Merging Active-Space and Renormalized Coupled-Cluster Methods via the CC(P;Q) Formalism, with Benchmark Calculations for Singlet–Triplet Gaps in Biradical Systems

Complete Active Space Iterative Coupled Cluster Theory

In this work, we investigate the possibility of improving multireference-driven coupled cluster (CC) approaches with an algorithm that iteratively combines complete active space (CAS) calculations with tailored CC and externally corrected CC. This is accomplished by establishing a feedback loop between the CC and CAS parts of a calculation through a similarity transformation of the Hamiltonian with those CC amplitudes that are not encompassed by the active space. We denote this approach as the complete active space iterative coupled cluster (CASiCC) ansatz. We investigate its efficiency and accuracy in the singles and doubles approximation by studying the prototypical molecules H4, H8, H2O, and N2. Our results demonstrate that CASiCC systematically improves on the single-reference CCSD and the externally corrected CCSD methods across entire potential energy curves while retaining modest computational costs. However, the tailored coupled cluster method shows superior performance in the strong correlation regime, suggesting that its accuracy is based on error compensation. We find that the iterative versions of externally corrected and tailored coupled cluster methods converge to the same results.

Renormalized Internally Contracted Multireference Coupled Cluster with Perturbative Triples

In this work, we combine the many-body formulation of the internally contracted multireference coupled cluster (ic-MRCC) method with Evangelista's multireference formulation of the driven similarity renormalization group (DSRG). The DSRG method can be viewed as a unitary multireference coupled cluster theory, which renormalizes the amplitudes based on a flow equation approach to eliminate numerical instabilities. We extend this approach by demonstrating that the unitary flow equation approach can be adapted for nonunitary transformations, rationalizing the renormalization of ic-MRCC amplitudes. We denote the new approach, the renormalized ic-MRCC (ric-MRCC) method. To achieve high accuracy with a reasonable computational cost, we introduce a new approximation to the Baker-Campbell-Hausdorff expansion. We fully consider the linear commutator while approximating the quadratic commutator, for which we neglect specific contractions involving amplitudes with active indices. Moreover, we introduce approximate perturbative triples to obtain the ric-MRCCSD[T] method. We demonstrate the accuracy of our approaches in comparison to advanced multireference methods for the potential energy curves of H8, F2, H2O, N2, and Cr2. Additionally, we show that ric-MRCCSD and ric-MRCSSD[T] match the accuracy of CCSD(T) for evaluating spectroscopic constants and of full configuration interaction energies for a set of small molecules.

Striking the right balance of encoding electron correlation in the Hamiltonian and the wavefunction ansatz

Multi-configurational electronic structure theory delivers the most versatile approximations to many-electron wavefunctions, flexible enough to deal with all sorts of transformations, ranging from electronic excitations, to open-shell molecules and chemical reactions. Multi-configurational models are therefore essential to establish universally applicable, predictive ab initio methods for chemistry. Here, we present a discussion of explicit correlation approaches which address the nagging problem of dealing with static and dynamic electron correlation in multi-configurational active-space approaches. We review the latest developments and then point to their key obstacles. Our discussion is supported by new data obtained with tensor network methods. We argue in favor of simple electron-only correlator expressions that may allow one to define transcorrelated models in which the correlator does not bear a dependence on molecular structure.

Relativistic Coupled Cluster with Completely Renormalized and Perturbative Triples Corrections

We have implemented noniterative triples corrections to the energy from coupled-cluster with single and double excitations (CCSD) within the 1-electron exact two-component (1eX2C) relativistic framework. The effectiveness of both the CCSD(T) and the completely renormalized (CR) CC(2,3) approaches are demonstrated by performing all-electron computations of the potential energy curves and spectroscopic constants of copper, silver, and gold dimers in their ground electronic states. Spin-orbit coupling effects captured via the 1eX2C framework are shown to be crucial for recovering the correct shape of the potential energy curves, and the correlation effects due to triples in these systems change the dissociation energies by about 0.1-0.2 eV or about 4-7%. We also demonstrate that relativistic effects and basis set size and contraction scheme are significantly more important in Au2 than in Ag2 or Cu2.

Machine Learning the Quantum Mechanical Wave Function

Strongly correlated systems have been challenging to computational chemists for a long time. To solve these systems, multireference methods have been developed over the years. Recently, with the fast development of machine learning and artificial intelligence methods, these methods have also influenced the quest for optimal wave function ansatz. Machine learning approaches have been used in many different flavors. From this perspective, we will discuss the different milestones achieved in the use of machine learning for solving the quantum many body problem.

Converging high-level coupled-cluster energetics via adaptive selection of excitation manifolds driven by moment expansions

A novel approach to rapidly converging high-level coupled-cluster (CC) energetics in an automated fashion is proposed. The key idea is an adaptive selection of excitation manifolds defining higher--than--two-body components of the cluster operator inspired by CC(P;Q) moment expansions. The usefulness of the resulting methodology is illustrated by molecular examples where the goal is to recover the electronic energies obtained using the CC method with a full treatment of singly, doubly, and triply excited clusters (CCSDT) when the noniterative triples corrections to CCSD fail.

Unitary Coupled Cluster: Seizing the Quantum Moment

Shallow, CNOT-efficient quantum circuits are crucial for performing accurate computational chemistry simulations on current noisy quantum hardware. Here, we explore the usefulness of noniterative energy corrections, based on the method of moments of coupled-cluster theory, for accelerating convergence toward full configuration interaction. Our preliminary numerical results relying on iteratively constructed ansätze suggest that chemically accurate energies can be obtained with substantially more compact circuits, implying enhanced resilience to gate and decoherence noise.

Explicitly Correlated Electronic Structure Calculations with Transcorrelated Matrix Product Operators

In this work, we present the first implementation of the transcorrelated electronic Hamiltonian in an optimization procedure for matrix product states by the density matrix renormalization group (DMRG) algorithm. In the transcorrelation ansatz, the electronic Hamiltonian is similarity-transformed with a Jastrow factor to describe the cusp in the wave function at electron-electron coalescence. As a result, the wave function is easier to approximate accurately with the conventional expansion in terms of one-particle basis functions and Slater determinants. The transcorrelated Hamiltonian in first quantization comprises up to three-body interactions, which we deal with in the standard way by applying robust density fitting to two- and three-body integrals entering the second-quantized representation of this Hamiltonian. The lack of hermiticity of the transcorrelated Hamiltonian is taken care of along the lines of the first work on transcorrelated DMRG [ J. Chem. Phys. 2020, 153, 164115] by encoding it as a matrix product operator and optimizing the corresponding ground state wave function with imaginary-time time-dependent DMRG. We demonstrate our quantum chemical transcorrelated DMRG approach at the example of several atoms and first-row diatomic molecules. We show that transcorrelation improves the convergence rate to the complete basis set limit in comparison to conventional DMRG. Moreover, we study extensions of our approach that aim at reducing the cost of handling the matrix product operator representation of the transcorrelated Hamiltonian.

A hybrid coupled cluster-machine learning algorithm: Development of various regression models and benchmark applications

The iterative solution of the coupled cluster equations exhibits a synergistic relationship among the various cluster amplitudes. The iteration scheme is analyzed as a multivariate discrete time propagation of nonlinearly coupled equations, which is dictated by only a few principal cluster amplitudes. These principal amplitudes usually correspond to only a few valence excitations, whereas all other cluster amplitudes are enslaved and behave as auxiliary variables [Agarawal et al., J. Chem. Phys. 154, 044110 (2021)]. We develop a coupled cluster-machine learning hybrid scheme where various supervised machine learning strategies are introduced to establish the interdependence between the principal and auxiliary amplitudes on-the-fly. While the coupled cluster equations are solved only to determine the principal amplitudes, the auxiliary amplitudes, on the other hand, are determined via regression as unique functionals of the principal amplitudes. This leads to significant reduction in the number of independent degrees of freedom during the iterative optimization, which saves significant computation time. A few different regression techniques have been developed, which have their own advantages and disadvantages. The scheme has been applied to several molecules in their equilibrium and stretched geometries, and our scheme, with all the regression models, shows a significant reduction in computation time over the canonical coupled cluster calculations without unduly sacrificing the accuracy.

Assessing the Accuracy of Tailored Coupled Cluster Methods Corrected by Electronic Wave Functions of Polynomial Cost

Tailored coupled cluster theory represents a computationally inexpensive way to describe static and dynamical electron correlation effects. In this work, we scrutinize the performance of various coupled cluster methods tailored by electronic wave functions of polynomial cost. Specifically, we focus on frozen-pair coupled cluster (fpCC) methods, which are tailored by pair-coupled cluster doubles (pCCD), and coupled cluster theory tailored by matrix product state wave functions optimized by the density matrix renormalization group (DMRG) algorithm. As test system, we selected a set of various small- and medium-sized molecules containing diatomics (N₂, F₂, C₂, CN⁺, CO, BN, BO⁺, and Cr₂) and molecules (ammonia, ethylene, cyclobutadiene, benzene, hydrogen chains, rings, and cuboids) for which the conventional single-reference coupled cluster singles and doubles (CCSD) method is not able to produce accurate results for spectroscopic constants, potential energy surfaces, and barrier heights. Most importantly, DMRG-tailored and pCCD-tailored approaches yield similar errors in spectroscopic constants and potential energy surfaces compared to accurate theoretical and/or experimental reference data. Although fpCC methods provide a reliable description for the dissociation pathway of molecules featuring single and quadruple bonds, they fail in the description of triple or hextuple bond-breaking processes or avoided crossing regions.

Near-Exact CCSDT Energetics from Rank-Reduced Formalism Supplemented by Non-iterative Corrections

We introduce a non-iterative energy correction, added on top of the rank-reduced coupled-cluster method with single, double, and triple substitutions, that accounts for excitations excluded from the parent triple excitation subspace. The formula for the correction is derived by employing the coupled-cluster Lagrangian formalism, with an additional assumption that the parent excitation subspace is closed under the action of the Fock operator. Owing to the rank-reduced form of the triple excitation amplitudes tensor, the computational cost of evaluating the correction scales as N⁷, where N is the system size. The accuracy and computational efficiency of the proposed method is assessed for both total and relative correlation energies. We show that the non-iterative correction can fulfill two separate roles. If the accuracy level of a fraction of kJ/mol is sufficient for a given system, the correction significantly reduces the dimension of the parent triple excitation subspace needed in the iterative part of the calculations. Simultaneously, it enables reproducing the exact CCSDT results to an accuracy level below 0.1 kJ/mol, with a larger, yet still reasonable, dimension of the parent excitation subspace. This typically can be achieved at a computational cost only several times larger than required for the CCSD(T) method. The proposed method retains the black-box features of the single-reference coupled-cluster theory; the dimension of the parent excitation subspace remains the only additional parameter that has to be specified.

High-level coupled-cluster energetics by merging moment expansions with selected configuration interaction

Inspired by our earlier semi-stochastic work aimed at converging high-level coupled-cluster (CC) energetics [J. E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017) and J. E. Deustua, J. Shen, and P. Piecuch, J. Chem. Phys. 154, 124103 (2021)], we propose a novel form of the CC(P; Q) theory in which the stochastic Quantum Monte Carlo propagations, used to identify dominant higher-than-doubly excited determinants, are replaced by the selected configuration interaction (CI) approach using the perturbative selection made iteratively (CIPSI) algorithm. The advantages of the resulting CIPSI-driven CC(P; Q) methodology are illustrated by a few molecular examples, including the dissociation of F₂ and the automerization of cyclobutadiene, where we recover the electronic energies corresponding to the CC calculations with a full treatment of singles, doubles, and triples based on the information extracted from compact CI wave functions originating from relatively inexpensive Hamiltonian diagonalizations.

An approximate coupled cluster theory via nonlinear dynamics and synergetics: The adiabatic decoupling conditions

The coupled cluster iteration scheme is analyzed as a multivariate discrete time map using nonlinear dynamics and synergetics. The nonlinearly coupled set of equations to determine the cluster amplitudes are driven by a fraction of the entire set of cluster amplitudes. These driver amplitudes enslave all other amplitudes through a synergistic inter-relationship, where the latter class of amplitudes behave as the auxiliary variables. The driver and the auxiliary variables exhibit vastly different time scales of relaxation during the iteration process to reach the fixed points. The fast varying auxiliary amplitudes are small in magnitude, while the driver amplitudes are large, and they have a much longer time scale of relaxation. Exploiting their difference in relaxation time scale, we employ an adiabatic decoupling approximation, where each of the fast relaxing auxiliary modes is expressed as a unique function of the principal amplitudes. This results in a tremendous reduction in the independent degrees of freedom. On the other hand, only the driver amplitudes are determined accurately via exact coupled cluster equations. We will demonstrate that the iteration scheme has an order of magnitude reduction in computational scaling than the conventional scheme. With a few pilot numerical examples, we would demonstrate that this scheme can achieve very high accuracy with significant savings in computational time.

High-level coupled-cluster energetics by Monte Carlo sampling and moment expansions: Further details and comparisons

We recently proposed a novel approach to converging electronic energies equivalent to high-level coupled-cluster (CC) computations by combining the deterministic CC(P;Q) formalism with the stochastic configuration interaction (CI) and CC Quantum Monte Carlo (QMC) propagations. This article extends our initial study [J. E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)], which focused on recovering the energies obtained with the CC method with singles, doubles, and triples (CCSDT) using the information extracted from full CI QMC and CCSDT-MC, to the CIQMC approaches truncated at triples and quadruples. It also reports our first semi-stochastic CC(P;Q) calculations aimed at converging the energies that correspond to the CC method with singles, doubles, triples, and quadruples (CCSDTQ). The ability of the semi-stochastic CC(P;Q) formalism to recover the CCSDT and CCSDTQ energies, even when electronic quasi-degeneracies and triply and quadruply excited clusters become substantial, is illustrated by a few numerical examples, including the F-F bond breaking in F₂, the automerization of cyclobutadiene, and the double dissociation of the water molecule.

Single-Root Multireference Brillouin-Wigner Perturbative Approach to Excitation Energies

The state-specific Brillouin-Wigner multireference perturbation theory [which employs Jeziorski-Monkhorst parametrization of the wave function] using improved virtual orbitals, denoted as IVO-BWMRPT, is applied to calculate excitation energies (EEs) for methylene, ethylene, trimethylenemethane, and benzyne systems exhibiting various degrees of diradical character. In IVO-BWMRPT, all of the parameters appearing in the wave function ansatz are optimized for a specific electronic state. For these systems, the IVO-BWMRPT method provides EEs that are in close agreement with the benchmark results and experiments, where available, indicating that the method does not introduce imbalance in the target-specific treatment of closed- and open-shell states involved. The good performance of the present methodology is primarily related to structural compactness of the formalism. Overall, present findings are encouraging for both further development of the approach and chemical applications on the energy differences of strongly correlated systems.

Tailored coupled cluster theory in varying correlation regimes

The tailored coupled cluster (TCC) approach is a promising ansatz that preserves the simplicity of single-reference coupled cluster theory while incorporating a multi-reference wave function through amplitudes obtained from a preceding multi-configurational calculation. Here, we present a detailed analysis of the TCC wave function based on model systems, which require an accurate description of both static and dynamic correlation. We investigate the reliability of the TCC approach with respect to the exact wave function. In addition to the error in the electronic energy and standard coupled cluster diagnostics, we exploit the overlap of TCC and full configuration interaction wave functions as a quality measure. We critically review issues, such as the required size of the active space, size-consistency, symmetry breaking in the wave function, and the dependence of TCC on the reference wave function. We observe that possible errors caused by symmetry breaking can be mitigated by employing the determinant with the largest weight in the active space as reference for the TCC calculation. We find the TCC model to be promising in calculations with active orbital spaces which include all orbitals with a large single-orbital entropy, even if the active spaces become very large and then may require modern active-space approaches that are not restricted to comparatively small numbers of orbitals. Furthermore, utilizing large active spaces can improve on the TCC wave function approximation and reduce the size-consistency error because the presence of highly excited determinants affects the accuracy of the coefficients of low-excited determinants in the active space.

Recent developments in the general atomic and molecular electronic structure system

A discussion of many of the recently implemented features of GAMESS (General Atomic and Molecular Electronic Structure System) and LibCChem (the C++ CPU/GPU library associated with GAMESS) is presented. These features include fragmentation methods such as the fragment molecular orbital, effective fragment potential and effective fragment molecular orbital methods, hybrid MPI/OpenMP approaches to Hartree-Fock, and resolution of the identity second order perturbation theory. Many new coupled cluster theory methods have been implemented in GAMESS, as have multiple levels of density functional/tight binding theory. The role of accelerators, especially graphical processing units, is discussed in the context of the new features of LibCChem, as it is the associated problem of power consumption as the power of computers increases dramatically. The process by which a complex program suite such as GAMESS is maintained and developed is considered. Future developments are briefly summarized.

Reaction Profiles and Kinetics for Radical-Radical Hydrogen Abstraction via Multireference Coupled Cluster Theory

Radical-radical abstractions in hydrocarbon oxidation chemistry are disproportionation reactions that are generally exothermic with little or no barrier yet are underappreciated and poorly studied. Such challenging multireference electronic structure problems are tackled here using the recently developed state-specific multireference coupled cluster methods Mk-MRCCSD and Mk-MRCCSD(T), as well as the companion perturbation theory Mk-MRPT2 and the established MRCISD, MRCISD+Q, and CASPT2 approaches. Reaction paths are investigated for five prototypes involving radical-radical hydrogen abstraction: H + BeH → H₂+ Be, H + NH₂ → H₂ + NH, CH₃ + C₂H₅ → CH₄ + C₂H₄, H + C₂H₅ → H₂ + C₂H₄, and H + HCO → H₂ + CO. Full configuration interaction (FCI) benchmark computations for the H + BeH, H + NH₂, and H + HCO reactions prove that Mk-MRCCSD(T) provides superior accuracy for the interaction energies in the entrance channel, with mean absolute errors less than 0.3 kcal mol^-1 and percentage deviations less than 10% over the fragment separations of relevance to kinetics. To facilitate combustion studies, energetics for the CH₃ + C₂H₅, H + C₂H₅, and H + HCO reactions were computed at each level of theory with correlation-consistent basis sets (cc-pVXZ, X = T, Q, 5) and extrapolated to the complete basis set (CBS) limit. These CBS energies were coupled with CASPT2 projected vibrational frequencies along a minimum energy path to obtain rate constants for these three reactions. The rigorous Mk-MRCCSD(T)/CBS results demonstrate unequivocally that these three reactions proceed with no barrier in the entrance channel, contrary to some earlier predictions. Mk-MRCCSD(T) also reveals that the economical CASPT2 method performs well for large interfragment separations but may deteriorate substantially at shorter distances.

A zeroth-order active-space frozen-orbital embedding scheme for multireference calculations

Multireference computations of large-scale chemical systems are typically limited by the computational cost of quantum chemistry methods. In this work, we develop a zeroth-order active space embedding theory [ASET(0)], a simple and automatic approach for embedding any multireference dynamical correlation method based on a frozen-orbital treatment of the environment. ASET(0) is combined with the second-order multireference driven similarity renormalization group and tested on several benchmark problems, including the excitation energy of 1-octene and bond-breaking in ethane and pentyldiazene. Finally, we apply ASET(0) to study the singlet-triplet gap of p-benzyne and 9,10-anthracyne diradicals adsorbed on a NaCl surface. Our results show that despite its simplicity, ASET(0) is a powerful and sufficiently accurate embedding scheme applicable when the coupling between the fragment and the environment is in the weak to medium regime.

Multireference Theories of Electron Correlation Based on the Driven Similarity Renormalization Group

The driven similarity renormalization group (DSRG) provides an alternative way to address the intruder state problem in quantum chemistry. In this review, we discuss recent developments of multireference methods based on the DSRG. We provide a pedagogical introduction to the DSRG and its various extensions and discuss its formal properties in great detail. In addition, we report several illustrative applications of the DSRG to molecular systems.

Novel strategy to implement active-space coupled-cluster methods

A new approach is presented for the efficient implementation of coupled-cluster (CC) methods including higher excitations based on a molecular orbital space partitioned into active and inactive orbitals. In the new framework, the string representation of amplitudes and intermediates is used as long as it is beneficial, but the contractions are evaluated as matrix products. Using a new diagrammatic technique, the CC equations are represented in a compact form due to the string notations we introduced. As an application of these ideas, a new automated implementation of the single-reference-based multi-reference CC equations is presented for arbitrary excitation levels. The new program can be considered as an improvement over the previous implementations in many respects; e.g., diagram contributions are evaluated by efficient vectorized subroutines. Timings for test calculations for various complete active-space problems are presented. As an application of the new code, the weak interactions in the Be dimer were studied.

Converging High-Level Coupled-Cluster Energetics by Monte Carlo Sampling and Moment Expansions

We propose a new approach to the determination of accurate electronic energies that are equivalent to the results of high-level coupled-cluster (CC) calculations. The approach is based on merging the CC(P;Q) formalism, which corrects energies obtained with an arbitrary truncation in the cluster operator, with the stochastic configuration interaction and CC ideas. The advantages of the proposed methodology are illustrated by molecular examples, where the goal is to recover the energetics obtained in the CC calculations with a full treatment of singly, doubly, and triply excited clusters.