Second-order Rayleigh-Schrodinger equation

In the second order, however, the EOM-CC and CCLR formulations differ due to the appearance of quadratic terms in Eq. (18). Starting from the CCLR second-order Rayleigh-Schrodinger equation projected onto the zeroth-order left-hand wave function, we obtain... 

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. 

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). 

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