Intruder states perturbation theory

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]. 

Comparison of the experimental data to the CASPT2 values shows that for the large majority of states the calculated value does not deviate more than 0.2 eV from the experimentally determined one. Exceptions are the 4A2g state in Co-O (1.79 vs. 2.14) and the lower d-d transitions in MnO, for which the calculated transition energies are too high. In none of these cases are there indications from the perturbation theory that intruder states artificially affect the calculated transition energies. Hence, it is unlikely that the discrepancy between calculated and experimental values can be attributed to an incorrect or incomplete treatment of the electron correlation effects. 

The CASSCF wavefiinction is used as reference function in a second-order estimate of the remaining dynamical correlation effects. All valence electrons were correlated in this step and also the 3s and 3p shells on copper. Relativistic corrections (the Darwin and mass-velocity terms) were added to all CASPT2 energies. They were obtained at the CASSCF level using first-order perturbation theory. A level-shift (typically 0.3 Hartree) was added to the zeroth order Hamiltonian in order to remove intruder states [30]. Transition moments were conputed with the CAS state-interaction method [31] at the CASSCF level. They were... 

In many multireference perturbation theories, the energy spectrum computed with the approximated Hamiltonian is ill conditioned. Functions outside the reference space become artificially degenerate with reference wavefunction. The phenomenon is known in the literature as intruder state, and it results in unphysical energy corrections and spurious bumps along a potential energy surface. Several schemes, such as ad hoc level shift parameter, have been proposed [97-100], yet... 

Fig. 4.6. The energy of the intruder level in relation to the ionisation threshold for the Ca-like isoelectronic sequence, showing how the doubly excited intruder state can be tracked down to the approximate position where the series perturbation is observed by using a combination of theory and experiment (after K.S. Bhatia et al. [145]).

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... 

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. 

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... 

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]) ... 

Fig. 2.4 In multireference Rayleigh-Schrodinger perturbation theory, states from outside the reference space, V, 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 < X < +1, which often have a physical origin, backdoor intruder states are frequently unphysical...

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. 

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. 

Witek HA, Choe Y, Finley IP, Hirao K (2002) Intruder state avoidance multireference Moller-Plesset perturbation theory. 1 Comput Chem 23 957... 

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