Rayleigh-Schrodinger perturbation theory method

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... 

E. Schrodinger, Ann. Phys. 80 (1926), 437. The quantal formalism substantially follows the classical method developed by Lord Rayleigh (Theory of Sound [1894]) and is commonly referred to as Rayleigh-Schrodinger perturbation theory. ... 

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... 

There have also been other attempts to evaluate these energy differences directly."" These methods utilize Rayleigh-Schrodinger perturbation theory to express the energies for both states with a common orbital basis. When the perturbation series for the two state energies are subtracted, it is found that there is a considerable cancellation of identical terms from the individual series. 

Clearly, standard Rayleigh-Schrodinger perturbation theory is not applicable and other perturbation methods have to be devised. Excellent surveys of the large and confusing variety of methods, usually called exchange perturbation theories , that have been developed are available [28, 65]. Here it is sufficient to note that the methods can be classified as either symmetric or symmetry-adapted . Symmetric methods start with antisymmetrized product functions in zeroth order and deal with the non-orthogonality problem in various ways. Symmetry-adapted methods start with non-antisymmetrized product functions and deal with the antisymmetry problem in some other way, such as antisymmetrization at each order of perturbation theory. 

M0ller and Plesset proposed an alternative way to tackle the problem of electron correlation [Moller and Plesset 1934] Their method is based upon Rayleigh-Schrodinger perturbation theory, in which the true Hamiltonian operator X is expressed as the sum of a zeroth-order Hamiltonian Xq (for which a set of molecular orbitals can be obtained) and a perturbation, f " ... 

Before we discuss the results obtained from the FD and the LS methods for the sd-shell, we show in Fig. 8 the behavior of the effective interaction through third order in Rayleigh-Schrodinger perturbation theory. This means that only folded diagrams of third order in the G-matrix are evaluated. 

As an alternative to BriUouin-Wigner perturbation theory, we may consider Rayleigh-Schrodinger perturbation theory, which is size consistent. In this method, the total energy is computed in a stepwise maimer ... 

In spite of this progress, problems remain and the description of electron correlation in molecules will remain an active field of research in the years ahead. The most outstanding problem is the development of robust theoretical apparatus for handling multi-reference treatments. Methods based on Rayleigh-Schrodinger perturbation theory suffer from the so-called intruder state problem. In recent years, it has been recognized that Brillouin-Wigner perturbation theory shows promise as a robust technique for the multi-reference problem which avoids the intruder state problem. 

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. 

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