Brillouin-Wigner perturbation theory development

Today, there remain a number of problems in molecular electronic structure theory. The most outstanding of these is undoubtedly the development of a robust theoretical apparatus for the accurate description of dissociative processes which usually demand the use of multi-reference functions. This requirement has recently kindled a renewal of interest in the Brillouin-Wigner perturbation theory and its application to such problems. This contribution describes the application of... 

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. 

A posteriori corrections can be developed for calculations performed by using the Brillouin-Wigner perturbation expansion. These a posteriori corrections can be obtained for the Brillouin-Wigner perturbation theory itself and, more importantly, for methods, such as limited configuration interaction or multi-reference coupled cluster theory, which can be formulated within the framework of a Brillouin-Wigner perturbation expansion. 

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... 

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. 

In this section, we present Brillouin-Wigner perturbation theory in both its single reference and its multireference form. This will serve both to emphasize the similarity of single reference and multireference formulations of Brillouin-Wigner perturbation theory and to establish notation for later sections. In section 3.1, we define the basic concepts of any perturbation theory. The definition of single and multireference spaces is considered in section 3.2 and the model wave function is described in section 3.3. The Brillouin-Wigner expansion is developed in section 3.4. 

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. 

Brillouin-Wigner perturbation theory was, however, used as a step in the development of an acceptable many-body perturbation theory most notably by Brandow [67] in his pioneering work on multi-reference formalisms for the many-body problem. In a review entitled Linked-Cluster Expansions for the Nuclear Many-Body Problem and published in 1967, B.H. Brandow writes ... 

In the second section of this chapter, we shall employ the partitioning technique to develop various types of perturbation theory, including Rayleigh-Schrddinger perturbation theory and Brillouin-Wigner perturbation theory. This involves the expansion of the inverse operators which occur in the effective Hamiltonian operator and other operators obtained by the partitioning technique. Different expansions lead to different types of perturbation theory. 

In Section 14.1, we developed Rayleigh-Schrodingerperturbation theory (RSPT), where the enei gy and the wave-function corrections are obtained in a noniterative form. In this exercise, we develop Brillouin—Wigner perturbation theory (BWPT), where the corrections are generated iteratively. 

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