Rayleigh-Schrodinger perturbation theory, second order energy

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. 

The method is based on the following procedure.33 All possible doubly excited configurations are generated from the Hartree-Fock function and their contributions to the second-order Rayleigh-Schrodinger perturbation theory energy computed. Approximately 100 of the most important are used for a Cl calculation, all singly... 

The second order interaction energy, according to Rayleigh-Schrodinger perturbation theory is given by ... 

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

We now apply standard Rayleigh-Schrodinger perturbation theory, using the Hartree-Fock determinant as the zero-order state, and expand the perturbed states in the set of excited determinants. This approach gives rise to Moller-Plesset perturbation theory [1]. To first order, we recover the Hartree-Fock energy hf and, to second order, we obtain the second-order Moller-Plesset (MP2) energy ... 

Explicit formulae for the energy and the state vector up to the second order of the Rayleigh-Schrodinger perturbation theory are presented in Table 1.7. Here the following series are assumed... 

This expression is similar to the second-order energy given by the Rayleigh-Schrodinger perturbation theory with addition of the first term. 

Fig. 11. The spectra for obtained with the bare G-matrix (//ifV) and Rayleigh-Schrodinger perturbation theory through second-order in the G-matrix (WifV). All three potentials have been employed. Energies in MeV.

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. 

One of the basic computational methods for the correlation energy is the MP2 method, which gives the result correct through the second order of the Rayleigh-Schrodinger perturbation theory (with respect to energy). 

On taking the Rayleigh-Schrodinger perturbation expansion for the energy beyond second-order, we find that so-called renormalization terms appear. In third-order Rayleigh-Schrodinger perturbation theory, the energy coefficient can be written in the form... 

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