Exact Exchange Research Articles

Implementation of time-dependent Hartree–Fock in real space

Abstract Time-dependent Hartree–Fock (TDHF) is one of the fundamental post-Hartree–Fock (HF) methods to describe excited states. In its Tamm-Dancoff form, equivalent to Configuration Interaction Singles, it is still widely used and particularly applicable to big molecules where more accurate methods may be unfeasibly expensive. However, it is rarely implemented in real space, mostly because of the expensive nature of the exact-exchange potential in real space. Compared to widely used Gaussian-type orbitals (GTO) basis sets, real space often offers easier implementation of equations and more systematic convergence of Rydberg states, as well as favorable scaling, effective domain parallelization, flexible boundary conditions, and ability to treat model systems. We implemented TDHF in the Octopus real-space code as a step toward linear-response hybrid time-dependent density-functional theory (TDDFT), other post-HF methods, and ensemble density-functional theory methods involving exact exchange. Calculation of HF’s non-local exact exchange is very expensive in real space. We overcome this limitation with Octopus’ implementation of Adaptively Compressed Exchange, and find the appropriate mixing scheme and starting point to complete the ground-state calculation in a practical amount of time, and thus enable TDHF. We compared our results to those from GTOs on a set of small molecules and confirmed close agreement of results, though with larger deviations than in the case of semi-local TDDFT. We find that convergence of TDHF demands a finer real-space grid than semi-local TDDFT. We also present the subtleties in benchmarking a real-space calculation against GTOs, relating to Rydberg and vacuum states.

Suppressing the gauge problem in local hybrid functionals without a calibration function: The choice of local mixing function.

Modern functionals based on the exact-exchange (EXX) energy density like local hybrid functionals (LHs) or range-separated LHs have recently received additional attention due to their advantages over established functionals when it comes to the local balance between self-interaction errors and static-correlation errors. A possible theoretical drawback of such functionals over the years has been the so-called gauge problem due to the inherent ambiguity of exchange-energy densities. Modern LHs like LH20t or more sophisticated functionals based thereon have been constructed using suitably optimized calibration functions (CFs) to minimize the mismatch of the semi-local and EXX energy densities. Here, we show that the unphysical contributions arising from the gauge problem may also be reduced significantly without a CF by tailoring the position-dependence of the EXX admixture (local mixing function, LMF) in a way to suppress spurious positive energy-density contributions locally in space. This is achieved by building the so-called x-LMFs upon the ratio between EXX and semi-local exchange-energy densities. The resulting LH24x functional provides similar accuracy, e.g., for the GMTKN55 test suite, as LH20t, but without introduction of a CF! We provide detailed comparative analyses of integrated energies and spatially resolved energy densities. The good performances of LHs for chemically relevant energy differences are to some extent due to the core nature of unphysical artifacts that cancel out efficiently.

The role of the exact Hartree-Fock exchange in the investigation of defects in crystalline systems.

The role of the exact Hartree-Fock (HF) exchange in determining the band gap and other properties of defects in crystalline solids is investigated. Two defects in diamond, VHd1 and VHq1 (one first neighbour of the vacancy is saturated with hydrogen, and the three unpaired electrons combine to give a quadruplet, 3 spin up, or a doublet, two spin up and one down), are used as test cases. The results obtained with a gradient corrected functional, PBE, one range separate hybrid, HSE06, two full range hybrids, B3LYP and PBE0, and the Hartree-Fock Hamiltonian are compared. The crucial role of the exact HF exchange emerging from this comparison is confirmed by a set of calculations with a variable functional, PBE(X), in which X is the percentage of HF exchange, which is varied from 0 to 100, where PBE(0) coincides with PBE, and PBE(25) with PBE0.

A deep equivariant neural network approach for efficient hybrid density functional calculations

Hybrid density functional calculations are essential for accurate description of electronic structure, yet their widespread use is restricted by the substantial computational cost. Here we develop DeepH-hybrid, a deep equivariant neural network method for learning the hybrid-functional Hamiltonian as a function of material structure, which circumvents the time-consuming self-consistent field iterations and enables the study of large-scale materials with hybrid-functional accuracy. Our extensive experiments demonstrate good reliability as well as effective transferability and efficiency of the method. As a notable application, DeepH-hybrid is applied to study large-supercell Moiré-twisted materials, offering the first case study on how the inclusion of exact exchange affects flat bands in magic-angle twisted bilayer graphene. The work generalizes deep-learning electronic structure methods to beyond conventional density functional theory, facilitating the development of deep-learning-based ab initio methods.

Bridging electronic and classical density-functional theory using universal machine-learned functional approximations.

The accuracy of density-functional theory (DFT) calculations is ultimately determined by the quality of the underlying approximate functionals, namely the exchange-correlation functional in electronic DFT and the excess functional in the classical DFT formalism of fluids. For both electrons and fluids, the exact functional is highly nonlocal, yet most calculations employ approximate functionals that are semi-local or nonlocal in a limited weighted-density form. Machine-learned (ML) nonlocal density-functional approximations show promise in advancing applications of both electronic and classical DFTs, but so far these two distinct research areas have implemented disparate approaches with limited generality. Here, we formulate a universal ML framework and training protocol to learn nonlocal functionals that combine features of equivariant convolutional neural networks and the weighted-density approximation. We prototype this new approach for several 1D and quasi-1D problems and demonstrate that functionals with exactly the same hyperparameters achieve excellent accuracy for a diverse set of systems, including the hard-rod fluid, the inhomogeneous Ising model, the exact exchange energy of electrons, the electron kinetic energy for orbital-free DFT, as well as for liquid water with 1D inhomogeneities. These results lay the foundation for a universal ML approach to approximate exact 3D functionals spanning electronic and classical DFTs.

Density isobar of water and melting temperature of ice: Assessing common density functionals.

We investigate the density isobar of water and the melting temperature of ice using six different density functionals. Machine-learning potentials are employed to ensure computational affordability. Our findings reveal significant discrepancies between various base functionals. Notably, even the choice of damping can result in substantial differences. Overall, the outcomes obtained through density functional theory are not entirely satisfactory across most utilized functionals. All functionals exhibit significant deviations either in the melting temperature or equilibrium volume, with most of them even predicting an incorrect volume difference between ice and water. Our heuristic analysis indicates that a hybrid functional with 25% exact exchange and van der Waals damping averaged between zero and Becke-Johnson dampings yields the closest agreement with experimental data. This study underscores the necessity for further enhancements in the treatment of van der Waals interactions and, more broadly, density functional theory to enable accurate quantitative predictions for molecular liquids.

Role of Exact Exchange and Empirical Dispersion in Density Functional Theory-Based Three-Body Noncovalent Interactions.

Total and three-body interaction energies are calculated for a benchmark set of three-body systems using a range of different types of density functional theory (DFT) methods, with the results compared to CCSD(T)/CBS results from the benchmark reference [Phys. Chem. Chem. Phys. 2023, 25, 28621-28637]. Inclusion of Hartree-Fock exchange, via either a global or range-separated hybrid approach or inclusion of empirical dispersion corrections, increases accuracy for total and three-body interactions. Basis set convergence testing shows that the aug-cc-pVTZ basis set is well converged with little to no change seen when using quadruple-ζ basis sets. The accuracy of the DFT methods is similar when calculating interaction energies for both global and local minimum structures. Overall, the CAM-B3LYP-D3BJ, B97D3, and ωB97XD functionals are recommended for calculating three-body interactions.

Hollow ion atomic structure and X-ray emission in dense hot plasmas

Hollow ion X-ray emission is of great interest in high-energy-density research, since negligible opacity allows studies from the interior of very dense objects. In this paper, ionization potential depressions of the isoelectronic sequences for single and double K-shell vacancies are obtained from a pure ab initio multiconfiguration Hartree–Fock simulation including exact exchange terms and finite temperature dense plasma effects. It is demonstrated that the simultaneous representation of these ab initio data in the form of a map of hollow ion X-ray transition energies enables identification of important steps in the matter evolution and ionization dynamics. Mapping along the isoelectronic sequence as a function of the pumping energy of a X-ray free electron laser also enables visualization of the impact of ionization potential depression on the pathways of hollow ion formation.

Extensive reference set and refined computational protocol for calculations of 57Fe Mössbauer parameters.

Mössbauer spectroscopy is a powerful technique for probing the local electronic structure of iron compounds, because it reports in an element-selective manner on both the oxidation state and coordination environment of the Fe ion. Computational prediction of the two main Mössbauer parameters, isomer shift (δ) and quadrupole splitting (ΔEQ), has long been targeted by quantum chemical studies, and useful protocols based on density functional theory have been proposed. Here we present an extensive curated reference set of Fe compounds that is considerably larger and more diverse than literature precedents. We make a distinction between low-temperature and high-temperature experimental subgroups. This set is employed for optimizing a refined computational protocol utilizing the scalar version of the exact 2-component (X2C) Hamiltonian with the finite nucleus approximation. Attention is devoted to having an accurate and flexible all-electron basis set for Fe. We assess the performance of several DFT methods that cover all representative families and rungs of functionals and find that hybrid functionals with ca. 25-30% exact exchange offer the best accuracy for isomer shifts. The work establishes a refined general protocol of wide applicability that achieves good performance for the prediction of isomer shifts in a wider variety of systems than before, but the limitations of DFT for quadrupole splittings are also highlighted. Finally, comparison of calculated values with high-temperature experimental results shows that the use of an empirical correction factor is required to account for the second-order Doppler shift and to achieve the same quality of correlation as with the low-temperature data.

Use of Multigrids to Reduce the Cost of Performing Interpolative Separable Density Fitting.

In this article, we present an interpolative separable density fitting (ISDF)-based algorithm to calculate the exact exchange in periodic mean field calculations. In the past, decomposing the two-electron integrals into the tensor hypercontraction (THC) form using ISDF was the most expensive step of the entire mean field calculation. Here, we show that by using a multigrid-ISDF algorithm, both the memory and the CPU cost of this step can be reduced. The CPU cost is brought down from cubic scaling to quadratic scaling with a low computational prefactor which reduces the cost by almost 2 orders of magnitude. Thus, in the new algorithm, the cost of performing ISDF is largely negligible compared to other steps. Along with the CPU cost, the memory cost of storing the factorized two-electron integrals is also reduced by a factor of up to 35. With the current algorithm, we can perform Hartree-Fock calculations on a diamond supercell containing more than 17,000 basis functions and more than 1500 electrons on a single node with no disk usage. For this calculation, the cost of constructing the exchange matrix is only a factor of 4 slower than the cost of diagonalizing the Fock matrix. Augmenting our approach with linear scaling algorithms can further speed up the calculations.

The Staggered Mesh Method: Accurate Exact Exchange Toward the Thermodynamic Limit for Solids.

In periodic systems, the Hartree-Fock (HF) exchange energy exhibits the slowest convergence of all HF energy components as the system size approaches the thermodynamic limit. We demonstrate that the recently proposed staggered mesh method for Fock exchange energy [Xing, Li, and Lin, Math. Comp., 2024], which is specifically designed to sidestep certain singularities in exchange energy evaluation, can expedite the finite-size convergence rate for the exact exchange energy across a range of insulators and semiconductors when compared to the regular and truncated Coulomb methods. This remains true even for two computationally cheaper versions of this new method, which we call non-SCF and split-SCF staggered mesh. Additionally, a sequence of numerical tests on simple solids showcases the staggered mesh method's ability to improve convergence toward the thermodynamic limit for band gaps, bulk moduli, equilibrium lattice dimensions, energies, and phonon force constants.

Application of the Adiabatic Connection Random Phase Approximation to Electron-Nucleus Hyperfine Coupling Constants.

The electron-nucleus hyperfine coupling constant is a challenging property for density functional methods. For accurate results, hybrid functionals with a large amount of exact exchange are often needed and there is no clear "one-for-all" functional which describes the hyperfine coupling interaction for a large set of nuclei. To alleviate this unfavorable situation, we apply the adiabatic connection random phase approximation (RPA) in its post-Kohn-Sham fashion to this property as a first test. For simplicity, only the Fermi-contact and spin-dipole terms are calculated within the nonrelativistic and the scalar-relativistic exact two-component framework. This requires to solve a single coupled-perturbed Kohn-Sham equation to evaluate the relaxed density matrix, which comes with a modest increase in computational demands. RPA performs remarkably well and substantially improves upon its Kohn-Sham (KS) starting point while also reducing the dependence on the KS reference. For main-group systems, RPA outperforms global, range-separated, and local hybrid functionals─at similar computational costs. For transition-metal compounds and lanthanide complexes, a similar performance as for hybrid functionals is observed. In contrast, related post-Hartree-Fock methods such as Møller-Plesset perturbation theory or CC2 perform worse than semilocal density functionals.

Exchange-correlation potential built on the derivative discontinuity of electron density.

Electronic structures are fully determined by the exchange-correlation (XC) potential. In this work, we develop a new method to construct reliable XC potentials by properly mixing the exact exchange and the local density approximation potentials in real space. The spatially dependent mixing parameter is derived based on the derivative discontinuity of electron density and is first-principle. We derived the equations for solving the mixing parameter and proposed an approximation to simplify these equations. Based on this approximation, this new method gives reasonable predictions for the ionization energies, fundamental gaps, and singlet-triplet energy differences for various molecular systems. The impact of the approximation on the constructed XC potentials is examined, and it is found that the quality of the XC potentials can be further improved by removing the approximation. This work demonstrates that the derivative discontinuity of electron density is a promising constraint for constructing high-quality XC potentials.

Training Machine-Learned Density Functionals on Band Gaps.

The systematic underestimation of band gaps is one of the most fundamental challenges in semilocal density functional theory (DFT). In addition to hindering the application of DFT to predicting electronic properties, the band gap problem is intimately related to self-interaction and delocalization errors, which make the study of charge transfer mechanisms with DFT difficult. To expand the range of available tools for addressing the band gap problem, we design an approach for machine learning density functionals based on Gaussian processes to explicitly fit single-particle energy levels. We also introduce nonlocal features of the density matrix that are expressive enough to fit these single-particle levels. Combining these developments, we train a machine-learned functional for the exact exchange energy that predicts molecular energy gaps and reaction energies of a wide range of molecules in excellent agreement with reference hybrid DFT calculations. In addition, while being trained solely on molecular data, our model predicts reasonable formation energies of polarons in solids, showcasing its transferability and robustness. We discuss how this approach can be generalized to full exchange-correlation functionals, thus paving the way to the design of state-of-the-art functionals for the prediction of electronic properties of molecules and materials.

New global minimum conformers for the Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document} and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document} clusters: low symmetric species featuring different active sites

ContextThe study of platinum (Pt) clusters and nanoparticles is essential due to their extensive range of potential technological applications, particularly in catalysis. The electronic properties that yield optimal catalytic performance at the nanoscale are significantly influenced by the size and structure of Pt clusters. This research aimed to identify the lowest-energy conformers for Pt18\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{18}$$\\end{document}, Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document}, and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document} species using Density Functional Theory (DFT). We discovered new low-symmetry conformers for Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document} and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document}, which are 3.0 and 1.0 kcal/mol more stable, respectively, than previously reported structures. Our study highlights the importance of using density functional approximations that incorporate moderate levels of exact Hartree-Fock exchange, alongside basis sets of at least quadruple-zeta quality. The resulting structures are asymmetric with varying active sites, as evidenced by sigma hole analysis on the electrostatic potential surface. This suggests a potential correlation between electronic structure and catalytic properties, warranting further investigation.MethodsAn equivariant graph neural network interatomic potential (NequIP) within the Atomic Simulation Environment suite (ASE) was used to provide initial geometries of the aggregates under study. DFT calculations were performed with the ORCA 5 package, using functional approximations that included Generalized Gradient Approximation (PBE), meta-GGA (TPSS, M06-L), hybrid (PBE0, PBEh), meta-GGA hybrid (TPSSh), and range-separated hybrid (ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}B97x) functionals. Def2-TZVP and Def2-QZVP as well as members of the cc-pwCVXZ-PP family to check basis set convergence were used. QTAIM calculations were performed using the AIMAll suite. Structures were visualized with the AVOGADRO code.

Geometric and electronic structures of Cs2BB′X6 double perovskites: The importance of exact exchange

A widely adopted computational protocol in contemporary materials research is to first relax materials' geometries using semilocal density functional approximations (DFA), and then determining their electronic band structures using the more expensive hybrid functionals. This procedure often works well, as the popular semilocal DFAs, such as the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation, yield rather good geometries for a wide range of materials. However, here we show that, for some of the lead-free halide double perovskites (HDPs) Cs2BB′X6 (B=Ag+, Na+; B′=In3+, Bi3+; X=Cl−, Br−), the validity of this common practice is questionable. We find that, for these HDPs, the geometrical structures, in particular, the B(B′)−X bond lengths predicted by PBE show large deviations from the experimental values. Additionally, the band gaps of some of these materials (specifically, the In-based HDPs) are sensitive to the B(B′)−X bond lengths. As a consequence, the band gaps obtained using the hybrid functionals (such as the Heyd-Scuseria-Ernzerhof functional) based on the PBE geometries can still be quite off, in particular, for HDPs with B′=In3+. The situation is significantly improved by using hybrid functionals with tuned portion of exact exchange, based on the geometries determined consistently under the same level of theory. The successes and failures of several popular exchange-correlation (XC) functionals are traced back to the so-called delocalization error, and can be quantitatively analyzed and understood via a three-atom linear-chain B−X−B′ molecular model. Finally, our findings provide a practical guide for choosing appropriate XC functionals for describing HDPs and point to a promising path for band structure engineering via doping and alloying. Published by the American Physical Society 2024

Reaction barriers at metal surfaces computed using the random phase approximation: Can we beat DFT in the generalized gradient approximation?

Reaction barriers for molecules dissociating on metal surfaces (as relevant to heterogeneous catalysis) are often difficult to predict accurately with density functional theory (DFT). Although the results obtained for several dissociative chemisorption reactions via DFT in the generalized gradient approximation (GGA), in meta-GGA, and for GGA exchange + van der Waals correlation scatter around the true reaction barrier, there is an entire class of dissociative chemisorption reactions for which GGA-type functionals collectively underestimate the reaction barrier. Little is known why GGA-DFT collectively fails in some cases and not in others, and we do not know whether other methods suffer from the same inconsistency. Here, we present barrier heights for dissociative chemisorption reactions obtained from the random phase approximation in the adiabatic-connection fluctuation-dissipation theorem (ACFDT-RPA) and from hybrid functionals with different amounts of exact exchange. By comparing the results obtained for the dissociative chemisorption reaction of H2 on Al(110) (where GGA-DFT collectively underestimates the barrier) and H2 on Cu(111) (where GGA-DFT scatters around the true barrier), we can gauge whether the inconsistent description of the systems persists for hybrid functionals and ACFDT-RPA. We find hybrid functionals to improve the relative description of the two systems, but to fall short of chemical accuracy. ACFDT-RPA improves the results further and leads to chemically accurate barriers for both systems. Together with an analysis of the density of states and the results from selected GGA, meta-GGA, and GGA exchange + van der Waals correlation functionals, these results allow us to discuss possible origins for the inconsistent behavior of GGA-based functionals for molecule-metal reaction barriers.

A new parameterization of the DFT/CIS method with applications to core-level spectroscopy.

Time-dependent density functional theory (TD-DFT) within a restricted excitation space is an efficient means to compute core-level excitation energies using only a small subset of the occupied orbitals. However, core-to-valence excitation energies are significantly underestimated when standard exchange-correlation functionals are used, which is partly traceable to systemic issues with TD-DFT's description of Rydberg and charge-transfer excited states. To mitigate this, we have implemented an empirically modified combination of configuration interaction with single substitutions (CIS) based on Kohn-Sham orbitals, which is known as "DFT/CIS." This semi-empirical approach is well-suited for simulating x-ray near-edge spectra, as it contains sufficient exact exchange to model charge-transfer excitations yet retains DFT's low-cost description of dynamical electron correlation. Empirical corrections to the matrix elements enable semi-quantitative simulation of near-edge x-ray spectra without the need for significant a posteriori shifts; this should be useful in complex molecules and materials with multiple overlapping x-ray edges. Parameter optimization for use with a specific range-separated hybrid functional makes this a black-box method intended for both core and valence spectroscopy. Results herein demonstrate that realistic K-edge absorption and emission spectra can be obtained for second- and third-row elements and 3d transition metals, with promising results for L-edge spectra as well. DFT/CIS calculations require absolute shifts that are considerably smaller than what is typical in TD-DFT.

A Visual Representation for Accurate Local Basis Set Construction and Optimization: A Case Study of SrTiO3 with Hybrid DFT Functionals

The linear combination of atomic orbitals (LCAO) method is advantageous for calculating important bulk and surface properties of crystals and defects in/on them. Compared to plane wave calculations and contrary to common assumptions, hybrid density functional theory (DFT) functionals are actually less costly and easier to implement in LCAO codes. However, choosing the proper basis set (BS) for the LCAO calculations representing Guassian-type functions is crucial, as the results depend heavily on its quality. In this study, we introduce a new basis set (BS) visual representation, which helps us (1) analyze the collective behavior of individual atoms’ shell exponents (s, p, and d), (2) better compare different BSs, (3) identify atom-type invariant relationships, and (4) suggest a robust method for building a local all-electron BS (denoted as BS1) from scratch for each atom type. To compare our BS1 with the others existing in the literature, we calculate the basic bulk properties of SrTiO3 (STO) in cubic and tetragonal phases using several hybrid DFT functionals (B3LYP, PBE0, and HSE06). After adjusting the exact Hartree–Fock (HF) exchange of PBEx, HSEx, and the state-of-the-art meta-GGA hybrid r2SCANx functionals, we find the r2SCAN15 and HSE27 for BS1, with the amount of exact HF exchange of 0.15 and 0.27, respectively, perform equally well for reproducing several most relevant STO properties. The proposed robust BS construction scheme has the advantage that all parameters of the obtained BS can be reoptimized for each new material, thus increasing the quality of DFT calculation predictions.

Efficient all-electron hybrid density functionals for atomistic simulations beyond 10 000 atoms.

Hybrid density functional approximations (DFAs) offer compelling accuracy for abinitio electronic-structure simulations of molecules, nanosystems, and bulk materials, addressing some deficiencies of computationally cheaper, frequently used semilocal DFAs. However, the computational bottleneck of hybrid DFAs is the evaluation of the non-local exact exchange contribution, which is the limiting factor for the application of the method for large-scale simulations. In this work, we present a drastically optimized resolution-of-identity-based real-space implementation of the exact exchange evaluation for both non-periodic and periodic boundary conditions in the all-electron code FHI-aims, targeting high-performance central processing unit (CPU) compute clusters. The introduction of several new refined message passing interface (MPI) parallelization layers and shared memory arrays according to the MPI-3 standard were the key components of the optimization. We demonstrate significant improvements of memory and performance efficiency, scalability, and workload distribution, extending the reach of hybrid DFAs to simulation sizes beyond ten thousand atoms. In addition, we also compare the runtime performance of the PBE, HSE06, and PBE0 functionals. As a necessary byproduct of this work, other code parts in FHI-aims have been optimized as well, e.g., the computation of the Hartree potential and the evaluation of the force and stress components. We benchmark the performance and scaling of the hybrid DFA-based simulations for a broad range of chemical systems, including hybrid organic-inorganic perovskites, organic crystals, and ice crystals with up to 30 576 atoms (101 920 electrons described by 244 608 basis functions).