Rayleigh-Schrodinger perturbation theory intruder state problem

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. 

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. 

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... 

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. 

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