Showing papers on "Coupled cluster published in 2022"

A quantum computing view on unitary coupled cluster theory

Abstract: We present a review of the Unitary Coupled Cluster (UCC) ansatz and related ansätze which are used to variationally solve the electronic structure problem on quantum computers. A brief history of coupled cluster (CC) methods is provided, followed by a broad discussion of the formulation of CC theory. This includes touching on the merits and difficulties of the method and several variants, UCC among them, in the classical context, to motivate their applications on quantum computers. In the core of the text, the UCC ansatz and its implementation on a quantum computer are discussed at length, in addition to a discussion on several derived and related ansätze specific to quantum computing. The review concludes with a unified perspective on the discussed ansätze, attempting to bring them under a common framework, as well as with a reflection upon open problems within the field.

All-Electron BSE@GW Method for K-Edge Core Electron Excitation Energies.

Abstract: We present an accurate computational approach to calculate absolute K-edge core electron excitation energies as measured by X-ray absorption spectroscopy. Our approach employs an all-electron Bethe-Salpeter equation (BSE) formalism based on GW quasiparticle energies (BSE@GW) using numeric atom-centered orbitals (NAOs). The BSE@GW method has become an increasingly popular method for the computation of neutral valence excitation energies of molecules. However, it was so far not applied to molecular K-edge excitation energies. We discuss the influence of different numerical approximations on the BSE@GW calculation and employ in our final setup (i) exact numeric algorithms for the frequency integration of the GW self-energy, (ii) G0W0 and BSE starting points with ∼50% of exact exchange, (iii) the Tamm-Dancoff approximation and (iv) relativistic corrections. We study the basis set dependence and convergence with common Gaussian-type orbital and NAO basis sets. We identify the importance of additional spatially confined basis functions as well as of diffuse augmenting basis functions. The accuracy of our BSE@GW method is assessed for a benchmark set of small organic molecules, previously used for benchmarking the equation-of-motion coupled cluster method [Peng et al., J. Chem. Theory Comput., 2015, 11, 4146], as well as the medium-sized dibenzothiophene (DBT) molecule. Our BSE@GW results for absolute excitation energies are in excellent agreement with the experiment, with a mean average error of only 0.63 eV for the benchmark set and with errors <1 eV for the DBT molecule.

Quantum Simulation of Molecular Electronic States with a Transcorrelated Hamiltonian: Higher Accuracy with Fewer Qubits.

Abstract: Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.

Fast Predictions of Reaction Barrier Heights: Toward Coupled-Cluster Accuracy.

Abstract: Quantitative estimates of reaction barriers are essential for developing kinetic mechanisms and predicting reaction outcomes. However, the lack of experimental data and the steep scaling of accurate quantum calculations often hinder the ability to obtain reliable kinetic values. Here, we train a directed message passing neural network on nearly 24,000 diverse gas-phase reactions calculated at CCSD(T)-F12a/cc-pVDZ-F12//ωB97X-D3/def2-TZVP. Our model uses 75% fewer parameters than previous studies, an improved reaction representation, and proper data splits to accurately estimate performance on unseen reactions. Using information from only the reactant and product, our model quickly predicts barrier heights with a testing MAE of 2.6 kcal mol-1 relative to the coupled-cluster data, making it more accurate than a good density functional theory calculation. Furthermore, our results show that future modeling efforts to estimate reaction properties would significantly benefit from fine-tuning calibration using a transfer learning technique. We anticipate this model will accelerate and improve kinetic predictions for small molecule chemistry.

The MOBH35 Metal–Organic Barrier Heights Reconsidered: Performance of Local-Orbital Coupled Cluster Approaches in Different Static Correlation Regimes

Abstract: We have revisited the MOBH35 (Metal-Organic Barrier Heights, 35 reactions) benchmark [Iron; , Janes, J. Phys. Chem. A, 2019, 123 (17), 3761-3781; ibid. 2019, 123, 6379-6380] for realistic organometallic catalytic reactions, using both canonical CCSD(T) and localized orbital approximations to it. For low levels of static correlation, all of DLPNO-CCSD(T), PNO-LCCSD(T), and LNO-CCSD(T) perform well; for moderately strong levels of static correlation, DLPNO-CCSD(T) and (T1) may break down catastrophically, and PNO-LCCSD(T) is vulnerable as well. In contrast, LNO-CCSD(T) converges smoothly to the canonical CCSD(T) answer with increasingly tight convergence settings. The only two reactions for which our revised MOBH35 reference values differ substantially from the original ones are reaction 9 and to a lesser extent 8, both involving iron. For the purpose of evaluating density functional theory (DFT) methods for MOBH35, it would be best to remove reaction 9 entirely as its severe level of static correlation makes it just too demanding for a test. The magnitude of the difference between DLPNO-CCSD(T) and DLPNO-CCSD(T1) is a reasonably good predictor for errors in DLPNO-CCSD(T1) compared to canonical CCSD(T); otherwise, monitoring all of T1, D1, max|tiA|, and 1/(εLUMO - εHOMO) should provide adequate warning for potential problems. Our conclusions are not specific to the def2-SVP basis set but are largely conserved for the larger def2-TZVPP, as they are for the smaller def2-SV(P): the latter may be an economical choice for calibrating against canonical CCSD(T). Finally, diagnostics for static correlation are statistically clustered into groups corresponding to (1) importance of single excitations in the wavefunction; (2a) the small band gap, weakly separated from (2b) correlation entropy; and (3) thermochemical importance of correlation energy, as well as the slope of the DFT reaction energy with respect to the percentage of HF exchange. Finally, a variable reduction analysis reveals that much information on the multireference character is provided by T1, IND/Itot, and the exchange-based diagnostic A100[TPSS].

Modeling of Multiresonant Thermally Activated Delayed Fluorescence Emitters─Properly Accounting for Electron Correlation Is Key!

Abstract: With the surge of interest in multiresonant thermally activated delayed fluorescent (MR-TADF) materials, it is important that there exist computational methods to accurately model their excited states. Here, building on our previous work, we demonstrate how the spin-component scaling second-order approximate coupled-cluster (SCS-CC2), a wavefunction-based method, is robust at predicting the ΔEST (i.e., the energy difference between the lowest singlet S1 and triplet T1 excited states) of a large number of MR-TADF materials, with a mean average deviation (MAD) of 0.04 eV compared to experimental data. Time-dependent density functional theory calculations with the most common DFT functionals as well as the consideration of the Tamm-Dancoff approximation (TDA) consistently predict a much larger ΔEST as a result of a poorer account of Coulomb correlation as compared to SCS-CC2. Very interestingly, the use of a metric to assess the importance of higher order excitations in the SCS-CC2 wavefunctions shows that Coulomb correlation effects are substantially larger in the lowest singlet compared to the corresponding triplet and need to be accounted for a balanced description of the relevant electronic excited states. This is further highlighted with coupled cluster singles-only calculations, which predict very different S1 energies as compared to SCS-CC2 while T1 energies remain similar, leading to very large ΔEST, in complete disagreement with the experiments. We compared our SCS-CC2/cc-pVDZ with other wavefunction approaches, namely, CC2/cc-pVDZ and SOS-CC2/cc-pVDZ leading to similar performances. Using SCS-CC2, we investigate the excited-state properties of MR-TADF emitters showcasing large ΔET2T1 for the majority of emitters, while π-electron extension emerges as the best strategy to minimize ΔEST. We also employed SCS-CC2 to evaluate donor-acceptor systems that contain a MR-TADF moiety acting as the acceptor and show that the broad emission observed for some of these compounds arises from the solvent-promoted stabilization of a higher-lying charge-transfer singlet state (S2). This work highlights the importance of using wavefunction methods in relation to MR-TADF emitter design and associated photophysics.

Twenty Years of Auxiliary-Field Quantum Monte Carlo in Quantum Chemistry: An Overview and Assessment on Main Group Chemistry and Bond-Breaking.

Abstract: In this work, we present an overview of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) approach from a computational quantum chemistry perspective and present a numerical assessment of its performance on main group chemistry and bond-breaking problems with a total of 1004 relative energies. While our benchmark study is somewhat limited, we make recommendations for the use of ph-AFQMC for general main-group chemistry applications. For systems where single determinant wave functions are qualitatively accurate, we expect the accuracy of ph-AFQMC in conjunction with a single-determinant trial wave function to be between that of coupled-cluster with singles and doubles (CCSD) and CCSD with perturbative triples (CCSD(T)). For these applications, ph-AFQMC should be a method of choice when canonical CCSD(T) is too expensive to run. For systems where multireference (MR) wave functions are needed for qualitative accuracy, ph-AFQMC is far more accurate than MR perturbation theory methods and competitive with MR configuration interaction (MRCI) methods. Due to the computational efficiency of ph-AFQMC compared to MRCI, we recommended ph-AFQMC as a method of choice for handling dynamic correlation in MR problems. We conclude with a discussion of important directions for future development of the ph-AFQMC approach.

Wavefunction embedding for molecular polaritons.

Abstract: Polaritonic chemistry relies on the strong light-matter interaction phenomena for altering the chemical reaction rates inside optical cavities. To explain and understand these processes, the development of reliable theoretical models is essential. While computationally efficient quantum electrodynamics self-consistent field (QED-SCF) methods, such as quantum electrodynamics density functional theory, need accurate functionals, quantum electrodynamics coupled cluster (QED-CC) methods provide a systematic increase in accuracy but at much greater cost. To overcome this computational bottleneck, herein we introduce and develop the QED-CC-in-QED-SCF projection-based embedding method that inherits all the favorable properties from the two worlds: computational efficiency and accuracy. The performance of the embedding method is assessed by studying some prototypical but relevant reactions, such as methyl transfer reaction, proton transfer reaction, and protonation reaction, in a complex environment. The results obtained with the new embedding method are in excellent agreement with more expensive QED-CC results. The analysis performed on these reactions indicates that the electron-photon correlation effects are local in nature and that only a small region should be treated at the QED-CC level for capturing important effects due to cavity. This work sets the stage for future developments of polaritonic quantum chemistry methods and will serve as a guideline for the development of other polaritonic embedding models.

Coupled Cluster Molecular Dynamics of Condensed Phase Systems Enabled by Machine Learning Potentials: Liquid Water Benchmark.

Abstract: Coupled cluster theory is a general and systematic electronic structure method, but in particular the highly accurate "gold standard" coupled cluster singles, doubles and perturbative triples, CCSD(T), can only be applied to small systems. To overcome this limitation, we introduce a framework to transfer CCSD(T) accuracy of finite molecular clusters to extended condensed phase systems using a high-dimensional neural network potential. This approach, which is automated, allows one to perform high-quality coupled cluster molecular dynamics, CCMD, as we demonstrate for liquid water including nuclear quantum effects. The machine learning strategy is very efficient, generic, can be systematically improved, and is applicable to a variety of complex systems.

NMR Coupling Constants Based on the Bethe-Salpeter Equation in the GW Approximation.

Abstract: We present the first steps to extend the Green's function GW method and the Bethe-Salpeter equation (BSE) to molecular response properties such as nuclear magnetic resonance (NMR) indirect spin-spin coupling constants. We discuss both a nonrelativistic one-component and a quasi-relativistic two-component formalism. The latter describes scalar-relativistic and spin-orbit effects and allows us to study heavy-element systems with reasonable accuracy. Efficiency is maintained by the application of the resolution of the identity approximation throughout. The performance is demonstrated using conventional central processing units (CPUs) and modern graphics processing units (GPUs) for molecules involving several thousand basis functions. Our results show that a large amount of Hartree-Fock exchange is vital to provide a sufficient Kohn-Sham starting point to compute the GW quasi-particle energies. As the GW-BSE approach is generally less accurate for triplet excitations or related properties such as the Fermi-contact interaction, the admixture of the Kohn-Sham correlation kernel through the contracted BSE (cBSE) method improves the results for NMR coupling constants. This leads to remarkable results when combined with the eigenvalue-only self-consistent variant (evGW) and Becke's half and half functional (BH&HLYP) or the CAM-QTP family. The developed methodology is used to calculate the Karplus curve of tin molecules, illustrating its applicability to extended chemically relevant molecules. Here, the GW-cBSE method improves upon the chosen BH&HLYP Kohn-Sham starting points.

Predicting properties of periodic systems from cluster data: A case study of liquid water.

Abstract: The accuracy of the training data limits the accuracy of bulk properties from machine-learned potentials. For example, hybrid functionals or wave-function-based quantum chemical methods are readily available for cluster data but effectively out of scope for periodic structures. We show that local, atom-centered descriptors for machine-learned potentials enable the prediction of bulk properties from cluster model training data, agreeing reasonably well with predictions from bulk training data. We demonstrate such transferability by studying structural and dynamical properties of bulk liquid water with density functional theory and have found an excellent agreement with experimental and theoretical counterparts.

Machine Learning of Coupled Cluster (T)-Energy Corrections via Delta (Δ)-Learning.

Abstract: Accurate thermochemistry is essential in many chemical disciplines, such as astro-, atmospheric, or combustion chemistry. These areas often involve fleetingly existent intermediates whose thermochemistry is difficult to assess. Whenever direct calorimetric experiments are infeasible, accurate computational estimates of relative molecular energies are required. However, high-level computations, often using coupled cluster theory, are generally resource-intensive. To expedite the process using machine learning techniques, we generated a database of energies for small organic molecules at the CCSD(T)/cc-pVDZ, CCSD(T)/aug-cc-pVDZ, and CCSD(T)/cc-pVTZ levels of theory. Leveraging the power of deep learning by employing graph neural networks, we are able to predict the effect of perturbatively included triples (T), that is, the difference between CCSD and CCSD(T) energies, with a mean absolute error of 0.25, 0.25, and 0.28 kcal mol-1 (R2 of 0.998, 0.997, and 0.998) with the cc-pVDZ, aug-cc-pVDZ, and cc-pVTZ basis sets, respectively. Our models were further validated by application to three validation sets taken from the S22 Database as well as to a selection of known theoretically challenging cases.

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple.

Abstract: Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnFn, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Coupled Cluster Benchmarking of Large Noncovalent Complexes in L7 and S12L as Well as the C60 Dimer, DNA-Ellipticine, and HIV-Indinavir.

Abstract: In this work, we report the benchmark binding energies of the seven complexes within the L7 data set, six host-guest complexes from the S12L data set, a C60 dimer, the DNA-ellipticine intercalation complex, and the largest system of the study, the HIV-indinavir system, which contained 343 atoms or 139 heavy atoms. The high-quality values reported were obtained via a focal point method that relies on the canonical form of second-order Møller-Plesset theory and the domain-based local pair natural orbital scheme for the coupled cluster with single double and perturbative triple excitations [DLPNO-CCSD(T)] extrapolated to the complete basis set (CBS) limit. The results in this work not only corroborate but also improve upon some previous benchmark values for large noncovalent complexes albeit at a relatively steep cost. Although local CCSD(T) and the largely successful fixed-node diffusion Monte Carlo (FN-DMC) have been shown to generally agree for small- to medium-size systems, a discrepancy in their reported binding energy values arises for large complexes, where the magnitude of the disagreement is a definite cause for concern. For example, the largest deviation in the L7 data set was 2.8 kcal/mol (∼10%) on the low end in C3GC. Such a deviation only grows worse in the S12L set, which showed a difference of up to 10.4 kcal/mol (∼25%) by a conservative estimation in buckycatcher-C60. The DNA-ellipticine complex also generated a disagreement of 4.4 kcal/mol (∼10%) between both state-of-the-art methods. The disagreement between local CCSD(T) and FN-DMC in large noncovalent complexes shows that it is urgently needed to have the canonical CCSD(T), the Monte Carlo CCSD(T), or the full configuration interaction quantum Monte Carlo approaches available to large systems on the hundred-atom scale to solve this dilemma. In addition, the performances of cheaper popular computational methods were assessed for the studied complexes with respect to DLPNO-CCSD(T)/CBS. r2SCAN-3c, B97M-V, and PBE0+D4 work well in large noncovalent complexes in this work, and GFN2-xTB performs well in π-π stacking complexes. B97M-V is the most reliable computationally efficient approach to predicting noncovalent interactions for large complexes, being the only one to have binding errors within the so-called 1 kcal/mol "chemical accuracy". The benchmark interaction energies of these host-guest complexes, molecular materials, and biological systems with electronic and medicinal implications provide crucial reference data for the improvement of current and future lower-cost methods.

Equation-of-motion cavity quantum electrodynamics coupled-cluster theory for electron attachment

Abstract: The electron attachment variant of equation-of-motion coupled-cluster theory (EOM-EA-CC) is generalized to the case of strong light-matter coupling within the framework of cavity quantum electrodynamics (QED). The resulting EOM-EA-QED-CC formalism provides an ab initio, correlated, and non-perturbative description of cavity-induced effects in many-electron systems that complements other recently proposed cavity-QED-based extensions of CC theory. Importantly, this work demonstrates that QED generalizations of EOM-CC theory are useful frameworks for exploring particle-non-conserving sectors of Fock space, thereby establishing a path forward for the simultaneous description of both strong electron-electron and electron-photon correlation effects.

Reconciling Local Coupled Cluster with Multireference Approaches for Transition Metal Spin-State Energetics

Abstract: Spin-state energetics of transition metal complexes remain one of the most challenging targets for electronic structure methods. Among single-reference wave function approaches, local correlation approximations to coupled cluster theory, most notably the domain-based local pair natural orbital (DLPNO) approach, hold the promise of bringing the accuracy of coupled cluster theory with single, double, and perturbative triple excitations, CCSD(T), to molecular systems of realistic size with acceptable computational cost. However, recent studies on spin-state energetics of iron-containing systems raised doubts about the ability of the DLPNO approach to adequately and systematically approximate energetics obtained by the reference-quality complete active space second-order perturbation theory with coupled-cluster semicore correlation, CASPT2/CC. Here, we revisit this problem using a diverse set of iron complexes and examine several aspects of the application of the DLPNO approach. We show that DLPNO-CCSD(T) can accurately reproduce both CASPT2/CC and canonical CCSD(T) results if two basic principles are followed. These include the consistent use of the improved iterative (T1) versus the semicanonical perturbative triple corrections and, most importantly, a simple two-point extrapolation to the PNO space limit. The latter practically eliminates errors arising from the default truncation of electron-pair correlation spaces and should be viewed as standard practice in applications of the method to transition metal spin-state energetics. Our results show that reference-quality results can be readily achieved with DLPNO-CCSD(T) if these principles are followed. This is important also in view of the applicability of the method to larger single-reference systems and multinuclear clusters, whose treatment of dynamic correlation would be challenging for multireference-based approaches.

A Mountaineering Strategy to Excited States: Revising Reference Values with EOM-CC4

Abstract: In the framework of the computational determination of highly accurate vertical excitation energies in small organic compounds, we explore the possibilities offered by the equation-of-motion formalism relying on the approximate fourth-order coupled-cluster (CC) method, CC4. We demonstrate, using an extended set of more than 200 reference values based on CC including up to quadruples excitations (CCSDTQ), that CC4 is an excellent approximation to CCSDTQ for excited states with a dominant contribution from single excitations with an average deviation as small as 0.003 eV. We next assess the accuracy of several additive basis set correction schemes, in which vertical excitation energies obtained with a compact basis set and a high-order CC method are corrected with lower-order CC calculations performed in a larger basis set. Such strategies are found to be overall very beneficial, though their accuracy depends significantly on the actual scheme. Finally, CC4 is employed to improve several theoretical best estimates of the QUEST database for molecules containing between four and six (nonhydrogen) atoms, for which previous estimates were computed at the CCSDT level.

Unitary Selective Coupled-Cluster Method

Abstract: Simulating molecules using the Variational Quantum Eigensolver method is one of the promising applications for NISQ-era quantum computers. Designing an efficient ansatz to represent the electronic wave function is crucial in such simulations. Standard unitary coupled-cluster with singles and doubles (UCCSD) ansatz tends to have a large number of insignificant terms that do not lower the energy of the system. In this work, we present a unitary selective coupled-cluster method, a way to construct a unitary coupled-cluster ansatz iteratively using a selection procedure with excitations up to fourth order. This approach uses the electronic Hamiltonian matrix elements and the amplitudes for excitations already present in the ansatz to find the important excitations of higher order and to add them to the ansatz. The important feature of the method is that it systematically reduces the energy error with increasing ansatz size for a set of test molecules. The main advantage of the proposed method is that the effort to increase the ansatz does not require any additional measurements on a quantum computer.

Transfer Learning for Affordable and High-Quality Tunneling Splittings from Instanton Calculations.

Abstract: The combination of transfer learning (TL) a low-level potential energy surface (PES) to a higher level of electronic structure theory together with ring-polymer instanton (RPI) theory is explored and applied to malonaldehyde. The RPI approach provides a semiclassical approximation of the tunneling splitting and depends sensitively on the accuracy of the PES. With second-order Møller-Plesset perturbation theory (MP2) as the low-level model and energies and forces from coupled cluster singles, doubles, and perturbative triples [CCSD(T)] as the high-level (HL) model, it is demonstrated that CCSD(T) information from only 25-50 judiciously selected structures along and around the instanton path suffice to reach HL accuracy for the tunneling splitting. In addition, the global quality of the HL-PES is demonstrated through a mean average error of 0.3 kcal/mol for energies up to 40 kcal/mol above the minimum energy structure (a factor of 2 higher than the energies employed during TL) and <2 cm-1 for harmonic frequencies compared with computationally challenging normal mode calculations at the CCSD(T) level.

Cluster expansions of multicomponent ionic materials: Formalism and methodology

Abstract: The cluster expansion (CE) method has seen continuous and increasing use in the study of configuration-dependent properties of crystalline materials. The original development of the CE method along with the underlying mathematical formalism and assumptions was focused on the study of metallic alloys. Since then the methodology has been actively and successfully used in the study of ionic materials as well. In this work, we present a cohesive reformulation of the mathematical formalism underlying the CE method based on a synthesis of its original formulation, several additions and extensions that have been proposed since, and a revised representation of its constituent mathematical objects. We then proceed to describe some of the formal implications of using the methodology for charge-neutral configurations in ionic systems. In particular, we discuss the reduction of the size of configuration spaces and the resulting linear dependencies that arise among correlation functions that span the larger unconstrained configuration space. Additionally, we explore the effects of long-range electrostatic interactions. We also demonstrate how the previously proposed use of a point electrostatic term successfully accounts for the majority of the longer-range electrostatic interactions, and leaves the cluster expansion terms to capture mostly short-range interactions. Finally, we present and discuss a variety of recently developed methodologies, including training structure selection, oxidation state assignment, structure mapping, and regression algorithms, that are necessary to address these formal mathematical notions for a practical implementation of the CE method in the study of multicomponent ionic materials.

Compressing Many-Body Fermion Operators under Unitary Constraints

Abstract: The most efficient known quantum circuits for preparing unitary coupled cluster states and applying Trotter steps of the arbitrary basis electronic structure Hamiltonian involve interleaved sequences of Fermionic Gaussian circuits and Ising interaction-type circuits. These circuits arise from factorizing the two-body operators generating those unitaries as a sum of squared one-body operators that are simulated using product formulas. We introduce a numerical algorithm for performing this factorization that has an iteration complexity no worse than single particle basis transformations of the two-body operators and often results in many times fewer squared one-body operators in the sum of squares, compared to the analytical decompositions. As an application of this numerical procedure, we demonstrate that our protocol can be used to approximate generic unitary coupled cluster operators and prepare the necessary high-quality initial states for techniques (like ADAPT-VQE) that iteratively construct approximations to the ground state.

Calculation of Metallocene Ionization Potentials via Auxiliary Field Quantum Monte Carlo: Toward Benchmark Quantum Chemistry for Transition Metals

Abstract: The accurate ab initio prediction of ionization energies is essential to understanding the electrochemistry of transition metal complexes in both materials science and biological applications. However, such predictions have been complicated by the scarcity of gas phase experimental data, the relatively large size of the relevant molecules, and the presence of strong electron correlation effects. In this work, we apply all-electron phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) utilizing multideterminant trial wave functions to six metallocene complexes to compare the computed adiabatic and vertical ionization energies with experimental results. We find that ph-AFQMC yields mean absolute errors (MAEs) of 1.69 ± 1.02 kcal/mol for the adiabatic energies and 2.85 ± 1.13 kcal/mol for the vertical energies. We also carry out density functional theory (DFT) calculations using a variety of functionals, which yields MAEs of 3.62-6.98 kcal/mol and 3.31-9.88 kcal/mol, as well as one variant of localized coupled cluster calculations (DLPNO-CCSD(T0) with moderate PNO cutoffs), which has MAEs of 4.96 and 6.08 kcal/mol, respectively. We also test the reliability of DLPNO-CCSD(T0) and DFT on acetylacetonate (acac) complexes for adiabatic energies measured in the same manner experimentally, and we find higher MAEs, ranging from 4.56 to 10.99 kcal/mol (with a different ordering) for DFT and 6.97 kcal/mol for DLPNO-CCSD(T0). Finally, by utilizing experimental solvation energies, we show that accurate reduction potentials in solution for the metallocene series can be obtained from the AFQMC gas phase results.

Molecular Auger decay rates from complex-variable coupled-cluster theory

Abstract: The emission of an Auger electron is the predominant relaxation mechanism of core-vacant states in molecules composed of light nuclei. In this non-radiative decay process, one valence electron fills the core vacancy while a second valence electron is emitted into the ionization continuum. Because of this coupling to the continuum, core-vacant states represent electronic resonances that can be tackled with standard quantum-chemical methods only if they are approximated as bound states, meaning that Auger decay is neglected. Here, we present an approach to compute Auger decay rates of core-vacant states from coupled-cluster and equation-of-motion coupled-cluster wave functions combined with complex scaling of the Hamiltonian or, alternatively, complex-scaled basis functions. Through energy decomposition analysis, we illustrate how complex-scaled methods are capable of describing the coupling to the ionization continuum without the need to model the wave function of the Auger electron explicitly. In addition, we introduce in this work several approaches for the determination of partial decay widths and Auger branching ratios from complex-scaled coupled-cluster wave functions. We demonstrate the capabilities of our new approach by computations on core-ionized states of neon, water, dinitrogen, and benzene. Coupled-cluster and equation-of-motion coupled-cluster theory in the singles and doubles approximation both deliver excellent results for total decay widths, whereas we find partial widths more straightforward to evaluate with the former method. We also observe that the requirements towards the basis set are less arduous for Auger decay than for other types of resonances so that extensions to larger molecules are readily possible.

Vertical ionization potential benchmark for unitary coupled-cluster and algebraic-diagrammatic construction methods.

Abstract: The performance of several methods for the calculation of vertical ionization potentials (IPs) or, more generally, electron-detachment energies based on unitary coupled-cluster (UCC) theory and the algebraic-diagrammatic construction (ADC) scheme is evaluated with respect to benchmark data computed at the level of equation-of-motion coupled-cluster theory, including single, double, and triple excitations (IP-EOM-CCSDT). Based on a statistical evaluation of about 200 electron-detached states of 41 molecules, the second-order methods IP-ADC(2) and IP-UCC2 show modest accuracies with IP-EOM-CCSDT as reference, exposing a mean signed error and a standard deviation of the error of -0.54 ± 0.50 and -0.49 ± 0.54 eV, respectively, accompanied by a mean absolute error (MAE) of 0.61 and 0.58 eV, respectively. The strict third-order IP-ADC method demonstrates an accuracy of 0.26 ± 0.35 eV (MAE = 0.35 eV), while the IP-UCC3 method is slightly more accurate with 0.24 ± 0.26 eV (MAE = 0.29 eV). Employing the static self-energy computed using the Dyson expansion method (DEM) improves the IP-ADC(3) performance to 0.27 ± 0.28 eV, with the mean absolute error of this method being 0.32 eV. However, employing the simpler improved fourth-order scheme Σ(4+) for the static self-energy provides almost identical results as the DEM. Based on the quality of the present benchmark results, it therefore appears not necessary to use the computationally more demanding DEM.

Electronic structure of strongly correlated systems: recent developments in multiconfiguration pair-density functional theory and multiconfiguration nonclassical-energy functional theory

Abstract: Strong electron correlation plays an important role in transition-metal and heavy-metal chemistry, magnetic molecules, bond breaking, biradicals, excited states, and many functional materials, but it provides a significant challenge for modern electronic structure theory. The treatment of strongly correlated systems usually requires a multireference method to adequately describe spin densities and near-degeneracy correlation. However, quantitative computation of dynamic correlation with multireference wave functions is often difficult or impractical. Multiconfiguration pair-density functional theory (MC-PDFT) provides a way to blend multiconfiguration wave function theory and density functional theory to quantitatively treat both near-degeneracy correlation and dynamic correlation in strongly correlated systems; it is more affordable than multireference perturbation theory, multireference configuration interaction, or multireference coupled cluster theory and more accurate for many properties than Kohn-Sham density functional theory. This perspective article provides a brief introduction to strongly correlated systems and previously reviewed progress on MC-PDFT followed by a discussion of several recent developments and applications of MC-PDFT and related methods, including localized-active-space MC-PDFT, generalized active-space MC-PDFT, density-matrix-renormalization-group MC-PDFT, hybrid MC-PDFT, multistate MC-PDFT, spin-orbit coupling, analytic gradients, and dipole moments. We also review the more recently introduced multiconfiguration nonclassical-energy functional theory (MC-NEFT), which is like MC-PDFT but allows for other ingredients in the nonclassical-energy functional. We discuss two new kinds of MC-NEFT methods, namely multiconfiguration density coherence functional theory and machine-learned functionals.

Iterative quantum phase estimation with variationally prepared reference state

Abstract: The iterative quantum phase estimation algorithm (IQPE) is theoretically appealing in its wide scope of being able to handle electronic correlation. However, the quality of the initial input state strongly enhances the probability of landing on the desired eigenstate. In this work, we systematically study two different parametrization schemes of the unitary coupled cluster (UCC) ansätz in the variational quantum eigensolver (VQE) framework toward the reference state preparation for IQPE. The efficacy of the UCC variants toward an appropriate state preparation is studied with prototypical H4 molecule on a circle. While the conventional UCC ansätz can lead to high success probability across various degrees of electronic complexity, a resource efficient minimally parametrized UCC ansätz consisting of active space excitations is shown to incorporate the essential static correlation in the reference state description. We demonstrate that such a carefully prepared initial state can significantly reduce the effects of noise due to sampling in the estimation of the desired eigenphase.

Coupled cluster downfolding methods: The effect of double commutator terms on the accuracy of ground-state energies

Abstract: Downfolding coupled cluster techniques have recently been introduced into quantum chemistry as a tool for the dimensionality reduction of the many-body quantum problem. As opposed to earlier formulations in physics and chemistry based on the concept of effective Hamiltonians, the appearance of the downfolded Hamiltonians is a natural consequence of the single-reference exponential parameterization of the wave function. In this paper, we discuss the impact of higher-order terms originating in double commutators. In analogy to previous studies, we consider the case when only one- and two-body interactions are included in the downfolded Hamiltonians. We demonstrate the efficiency of the many-body expansions involving single and double commutators for the unitary extension of the downfolded Hamiltonians on the example of the beryllium atom, and bond-breaking processes in the Li2 and H2O molecules. For the H2O system, we also analyze energies obtained with downfolding procedures as functions of the active space size.

AGP-based unitary coupled cluster theory for quantum computers

Abstract: Electronic structure methods typically benefit from symmetry breaking and restoration, specially in the strong correlation regime. The same goes for ansätze on a quantum computer. We develop a unitary coupled cluster method based on the antisymmetrized geminal power (AGP)—a state formally equivalent to the number-projected Bardeen–Cooper–Schrieffer wavefunction. We demonstrate our method for the single-band Fermi–Hubbard Hamiltonian in one and two dimensions. We also explore post-selection as a state preparation step to obtain correlated AGP and prove that it scales no worse than O(M) in the number of measurements, thereby making it a less expensive alternative to gauge integration to restore particle number symmetry.

Variational quantum eigensolver simulations with the multireference unitary coupled cluster ansatz: a case study of the C2v quasi-reaction pathway of beryllium insertion into a H2 molecule.

Abstract: Variational quantum eigensolver (VQE)-based quantum chemical calculations have been extensively studied as a computational model using noisy intermediate-scale quantum devices. The VQE uses a parametrized quantum circuit defined through an "ansatz" to generate approximated wave functions, and the appropriate choice of an ansatz is the most important step. Because most chemistry problems focus on the energy difference between two electronic states or structures, calculating the total energies in different molecular structures with the same accuracy is essential to correctly understand chemistry and chemical processes. In this context, the development of ansatzes that are capable of describing electronic structures of strongly correlated systems accurately is an important task. Here we applied a conventional unitary coupled cluster (UCC) and a newly developed multireference unitary coupled cluster with partially generalized singles and doubles (MR-UCCpGSD) ansatzes to the quasi-reaction pathway of Be insertion into H2, LiH molecule under covalent bond dissociation, and a rectangular tetra-hydrogen cluster known as a P4 cluster; these are representative systems in which the static electron correlation effect is prominent. Our numerical simulations revealed that the UCCSD ansatz exhibits extremely slow convergence behaviour around the point where an avoided crossing occurs in the Be + H2 → BeH2 reaction pathway, resulting in a large discrepancy of the simulated VQE energy from the full-configuration interaction (full-CI) value. By contrast, the MR-UCCpGSD ansatz can give more reliable results with respect to total energy and the overlap with the full-CI solution, insisting the importance of multiconfigurational treatments in the calculations of strongly correlated systems. The MR-UCCpGSD ansatz allows us to compute the energy with the same accuracy regardless of the strength of multiconfigurational character, which is an essential property to discuss energy differences of various molecular systems.

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

Abstract: Abstract 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.