Coupled Cluster Downfolding Theory: towards universal many-body algorithms for dimensionality reduction of composite quantum systems in chemistry and materials science

The single reference coupled cluster (CC) approach to the many-electron correlation problem is examined from the viewpoint of the method of moments (MM). This yields generally an inconsistent (overcomplete) set of equations for cluster amplitudes, which can be solved either in the least squares sense or by selective projection process restricting the number of equations to that of the unknowns. These resulting generalized MM-CC equations always contain the standard CC equations as a special case. Since, in the MM-CC formalism, the Schrodinger equation will be approximately satisfied on a subspace spanned by non-canonical configurations, this procedure may be helpful in extending the standard single reference CC theory to quasi-degenerate situations. To examine the potential usefulness of this idea, we explore the linear version of the CC approach for systems with a quasi-degenerate reference, in which case the standard linear theory is plagued with singularities due to the intruder states. Implications of this analysis for the structure of the wavefunction are also briefly discussed.

A new approximation strategy, split-amplitude strategy, useful within the framework of the coupled-cluster (CC) methodology is proposed. It consists in representing the individual cluster amplitudes as a sum of two components, one of fixed value, which may be obtained from external sources, and the other determined from a set of modified CC equations. This approach provides new possibilities of absorbing information concerning the values of cluster amplitudes from independent calculations. By properly choosing the fixed amplitude components, one may substantially reduce the magnitudes of the most significant amplitudes to be determined for the state considered, which in turn causes that the known approximation procedures are more justifiable when applied to the modified CC equations than to the equations of the standard CC approaches. The split-amplitude strategy has been employed to setting up several almost-linear CC (AL-CC) approaches of a single reference type corresponding to the basic CC methods. These low-cost approaches seem to be useful in theories of processes involving nondynamically correlated (quasidegenerate) states. The AL-CC methods have been applied to the ground states for various geometries of the following molecular systems: H8, H2O, BH, and HF. It is found that the energies obtained for a given AL-CC approach are very close to those for the corresponding basic CC method, which is true even for strongly quasidegenerate states.

The performance of various coupled-cluster (CC) approaches using both single and multideterminantal references is investigated for the (quasi-)degenerate states of molecular systems, where inclusion of higher excitations (or equivalently nondynamic correlation) proves to be needed. The prototype system H8 represents an adequate model for our study, where we can vary the degree of degeneracy from a completely degenerate situation to a nondegenerate one in a continuous way. To obtain a reliable benchmark for our CC results, the full configuration interaction (FCI) and large-scale complete active space configuration interaction (CAS CI) calculations, respectively, are performed for a variety of geometries and states. The convergence of the approximate single reference CC approaches is found to be extremely sensitive to the level of degeneracies involved. In the nondegenerate case the standard CC method with single and double excitations is found to be quite satisfactory; in the (quasi-)degenerate situations, however, the inclusion of triple excitations and noniterative quadruple excitations is needed to furnish semiquantitative values of correlation energies. The alternative treatment of nondynamic correlation using a multideterminantal Hilbert space coupled-cluster (MRCC) method demonstrates the power of this approach, which provides a balanced description of both dynamic and nondynamic correlation in the degenerate region for all the investigated states of H8. Its convergence for nondegenerate situations, however, is less satisfactory, being affected by an intruder state problem.

We consider a fermionic system for which there exist a single-reference configuration-interaction (CI) expansion of the ground state wave function that converges, albeit not necessarily rapidly, with respect to excitation number. We show that, if the coefficients of Slater determinants (SD) with $l\leq k$ excitations can be defined with a number of free parameters $N_{\leq k}$ bounded polynomially in $k$, the ground state energy $E$ only depends on a small fraction of all the wave function parameters, and is the solution of equations of the coupled-cluster (CC) form. This generalizes the standard CC method, for which $N_{\leq k}$ is bounded by a constant. Based on that result and low-rank tensor decompositions (LRTD), we discuss two possible extensions of the CC approach for wave functions with general polynomial bound for $N_{\leq k}$. First, one can use LRTD to represent the amplitudes of the CC cluster operator $T$ which, unlike in the CC case, is not truncated with respect to excitation number, and the energy and tensor parameters are given by a LRTD-adapted version of standard CC equations. Second, the LRTD can also be used to directly parametrize the CI coefficients, which involves different equations of the CC form. We derive those equations for up to quadruple-excitation coefficients, using a different type of excitation operator in the CC wave function ansatz, and a Hamiltonian representation in terms of excited particle and hole operators. We complete the proposed CC extensions by constructing compact tensor representations of coefficients, or $T$-amplitudes, using superpositions of tree tensor networks which take into account different possible types of entanglement between excited particles and holes. Finally, we discuss why the proposed extensions are theoretically applicable at larger coupling strengths than those treatable by the standard CC method.

New single-reference coupled-cluster (CC) methods that can be applied to quasi-degenerate electronic states and molecular potential energy surfaces involving bond breaking and that can provide the virtually exact results for many-electron systems are reviewed. Three types of approaches are discussed: (i) the renormalized CC methods, which represent the approximate variants of the more general formalism of the method of moments of coupled-cluster equations (MMCC), (ii) the extended CC (ECC) methods, and (iii) the generalized CC (GCC) theory, in which many-electron wave functions are represented by the exponential cluster expansions involving general two-body operators. The main idea of the renormalized CC and other MMCC theories is that of the noniterative energy corrections which, when added to the energies obtained in the standard CC calculations, such as CCSD (CC singles and doubles), recover the exact or virtually exact energies. It is demonstrated that the completely renormalized CCSD(T) and CCSD(TQ) methods and their quadratic MMCC analogs represent “blackbox” approaches, which are capable of removing the failing of the standard CCSD, CCSD(T), and similar methods at larger internuclear separations. The ECC methods, in which products involving low-order cluster components mimic the effects of higher-order clusters, are based on the idea of optimizing two cluster operators. It is shown that the ECC methods with singles and doubles (ECCSD), including the quadratic and bilinear ECCSD theories, provide great improvements in the poor description of multiple bond breaking by the standard CC approaches. Finally, we provide strong arguments that the GCC theory may represent the exact many-body formalism. This result may have a significant impact on future quantum calcalatems for many-electron and other pairwise interacting many-fermion systems.

ABSTRACTWe have recently proposed the CC(P;Q) methodology that provides a systematic approach to correcting the energies obtained in active-space coupled-cluster (CC) calculations for the remaining, mostly dynamical, correlations. We report the development of the CC(t,q;3) and CC(t,q;3,4) methods, which use the CC(P;Q) formalism to correct the energies obtained with the CC approach with singles, doubles and active-space triples and quadruples, termed CCSDtq, for the remaining triples [CC(t,q;3)] or the remaining triples and quadruples [CC(t,q;3,4)] missing in CCSDtq. By examining the double dissociation of water, the Be + H2 → HBeH insertion, and the singlet–triplet gap in the (HFH)− system, we demonstrate that the CC(t,q;3) and CC(t,q;3,4) methods, particularly the latter one, offer significant improvements in the CCSDtq results, reproducing the total and relative energies obtained with the CC approach with singles, doubles, triples and quadruples (CCSDTQ) to within fractions of a millihartree, at the tiny fraction of the computational effort involved in the CCSDTQ calculations, even when electronic quasi-degeneracies become substantial. We also show that the previously examined CC(t;3) approach, which corrects the active-space CCSDt energies for the triples missing in CCSDt, accurately reproduces the CCSDT energetics.

The objective of this paper is to provide an overview of various multi-reference (MR) coupled-cluster (CC) approaches, particularly those relating to our own research. Although MR CC methods have been around for almost three decades and much work has been expended on their development and implementation, no general purpose codes are presently available. In view of the complexity, inherent difficulties, and computational demands of both genuine valence and state universal (VU and SU) MR CC methods, attention has been directed towards the state selective or state specific (SS) approaches that focus on one state at a time. These methods are based on either the genuine MR CC formalism or on a single-reference (SR) CC Ansatz, in which higher-than-pair clusters are accounted for by relying on basic ideas of general MR approaches. This is achieved either internally by relying on CC or MBPT formalism or by exploiting some external source providing approximate values of these clusters and accomplished by either correcting equations yielding the cluster amplitudes or directly by evaluating the corrections to the CCSD energy. Nowadays there exists a whole plethora of such various approaches for handling of quasi-degenerate states with a various degree of MR character and our goal is to outline their basic features and comment on their pro’s and con’s, their usefulness and weaknesses, as well as point out their mutual relationship.

The basic aspects of coupled cluster (CC) theories are reviewed from the perspective of its applicability to molecular systems with strong many-body correlation effects. In practice strong correlation refers to systems where the corresponding wavefunctions are characterized by multiconfigurational character corresponding to collective excitations from the reference function/functions. Several CC formalisms specifically designed to tackle these situations are discussed. These include single reference CC methodologies accounting for high-rank excitations and multireference CC approaches. Special attention is paid to non-iterative methods, which provide a widely accepted compromise between accuracy and numerical cost. We also discuss major theoretical and computational challenges which have to be addressed for the future developments of CC methodologies.

The recently developed new approach to the many-electron correlation problem in atoms and molecules, termed the method of moments of coupled-cluster (CC) equations (MMCC), is reviewed. The ground-state MMCC formalism and its extension to excited electronic states via the equation-of-motion coupled-cluster (EOMCC) approach are discussed. The main principle of all MMCC methods is that of the non-iterative energy corrections which, when added to the ground- and excited-state energies obtained in the standard CC calculations, such as CCSD or EOMCCSD, recover the exact, full configuration interaction (CI) energies. Three types of the MMCC approximations are reviewed in detail: (i) the CI-corrected MMCC methods, which can be applied to ground and excited states; (ii) the renormalized and completely renormalized CC methods for ground states; and (iii) the quasi-variational MMCC approaches for the ground-state problem, including the quadratic MMCC models. It is demonstrated that the MMCC formalism provides a new theoretical framework for designing 'black-box' CC approaches that lead to an excellent description of entire potential energy surfaces of ground- and excited-state molecular systems with an ease of use of the standard single-reference methods. The completely renormalized (CR) CCSD(T) and CCSD(TQ) methods and their quadratic and excited-state MMCC analogues remove the failing of the standard CCSD, CCSD(T), EOMCCSD and similar methods at larger internuclear separations and for states that normally require a genuine multireference description. All theoretical ideas are illustrated by numerical examples involving bond breaking, excited vibrational states, reactive potential energy surfaces and difficult cases of excited electronic states. The description of the existing and well-established variants of the MMCC theory, such as CR-CCSD(T), is augmented by the discussion of future prospects and potentially useful recent developments, including the extension of the black-box CR-CCSD(T) method to excited states.

In this work we analyze hermitian aspects of methods which can offer improved numerical approximations, simpler computational evaluations or other benefits. There is a route toward approximations to the correlation problem that is neither purely perturbative, nor infinite order as is Coupled Cluster (CC), and this has certain hermitian aspects that are of interest. As perturbation theory (PT) is hermitian, we consider CCPT. The CCPT approach might be offered as an alternative to the infinite order CC approach for the reference state correlation problem itself with some advantages like, potentially, e.g., a more rapid convergence to a satisfactory answer in low order. The CCPT methods up to fifth order with the inclusion of the connected quadruple excitations have been formulated and implemented. We illustrate its results by comparison with several full configuration interaction values. For comparison purposes we also compare our new results with results obtained using the standard CC and MBPT variants like: CCSD, CCSD(T), CCSDT, CCSDTQ, MBPT2, MBPT3, MBPT4, MBPT5 and MBPT6.

Coupled cluster study of the triple bond

A comparison of the renormalized and active-space coupled-cluster methods: Potential energy curves of BH and F 2

Standard multireference (MR) coupled cluster (CC) approaches are based on the effective Hamiltonian formalism and generalized Bloch equation. Their implementation, relying on the valence universal or state universal cluster Ansatz, is very demanding and their practical exploitation is often plagued with intruder state and multiple solution problems. These problems are avoided in the so-called state selective or state specific (SS) MR approaches that concentrate on one state at a time. To preserve as much as possible the flexibility and generality offered by the general MR CC approaches, yet obtaining a reliable and manageable algorithm, we propose a novel SS strategy providing a size-extensive CC formalism, while exploiting the MR model space and the corresponding excited state manifold. This strategy involves three steps: (i) The construction of a variational configuration interaction (CI) wave function within the singly (S) and doubly (D) excited state manifold, (ii) the cluster analysis of this CI wave function providing the information about the higher than pair cluster amplitudes, and (iii) the exploitation of these amplitudes in the so-called externally corrected CCSD procedure. This approach is referred to as the reduced MR (RMR) SS CCSD method and is implemented at the ab initio level and applied to several model systems for which the exact full CI results are available. These include two four electron H4 systems (usually referred to as the H4 and S4 models), an eight electron H8 model and the singlet-triplet separation problem in CH2. It is shown that the RMR CCSD approach produces highly accurate results, is free from intruder state problems, is very general and effective and applicable to both closed and open shell systems.

The new state-selective (SS) multireference (MR) coupled-cluster (CC) method exploiting the single-reference (SR) particle-hole formalism, which we have introduced in our recent paper [P. Piecuch, N. Oliphant, and L. Adamowicz, J. Chem. Phys. 99, 1875 (1993)], has been implemented and the results of the pilot calculations for the minimum basis-set (MBS) model composed of eight hydrogen atoms in various geometrical arrangements are presented. This model enables a continuous transition between degenerate and nondegenerate regimes. Comparison is made with the results of SR CC calculations involving double (CCD), single and double (CCSD), single, double, and triple (CCSDT), and single, double, triple, and quadruple (CCSDTQ) excitations. Our SS CC energies are also compared with the results of the Hilbert space, state-universal (SU) MR CC(S)D calculations, as well as with the MR configuration interaction (CI) results (with and without Davidson-type corrections) and the exact correlation energies obtained using the full CI (FCI) method. Along with the ground-state energies, we also analyze the resulting wave functions by examining some selected cluster components. This analysis enables us to assess the quality of the resulting wave functions. Our SS CC theory truncated at double excitations, which emerges through selection of the most essential clusters appearing in the full SR CCSDTQ formalism [SS CCSD (TQ) method] provides equally good results in nondegenerate and quasidegenerate regions. The difference between the ground-state energy obtained with the SS CCSD(TQ) approach and the FCI energy does not exceed 1.1 mhartree over all the geometries considered. This value compares favorably with the maximum difference of 2.8 mhartree between the SU CCSD energies and the FCI energies obtained for the same range of geometries. The SS CCSD(T) method, emerging from the SR CCSDT theory through selection of the most essential clusters, is less stable, since it neglects very important semi-internal quadruple excitations. Unlike the genuine multideterminantal SU CC formalism, our SS CC approach is not affected by the intruder state problem and its convergence remains satisfactory in nondegenerate and quasidegenerate regimes.

Strongly correlated (SC) systems present significant challenges for classical quantum chemistry methods. Quantum computing, particularly the variational quantum eigensolver (VQE), offers a promising framework to address these challenges by inherently supporting exponentially large configuration spaces. However, its application to SC systems remains limited due to the single-reference nature of the widely used ansatzes such as unitary coupled cluster (UCC). To address this challenge, we propose the generalized valence bond-based unitary block correlated coupled cluster (GVB-UBCCC) method. This novel ansatz incorporates the multiconfigurational nature of generalized valence bond (GVB) and the accuracy of block correlated coupled cluster (BCCC) methods, making it well-suited for SC systems. We have implemented the GVB-UBCCC method with up to two-block correlation (GVB-UBCCC2) and applied it to investigate ground-state energies for several SC systems, including H4, the water dimer, N2H2, and S6, at most described by 24 qubits. Our approach demonstrates that for these systems, GVB-UBCCC2 can achieve more accurate ground-state energies than UCCSD in most cases while requiring only O(N2) quantum gates and parameters, as opposed to the O(N4) scaling of UCCSD. The results highlight the effectiveness and potential advantages of GVB-UBCCC in SC systems.