Second-order approximation, constrained

Flere the vector variable is a Gaussian vector with associated width matrix given by 0 (0, q ). The result above from Paper III is the simplest generalization of the expression obtained in Paper I for coordinate-dependent operators. The expression reveals the role played by the centroid-constrained correlation function matrix [Eq. (2.57)] in defining the effective width factor in phase space for the centroid quasiparticle. A more careful treatment of the operator ordering problems demonstrates that the derivation of the equations above involves additional approximation beyond second-order truncation of the cumulant expansion [59]. 

Actually the form of working equation of the first order cluster amplitudes and hence the perturbative scheme depend on the treatment of the four above-mentioned terms. One can treat all the four terms consistently in the same partitioning scheme. This approach is rather inflexible, since this necessarily constrains us to use only a very specific partitioning strategy, viz. the one proposed by Dyall [48]. We follow here a somewhat hybrid strategy where we treat the first and the second terms strictly by perturbation theory, while we treat the third and fourth terms on the same footing, but not in the sense of strict perturbation. Since it is natural to have the unperturbed state-energy Eq appear in the denominator in the RS version, we approximated Hij y in these terms by in since this leads to ... 

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