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

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^-1HΩ 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.