Rayleigh-Schrodinger perturbation wave function

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. 

If the solutions (energies and wave functions P ) of the Schrodinger equation for the unperturbed system Tf(°) P = F,1,01 4/jl°l are known, and the operator form of the perturbation, Hp, can be specified, the Rayleigh-Schrodinger perturbation theory will provide a description of the perturbed system in terms of the unperturbed system. Thus, for the perturbed system, the SE is... 

The Rayleigh-Schrodinger perturbation expansion for the exact wave function to the first order is given by... 

If the distance between the interacting systems A and B is sufficiently large to enable the overlap of the respective electronic clouds to be disregarded, then the interaction energy may be calculated from the Rayleigh- Schrodinger perturbation theory, using as a basis of functions to describe the wave function of AB... 

Ho is the normal electronic Hamilton operator, and the perturbations are described by the operators Pj and P2, with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrodinger perturbation theory (section 4.8) gives the... 

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... 

In order to develop the Rayleigh-Schrodinger perturbation expansion for the energy and the wave function, we define the resolvent... 

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

In this section we will investigate the nature of the exact perturbed wave function through Rayleigh-Schrodinger perturbation theory. We will employ intermediate normalization... 

In Rayleigh-Schrodinger perturbation theory one starts with a nonrelativistic solution for the state wave functions F, and their energies E, ... 

Reduced Resolvent or the Almosf Inverse of MBPT Machinery Part 1 Energy Equation MBPT Machinery Part 2 Wave Function Equation BriUouin-Wigner Perturbation Theory Rayleigh-Schrodinger Perturbation Theory... 

In Brillouin-Wigner perturbation theory a power series expansion is made for the wave function (A) while in the more familiar Rayleigh-Schrodinger perturbation theory a power series expansion is made for both the wave function and the energy a (A). Furthermore, we require that for A 7 0, lintruder states arise. 

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