Some recent advances in the area of multi-reference coupled-cluster theory of the state-universal type are overviewed. An emphasis is placed on the following new developments: (i) the idea of combining the state-universal multi-reference coupled-cluster singles and doubles method (SUMRCCSD) with the multi-reference many-body perturbation theory (MRMBPT), in which cluster amplitudes of the SUMRCCSD formalism that carry only core and virtual orbital indices are replaced by their first-order MRMBPT estimates; and (ii) the idea of combining the recently proposed method of moments of coupled-cluster equations with the SUMRCC formalism. It is demonstrated that the new SUMRCCSD(1) method, obtained by approximating the SUMRCCSD cluster amplitudes carrying only core and virtual orbital indices by their first-order MRMBPT values, provides the results that are comparable to those obtained with the complete SUMRCCSD approach.
Some recent advances in the area of multi-reference coupled-cluster theory of the state-universal type are overviewed. An emphasis is placed on the following new developments: (i) the idea of combining the state-universal multi-reference coupled-cluster singles and doubles method (SUMRCCSD) with the multi-reference many-body perturbation theory (MRMBPT), in which cluster amplitudes of the SUMRCCSD formalism that carry only core and virtual orbital indices are replaced by their first-order MRMBPT estimates; and (ii) the idea of combining the recently proposed method of moments of coupled-cluster equations with the SUMRCC formalism. It is demonstrated that the new SUMRCCSD(1) method, obtained by approximating the SUMRCCSD cluster amplitudes carrying only core and virtual orbital indices by their first-order MRMBPT values, provides the results that are comparable to those obtained with the complete SUMRCCSD approach.
The traditional state universal multi-reference coupled cluster (SUMRCC) theory uses the Jeziorski-Monkhorst (JM) based Ansatz of the wave operator: Ω = Σ(μ)Ω(μ)|φ(μ)><φ(μ)|, where Ω(μ) = exp(T(μ)) is the cluster representation of the component of Ω inducing virtual excitations from the model function φ(μ). In the first formulations, φ(μ)s were chosen to be single determinants and T(μ)s 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, {T(μ)}, which even at singles-doubles truncation, generates non-commuting cluster operators. We propose the use of normal-ordered exponential parameterization for Ω:Σ(μ){exp(T(μ))}|φ(μ)><φ(μ)|, where {} denotes the normal ordering with respect to a common closed shell vacuum which makes the "direct term" of the SUMRCC equations terminate at the quartic power. We choose our model functions {φ(μ)} 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 {T(μ)} such that no redundancies in the virtual functions {χ(μ)(l)} are involved. Using example applications for electron detached/attached and h-p excited states relative to a closed shell ground state we discuss 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 captures the most important aspects of physics with a considerably simpler scheme.
Multi-reference methods for the description of quasi-degenerate ground states and excited states of many-electron systems have been one of the main objectives of scientific activity at the Quantum Theory Project (QTP) University of Florida in Gainesville. The pioneering works by Dr. Per-Olov Löwdin on the partitioning technique, and important contributions to the development of multi-reference methods made by Dr. Monkhorst and Dr. Bartlett must be mentioned here. Dr. Bartlett, his group and its visitors have been especially active in this field. In this paper we describe some of the multi-reference developments, problems that have been faced and solved as well as those which still require solving. The focus is on multi-reference many-body perturbation theory and coupled-cluster approaches. We present recent advances and future perspectives in the development of these methods. Special attention is paid to two multi-reference approaches that seem very promising and have been either initiated or developed at QTP.
The purpose of this paper is to introduce a quasiparticle-based multi-reference coupled-cluster (MRCC) approach. The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference (MR) basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. On the basis of these quasiparticles, a generalization of the normal-ordered operator products for the MR case can be introduced as an alternative to the approach of Mukherjee and Kutzelnigg [Recent Prog. Many-Body Theor. 4, 127 (1995); Mukherjee and Kutzelnigg, J. Chem. Phys. 107, 432 (1997)]. Based on the new normal ordering any quasiparticle-based theory can be formulated using the well-known diagram techniques. Beyond the general quasiparticle framework we also present a possible realization of the unitary transformation. The suggested transformation has an exponential form where the parameters, holding exclusively active indices, are defined in a form similar to the wave operator of the unitary coupled-cluster approach. The definition of our quasiparticle-based MRCC approach strictly follows the form of the single-reference coupled-cluster method and retains several of its beneficial properties. Test results for small systems are presented using a pilot implementation of the new approach and compared to those obtained by other MR methods.
A general coupled-cluster method valid for arbitrary multideterminantal reference states is formulated. The resulting cluster expansion for the wave function is a generalization of that introduced by Silverstone and Sinano\ifmmode \check{g}\else \v{g}\fi{}lu and applied by Sinano\ifmmode \check{g}\else \v{g}\fi{}lu and collaborators. The connected nature of the cluster operators and the effective interaction is proven in the case when the reference space is complete, i.e., is invariant under unitary transformations of partly occupied orbitals. For incomplete reference spaces the disconnected terms appearing in the effective interaction are properly generated by the coupled-cluster theory. Approximate schemes for solving coupled-cluster equations are proposed and their relation with perturbation theory is briefly discussed.
Many-body perturbation theories (MBPT) (both single and multireference), by design, are capable of representing electron correlation very well. However, the practical implementation of these methods for many-electron systems beyond a certain order is extremely difficult, if not impossible. For instance, there are hardly any calculations beyond third order at the multireference and fifth order at the single reference of perturbation theories. Thus, these methods are typically limited to systems where the higher-order electron correlation effect is not significant. A practical solution to this problem is the coupled cluster theory, which was first introduced by Coester [20] and Coester and Kümmel [21] in nuclear physics and was later adopted in the electron structure theory by Čížek [23, 106] and Paldus and Čížek [107]. The coupled-cluster (CC) theory is a non-perturbative all-order theory. This stateof-the-art theory has now been recognized as the basis of modern quantitative quantum chemistry and atomic physics for high-precision calculations. Its superior accuracy is rooted in the exponential ansatz for the wave function, which provides an energy expression that scales correctly with the number of electrons and under certain provisions leads to a faster convergence of the many-body expansion.
The Unitary Group Adapted State Universal Multireference Coupled Cluster (UGA-SUMRCC) theory, recently developed by us (J. Chem. Phys.2012, 137, 074104), contains exactly the right number of linearly independent cluster operators. This avoids any redundancy of the excitation manifold in a way exactly paralleling the traditional spin-orbital based SUMRCC. The choice of the linearly independent cluster operators inducing the same change of orbital occupancy becomes increasingly cumbersome if we go over to the cases of active CSFs with more than two active quasiparticles. In the present development, we explore several aspects of the UGA-SUMRCC theory: (a) The first is a variant where we have deliberately incorporated redundancy of the cluster amplitudes to simplify the working equations and have shown that it can serve as a very good approximation to the parent UGA-SUMRCC theory for states with more than two valence occupancies. This in turn suggests that it could be a useful avenue to pursue for arbitrary mh-np situation since the working equations assume simpler algebraic structure in such cases. (b) The analyses of the aspects of size extensivity are known to involve greater complexity if they involve various reduced density matrices (RDMs), since the RDMs are not size-extensive quantities. We have presented the proof for UGA-SUMRCC starting from equations containing h-p RDMs via a decomposition involving products of size-extensive cumulants and argue that it has relevance for general cases beyond the h-p model spaces. (c) A useful extension of UGA-SUMRCC lies in formulating the theory for direct calculations of energy differences of spectroscopic interest such as excitation energies, ionization potentials, and electron affinities relative to a closed shell ground state, thus providing attractive alternatives to other allied methods such as SAC-CI, CC-LRT, EOM-CC, STEOM-CC, or ADC. This extension, called UGA-based Quasi-Fock MRCC by us, also leads to exact cancellation of common correlation terms between the initial and final states. Taking a cue from the hierarchical development in Fock-space theories but keeping in mind the advantages of a state-universal (equivalently called a valence specific) theory, our formulation proposes a spin-adapted, accurate, and compact scheme for studying such energy differences. Our results demonstrate superior performance of the method as compared to EOM-CC.