Multi-reference intruder state

The effects of intruder states are generally more severe for molecules than for atoms, due to more dense energy levels. Therefore, even if there are ways of avoiding - or at least reducing - the effect of intruder states in the multi-reference approach, it is when the interest lies entirely in one or a few particular states, more advantageous to study one state at a time in a state-specific approach. 

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 exact 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-Schrddinger 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 [105] write (see also [106]) ... 

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. 

Figure 1.1. In multi-reference Rayleigh-Schrodinga- perturbation theory, states from outside the refe-ence space, P, which assume an energy below that of any state among the reference set when the perturbation is switched on, are termed intruder states .

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... 

We shall consider in turn each of these widely used approximation methods in this chapter. We shall provide an account of the Brillouin-Wigner formulation of each of these methods in a self-contained manner so that extensive cross referencing can be avoided. We shall establish the value of the Brillouin-Wigner method in the study of problems requiring a multi-reference formalism for a broad range of theoretical approaches. In this way, any problems associated with intruder states can be avoided. A posteriori corrections can be introduced to remove terms which scale in a non linear fashion with particle number. We shall not, for example, consider in any detail hybrid... 

Although the past 20 years have witnessed a great progress in the Hilbert space multi-reference coupled cluster methods (see, for example, the work of Mukherjee and Pal [99],Paldus [101], Jeziorski and Paldus [102], Jankowski et al. [103],Paldus et al. [104], Paldus et al. [105], Meissner et al. [106], Kucharski and Bartlett [107], Balkovd et al. [108], Baikova and Bartlett [109], Balkovd et al. [110], Baikova et al. [Ill], Berkovic and Kaldor [112]) only a few applications of this approach have been reported, mostly oriented to the simple model systems exploiting a lowdimensional model space. Among the reasons for this paucity of applications are the choice of an appropriate model space, convergence difficulties arising from intruder state problems and from multiple solutions of non-linear coupled cluster equations. 

The application of the Brillouin-Wigner coupled cluster theory to the multireference function electron correlation problem yields two distinct approaches (i) the multi-root formalism which was discussed in Section 4.2.2 and (ii) the single-root formalism described in the previous subsections of this section. Section 4.2.3. The multiroot multi-reference Brillouin-Wigner coupled cluster formalism reveals insights into other formulations of the multi-reference coupled cluster problem which often suffer from the intruder state problem which, and in practice, may lead to spurious... 

In Brillouin-Wigner coupled cluster theory, the simple a posteriori correction described above is exact in the case of the single-reference formalism. In the state-specific multi-reference Brillouin-Wigner coupled cluster theory, the simple a posteriori correction is approximate. An iterative correction for lack of extensivity has been studied by Kttner [38], but this reintroduces the intruder state problem. 

The multi-reference configuration interaction (mrci) method with single and doubles (mr-cisd) is one of the few quantum chemical methods which are used in routine calculations for systems requiring a multi-reference description. The main reason for that is its formal and computational simplicity and resistance to the intruder-state problem which frequently occurs in other multi-reference-type calculations. 

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