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

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 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 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 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 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 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 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.