Rayleigh-Schrodinger power series

Erom eq. (1.13) we can develop a series expansion for the exact eigenfunction in powers of A with coefficients depending on the perturbed energy rather than the energy of the model Em, as would be the case in the more familiar Rayleigh-Schrodinger perturbation series. Iterating this basic formula, we find... 

The mathematical procedure that we present here for solving equation (9.15) is known as Rayleigh-Schrodinger perturbation theory. There are other procedures, but they are seldom used. In the Rayleigh-Schrodinger method, the eigenfunctions tpn and the eigenvalues E are expanded as power series in A... 

The Lennard-Jones, Brillouin, Wigner perturbation expansion is not a simple power series in A since each depends on the exact energy, S. Each energy coefficient in the Rayleigh-Schrodinger perturbation expansion consists of a principal term of the form... 

We can develop a series expansion for in powers of X. The coefficients in this expansion depend on the exact perturbed energy rather than, as is the case in the more familiar Rayleigh-Schrodinger expansion, on the unperturbed energy7 . Iterating the basic formula (2.21), we obtain... 

The perturbation approach originally proposed by Mpller and Plesset [26] (MP method) provides a simpler and less time-consuming scheme for computing the electron correlation effect. Within this scheme, the full Cl Hamiltonian is treated as a perturbed Hamiltonian and the energy and the wavefunction are expanded in power series following the 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. 

The Rayleigh-Schrodinger perturbation theory gives an explicit power series in A for the characteristic values Fn and the characteristic functions (j> of a Heimitean... 

It should be noted that the expansion (1.18) is not a simple power series expansion in A, since the ,u which appears in the denominators of also depends on A. However, if the terms 1/ ( - Em) are expanded in powers of A, then the usual Rayleigh-Schrodinger series for 2. is recovered which is, as is well known, a simple power series in A. 

The Rayleigh-Schrddinger perturbation theory is developed by substituting the power series for the wave function (1.22) and that for the energy (1.21) into the Schrodinger equation (1.3) in the form... 

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