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Jordan–Wigner Transformation for the Description of Strong Correlation in Fermionic Systems
Seniority is a useful way of organizing Hilbert space for strongly correlated systems. The exact zero-seniority wave function, doubly occupied configuration interaction (DOCI), provides accurate results (given the right orbitals) for many strongly correlated electronic systems but has a combinatorial computational cost. In many cases, pair coupled cluster doubles provide a polynomial-cost approximation that closely reproduces the energies of DOCI, but it breaks down in some cases and, as shown herein, it does not provide particularly good density matrices. In this article, we demonstrate that by using the Jordan–Wigner transformation to turn the seniority zero problem back into a Fermionic one, we can provide mean-field variational results of DOCI quality for the Hubbard model and a few small molecular dissociation examples, with polynomial cost, both for the energies and for density matrices, all while being protected from collapse. This success is rooted in the proof we provide, showing that the Hartree–Fock wave function on the Jordan–Wigner-transformed Hamiltonian transforms back to variational coupled cluster doubles in the seniority zero representation, but restricted to have determinant rather than permanent amplitude coefficients, without compromising its overall accuracy.
Enhancing T c in a composite superconductor/metal bilayer system: A dynamical cluster approximation study
It has been proposed that the superconducting transition temperature T c of an unconventional superconductor with a large pairing scale but strong phase fluctuations can be enhanced by coupling it to a metal. However, the general efficacy of this approach across different parameter regimes remains an open question. Using the dynamical cluster approximation, we study this question in a system composed of an attractive Hubbard layer in the intermediate coupling regime, where the magnitude of the attractive Coulomb interaction |U| is slightly larger than the bandwidth W, hybridized with a noninteracting metallic layer. We find that while the superconducting transition becomes more mean-field-like with increasing interlayer hopping, the superconducting transition temperature T c exhibits a nonmonotonic dependence on the strength of the hybridization t ⊥ . Furthermore, this behavior arises from a reduction of the effective pairing interaction in the correlated layer that outcompetes the growth in the intrinsic pair-field susceptibility induced by the coupling to the metallic layer. We find that the largest T c inferred here for the composite system is comparable to the maximum value currently estimated for the isolated negative-U Hubbard model.
Fermi–Löwdin orbital self-interaction correction of adsorption energies on transition metal ions
Density functional theory (DFT)-based descriptions of the adsorption of small molecules on transition metal ions are prone to self-interaction errors. Here, we show that such errors lead to a large over-estimation of adsorption energies of small molecules on Cu + , Zn + , Zn 2+ , and Mn + in local spin density approximation (LSDA) and Perdew, Burke, Ernzerhof (PBE) generalized gradient approximation calculations compared to reference values computed using the coupled-cluster with single, doubles, and perturbative triple excitations method. These errors are significantly reduced by removing self-interaction using the Perdew–Zunger self-interaction correction (PZ-SIC) in the Fermi–Löwdin Orbital (FLO) SIC framework. In the case of FLO-PBE, typical errors are reduced to less than 0.1 eV. Furthermore, analysis of the results using DFT energies evaluated on self-interaction-corrected densities [DFT(@FLO)] indicates that the density-driven contributions to the FLO-DFT adsorption energy corrections are roughly the same size in DFT = LSDA and PBE, but the total corrections due to removing self-interaction are larger in LSDA.
Equation-of-Motion Coupled-Cluster Cumulant Green’s Function for Excited States and X-ray Spectra
Green’s function methods within many-body perturbation theory provide a general framework for treating electronic correlations in excited states and spectra. Conventional methods using the Dyson equation or the cumulant expansion are typically based on the GW self-energy approximation. In order to extend this approximation in molecular systems, a non-perturbative real-time coupled-cluster cumulant Green’s function approach has been introduced, where the cumulant is obtained as the solution to a set of coupled first order, non-linear differential equations. This approach naturally includes non-linear corrections to conventional cumulant Green’s function techniques where the cumulant is linear in the GW self-energy. The method yields the spectral function associated with the core Green’s function, which is directly related to the x-ray photoemission spectra (XPS) of molecular systems. The approach also yields very good results for binding energies and satellite excitations. The x-ray absorption spectrum (XAS) is then obtained as a convolution of the core spectral function and an effective one-body absorption spectrum. Here this approach is extended to include the full coupled-cluster-singles (CCS) core Green’s function by including the complete form of the non-linear contributions to the cumulant as well as all single, double, and triple cluster excitations in the CC amplitude equations. This approach naturally builds in orthogonality and shake-up effects analogous to those in the Mahan-Noizeres-de Dominicis edge singularity corrections that enhance the XAS near the edge. Themethod is illustrated for the XPS and XAS of NH 3 .
TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations
TURBOMOLE is a collaborative, multi-national software development project aiming to provide highly efficient and stable computational tools for quantum chemical simulations of molecules, clusters, periodic systems, and solutions. The TURBOMOLE software suite is optimized for widely available, inexpensive, and resource-efficient hardware such as multi-core workstations and small computer clusters. TURBOMOLE specializes in electronic structure methods with outstanding accuracy–cost ratio, such as density functional theory including local hybrids and the random phase approximation (RPA), GW-Bethe–Salpeter methods, second-order Møller–Plesset theory, and explicitly correlated coupled-cluster methods. TURBOMOLE is based on Gaussian basis sets and has been pivotal for the development of many fast and low-scaling algorithms in the past three decades, such as integral-direct methods, fast multipole methods, the resolution-of-the-identity approximation, imaginary frequency integration, Laplace transform, and pair natural orbital methods. This review focuses on recent additions to TURBOMOLE’s functionality, including excited-state methods, RPA and Green’s function methods, relativistic approaches, high-order molecular properties, solvation effects, and periodic systems. A variety of illustrative applications along with accuracy and timing data are discussed. Moreover, available interfaces to users as well as other software are summarized. TURBOMOLE’s current licensing, distribution, and support model are discussed, and an overview of TURBOMOLE’s development workflow is provided. Challenges such as communication and outreach, software infrastructure, and funding are highlighted.
A Massively Parallel Implementation of the CCSD(T) Method Using the Resolution-of-the-Identity Approximation and a Hybrid Distributed/Shared Memory Parallelization Model
In this work, a parallel algorithm is described for the coupled-cluster singles and doubles method augmented with a perturbative correction for triple excitations [CCSD(T)] using the resolution-of-the-identity (RI) approximation for two-electron repulsion integrals (ERIs). The algorithm bypasses the storage of four-center ERIs by adopting an integral-direct strategy. The CCSD amplitude equations are given in a compact quasi-linear form by factorizing them in terms of amplitude-dressed three-center intermediates. A hybrid MPI/OpenMP parallelization scheme is employed, which uses the OpenMP-based shared memory model for intranode parallelization and the MPI-based distributed memory model for internode parallelization. Parallel efficiency has been optimized for all terms in the CCSD amplitude equations. Two different algorithms have been implemented for the rate-limiting terms in the CCSD amplitude equations that entail and -scaling computational costs, where N O and N V denote the number of correlated occupied and virtual orbitals, respectively. One of the algorithms assembles the four-center ERIs requiring N V 4 and N O 2 N V 2 -scaling memory costs in a distributed manner on a number of MPI ranks, while the other algorithm completely bypasses the assembling of quartic memory-scaling ERIs and thus largely reduces the memory demand. It is demonstrated that the former memory-expensive algorithm is faster on a few hundred cores, while the latter memory-economic algorithm shows a better strong scaling in the limit of a few thousand cores. The program is shown to exhibit a near-linear scaling, in particular for the compute-intensive triples correction step, on up to 8000 cores. The performance of the program is demonstrated via calculations involving molecules with 24–51 atoms and up to 1624 atomic basis functions. As the first application, the complete basis set (CBS) limit for the interaction energy of the π-stacked uracil dimer from the S66 data set has been investigated. This work reports the first calculation of the interaction energy at the CCSD(T)/aug-cc-pVQZ level without local orbital approximation. The CBS limit for the CCSD correlation contribution to the interaction energy was found to be -8.01 kcal/mol, which agrees very well with the value -7.99 kcal/mol reported by Schmitz, Hättig, and Tew [ Phys. Chem. Chem. Phys. 2014 , 16 , 22167-22178]. The CBS limit for the total interaction energy was estimated to be -9.64 kcal/mol.
Density Functionals for Hydrogen Storage: Defining the H2Bind275 Test Set with Ab Initio Benchmarks and Assessment of 55 Functionals
Efficient and high-capacity storage materials are indispensable for a hydrogen-based economy. In silico tools can accelerate the process of discovery of new adsorbent materials with optimal hydrogen adsorption enthalpies. Density functional theory is well-poised to become a very useful tool for enabling high-throughput screening of potential materials. In this work, we have identified density functional approximations that provide good performance for hydrogen binding applications following a two-pronged approach. First, we have compiled a data set (H2Bind275) that comprehensively represents the hydrogen binding problem capturing the chemical and mechanistic diversity in the binding sites encountered in hydrogen storage materials. We have also computed reference interaction energies for this data set using coupled-cluster theory. Second, we have assessed the performance of 55 density functional approximations for predicting H 2 interaction energies and have identified two hybrid density functionals (ωB97X-V and ωB97M-V), two double hybrid density functionals (DSD-PBEPBE-D3(BJ) and PBE0-DH), and one semilocal density functional (B97M-V) as the best performing ones. We have recommended the addition of empirical dispersion corrections to systematically underbinding density functionals such as revPBE, BLYP, and B3LYP for improvements in performance at negligible additional cost. We have also recommended the usage of the def2-TZVPP basis set as it represents a good compromise between accuracy and cost, limiting the finite basis set errors to less than 1 kJ/mol.
Definitive Assessment of the Accuracy, Variationality, and Convergence of Relativistic Coupled Cluster and Density Matrix Renormalization Group in 100-Orbital Space
Accuracy, variationality, and convergence underpin the reliability of modern electronic structure methods, yet definitive benchmarks in the relativistic regime remain elusive due to the absence of numerically exact full configuration interaction (CI) references. Recent algorithmic advances in the CI framework, enabled by the small-tensor-product (STP) decomposition approach, have dramatically extended the tractable size of the configuration space, making numerically exact CI calculations feasible in large active spaces previously beyond reach. In this paper, we employ the recently developed STP-CI framework to perform large-scale numerically exact CI calculations and directly benchmark relativistic coupled cluster and density matrix renormalization group methods. Definitive benchmarking of approximate relativistic electronic structure methods is ensured through the application of the gap theorem, which provides rigorous error bounds on the CI reference and establishes a controlled standard for assessing accuracy, variationality, and convergence.
Enhancing the accuracy of XPS calculations: Exploring hybrid basis set schemes for CVS-EOMIP-CCSD calculations
Reliable computational methodologies and basis sets for modeling x-ray spectra are essential for extracting and interpreting electronic and structural information from experimental x-ray spectra. In particular, the trade-off between numerical accuracy and computational cost due to the size of the basis set is a major challenge, since molecular orbitals undergo extreme relaxation in the core-hole state. To gain clarity on the changes in electronic structure induced by the formation of a core-hole, the use of sufficiently flexible basis for expanding the orbitals, particularly for the core region, has been shown to be essential. This work focuses on the refinement of core-hole ionized state calculations using the equation-of-motion coupled cluster family of methods through an extensive analysis on the effectiveness of “hybrid” and mixed basis sets. In this investigation, we utilize the CVS-EOMIP-CCSD method in combination and construct hybrid basis sets piecewise from readily available Dunning’s correlation consistent basis sets in order to calculate x-ray ionization energies (IEs) for a set of small gas phase molecules. Our results provide insights into the impact of basis sets on the CVS-EOMIP-CCSD calculations of K-edge IEs of first-row p-block elements. Furthermore, these insights enable us to understand more about the basis set dependence of the core IEs computed and allow us to establish a protocol for deriving reliable and cost-effective theoretical estimates for computing IEs of small molecules containing such elements.
27 AlNMR chemical shift of Al(OH)$^{-}_{4}$ calculated from first principles: Assessment of error cancellation in chemically distinct reference and target systems
Predicting accurate nuclear magnetic resonance chemical shieldings relies upon cancellation of different types of errors between the theoretically calculated shielding constant of the analyte of interest and the reference. Often, the intrinsic error in computed shieldings due to basis sets, approximations in the Hamiltonian, description of the wave function, and dynamic effects is nearly identical between the analyte and reference, yet if the electronic structure or sensitivity to local environment differs dramatically, this cannot be taken for granted. Detailed prior work has examined the octahedral trivalent cation Al(H 2 O)$^{3+}_{6}$, accounting for ab initio intrinsic errors. However, the use of this species as a reference for the chemically distinct tetrahedral anion Al(OH)$^{-}_{4}$ requires an understanding of how these errors cancel, in order to define the limits of accurately predicting 27 Al chemical shielding in Al(OH)$^{-}_{4}$. In this work, we estimate the absolute shielding of the 27 Al nucleus in Al(OH)$^{-}_{4}$ at the coupled cluster level (515.1± 5.3 ppm). Shielding sensitivity to the choice of method approximation and atomic basis sets used has been evaluated. Solvent and thermal effects are assessed through ensemble averaging techniques using ab-initio molecular dynamics. The contribution of each type of intrinsic error is assessed for the Al(H 2 O)$^{3+}_{6}$ and Al(OH)$^{-}_{4}$ ions, revealing significant differences that fundamentally hamper the ability to accurately calculate the 27 Al chemical shift of Al(OH)$^{-}_{4}$ from first principles.
Cluster perturbation theory X: A parallel implementation of Lagrangian perturbation series for the coupled cluster singles and doubles ground-state energy through fifth order
We describe an efficient implementation of cluster perturbation and Møller-Plesset Lagrangian energy series through fifth order that target the coupled cluster singles and doubles energy utilizing the resolution of the identity approximation. We illustrate the computational performance of the implementation by performing ground state energy calculations on systems with up to 1200 basis functions using a single node and by comparison to conventional CCSD calculations. We further show that our hybrid MPI/OMP parallel implementation that also utilizes graphical processing units can be used to obtain fifth order energies on systems with almost 1200 basis functions with a 90 minute "time to solution" running on Frontier at Oak Ridge National Laboratory.
Spatially Accelerated Winding Numbers for Curved Geometry
The generalized winding number (GWN) is a scalar field that supports robust containment queries on curved geometry, including non-watertight, overlapping, and nested boundary representations. While queries can be easily parallelized over samples, direct evaluation on parametric curves and surfaces remains costly for large and complex models. Fast, state-of-the-art GWN approaches leverage a spatial index to approximate the GWN, typically coupled with a Taylor expansion which approximates the GWN contribution for far clusters of geometric primitives. However, such methods operate only on discrete inputs such as triangle meshes and point clouds, and would introduce containment errors near boundaries if applied to curved input. We extend support for fast GWN evaluation over arbitrary collections of NURBS curves in 2D and trimmed NURBS patches in 3D via a Bounding Volume Hierarchy that stores efficiently precomputed moment data in the hierarchy nodes. When querying the hierarchy, approximations for far clusters are used alongside direct evaluation for nearby NURBS primitives, achieving sub-linear complexity while preserving the geometric features in the vicinity of the query point. Central to our performance improvements is an adaptive subdivision strategy for NURBS primitives during a preprocessing phase, creating better spatial partitions while retaining the same accuracy for containment decisions as a direct evaluation. We demonstrate the performance and accuracy of our approach across a large collection of 2D and 3D datasets.
Understanding the chemical bonding of ground and excited states of HfO and HfB with correlated wavefunction theory and density functional approximations
Knowledge of the chemical bonding of HfO and HfB ground and low-lying electronic states provides essential insights into a range of catalysts and materials that contain Hf–O or Hf–B moieties. Here, we carry out high-level multi-reference configuration interaction theory and coupled cluster quantum chemical calculations on these systems. We compute full potential energy curves, excitation energies, ionization energies, electronic configurations, and spectroscopic parameters with large quadruple-ζ and quintuple-ζ quality correlation consistent basis sets. We also investigate equilibrium chemical bonding patterns and effects of correlating core electrons on property predictions. Differences in the ground state electron configuration of HfB(X 4 Σ - ) and HfO(X 1 Σ + ) lead to a significantly stronger bond in HfO than HfB, as judged by both dissociation energies and equilibrium bond distances. We extend our analysis to the chemical bonding patterns of the isovalent HfX (X = O, S, Se, Te, and Po) series and observe similar trends. We also note a linear trend between the decreasing value of the dissociation energy (D e ) from HfO to HfPo and the singlet–triplet energy gap (ΔE S–T ) of the molecule. Finally, we compare these benchmark results to those obtained using density functional theory (DFT) with 23 exchange–correlation functionals spanning multiple rungs of “Jacob’s ladder.” When comparing DFT errors to coupled cluster reference values on dissociation energies, excitation energies, and ionization energies of HfB and HfO, we observe semi-local generalized gradient approximations to significantly outperform more complex and high-cost functionals.
Magnetic phase diagram of a two-orbital model for bilayer nickelates with varying doping
Motivated by the recently discovered high-T c bilayer nickelate superconductor La 3 Ni 2 O 7 , we comprehensively research a bilayer 2×2×2 cluster for different electronic densities n by using the Lanczos method. We also employ the random-phase approximation to quantify the first magnetic instability with increasing Hubbard coupling strength, also varying n. Based on the spin structure factor S(q), we have obtained a rich magnetic phase diagram in the plane defined by n and U/W, at fixed Hund coupling, where U is the Hubbard strength and W the bandwidth. We have observed numerous states, such as A-AFM, Stripes, G-AFM, and C-AFM. At half-filling, n=2 (two electrons per Ni site, corresponding to N=16 electrons), the canonical superexchange interaction leads to a robust G-AFM state (π,π,π) with antiferromagnetic couplings both in-plane and between layers. By increasing or decreasing electronic densities, ferromagnetic tendencies emerge from the “half-empty” and “half-full” mechanisms, leading to many other interesting magnetic tendencies. In addition, the spin-spin correlations become weaker both in the hole or electron doping regions compared with half-filling. At n=1.5 (or N=12), density corresponding to La 3 Ni 2 O 7 , we obtained the “Stripe 2” ground state (antiferromagnetic coupling in one in-plane direction, ferromagnetic coupling in the other, and antiferromagnetic coupling along the z axis) in the 2×2×2 cluster. In addition, we obtained a much stronger AFM coupling along the z axis than the magnetic coupling in the xy plane. The random-phase approximation calculations with varying n give very similar results as Lanczos, even though both techniques are based on quite different procedures. Additionally, a state with q/π=(0.6,0.6,1) close to the E-phase wavevector is found in our RPA calculations by slightly reducing the filling to n=1.25, possibly responsible for the E-phase SDW recently observed in experiments. In conclusion, our predictions can be tested by chemically doping La 3 Ni 2 O 7 .
Bridging reaction theory and nuclear structure in $π^±-$ 48 Ca scattering
Here, we extend the pion-nucleus multiple-scattering framework to include detailed second-order rescattering dynamics for nuclei with nonzero isospin. To account for intermediate charge-exchange and nucleon spin-flip effects, we develop a scattering potential that depends on the one- and two-body densities of the target nucleus. We compute one-body densities from coupled-cluster theory and two-body densities within the Hartree-Fock approximation. To estimate theoretical uncertainties, we employ modern nuclear Hamiltonians derived from chiral effective field theory. While the sensitivity to nuclear structure details is mild, second-order corrections are found to be sizable and essential for accurately reproducing differential cross sections measured in 𝜋 ± − 48 Ca elastic scattering within the Δ(1232)-resonance region.
Spin-singlet topological superconductivity in the attractive Rashba-Hubbard model
Fully gapped, spin-singlet superconductors with antisymmetric spin-orbit coupling in a Zeeman magnetic field provide a promising route to realize superconducting states with non-Abelian topological order and therefore fault-tolerant quantum computation. Here we use a quantum Monte Carlo dynamical cluster approximation to study the superconducting properties of a doped two-dimensional attractive Hubbard model with Rashba spin-orbit coupling in a Zeeman magnetic field. We generally find that the Rashba coupling has a beneficial effect towards s-wave superconductivity. In the presence of a finite Zeeman field, when superconductivity is suppressed by Pauli pair breaking, the Rashba coupling counteracts the spin imbalance created by the Zeeman field by mixing the spins, and thus restores superconductivity at finite temperatures. Further, we show that this favorable effect of the spin-orbit coupling is traced to a spin-flip driven enhancement of the amplitude for the propagation of a pair of electrons in time-reversed states. Moreover, by inspecting the Fermi surface of the interacting model, we show that for sufficiently large Rashba coupling and Zeeman field, the superconducting state is expected to be topologically nontrivial.
An Attractive Way to Correct for Missing Singles Excitations in Unitary Coupled Cluster Doubles Theory
Coupled cluster methods based exclusively on double excitations are comparatively “cheap” and interesting model chemistries, as they are typically able to capture the bulk of the dynamic electron correlation effects. The trade-off in such approximations is that the effect of neglected excitations, particularly single excitations, can be considerable. Using standard and electron-pair-restricted T 2 operators to define two flavors of unitary coupled cluster doubles (UCCD) methods, we investigate the extent to which missing single excitations can be recovered from low-order corrections in many-body perturbation theory (MBPT) within the unitary coupled cluster (UCC) formalism. Here, our analysis includes the derivations of finite-order UCC energy functionals, which are used as a basis to define perturbative estimates of missed single excitations. This leads to the novel UCCD[4S] and UCCD[6S] methods, which consider energy corrections for missing single excitations through fourth- and sixth-order in MBPT, respectively. We also apply the same methodology to the electron-pair-restricted ansatz, but the improvements are only marginal. Our findings show that augmenting UCCD with these post hoc perturbative corrections can lead to UCCSD-quality results.