Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. IV. Perturbative analysis

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

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The k-particle irreducible Brillouin conditions IBCk and the k-particle irreducible contracted Schrodinger equations ICSEk for a closed-shell state are analyzed in terms of a Moller-Plesset-type perturbation expansion. The zeroth order is Hartree-Fock. From the IBC2 i.e., from the two-particle IBC to first order in the perturbation parameter μ, one gets the leading correction λ2 to the two-particle cumulant λ2 correctly. However, in order to construct the second-order energy E2, one also needs the second-order diagonal correction γD 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 ICSE1 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 ICSE3 to get λ3, which is needed in the ICSE2 to get λ2, which, in turn, is needed in the ICSE1 to get γ. Generally the (k+1)-particle approximation is needed to obtain Ek correctly. One gains something, if one replaces the standard hierarchy, in which one solves the ICSEk, ignoring λk+1 and λk+2, by a renormalized hierarchy, in which only λk+2 is ignored, and λk+1 is expressed in terms of the λp of lower particle rank via the partial trace relation for λk+2. Then the k-particle approximation is needed to obtain Ek correctly. This is still poorer than coupled-cluster theory, where the k-particle approximation yields Ek+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 γD in terms of λ2. 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 λ2 in terms of a localized representation. The perturbative analysis does not encourage the use of a k-particle hierarchy based on the ICSEk (or on their reducible counterparts, the CSEk), it rather favors the approach in terms of the unitary transformation, where the k-particle approximation yields the energy correct up to E2k-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 ICSEk hierarchy, in favor of a quadratically convergent one based on successive unitary transformations.

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