Intruder states correction

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. 

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]. 

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. 

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... 

C can be written as C -I- o(e) provided that s is small compared to the eigenvalues of Hg, which will be the case if there are no intruder states. If the intruder states have been removed by the level shift, it is possible to make an approximate a posteriori correction of the second-order energy to an assumed unshifted calculation without the intruder state. The correction is obtained from Eq. (26) by replacing C with C,... 

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. 

To avoid the arbitrariness noticed in (i), one may introduce a spectroscopic basis set of AOs spanning only the valence and lowest Rydberg states of the atoms. But it is known that the molecular construction implies some orbital contraction, which can hardly be mimicked from spectroscopic AOs. It is also very important when an atom A approaches an atom B that the basis set of the atom B involves same oscillating spatially contracted functions which are used by the outer electron orbitals of A to minimize their repulsion with the electrons of B. It seems almost compulsory for a correct molecular treatment that the atomic information concerns the energy of unbound spatially concentrated atomic distributions. Then the occurrence of the second problem (i.e. the occurrence of core excited intruder states) is not excluded. 

The NEVPT approach has a list of remarkable qualitative properties (size-consistency, invariance to the rotation of active orbitals, absence of intruder states, first-order correction to the wave function is a pure spin state) which indicates that this as a serious candidate when an MR problem should be solve. Although with all these properties the QMBPT2 method cannot compete (e.g., it is obviously not invariant to the rotation of active orbitals), the relative accuracy of these methods is still an interesting question. 

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... 

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. 

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