Rayleigh-Schrodinger perturbation theory Brillouin-Wigner

Brillouin-Wigner perturbation theory Rayleigh-Schrodinger perturbation theory... 

Elimination of the size-extensivity error in the MRBWCC theory was suggested by Hubac and Wilson [68]. The Brillouin-Wigner perturbation theory has been known notoriously as a method not furnishing size extensivity. There was therefore every reason to believe that the size-extensivity error in MR BWCCSD originated from the use of the Brillouin-Wigner type resolvent (15) instead of the Rayleigh-Schrodinger... 

The renewal of interest in Brillouin-Wigner perturbation theory for many-body systems seen in recent years, is driven by the need to develop a robust multi-reference theory. Multi-reference formalisms are an important prerequisite for theoretical descriptions of dissociative phenomena and of many electronically excited states. Brillouin-Wigner perturbation theory is seen as a remedy to a problem which plagues multi-reference Rayleigh-Schrodinger perturbation theory the so-called intruder state problem. 

The Rayleigh-Schrodinger peiturbation theory can be derived from the Lennard-Jones-Brillouin-Wigner perturbation theory by expanding the energy-dependent denominators which occur in the latter... 

In the work of Brandow [10], Brillouin-Wigner perturbation theory is used as a step in the theoretical development of first Rayleigh-Schrodinger perturbation theory and then the many-body perturbation theory. In the a posteriori correction developed by the present authors in a paper entitled On the use of Brillouin-Wigner... 

Brillouin-Wigner perturbation theory is employed as a computational technique -a technique which avoids the intruder state problem - and then the relation between the Brillouin-Wigner and Rayleigh-Schrodinger propagators is used to correct the calculation for lack of extensivity. 

Rayleigh-Schrodinger and Brillouin-Wigner Perturbation Theories and A Posteriori Many-Body Corrections... 

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. 

Let us go back to our problem we want to have Eq on the left-hand side of the last equation, while - for the time being - Eq occurs on the right-hand sides of both equations. To exit the situation we will treat Eq occurring on the right-hand side as a parameter manipulated in such a way as to obtain equality in both above equations. We may do it in two ways. One leads to Brillouin-Wigner perturbation theory, the other to 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. 

Brillouin-Wigner perturbation theory can be developed for both the single reference function case and the multireference function case using a common formalism. This contrasts with the situation for Rayleigh-Schrodinger perturbation theory. We shall, therefore, consider the single reference and multireference formalisms together. 

For the ground state a = 0) the denominators of Brillouin-Wigner perturbation theory are related to those of Rayleigh-Schrodinger pertm-bation theory by the identity... 

In this section, we shall first provide an elementary derivation of the Brillouin-Wigner perturbation theory and then present a comparable introduction to the Rayleigh-Schrodinger perturbation theory. The section concludes with a comparison of the advantages and disadvantages of the two theories. 

In the preceding section, we have given elementary presentations of both the Brillouin-Wigner perturbation theory and the Rayleigh-Schrodinger perturbation... 

Brillouin-Wigner perturbation theory has five main advantages over the Rayleigh-Schrodinger perturbation theory. These are ... 

Multi-reference Brillouin-Wigner theory overcomes the intruder state problem because the energy is contained in the denominator factors. Calculations are therefore state-specific , that is, they are performed for one state at a time. This is in contrast to multi-reference Rayleigh-Schrodinger perturbation theory, which is applied to a manifold of states simultaneously. Multi-reference Brillouin-Wigner perturbation theory is applied to a single state. Wenzel and Steiner [77] write ... 

Whereas the multi-reference Rayleigh-Schrodinger perturbation theory approximates a manifold of states simultaneously, the multi-reference Brillouin-Wigner perturbation theory approach is applied to a single state - it is said to be state-specific . The multi-reference Brillouin-Wigner perturbation theory avoids the intruder state problem. If a particular Brillouin-Wigner-based formulation is not a valid many-body method, then a posteriori correction can be applied. This correction is designed to restore the extensivity of the method. This extensivity may be restored approximately... 

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