Intruder state

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

Intruder state problem is obviated by averaging of active orbital energies. 

While each approach has its own peculiarities, one common obstacle arises due to the so-called intruder states. These are the states from the orthogonal complement Mo of Mo, whose energy falls within the interval of energies characterizing the reference configurations spanning Mo or lies... 

We have carried out several applications showing the promise of this procedure [63,64], as well as addressed the question of the size-consistency and size-extensivity [65-67], to which we wish to turn our attention again in this paper. Finally, we have also extended the idea of externally corrected (ec) SR CCSD methods [68-70] (see also Refs. [21,24]) to the MR case, introducing the (N, M)-CCSB method [71], which exploits an Preference (A R) CISD wave functions as a source of higher-than-pair clusters in an M-reference SU CCSD method. Both the CMS and (N, M)-CCSD allow us to avoid the undesirable intruder states, while providing very encouraging results. 

It turns out that MR CISD represents again the most suitable source of the required higher-order clusters. Carefully chosen small reference space MR CISD involves a very small, yet representative, subset of such cluster amplitudes. Moreover, in this way we can also overcome the eventual intruder state problems by including such states in MR CISD, while excluding them from CMS SU CCSD. In other words, while we may have to exclude some references from Ado in order to avoid intruders, we can safely include them in the MR CISD model space Adi. In fact, we can even choose the CMS for Adi. Thus, designating the dimensions of Ado and Adi spaces by M and N, respectively, we refer to the ec SU CCSD method employing an NR-CISD as the external source by the acronym N, M)-CCSD. Thus, with this notation, we have that (N, 1)-CCSD = NR-RMR CCSD and (0, M)-CCSD = MR SU CCSD. Also, (0,1)-CCSD = SR CCSD. For details of this procedure and its applications we refer the reader to Refs. [63,64,71]. 

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. 

In spite of the intruder state problem, there is a range a systems where (relativistic) MBPT can produce reliable and accurate results for energies and width of resonance states. A few examples are given below. 

D. Nikolic, E. Lindroth, Intermediate Hamiltonian to avoid intruder state problems for doubly excited states, J. Phys. B 37 (2004) L285. 

Toward a Shell-Model Description of Intruder States and the Onset of Deformation... 

Basing on the nuclear shell-model and concentrating on the monopole,pairing and quadrupole corrections originating from the nucleon-nucleon force,both the appearance of low-lying 0+ intruder states near major closed shell (Z=50, 82)and sub-shell regions (Z=40,64) can be described.Moreover,a number of new facets related to the study of intruder states are presented. 

Figure 1. Schematic divisions of nuclei in three major regions (I) region near doubly closed shells (II) region of strongly deformed nuclei (Ill) region of intruder states.

In a schematic presentation(see fig.2) of a single-closed shell nucleus, we take as the lowest intruder state (0+state) a proton (tt) 2p-2h configuration... 

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