Aspects of size extensivity in unitary group adapted multi-reference coupled cluster theories: the role of cumulant decomposition of spin-free reduced density matrices

Abstract

We present in this paper a comprehensive study of the various aspects of size extensivity of a set of unitary group adapted multi-reference coupled cluster (UGA-MRCC) theories recently developed by us. All these theories utilize a Jeziorski– Monkhorst (JM) inspired spin-free cluster Ansatz of the form \(\mid\Psi\rangle=\sum\nolimits_\mu\Omega_\mu\mid\phi_\mu\rangle c_\mu\;\mathrm{with}\; \Omega_\mu\;=\;\left\{ \mathrm{exp}(T_\mu)\right\} \), where Tμ is expressed in terms of spin-free generators of the unitary group U(n) for n-orbitals with the associated cluster amplitudes. {···} indicates normal ordering with respect to the common closed shell core part, |0〉, of themodel functions, \({\phi_\mu}\) which is taken as the vacuum. We argue and emphasize in the paper that maintaining size extensivity of the associated theories is consequent upon (a) connectivity of the composites, Gμ, containing the Hamiltonian H and the various powers of T connected to it, (b) proving the connectivity of the MRCC equations which involve not only Gμs but also the associated connected components of the spin-free reduced density matrices (RDMs) obtained via their cumulant decomposition and (c) showing the extensivity of the cluster amplitudes for non-interacting groups of orbitals and eventually of the sizeconsistency of the theories in the fragmentation limits. While we will discuss the aspect (a) above rather briefly, since this was amply covered in our earlier papers, the aspect (b) and (c), not covered in detail hitherto, will be covered extensively in this paper. The UGA-MRCC theories dealt with in this paper are the spin-free analogs of the state-specific and state-universal MRCC developed and applied by us recently.We will explain the unfolding of the proof of extensivity by analyzing the algebraic structure of the working equations, decomposed into two factors, one containing the composite Gμ that is connected with the products of cumulants arising out of the cumulant decomposition of the RDMs and the second term containing some RDMs which is disconnected from the first and can be factored out and removed. This factorization ultimately leads to a set of connected MRCC equations. Establishing the extensivity and size-consistency of the theories requires careful separation of truly extensive cumulants from the ones which are a measure of spin correlation and are thus connected but not extensive.Wehave discussed in detail, using diagrams, the factorization procedure and have used suitable example diagrams to amplify the meanings of the various algebraic quantities of any diagram. Weconclude the paper by summarizing our findings and commenting on further developments in the future.