Multi-reference Coupled Cluster Theory

Profiling a potential energy surface (PES), all the way to dissociate a molecular state into particular fragments and to display real or avoided crossings,requires a multireference description and themaintenance of size-consistency. The many body methods, which suit this purpose, should thus be size-extensive. Size-extensive theories, which are invariant with respect to transformation among active orbitals are, in principle, size- consistent. Relatively cheaper size-extensive theories, which do not possess this invariance, can still be size-consistent if the active orbitals are localized on the asymptotic fragments. Such methods, if perturbative in nature, require the use of an unperturbed Hamiltonian, which has orbital invariance with respect to the transformation within active, core, and virtual orbitals. The principal focus of this paper is to numerically realize size-consistency with localized active orbitals using our recently developed orbitally noninvariant Unitary Group Adapted State Specific Multireference second order Perturbation Theory (UGA-SSMRPT2) as a prototype method. Our findings expose certain generic potential pitfalls of size- extensive but orbitally noninvariant MRPT theories, which are mostly related to the inability of reaching proper localized active orbitals in the fragments due to the artifacts of the orbital generation procedure. Despite the invariance of the zeroth order CAS function, lack of invariance of the MRPT itself then leads to size-inconsistency. In particular, reaching symmetry broken fragment active orbitals is an issue of concern where suitable state-averaging might ameliorate the problem, but then one has to abandon full orbital optimization. Additionally, there can be situations where the orbitals of the fragment reached as an asymptote of the supermolecule are not the same as those obtained from the optimization of the fragments individually and will require additional transformation. Moreover, for a certain PES, one may either abandon the use of optimized orbitals for that state to preserve proper symmetry and degeneracy in the fragment orbitals or be satisfied with the use of optimized orbitals, which generate broken symmetric orbitals in the fragmentation limit. All these pathologies are illustrated using the PES of various electronic states of multiply bonded systems like N2,C 2H2, HCN, C 2, and O 2. Subject to such proviso, the UGA-SSMRPT2 turns out to be an excellent theory for studying the PES leading to fragmentation of strongly correlated systems satisfying the requirements of size- consistency with localized active orbitals. An unexpected spin-off of our studies is the realization that the size-inextensive MRMP2, which bears a close structural similarity with our theory, might under certain situations display size-consistency. We analyze this feature concretely in our paper. Our studies may serve as a benchmark for monitoring numerically the size-consistency of any state specific multireference theory which is size-extensive but not invariant with respect to transformation of active orbitals.

We present here the formulations and implementations of Mukherjee's State-Specific and State-Universal Multi-reference Coupled Cluster theories, which are explicitly spin free being obtained via the Unitary Group Adapted (UGA) approach, and thus, do not suffer from spin-contamination. We refer to them as UGA-SSMRCC and UGA- SUMRCC respectively. We propose a new multi-exponential cluster Ansatz analogous to but different from the one suggested by Jeziorski and Monkhorst (JM). Unlike the JM Ansatz, our choice involves spin-free unitary generators for the cluster operators and we replace the traditional exponential structure for the wave-operator by a suitable normal ordered exponential. We sketch the consequences of choosing our Ansatz, which leads to fully spin-free finite power series structure of the direct term of the MRCC equations. The UGA-SUMRCC follows from a suitable hierarchical generation of the cluster amplitudes of increasing rank, while the UGA-SSMRCC requires suitable sufficiency conditions to arrive at a well-defined set of equations for the cluster amplitudes. We discuss two distinct and inequivalent sufficiency conditions and their pros and cons. We also discuss a variant of the UGA-SSMRCC, where the number of cluster amplitudes can be drastically reduced by internal contraction of the two-body inactive cluster amplitudes. These are the most numerous, and thus a spin-free internally contracted description will lead to a high speed-up factor. We refer to this as ICID-UGA-SSMRCC. Essentially the same mathematical manipulations provide us with the UGA-SUMRCC theory as well. Pilot numerical results are presented to indicate the promise and the efficacy of all the three methods.

The traditional state universal multi-reference coupled cluster (SUMRCC) theory uses the Jeziorski- Monkhorst (JM) based Ansatz of the wave operator:  =μμ|φμφμ|, where μ =exp(Tμ) is the cluster representation of the component of  inducing virtual excitations from the model func- tion φμ. In the first formulations, φμs were chosen to be single determinants and Tμs were defined in terms of spinorbitals. This leads to spin-contamination for the non-singlet cases. In this paper, we propose and implement an explicitly spin-free realization of the SUMRCC theory. This method uses spin-free unitary generators in defining the cluster operators, {Tμ}, which even at singles-doubles truncation, generates non-commuting cluster operators. We propose the use of normal-ordered ex- ponential parameterization for :μ{exp(Tμ)}|φμφμ|, where {} denotes the normal ordering with respect to a common closed shell vacuum which makes the “direct term” of the SUMRCC equa- tions terminate at the quartic power. We choose our model functions {φμ} as unitary group adapted (UGA) Gel’fand states which is why we call our theory UGA-SUMRCC. In the spirit of the original SUMRCC, we choose exactly the right number of linearly independent cluster operators in {Tμ} such that no redundancies in the virtual functions {χl μ} are involved. Using example applications for electron detached/attached and h-p excited states relative to a closed shell ground state we dis- cuss how to choose the most compact and non-redundant cluster operators. Although there exists a more elaborate spin-adapted JM-like ansatz of Datta and Mukherjee (known as combinatoric open- shell CC (COS-CC), its working equations are more complex. Results are compared with those from COS-CC, equation of motion coupled cluster methods, restricted open-shell Hartree-Fock coupled cluster, and full configuration interaction. We observe that our results are more accurate with respect to most other theories as a result of the use of the cluster expansion structure for our wave operator. Our results are comparable to those from the more involved COS-CC, indicating that our theory cap- tures the most important aspects of physics with a considerably simpler scheme. © 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4742058]