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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

Chemical Physics
Publication Date
DOI: 10.1016/j.chemphys.2008.12.008
  • Valence Universal Multi-Reference Coupled Cluster Theory (Vu-Mrcc)
  • Incomplete Model Space (Ims)
  • Intermediate Normalization (In)
  • Excitation Operator
  • Closed Operator
  • Effective Hamiltonian


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 H Ω would be in conflict with Ω q - op = 1 q - op , the model space projection of Ω , P Ω P = 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.

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