Coupled-cluster Formalism Research Articles

Ordering Effect of Charge-Charge Repulsion in Doped Antiferromagnetic Lattices: A Coupled Cluster Study.

In quantum chemistry, single-reference Coupled Cluster theory, and its refinements introduced by Bartlett, has become a "gold-standard" predictive method for taking into account electronic correlations in molecules. In this article, we introduce a new formalism based on a Coupled Cluster expansion of the wave function that is suited to describe model periodic systems and apply this methodology to the case of hole-doped antiferromagnetic two-dimensional (2D)-square spin-lattices as a proof of concept. More precisely, we focus our study on 1/5 and 1/7 doping ratios and discuss the possible ordering effect due to large hole-hole repulsion. Starting from one of the equivalent single determinants exhibiting a full spin alternation and the most remote location of the holes as a single reference, the method incorporates some corrections to the traditional Coupled Cluster formalism to take into account the nonadditivity of excitation energies to multiply excited determinants. The amplitudes of the excitations, which are possible on the excited determinants but impossible on the reference, are evaluated perturbatively, while their effect is treated as a dressing in the basic equations. The expansion does not show any sign of divergence of the wave operator. Finally, the probabilities of holes moving toward the first- and second-neighboring sites are reported, which confirms the importance of the hole-hole repulsion and offers a picture of how stripes expand around its central line in the "stripe phases" observed in cuprates.

Benchmarking Ionization Potentials from pCCD Tailored Coupled Cluster Models.

The ionization potential (IP) is an important parameter providing essential insights into the reactivity of chemical systems. IPs are also crucial for designing, optimizing, and understanding the functionality of modern technological devices. We recently showed that limiting the CC ansatz to the seniority-zero sector proves insufficient in predicting reliable and accurate ionization potentials within an IP equation-of-motion coupled-cluster formalism. Specifically, the absence of dynamical correlation in the seniority-zero pair coupled cluster doubles (pCCD) model led to unacceptably significant errors of approximately 1.5 eV. In this work, we aim to explore the impact of dynamical correlation and the choice of the molecular orbital basis (canonical vs localized) in CC-type methods targeting 230 ionized states in 70 molecules, comprising small organic molecules, medium-sized organic acceptors, and nucleobases. We focus on pCCD-based approaches as well as the conventional IP-EOM-CCD and IP-EOM-CCSD. Their performance is compared to the CCSD(T) or CCSDT equivalent and experimental reference data. Our statistical analysis reveals that all investigated frozen-pair coupled cluster methods exhibit similar performance, with differences in errors typically within chemical accuracy (1 kcal/mol or 0.05 eV). Notably, the effect of the molecular orbital basis, such as canonical Hartree-Fock or natural pCCD-optimized orbitals, on the IPs is marginal if dynamical correlation is accounted for. Our study suggests that triple excitations are crucial in achieving chemical accuracy in IPs when modeling electron detachment processes with pCCD-based methods.

Electron density-based protocol to recover the interacting quantum atoms components of intermolecular binding energy.

A new approach for obtaining interacting quantum atoms-defined components of binding energy of intermolecular interactions, which bypasses the use of standard six-dimensional integrals and two-particle reduced density matrix (2-RDM) reconstruction, is proposed. To examine this approach, three datasets calculated within the density functional theory framework using the def2-TZVP basis have been explored. The first two, containing 53 weakly bound bimolecular associates and 13 molecular clusters taken from the crystal, were used in protocol refinement, and the third one containing other 20 bimolecular and three cluster systems served as a validation reference. In addition, to verify the performance of the proposed approach on an exact 2-RDM, calculations within the coupled cluster formalism were performed for part of the first set systems using the cc-pVTZ basis set. The process of optimization of the proposed parametric model is considered, and the role of various energy contributions in the formation of non-covalent interactions is discussed with regard to the obtained trends.

Entanglement coupled cluster theory: Exact spin-adaptation.

We present a novel framework for spin-adapted coupled cluster theory. The approach exploits the entanglement of an open-shell molecule with electrons in a non-interacting bath. Together, the molecule and the bath form a closed-shell system, and electron correlation can be included using the standard spin-adapted closed-shell coupled cluster formalism. A projection operator, which enforces conditions on the electrons in the bath, is used to obtain the desired state of the molecule. This entanglement coupled cluster theory is outlined, and proof-of-concept calculations for doublet states are reported. The approach is further extendable to open-shell systems with other values of the total spin.

Sub-system self-consistency in coupled cluster theory.

In this article, we provide numerical evidence indicating that the single-reference coupled-cluster (CC) energies can be calculated alternatively to their copybook definition. We demonstrate that the CC energy can be reconstructed by diagonalizing the effective Hamiltonians describing correlated sub-systems of the many-body system. In the extreme case, we provide numerical evidence that the CC energy can be reproduced through the diagonalization of the effective Hamiltonian describing sub-system composed of a single electron. These properties of the CC formalism can be exploited to design protocols to define effective interactions in sub-systems used as probes to calculate the energy of the entire system and introduce a new type of self-consistency for approximate CC approaches.

Excited-state response theory within the context of the coupled-cluster formalism

Time-dependent response theories are foundational to the development of algorithms that determine quantum properties of electronic excited states of molecules and periodic systems. They are employed in wave-function, density-functional, and semiempirical methods, and are applied in an incremental order: linear, quadratic, cubic, etc. Linear response theory is known to produce electronic transitions from ground to excited state, and vice versa. In this work, a linear-response approach, within the context of the coupled cluster formalism, is developed to offer transition elements between different excited states (including permanent elements), and related properties. Our formalism, second linear response theory, is consistent with quadratic response theory, and can serve as an alternative to develop and study excited-state theoretical methods, including pathways for algorithmic acceleration. This work also formulates an extension of our theory for general propagations under non-linear external perturbations, where the observables are given by linked expressions which can predict their time-evolution under arbitrary initial states and could serve as a means of constructing general state propagators. A connection with the physics of wavefunction theory is developed as well, in which dynamical cluster operator amplitudes are related to wavefunction linear superposition coefficients.

Exploring alternative approaches to improve the convergence pattern in solving the coupled-cluster equations

The perturbational parameter λ formally appearing in the coupled-cluster (CC) and perturbation theory (PT) frameworks is included explicitly in the Hamiltonian and the CC formalism. Using the Møller–Plesset (MP) partitioning, properties of the resulting λ-dependent amplitude functions are discussed. Truncating the cluster expansion at singles and doubles (CCSD), we use these properties and knowledge of the CCSD amplitudes at values to estimate the CCSD amplitudes without solving the relevant equations at (corresponding to the original, physical problem). This extrapolation scheme is explored, with emphasis on convergence improvement of CC iterations. The approximate amplitudes generated by this procedure are used as the starting point for the CCSD iterations, an alternative to the first order MP (MP1) set. Numerical examples are presented which show that the technique combined with the direct inversion in the iterative subspace (DIIS) method is especially helpful in situations where the standard CCSD iterations converge slowly or not at all. We report cases where starting from MP1, the solution process has an unfavourable convergence pattern even with DIIS applied, while the amplitudes generated by our method provide a much better starting point, overcoming the divergence issues of the original sequence.

When Gold Is Not Enough: Platinum Standard of Quantum Chemistry with N7 Cost.

In this paper, we extend the rank-reduced coupled-cluster formalism to the calculation of non-iterative energy corrections due to quadruple excitations. There are two major components of the proposed formalism. The first is an approximate compression of the quadruple excitation amplitudes using the Tucker format. The second is a modified functional used for the evaluation of the corrections which gives exactly the same results for the exact amplitudes, but is less susceptible to errors resulting from the aforementioned compression. We show, both theoretically and numerically, that the computational cost of the proposed method scales as the seventh power of the system size. Using reference results for a set of small molecules, the method is calibrated to deliver relative accuracy of a few percent in energy corrections. To illustrate the potential of the theory, we calculate the isomerization energy of ortho/meta benzyne (C6H4) and the barrier height for the Cope rearrangement in bullvalene (C10H10). The method retains a near-black-box nature of the conventional coupled-cluster formalism and depends on only one additional parameter that controls the accuracy.

Angular-momentum projection in coupled-cluster theory: Structure of Mg34

Single-reference coupled-cluster theory is an accurate and affordable computational method for the nuclear many-body problem. For open-shell nuclei, the reference state typically breaks rotational invariance and angular momentum must be restored as a good quantum number. We perform angular-momentum projection after variation and employ the disentangled coupled-cluster formalism and a Hermitian approach. We compare our results with benchmarks for $^8$Be and $^{20}$Ne using a two-nucleon interaction from chiral effective field theory and for $pf$-shell nuclei within the traditional shell model. We compute the rotational band in the exotic nucleus $^{34}$Mg and find agreement with data.

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

The recently introduced coupled cluster (CC) downfolding techniques for reducing the dimensionality of quantum many-body problems recast the CC formalism in the form of the renormalization procedure allowing, for the construction of effective (or downfolded) Hamiltonians in small-dimensionality sub-space, usually identified with the so-called active space, of the entire Hilbert space. The resulting downfolded Hamiltonians integrate out the external (out-of-active-space) Fermionic degrees of freedom from the internal (in-the-active-space) parameters of the wave function, which can be determined as components of the eigenvectors of the downfolded Hamiltonians in the active space. This paper will discuss the extension of non-Hermitian (associated with standard CC formulations) and Hermitian (associated with the unitary CC approaches) downfolding formulations to composite quantum systems commonly encountered in materials science and chemistry. The non-Hermitian formulation can provide a platform for developing local CC approaches, while the Hermitian one can serve as an ideal foundation for developing various quantum computing applications based on the limited quantum resources. We also discuss the algorithm for extracting the semi-analytical form of the inter-electron interactions in the active spaces.

A new intermediate Hamiltonian Fock-space coupled-cluster formalism for the three-valence sector

The single-reference coupled-cluster (CC) method is one of the most efficient approaches for describing electron correlation effects in atoms and molecules but its successful application is limited to states dominated by a single Slater determinant. In cases where several determinants dominate the wave function expansion, the multi-reference (MR) CC formulations are required. They usually use the effective Hamiltonian formalism which allows dealing with two types of electron correlation appearing in the MR cases: nondynamic and dynamic. There are two MR-CC formulations of this type: the Hilbert space CC method and the Fock-space (FS) one. In the paper, a version of the FS-CC method designated for the description of systems with three-valence electrons is being considered. The idea of the intermediate Hamiltonian (IH) reformulation of the standard FS-CC method, which simplifies the calculations and avoids the intruder state problem, is discussed in this context. A new IH-FS-CC approach is presented which takes advantage of the redefinition of the normal ordering of the FS cluster expansion introduced by Lindgren. The redefinition significantly simplifies the IH-FS-CC formalism and lowers the computational cost of the calculations. The advantages of the new IH formulation are discussed in detail for the case of the FS-CC method in which the lowest-order cluster operators are included in all sectors of the Fock-space, including the three-valence one.

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.

Inner-shell photoabsorption and photoionisation cross-sections of valence excited states from asymmetric-Lanczos equation-of-motion coupled cluster singles and doubles theory

The asymmetric-Lanczos-based equation-of-motion coupled cluster formalism to compute photoabsorption and photoionisation cross-sections of valence excited states [B.N.C. Tenorio, M.A.C. Nascimento, A.B. Rocha, and S. Coriani, J. Chem. Phys. 151 (18), 184106 (2019). doi:10.1063/1.5125125] has been adapted to enable the calculation of photoabsorption and photoionisation spectral signatures from inner-shell electrons of such states. Since excited-state properties depend on both the electronic character of the molecular wavefunction and on the nuclear dynamics, we computed the photoionisation spectra using both ground and excited state optimised geometries. The total cross-section profiles were generated for the first two electronically excited states of water, ammonia, ethylene, and uracil by two different methodologies: an analytic continuation procedure based on the Padé approximants and by the Stieltjes imaging procedure. A comparison with literature results, whenever available, is presented. Remarkable differences were observed between the results of the core-ionisation cross-sections of the valence excited states yielded by the two quadrature approaches, at variance from previous studies on the valence photoionisation cross-sections of ground states and of valence excited states. Their origin remains unclear.

Dimensionality reduction of the many-body problem using coupled-cluster subsystem flow equations: Classical and quantum computing perspective

We discuss reduced-scaling strategies employing the recently introduced subsystem embedding subalgebra (SES) coupled-cluster (CC) formalism to describe quantum many-body systems. These strategies utilize properties of the SES CC formulations where the equations describing certain classes of subsystems can be integrated into computational flows composed of coupled eigenvalue problems of reduced dimensionality. Additionally, these flows can be determined at the level of the CC ansatz by the inclusion of selected classes of cluster amplitudes, which define the wave-function ``memory'' of possible partitioning of the many-body system into constituent subsystems. One of the possible ways of solving these coupled problems is through implementing procedures where the information is passed between the subsystems in a self-consistent manner. As a special case we consider local flow formulations where the local character of correlation effects can be closely related to the properties of subsystem embedding subalgebras employing a localized molecular basis. We also generalize flow equations to the time domain and to downfolding methods utilizing a double-exponential unitary CC ansatz, where the reduced dimensionality of constituent subproblems offers a possibility of efficient utilization of limited quantum resources in modeling realistic systems.

Sub-system quantum dynamics using coupled cluster downfolding techniques.

In this paper, we discuss extending the sub-system embedding sub-algebra coupled cluster formalism and the double unitary coupled cluster (DUCC) ansatz to the time domain. An important part of the analysis is associated with proving the exactness of the DUCC ansatz based on the general many-body form of anti-Hermitian cluster operators defining external and internal excitations. Using these formalisms, it is possible to calculate the energy of the entire system as an eigenvalue of downfolded/effective Hamiltonian in the active space, which is identifiable with the sub-system of the composite system. It can also be shown that downfolded Hamiltonians integrate out Fermionic degrees of freedom that do not correspond to the physics encapsulated by the active space. In this paper, we extend these results to the time-dependent Schrödinger equation, showing that a similar construct is possible to partition a system into a sub-system that varies slowly in time and a remaining sub-system that corresponds to fast oscillations. This time-dependent formalism allows coupled cluster quantum dynamics to be extended to larger systems and for the formulation of novel quantum algorithms based on the quantum Lanczos approach, which has recently been considered in the literature.

Potential Energy Surface for the CH4-H2 van der Waals Interaction.

Methane is an ubiquitous molecule, present as a minor component in many environments, including the Earth and planet atmospheres. Its van der Waals interaction with the main gases is an important ingredient for the understanding of radiative properties for those atmospheres. We present here the first precise determination of the interaction between CH4 and H2. We compute the interaction in an ab initio coupled cluster formalism, with extended atomic bases. We compare a pure CCSD(T) approach to an explicitly correlated CCSD(T)-F12a formalism. The full geometry of scattering two rigid molecules is used, resulting in a potential energy surface depending on 5 degrees of freedom. The long-range part of the ab initio computation is compared with an analytic multipolar expansion. The potential is fit onto a suitable formalism for subsequent scattering dynamics. The potential well is computed at -92.92 cm-1, an intermediate value if compared with other methane-diatomic molecule interactions.

Relativistic Real-Time Time-Dependent Equation-of-Motion Coupled-Cluster

We present a relativistic time-dependent equation-of-motion coupled-cluster with single and double excitations (TD-EOM-CCSD) formalism. Unlike other explicitly time-dependent quantum chemical methods, the present approach considers the time correlation function of the dipole operator, as opposed to the expectation value of the time-dependent dipole moment. We include both scalar relativistic effects and spin-orbit coupling variationally in this scheme via the use of the exact two-component (X2C) wave function as the reference that enters the coupled-cluster formalism. In order to evaluate the accuracy of X2C-TD-EOM-CCSD, we compare zero-field splitting in atomic absorption spectra of open-shell systems (Na, K, Mg+, and Ca+) with values obtained from experiment. In closed-shell species (Na+, K+, Mg2+, and Ca2+), we observe singlet-triplet mixing in the X2C-TD-EOM-CC calculations, which results from the use of the X2C reference. The effects of the X2C reference are evaluated by comparing spectra derived from X2C-TD-EOM-CC calculations to those from TD-EOM-CC calculations using a complex generalized Hartree-Fock ([Formula: see text]-GHF) reference.

Real-time density-matrix coupled-cluster approach for closed and open systems at finite temperature.

We extend the coupled-cluster method to correlated quantum dynamics of both closed and open systems at finite temperatures using the thermofield formalism. The approach expresses the time-dependent density matrix in an exponential ansatz and describes time-evolution along the Keldysh path contour. A distinct advantage of the approach is exact trace-preservation as a function of time, ensuring conservation of probability and particle number. Furthermore, the method avoids the computation of correlated bra-states, simplifying the computational implementation. We develop the method in a thermal quasiparticle representation, which allows seamless connection to the projection method and diagrammatic techniques of the traditional coupled-cluster formalism. For comparison, we also apply the thermofield framework to the density-matrix renormalization-group method to obtain reference results for closed and open systems at finite temperature. We test the singles and doubles approximation to the density-matrix coupled-cluster method on the correlated electronic dynamics of the single-impurity Anderson model, demonstrating that the new method successfully captures the correlated dynamics of both closed systems at finite temperature and driven-dissipative open systems. This encouraging performance motivates future applications to nonequilibrium quantum many-body dynamics in realistic systems.

Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms.

In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC (DUCC) formalism, the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects-used to define the effective Hamiltonian-from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of CAS representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing. Given growing interest in quantum computing, we provide energies for H2 and Be systems obtained with the quantum phase estimator algorithm available in the Quantum Development Kit for the approximate DUCC Hamiltonians.

Linear and quadratic internally contracted multireference coupled-cluster approximations.

Linear and quadratic approximations to the internally contracted multireference coupled-cluster (icMRCC) method are implemented and analyzed by using the linked and unlinked coupled-cluster formalisms. This includes methods based on perturbation theory as well as the coupled-electron pair approximation, CEPA(0). The similarities and differences between all the approximations serve to highlight and provoke discussion about methodological peculiarities of the icMRCC ansatz. When calculating potential energy curves (PECs), discontinuities are observed for the linear icMRCC energies. Using a diagrammatic representation, the terms that cause but also reduce these discontinuities are identified. For benchmarking test cases such as calculating PECs, singlet-triplet splittings, and barrier heights, the multireference CEPA(0) approximation performs well; however, it suffers from a lack of size consistency and so cannot represent a step forward to the goal of developing a computationally cheap and accurate icMRCC method.