Kutzelnigg, Werner ; Mukherjee, Debashis
(2004)
*Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. IV. Perturbative analysis*
Journal of Chemical Physics, 120
(16).
7350_1-7350_19.
ISSN 0021-9606

Full text not available from this repository.

Official URL: http://link.aip.org/link/?JCPSA6/120/7350/1

Related URL: http://dx.doi.org/10.1063/1.1652490

## Abstract

The k-particle irreducible Brillouin conditions IBC_{k} and the k-particle irreducible contracted Schrodinger equations ICSE_{k} for a closed-shell state are analyzed in terms of a Moller-Plesset-type perturbation expansion. The zeroth order is Hartree-Fock. From the IBC_{2} i.e., from the two-particle IBC to first order in the perturbation parameter μ, one gets the leading correction λ_{2} to the two-particle cumulant λ_{2} correctly. However, in order to construct the second-order energy E_{2}, one also needs the second-order diagonal correction γ_{D} to the one-particle density matrix ν. This can be obtained: (i) from the idempotency of the n-particle density matrix, i.e., essentially from the requirement of n-representability; (ii) from the ICSE_{1} or (iii) by means of perturbation theory via a unitary transformation in Fock space. Method (ii) is very unsatisfactory, because one must first solve the ICSE_{3} to get λ_{3}, which is needed in the ICSE_{2} to get λ_{2}, which, in turn, is needed in the ICSE_{1} to get γ. Generally the (k+1)-particle approximation is needed to obtain E_{k} correctly. One gains something, if one replaces the standard hierarchy, in which one solves the ICSE_{k}, ignoring λ_{k+1} and λ_{k+2}, by a renormalized hierarchy, in which only λ_{k+2} is ignored, and λ_{k+1} is expressed in terms of the λ_{p} of lower particle rank via the partial trace relation for λ_{k+2}. Then the k-particle approximation is needed to obtain E_{k} correctly. This is still poorer than coupled-cluster theory, where the k-particle approximation yields E_{k+1}. We also study the possibility to use some simple necessary n-representability conditions, based on the non-negativity of γ and two related matrices, in order to get estimates for γ_{D} in terms of λ_{2}. In general these estimates are rather weak, but they can become close to the best possible bounds in special situations characterized by a very sparse structure of λ_{2} in terms of a localized representation. The perturbative analysis does not encourage the use of a k-particle hierarchy based on the ICSE_{k} (or on their reducible counterparts, the CSE_{k}), it rather favors the approach in terms of the unitary transformation, where the k-particle approximation yields the energy correct up to E_{2k-1}. The problems that arise are related to the unavoidable appearance of exclusion-principle violating cumulants. The good experience with perturbation theory in terms of a unitary transformation suggests that one should abandon a linearly convergent iteration scheme based on the ICSE_{k} hierarchy, in favor of a quadratically convergent one based on successive unitary transformations.

Item Type: | Article |
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Source: | Copyright of this article belongs to American Institute of Physics. |

ID Code: | 21915 |

Deposited On: | 23 Nov 2010 09:02 |

Last Modified: | 05 Mar 2011 11:30 |

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