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Elsevier, Chemical Physics, 1-3(356), p. 54-63

DOI: 10.1016/j.chemphys.2008.12.008

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Use of a convenient size-extensive normalization in multi-reference coupled cluster (MRCC) theory with incomplete model space: a novel valence universal MRCC formulation

Journal article published in 2009 by Rahul Maitra, Dipayan Datta ORCID, Debashis Mukherjee
This paper was not found in any repository, but could be made available legally by the author.
This paper was not found in any repository, but could be made available legally by the author.

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Abstract

We present in this paper a size-extensive formulation of a valence universal multi-reference coupled cluster (VU-MRCC) theory which uses a general incomplete model space (IMS). The earlier formulations by Mukherjee [D. Mukherjee, Chem. Phys. Lett. 125 (1986) 207] led to size-extensive H eff which was both connected and 'closed', thereby leading to size-extensive energies. However, this necessitated abandoning the intermediate normalization (IN) for the valence universal wave-operator Ω when represented as a normal ordered exponential cluster Ansatz Ω={exp(S)} with S as the cluster operator. The lack of IN stemmed from the excitation operator S q-op which leads to excitations into the complementary model space by their action on at least one model function. The powers of S q-op can in general bring a model function Φ i back to another model function Φ j , and this is the reason why Ω does not respect IN. S q-op are all labelled by active orbitals only. To achieve connectivity of H eff , it must be a 'closed' operator. A closed operator is one which always produces a model function by its action on another model function. Since the decoupling conditions L q-op =0, and L op =0 for the transformed operator L=Ω -1 HΩ would be in conflict with Ω q-op =1 q-op , the model space projection of Ω, PΩP=P cannot be maintained for the normal ordered Ansatz. This leads to a somewhat awkward expression for H eff . Bera et al. [N. Bera, S. Ghosh, D. Mukherjee, S. Chattopadhyay, J. Phys. Chem. A 109 (2005) 11462] recently tried to simplify the expression for H eff , and accomplished this by introducing suitable counter-terms X cl in Ω to enforce Ω cl =1 cl . We show in this paper that H eff in this formulation leads to a disconnected H eff , though it is equivalent by a similarity transformation to a connected effective hamiltonian H¯ eff . Guided by the insight gleaned from this demonstration, we have proposed in this paper a new form of the wave-operator which never generates any powers of S q-op , which is closed. This 'externally projected' wave-operator does not need counter-terms X cl and automatically ensures Ω cl =1 cl , thereby yielding directly a closed connected H¯ eff . The desirable features of the traditional normal ordered Ansatz, such as the valence universality, subsystem embedding conditions hierarchical decoupling of the VU-MRCC equations for decreasing valence ranks are all satisfied by this new Ansatz for the wave-operator.