit when 

To find the conditions that (pf and must satisfy, we return to the first-order perturbation equation (12-11) but with (pf in place of rpi.  

Multiplying from the left by 4 f and integrating gives (taking (pf and (pf to be orthogonal and Hq hermitian) 

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We will now give an example of perturbation theory for a degenerate state. 

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