Reappraisal of the Role of Size-Extensive Normalization for Multireference Coupled Cluster (MRCC) Theory Using General Model Space:  A Valence Universal MRCC Approach

Abstract

We present a brief description of a valence-universal multireference coupled cluster (VU-MRCC) theory that can handle completely general incomplete model spaces, remaining close to the intermediate normalization (IN) condition for omega as much as possible without violating extensivity and without the use of a post facto correction. In this formalism, the connectedness of the cluster operators as well as effective Hamiltonian and hence the extensivity of the corresponding roots is achieved by invoking appropriate decoupling conditions on the special type of wave operator omega = {exp(S + X(cl))} satisfying the Bloch equations in the Fock-space S in an excitation operator and X is a closed operator (denoted by cl). This special type of wave-operator leads to a unique partition of the excitations from the model space into those generated by the cluster operators (open and quasi-open) and those generated by the effective Hamiltonian (closed). In this formulation, for every X(cl), there is a counterterm from {exp(S)}(cl) canceling each other. This leads to a connected expressions for cluster amplitudes, using the constraint omega(cl) = 1. The new form of the effective Hamiltonian preserves the extensivity of the target energies. Our analysis implies that IN for omega is a valid size-extensive normalization for certain special IMS such as the quasi-complete model space and the isolated incomplete model space.

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

“…To get around this difficulty Bera et al [81] suggested quite an intricate combinatoric analysis which in our opinion is really not necessary to arrive at a connected H eff . There is in fact much simpler and straightforward way to formulate the problem where the normal ordered Ansatz for X is replaced by a new Ansatz and no counter-term is needed at all.…”

“…This analysis is important in the sense that it indicates how a new Ansatz for X without the use of the closed operator X cl leads in a straight forward manner to a new H eff which is related by a similarity transformation to the one formulated by Bera et al [81]and at the same time ensures X cl ¼ 1 cl . The latter formulation will be presented in Section 3.…”

“…This can be accomplished by using certain closed counter-terms X cl which can be determined by imposing X cl ¼ 1 cl . Such a formulation in the VU-MRCC context has recently been proposed by Bera et al [81]. Curiously, the use of this counter-term does not necessarily lead to an explicitly connected H eff and Bera et al tried to circumvent this problem by invoking a rather involved transformation of associated H eff .…”

“…Curiously, the use of this counter-term does not necessarily lead to an explicitly connected H eff and Bera et al tried to circumvent this problem by invoking a rather involved transformation of associated H eff . Since this formulation is instructive to develop the much simpler and compact new VU-MRCC formulation we are after, let us digress at this point to present succinctly the approach of Bera et al [81]. If we introduce the Ansatz …”

“…In particular it is possible to envisage an Ansatz for X for which X cl ¼ 1 cl , such that one expects H eff ¼ f HX z}|{ g cl where z}|{ symbolizes all connected quantities obtained by contracting operators of H with those in X. This seems eminently possible since, although S qÀop fixes X qÀop by the decoupling condition, X cl is still at our disposal -being not related to the decoupling condition and X cl ¼ 1 cl is a sensible size-extensive normalization which is expected to yield connected H eff having a structure exactly analogous to that in a CMS [81]. Demonstration of the existence of a wave-operator with precisely such a relation forms the major content of this paper.…”