Multi-reference 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. 

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 ... 

Figure 1.2. In multi-reference Rayleigh-Schrodinger perturbation theory, states from outside the reference space, IP, which assume an energy below that of any state among the reference set for —1 < A < 0 are termed backdoor intruder states. Unlike the intruder states corresponding to 0 < A < +1, which often have a physical origin, backdoor intruder states are frequently unphysical.

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... 

MR-BWPT multi-reference Brillouin-Wigner perturbation theory MR-RSPT multi-reference Rayleigh-Schrodinger perturbation theory... 

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. 

Abstract The partitioning technique is described in some detail. The concept of a model space is introduced and the wave operator and the reaction operator are defined. Using these ideas, both the Rayleigh-Schrodinger and the Brillouin-Wigner expansions are developed, first for the case of a single-reference function and then for the multi-reference case. 

Equations (2.222) and (2.229) are the first two terms in the general Rayleigh-Schrodinger perturbation expansion for the multi-reference case. 

The formulation of a multi-reference bwcc theory can now proceed in two distinct ways. In the first option, we can formulate a multi-root version of the multi-reference BWCC theory which yields all roots of the d-dimensional 9 space simultaneously. This is the approach employed in most multi-reference coupled cluster formulations which are based on the Rayleigh-Schrodinger expansion. In the second option, we can use the state-specific wave operator (4.59) and formulate a state-specific (or single root) version of multi-reference bwcc theory [10]. 

The state-selective approach to the multi-reference problem was further developed by Banerjee and Simons [118], by Laidig and Bartlett [119], by Hoffmann and Simons [120], by Li and Paldus [121, 122] and by Jeziorski, Paldus and Jankowski [123] who formulated extensive open-shell cc theory, based on the unitary group approach (uga) formalism. The Rayleigh-Schrodinger formulation of a state-selective approach to the multi-reference correlation problem has been developed more recently by Mukherjee and his collaborators [124-130] and also by Schaefer and his colleagues [131-133]. 

We are now in a position to develop an a posteriori correction to state-specific limited multi-reference configuration interaction in the case of a p-state reference function. The following identity relates the Brillouin-Wigner and the Rayleigh-Schrodinger denominators [18,76,77] for the ground state (a = 0) ... 

The relationship between single-reference Brillouin-Wigner perturbation theory and its Rayleigh-Schrodinger counterpart is well known, but for completeness we include a brief account of the single-reference case in Section 4.4.1 before turning to the multi-reference case in Section 4.4.2. 

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