Degenerate-Level Perturbation Theory by Inspection

Degenerate and nondegenerate perturbation theories share the feature that the first-order correction to the energy is the expectation value of the perturbation operator using the unperturbed wavefunction. Often we can evaluate or estimate this expectation value by inspection. The special problem for degenerate wavefunctions or orbitals is that there is an infinite number of ways to express the unperturbed wavefunctions, and only one of these ways is proper for finding The mathematical procedure for finding the proper set of functions tends to complicate and obscure the situation. However, it is often easy to deduce the proper unperturbed wavefunctions by inspection. 

The clue for doing this comes from Eq. (12-76). It is apparent that this equation, which leads to the first-order corrections to the energy and to the proper unperturbed (or zeroth-order) wavefunctions, is exactly like the secular determinantal equation we would use for a variational calculation. Since the secular equation comes from a calculus-based determination of energy extrema, we know that the energies that come out of Eq. (12-76) must be the minimum and maximum possible values and that the proper zeroth-order functions must be those that give the maximum and minimum expectation values for H. This leads immediately to a powerful insight For a given 

The two perturbations described above bring up a final important point. Suppose we have two different perturbations occurring simultaneously in a system involving degenerate MOs or state functions. The question then arises as to whether the proper 

Chapter 12 Time-Independent Raylelgh-Schrddinger Perturbation Theory 

EXAMPLE 12-7 Consider the cyclopentadienyl radical in the HMO approximation. Assume it to be planar and pentagonal. (See Appendix 6 for data.) If 3 is perturbed to become a O.lOOOyS, what are the MO energy changes, to first order  

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Section 12-8 Degenerate-Level Perturbation Theory by Inspection... 

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