Perturbation theories intruder state problem

Hamiltonian has been diagonalized, a correction is applied for the appearance of the level shift in the denominators of the expressions for E,2). This level shift method has been applied successfully to a wide variety of problems in the field of spectroscopy and can be considered as a pragmatic solution to the intruder state problem inherent to perturbation theory. All CASSCF / CASPT2 calculations presented here have been done with MOLCAS-4 [27]. 

There has been a revival of interest in the Brillouin-Wigner perturbation theory since it is seen as a possible remedy to the intruder state problem. As... 

Hubac and his co-workers222"231 have explored the use of Brillouin-Wigner perturbation theory in solving the coupled cluster equations. For the case of a single reference function, this approach is entirely equivalent to other formulations of the coupled cluster equations. However, for the multireference case, the Brillouin-Wigner coupled cluster theory shows some promise in that it appears to alleviate the intruder state problem. No doubt perturbative analysis will help to gain a deeper understanding of this approach. 

Some of the more important aspects of the theory behind the method are described in the review. In particular, the choice of the zeroth-order Hamiltonian is discussed together with the intruder-state problem and its solution. A generalization of the method to a multistate perturbation approach is suggested. Problems specifically related to spectroscopic applications are discussed, such as the choice of the active space and the treatment of solvent effects. 

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]) ... 

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. 

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. 

Although this approach often resolves the intruder state problem very efficiently, a methodologically more satisfying route is taken in the n-electron valence state second-order perturbation theory (NEVPT2). By including two-electron interactions in this perturbative scheme does not suffer from the intrader state problem, except in some pathological cases. The zeroth-order Hamiltonian proposed by Dyall [16] reads... 

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... 

The formal K matrix theory for a Rydberg manifold perturbed by a broad intruder state was setup by Lane [393]. The results have been related in detail to the (/-reversal problem by Connerade et al. [380], whose... 

From Table 5 we see that the Bonn A potential is the one which gives the strongest binding in the ground state, irrespective of the approximation used for the 0-box. An LS calculation with a second-order 0-box results in the best reproduction of the data, when employing the Bonn A potential. The problem of the JT = 10 state we discussed in connection with second-order perturbation theory has vanished. However, in order to be consistent, third-order contributions have to be included. In this case, potential A introduces too much binding for the JT = 10 state, in line with our previous comments on result for the sd-shell. If one were to account for so-called intruder states, potential B is seemingly the most appropriate candidate for nuclear structure... 

We shall provide an overview of the applications that have been made over the period being review which demonstrate the many-body Brillouin-Wigner approach for each of these methods. By using Brillouin-Wigner methods, 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 methods such as the widely used ccsd(t) which employs ccsd theory together with a perturbative estimate of the triple excitation component of the correlation energy. 

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