Generalized antisymmetric ordered products, generalized normal ordered products, ordered and ordinary cumulants and their use in many electron correlation problem

• Published in: Computational and theoretical chemistry 2013-01, Vol.1003, p.62-70
• Main Authors: Sinha, Debalina, Maitra, Rahul, Mukherjee, Debashis
• Format: Article
• Language: English
• Subjects: Cumulants; Generalized normal ordering; Internally Contracted MRCC
• ISSN: 2210-271X
• DOI: 10.1016/j.comptc.2012.09.035

[Display omitted] ► We show the notion of generalized antisymmetrized product, generalized normal ordering (GNO) and cumulants. ► We show that all cumulants with rank greater than two have the antisymmetric property. ► Invoking an exponential parametrization of Ω in GNO, we formulate a compact Internally Contracted MRCC. ► The newly formulated Internally Contracted MRCC theory is manifestly size-extensive and orbital invariant. The use of cumulants in the direct determination of the 2-particle reduced density matrix (2-RDM), Γ2, via reconstruction schemes where Γ3 and Γ4 are expressed in terms of Γ2 and cumulants to ‘close’ the hierarchy of density equations of Nakatsuji has been systematically developed for about two decades. A challenging aspect of such developments is the imposition of the N-representability conditions on Γ2, all of which are not known. Some reasonable sufficiency conditions and the use of the so-called ‘weak positivity conditions’ have proved to be fruitful but a lot more remains to be done. There is another parallel development involving cumulants where certain ‘model cumulants’ are extracted from a multi-reference model function Ψ0 and dynamical correlation is introduced via a wave operator Ω, acting on Ψ0. The non-dynamical correlation is reflected in the various model cumulants. Such formulations are accomplished in the most compact manner if one uses the notion of generalized normal ordering (GNO) and generalized Wick’s theorem (GWT) with respect to Ψ0. The product of operators in GNO has vanishing expectation value with respect to Ψ0. The GWT expresses a product of n creation/annihilation operators as a sum of m operators in GNO (m⩽n) with products of various cumulants along with appropriate phases. This approach, unlike the reconstruction schemes, uses the model cumulants as intermediaries only which are N-representable by construction. In this paper we will introduce the notion of generalized antisymmetrized ordered products (GOPs) and show how a generalized normal ordering for a product of arbitrary number of creation and annihilation operators can naturally emerge as the limit of a hierarchy of GOP. We argue in this paper that the use of GOP leads naturally to the generalized Wick’s theorem where the normal ordered products are antisymmetric under permutations and have vanishing expectation values with respect to Ψ0. We will also show that, except the pair cumulants, all the higher body cumulants have the antisymmetric proper