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

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 Heff 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 Sq-op which leads to excitations into the complementary model space by their action on at least one model function. The powers of Sq-op can in general bring a model function ϕi back to another model function ϕj, and this is the reason why Ω does not respect IN. Sq-op are all labelled by active orbitals only. To achieve connectivity of Heff, 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 Lq-op=0, and Lop=0 for the transformed operator L=Ω-1HΩ would be in conflict with Ωq-op=1q-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 Heff. 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 Heff, and accomplished this by introducing suitable counter-terms Xcl in Ω to enforce Ωcl=1cl. We show in this paper that Heff in this formulation leads to a disconnected Heff, 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 Sq-op, which is closed. This 'externally projected' wave-operator does not need counter-terms Xcl and automatically ensures Ωcl=1cl, 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.