First-order Rayleigh-Schrodinger equation

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... 

HMO theory gives particularly simple and intuitively appealing results upon application of Rayleigh Schrodinger perturbation theory and we shall take advantage of this to interpret trends and make predictions (see, in particular, Section 4.6).d The equations for first- and second-order perturbation given below are derived in the Appendix (Section 4.11). 

Some general remarks on the perturbation theory and the many-body approach are now in order. Let us review first the essence of the nondegenerate Rayleigh-Schrodinger perturbation theory. Consider the time-independent Schrodinger equation for the ground state ... 

---