The state-selective coupled cluster method for quasi-degenerate electronic states

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.

Quantum Flow Algorithms for Simulating Many-Body Systems on Quantum Computers

We conducted quantum simulations of strongly correlated systems using the quantum flow (QFlow) approach, which enables sampling large subspaces of the Hilbert space through coupled variational problems in reduced dimensionality active spaces. Our QFlow algorithms significantly reduce circuit complexity and pave the way for scalable and constant-circuit-depth quantum computing. Our simulations show that QFlow can optimize the collective number of wave function parameters without increasing the required qubits using active spaces having an order of magnitude fewer number of parameters.

A hybrid stochastic configuration interaction-coupled cluster approach for multireference systems

The development of multireference coupled cluster (MRCC) techniques has remained an open area of study in electronic structure theory for decades due to the inherent complexity of expressing a multiconfigurational wavefunction in the fundamentally single-reference coupled cluster framework. The recently developed multireference-coupled cluster Monte Carlo (mrCCMC) technique uses the formal simplicity of the Monte Carlo approach to Hilbert space quantum chemistry to avoid some of the complexities of conventional MRCC, but there is room for improvement in terms of accuracy and, particularly, computational cost. In this paper, we explore the potential of incorporating ideas from conventional MRCC-namely, the treatment of the strongly correlated space in a configuration interaction formalism-to the mrCCMC framework, leading to a series of methods with increasing relaxation of the reference space in the presence of external amplitudes. These techniques offer new balances of stability and cost against accuracy, as well as a means to better explore and better understand the structure of solutions to the mrCCMC equations.

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.

Near-Linear Scaling in DMRG-Based Tailored Coupled Clusters: An Implementation of DLPNO-TCCSD and DLPNO-TCCSD(T)

We present a new implementation of density matrix renormalization group based tailored coupled clusters method (TCCSD), which employs the domain-based local pair natural orbital approach (DLPNO). Compared to the previous local pair natural orbital (LPNO) version of the method, the new implementation is more accurate, offers more favorable scaling, and provides more consistent behavior across the variety of systems. On top of the singles and doubles, we include the perturbative triples correction (T), which is able to retrieve even more dynamic correlation. The methods were tested on three systems: tetramethyleneethane, oxo-Mn(Salen), and iron(II)-porphyrin model. The first two were revisited to assess the performance with respect to LPNO-TCCSD. For oxo-Mn(Salen), we retrieved between 99.8 and 99.9% of the total canonical correlation energy which is an improvement of 0.2% over the LPNO version in less than 63% of the total LPNO runtime. Similar results were obtained for iron(II)-porphyrin. When the perturbative triples correction was employed, irrespective of the active space size or system, the obtained energy differences between two spin states were within the chemical accuracy of 1 kcal/mol using the default DLPNO settings.

Perturbative triples correction to domain-based local pair natural orbital variants of Mukherjee's state specific coupled cluster method

In this article we report an implementation of the perturbative triples correction to Mukherjee's state-specific multireference coupled cluster method based on the domain-based pair natural orbital approach (DLPNO-MkCC). We tested the performance of DLPNO-MkCCSD(T) in calculations involving tetramethyleneethane and isomers of naphthynes. These tests show that more than 97% of triples energy was recovered with respect to the canonical MkCCSD(T) method, which together with the DLPNO-MkCCSD part accounts for about 99.70-99.85% of the total correlation energy. The applicability of the method was demonstrated on calculations of singlet-triplet gaps for several large systems: triangulene, dynemicin A, and a beryllium complex.

Computerized implementation of higher-order electron-correlation methods and their linear-scaling divide-and-conquer extensions

We have implemented a linear-scaling divide-and-conquer (DC)-based higher-order coupled-cluster (CC) and Møller-Plesset perturbation theories (MPPT) as well as their combinations automatically by means of the tensor contraction engine, which is a computerized symbolic algebra system. The DC-based energy expressions of the standard CC and MPPT methods and the CC methods augmented with a perturbation correction were proposed for up to high excitation orders [e.g., CCSDTQ, MP4, and CCSD(2)_TQ ]. The numerical assessment for hydrogen halide chains, polyene chains, and first coordination sphere (C1) model of photoactive yellow protein has revealed that the DC-based correlation methods provide reliable correlation energies with significantly less computational cost than that of the conventional implementations. © 2017 Wiley Periodicals, Inc.

Active-space symmetry-adapted-cluster configuration-interaction and equation-of-motion coupled-cluster methods for high accuracy calculations of potential energy surfaces of radicals

The electron-attached (EA) and ionized (IP) symmetry-adapted-cluster configuration-interaction (SAC-CI) methods and their equation-of-motion coupled-cluster (EOMCC) analogs provide an elegant framework for studying open-shell systems. As shown in this study, these schemes require the presence of higher-order excitations, such as the four-particle-three-hole (4p-3h) or four-hole-three-particle (4h-3p) terms, in the electron attaching or ionizing operator R in order to produce accurate ground- and excited-state potential energy surfaces of radicals along bond breaking coordinates. The full inclusion of the 4p-3h/4h-3p excitations in the EA/IP SAC-CI and EOMCC methods leads to schemes which are far too expensive for calculations involving larger radicals and realistic basis sets. In order to reduce the large costs of such schemes without sacrificing accuracy, the active-space EA/IP EOMCC methodology [J. R. Gour et al., J. Chem. Phys. 123, 134113 (2005)] is extended to the EA/IP SAC-CI approaches with 4p-3h/4h-3p excitations. The resulting methods, which use a physically motivated set of active orbitals to pick out the most important 3p-2h/3h-2p and 4p-3h/4h-3p excitations, represent practical computational approaches for high-accuracy calculations of potential energy surfaces of radicals. To illustrate the potential offered by the active-space EA/IP SAC-CI approaches with up to 4p-3h/4h-3p excitations, the results of benchmark calculations for the potential energy surfaces of the low-lying doublet states of CH and OH are presented and compared with other SAC-CI and EOMCC methods, and full CI results.

Efficient formulation and computer implementation of the active-space electron-attached and ionized equation-of-motion coupled-cluster methods

The efficient, general-purpose implementations of the active-space electron-attached (EA) and ionized (IP) equation-of-motion coupled-cluster (EOMCC) methods including up to 3p-2h and 3h-2p excitations, called EA-EOMCCSDt and IP-EOMCCSDt, respectively, are discussed. The details of the algorithm that enables one to achieve a high degree of code vectorization for the active-space methods and the factorized forms of the EA- and IP-EOMCCSDt equations that maximize the benefits of using active orbitals in the process of selecting the dominant 3p-2h and 3h-2p excitations are presented. The results of benchmark calculations for the low-lying doublet and quartet states of the CH and SH radicals reveal that the active-space EA-EOMCCSDt and IP-EOMCCSDt methods are capable of producing results for the electronic excitations in open-shell systems that match the high accuracy of EA- and IP-EOMCC calculations with a full treatment of 3p-2h and 3h-2p excitations, even when the excited states of interest display a manifestly multideterminantal nature, with the costs that can be on the same order of those characterizing the basic EOMCC singles and doubles approach.

Active-space equation-of-motion coupled-cluster methods for excited states of radicals and other open-shell systems: EA-EOMCCSDt and IP-EOMCCSDt

The previously developed active-space coupled-cluster (CC) and equation-of-motion (EOM) CC methods are extended to radicals and other open-shell systems by combining them with the electron attached (EA) and ionized (IP) EOMCC approaches. As illustrated by the calculations for the CH and OH radicals, the resulting EA-EOMCCSDt and IP-EOMCCSDt theories are capable of providing a highly accurate description of the electronic spectra of radical systems, including excited states displaying a manifestly multideterminantal nature, with the low costs that are not much greater that those characterizing the standard EOMCC singles and doubles method.

Active-space coupled-cluster study of electronic states of Be3

The active-space coupled-cluster (CC) and equation-of-motion (EOM) CC methods with all single and double excitations with triple excitations defined via active orbitals (CCSDt, EOMCCSDt), as implemented with TENSOR CONTRACTION ENGINE, are applied to the challenging Be3 system, which is characterized by a large number of low-lying excited states dominated by two-electron transitions and significant high-order correlation effects in the ground electronic state. It is demonstrated that the CCSDt and EOMCCSDt methods provide an excellent description of complicated electronic quasidegeneracies present in the Be3 cluster. Different strategies for defining triple excitations within the CCSDtEOMCCSDt approach are discussed.

Comparison of low-order multireference many-body perturbation theories

Tests have been made to benchmark and assess the relative accuracies of low-order multireference perturbation theories as compared to coupled cluster (CC) and full configuration interaction (FCI) methods. Test calculations include the ground and some excited states of the Be, H(2), BeH(2), CH(2), and SiH(2) systems. Comparisons with FCI and CC calculations show that in most cases the effective valence shell Hamiltonian (H(v)) method is more accurate than other low-order multireference perturbation theories, although none of the perturbative methods is as accurate as the CC approximations. We also briefly discuss some of the basic differences among the multireference perturbation theories considered in this work.

The XΣg+1, BΔg1, and B′Σg+1 states of C2: A comparison of renormalized coupled-cluster and multireference methods with full configuration interaction benchmarks

Unusual bonding and electronic near degeneracies make the lowest-lying singlet states of the C2 molecule particularly challenging for electronic structure theory. Here we compare two alternative approaches to modeling bond-breaking reactions and excited states: sophisticated multireference configuration interaction and multireference perturbation theory methods, and a more "black box," single-reference approach, the completely renormalized coupled-cluster method. These approximate methods are assessed in light of their ability to reproduce the full configuration interaction potential energy curves for the X1Sigmag+, B1Deltag, and B' 1Sigmag+ states of C2, which are numerically exact solutions of the electronic Schrodinger equation within the space spanned by a 6-31G* basis set. Both the multireference methods and the completely renormalized coupled-cluster approach provide dramatic improvements over the standard single-reference methods. The multireference methods are nearly as reliable for this challenging test case as for simpler reactions which break only single bonds. The completely renormalized coupled-cluster approach has difficulty for large internuclear separations R in this case, but over the wide range of R=1.0-2.0 A, it compares favorably with the more complicated multireference methods.

Automated generation of coupled-cluster diagrams: Implementation in the multireference state-specific coupled-cluster approach with the complete-active-space reference

An algorithm for generation of the spin-orbital diagrammatic representation, the corresponding algebraical formulas, and the computer code of the coupled-cluster (CC) method with an arbitrary level of the electronic excitations has been developed. The method was implemented in the general case as well as for specific application in the state-specific multireference coupled-cluster theory (SSMRCC) based on the concept of a "formal reference state." The algorithm was tested in SSMRCC calculations describing dissociation of a single bond and in calculations describing simultaneous dissociation of two single bonds--the problem requiring up to six-particle excitations in the CC operator.

Combined coupled-cluster and many-body perturbation theories

Various approximations combining coupled-cluster (CC) and many-body perturbation theories have been derived and implemented into the parallel execution programs that take into account the spin, spatial (real Abelian), and permutation symmetries and that are applicable to closed- and open-shell molecules. The implemented models range from the CCSD(T), CCSD[T], CCSD(2)(T), CCSD(2)(TQ), and CCSDT(2)(Q) methods to the completely renormalized (CR) CCSD(T) and CCSD[T] approaches, where CCSD (CCSDT) stands for the CC method with connected single and double (single, double, and triple) cluster operators, and subscripted or parenthesized 2, T, and Q indicate the perturbation order or the excitation ranks of the cluster operators included in the corrections. The derivation and computer implementation have been automated by the algebraic and symbolic manipulation program TENSOR CONTRACTION ENGINE (TCE). The TCE-synthesized subroutines generate the tensors with the highest excitation rank in a blockwise manner so that they need not be stored in their entirety, while enabling the efficient reuse of other precalculated intermediate tensors defined by prioritizing the memory optimization as well as operation minimization. Consequently, the overall storage requirements for the corrections due to connected triple and quadruple cluster operators scale as O(n(4)) and O(n(6)), respectively (n being a measure of the system size). For systems with modest multireference character of their wave functions, we found that the order of accuracy is CCSD<CR-CCSD(T) approximately CCSD(2)(T) approximately CCSD(T)<CCSDT approximately CCSD(2)(TQ)<CCSDT(2)(Q), whereas CR-CCSD(T) is more effective in cases of larger quasidegeneracy. The operation costs of the TCE-generated CCSD(2)(TQ) and CCSDT(2)(Q) codes scale as rather steep O(n(9)), while the TCE-generated CCSD(T), CCSD(2)(T), and CR-CCSD(T) codes are near operation minimum [a noniterative O(n(7))]. The perturbative correction part of the CCSD(T)/cc-pVDZ calculations for azulene exhibited a 45-fold speedup upon a 64-fold increase in the number of processors from 8 to 512.