Rayleigh-Schrodinger perturbation theory first-order corrections

It should be apparent that the expressions for the wave functions after interaction [equations (3.38) and (3.39)] are equivalent to the Rayleigh-Schrodinger perturbation theory (RSPT) result for the perturbed wave function correct to first order [equation (A.109)]. Similarly, the parallel between the MO energies [equations (3.33) and (3.34)] and the RSPT energy correct to second order [equation (A. 110)] is obvious. The missing first-order correction emphasizes the correspondence of the first-order corrected wave function and the second-order corrected energy. Note that equations (3.33), (3.34), (3.38), and (3.39) are valid under the same conditions required for the application of perturbation theory, namely that the perturbation be weak compared to energy differences. 

Up to this point we are still dealing with undetermined quantities, energy and wave function corrections at each order. The first-order equation is one equation with two ImEnowns. Since the solutions to the unperturbed Schrodinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrodinger perturbation theory, and the equation in (4.32) becomes... 

Applying standard Rayleigh-Schrodinger perturbation theory, the first-order wave function and second-order energy correction F 2 are... 

Ho is the normal electronic Harmltonian operator, and the perturbations are described by the operators Pi and P2, with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrodinger perturbation theory (Section 4.8) gives the first- and second-order energy corrections. 

Using standard Rayleigh-SchrOdinger perturbation theory, the first-order correction to the orbital energies of the unperturbed [4m] carbocycle caused by insertion of the new bonds can be shown to be... 

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