Size extensive formulations Hamiltonian

Coupled Cluster based size-extensive intermediate hamiltonian formalisms were developed by our group [33-35] by way of transcribing a size-extensive CC formulation in an incomplete model space in the framework of intermediate hamiltonians. In this method, there are cluster operators correlating the main model space. There are no cluster operators for the intermediate space. This formulation thus is conceptually closer to the perturbative version of Kirtman... 

We define an operator as closed , if its action on any model function G P produces only internal excitations within the IMS. An operator is quasi-open , if there exists at least one model function which gets excited to the complementary model space R by its action. Obviously, both closed and quasi-open operators are all labeled by only active orbitals. An operator is open , if it involves at least one hole or particle excitation, leading to excitations to the g-space by acting on any P-space function. It was shown by Mukheijee [28] that a size-extensive formulation within the effective Hamiltonians is possible for an IMS, if the cluster operators are chosen as all possible quasi-open and open excitations, and demand that the effective Hamiltonian is a closed operator. Mukhopadhyay et al. [61] developed an analogous Hilbert-space approach using the same idea. We note that the definition of the quasi-open and closed operators depends only on the IMS chosen by us, and not on any individual model function. 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

In the spin-free formulations of the UGA-MRCC theories, the use of CSFs entails that both the MRCC equations are in matrix form and the associated effective Hamiltonians will involve various w-body spin-free reduced density matrices (n-RDMs). n-RDMs are product separable and hence not size-extensive. From now on, we will refer to the spin-free RDMs as simply the ROMs. When spinorbital-based RDMs are discussed, we will explicitly indicate this. So, no confusions should arise. It is non-trivial to establish the extensivity of both the cluster operators and the effective Hamiltonian in spite of the occurrence of these n-RDMs. This paper will briefly review the formulation of the UGA-MRCC theories mentioned above and will present a comprehensive account of the aspects of connectivity which leads to extensivity. Although in some of our earlier papers [47] we sketched how size extensivity emerges after the cumulant decomposition of the n-RDMs, we will present here a detailed and thorough analysis of the underlying issues. 

The key point in establishing the size-extensivity of coupled-eluster theory in the linked formulation is to note that the similarity-transformed Hamiltonian of the compound system... 

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