Double hybrid DFT calculations with Slater type orbitals

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

Hybrid density functional approximations (DFAs) offer compelling accuracy for ab initio 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).

Exploring CPS-Extrapolated DLPNO-CCSD(T1) Reference Values for Benchmarking DFT Methods on Enzymatically Catalyzed Reactions

Domain-based local pair natural orbital coupled-cluster singles doubles with perturbative triples [DLPNO-CCSD(T)] is regularly used to calculate reliable benchmark reference values at a computational cost significantly lower than that of canonical CCSD(T). Recent work has shown that even greater accuracy can be obtained at only a small additional cost through extrapolation to the complete PNO space (CPS) limit. Herein, we test two levels of CPS extrapolation, CPS(5,6), which approximates the accuracy of standard TightPNO, and CPS(6,7), which surpasses it, as benchmark values to test density functional approximations (DFAs) on a small set of organic and transition-metal-dependent enzyme active site models. Between the different reference levels of theory, there are changes in the magnitudes of the absolute deviations for all functionals, but these are small and there is minimal impact on the relative rankings of the tested DFAs. The differences are more significant for the metalloenzymes than the organic enzymes, so we repeat the tests on our entire ENZYMES22 set of organic enzyme active site models [Wappett, D. A.; Goerigk, L. J. Phys. Chem. A 2019, 123, 7057-7074] to confirm that using the CPS extrapolations for the reference values has negligible impact on the benchmarking outcomes. This means that we can particularly recommend CPS(5,6) as an alternative to standard TightPNO settings for calculating reference values, increasing the applicability of DLPNO-CCSD(T) in benchmarking reaction energies and barrier heights of larger models of organic enzymes. DLPNO-CCSD(T1)/CPS(6,7) energies for ENZYMES22 are finally presented as updated reference values for the set, reflecting the recent improvements in the method.

Chemically accurate singlet-triplet gaps of organic chromophores and linear acenes by the random phase approximation and σ-functionals

Predicting the energy differences between different spin-states is challenging for many widely used ab initio electronic structure methods. We here assess the ability of the direct random phase approximation (dRPA), dRPA plus two different screened second-order exchange (SOX) corrections, and σ-functionals to predict adiabatic singlet-triplet gaps. With mean absolute deviations of below 0.1 eV to experimental reference values, independent of the Kohn-Sham starting point, dRPA and σ-functionals accurately predict singlet-triplet gaps of 18 organic chromophores. The addition of SOX corrections to dRPA considerably worsens agreement with experiment, adding to the mounting evidence that dRPA+SOX methods are not generally applicable beyond-RPA methods. Also for a series of linear acene chains with up to ten fused rings, dRPA, and σ-functionals are in excellent agreement with coupled-cluster single double triple reference data. In agreement with advanced multi-reference methods, dRPA@PBE and σ-functional@PBE predict a singlet ground state for all chain lengths, while dRPA@PBE0 and σ-functional@PBE0 predict a triplet ground state for longer acenes. Our work shows dRPA and σ-functionals to be reliable methods for calculating singlet-triplet gaps in aromatic molecules.

Regularized and Opposite Spin-Scaled Functionals from Møller-Plesset Adiabatic Connection─Higher Accuracy at Lower Cost

Noncovalent interactions (NCIs) play a crucial role in biology, chemistry, material science, and everything in between. To improve pure quantum-chemical simulations of NCIs, we propose a methodology for constructing approximate correlation energies by combining an interpolation along the Møller-Plesset adiabatic connection (MP AC) with a regularization and spin-scaling strategy applied to MP2 correlation energies. This combination yields cosκos-SPL2, which exhibits superior accuracy for NCIs compared to any of the individual strategies. With the N4 formal scaling, cosκos-SPL2 is competitive or often outperforms more expensive dispersion-corrected double hybrids for NCIs. The accuracy of cosκos-SPL2 particularly shines for anionic halogen bonded complexes, where it surpasses standard dispersion-corrected DFT by a factor of 3 to 5.

The numerical evaluation of Slater integrals on graphics processing units

This article presents SlaterGPU, a graphics processing unit (GPU) accelerated library that uses OpenACC to numerically compute Slater-type orbital (STO) integrals. The electron repulsion integrals (ERI) are computed under the RI approximation using the Coulomb potential of the Slater basis function. To fully realize the performance capabilities of modern GPUs, the Slater integrals are evaluated in mixed-precision, resulting in speedups for the ERIs of over 80×. Parallelization on multiple GPUs allows for integral throughput of over 3 million integrals per second. This places STO integral throughput within reach of single-threaded, conventional Gaussian integration schemes. To test the quality of the integrals, the fluorine exchange reaction barrier in fluoromethane was computed using heat-bath configuration interaction (HBCI). In addition, the singlet-triplet gap of cyclobutadiene was examined using HBCI in a triple- ζ $$ \zeta $$ , polarized basis set. These benchmarks demonstrate the library's ability to generate the full set of integrals necessary for configuration interaction with up to 6 h $$ 6h $$ functions in the auxiliary basis.

Optimization of the r2SCAN-3c Composite Electronic-Structure Method for Use with Slater-Type Orbital Basis Sets

The "Swiss army knife" composite density functional electronic-structure method r²SCAN-3c (J. Chem. Phys. 2021, 154, 064103) is extended and optimized for the use with Slater-type orbital basis sets. The meta generalized-gradient approximation (meta-GGA) functional r²SCAN by Furness et al. is combined with a tailor-made polarized triple-ζ Slater-type atomic orbital (STO) basis set (mTZ2P), the semiclassical London dispersion correction (D4), and a geometrical counterpoise (gCP) correction. Relativistic effects are treated explicitly with the scalar-relativistic zeroth-order regular approximation (SR-ZORA). The performance of the new implementation is assessed on eight geometry and 74 energy benchmark sets, including the extensive GMTKN55 database as well as recent sets such as ROST61 and IONPI19. In geometry optimizations, the STO-based r²SCAN-3c is either on par with or more accurate than the hybrid density functional approximation M06-2X-D3(0)/TZP. In energy calculations, the overall accuracy is similar to the original implementation of r²SCAN-3c with Gaussian-type atomic orbitals (GTO), but basic properties, intermolecular noncovalent interactions, and barrier heights are better described with the STO approach, resulting in a lower weighted mean absolute deviation (WTMAD-2(STO) = 7.15 vs 7.50 kcal mol^-1 with the original method) for the GMTKN55 database. The STO-optimized r²SCAN-3c outperforms many conventional hybrid/QZ approaches in most common applications at a fraction of their cost. The reliable, robust, and accurate r²SCAN-3c implementation with STOs is a promising alternative to the original implementation with GTOs and can be generally used for a broad field of quantum chemical problems.

Stability of the polyynic form of C18, C22, C26, and C30 nanorings: a challenge tackled by range-separated double-hybrid density functionals

We calculate the relative energy between the cumulene and polyyne structures of a set of C_4k+2 (k = 4-7) rings (C₁₈, C₂₂, C₂₆, and C₃₀ prompted by the recent synthesis of the cyclo[18]carbon (or simply C₁₈) compounds. Reference results were obtained by a costly Quantum Monte-Carlo (QMC) approach, providing thus very accurate values allowing to systematically compare the performance of a variety of wavefunction methods [(i.e., MP2, SCS-MP2, SOS-MP2, DLPNO-CCSD, and DLPNO-CCSD(T)] as well as DFT approaches, applying for the latter a diversity of density functionals covering global and range-separated hybrid and double-hybrid models. The influence of the use of a range-separation scheme for density functionals, for both hybrid and double-hybrid expressions, is discussed according to its key role. Overall, range-separated double-hybrid functionals (e.g., RSX-QIDH) behave very accurately and provide competitive results compared with DLPNO-CCSD(T), at a more reasonable computational cost.

Efficient workflow for the investigation of the catalytic cycle of water oxidation catalysts: Combining GFN-xTB and density functional theory

Photocatalytic water oxidation remains the bottleneck in many artificial photosynthesis devices. The efficiency of this challenging process is inherently linked to the thermodynamic and electronic properties of the chromophore and the water oxidation catalyst (WOC). Computational investigations can facilitate the search for favorable chromophore‐catalyst combinations. However, this remains a demanding task due to the requirements on the computational method that should be able to correctly describe different spin and oxidation states of the transition metal, the influence of solvation and the different rates of the charge transfer and water oxidation processes. To determine a suitable method with favorable cost/accuracy ratios, the full catalytic cycle of a molecular ruthenium based WOC is investigated using different computational methods, including density functional theory (DFT) with different functionals (GGA, Hybrid, Double Hybrid) as well as the semi‐empirical tight binding approach GFN‐xTB. A workflow with low computational cost is proposed that combines GFN‐xTB and DFT and provides reliable results. GFN‐xTB geometries and frequencies combined with single‐point DFT energies give free energy changes along the catalytic cycle that closely follow the full DFT results and show satisfactory agreement with experiment, while significantly decreasing the computational cost. This workflow allows for cost efficient determination of energetic, thermodynamic and dynamic properties of WOCs.

A Computational Protocol Combining DFT and Cheminformatics for Prediction of pH-Dependent Redox Potentials

Discovering new materials for energy storage requires reliable and efficient protocols for predicting key properties of unknown compounds. In the context of the search for new organic electrolytes for redox flow batteries, we present and validate a robust procedure to calculate the redox potentials of organic molecules at any pH value, using widely available quantum chemistry and cheminformatics methods. Using a consistent experimental data set for validation, we explore and compare a few different methods for calculating reaction free energies, the treatment of solvation, and the effect of pH on redox potentials. We find that the B3LYP hybrid functional with the COSMO solvation method, in conjunction with thermal contributions evaluated from BLYP gas-phase harmonic frequencies, yields a good prediction of pH = 0 redox potentials at a moderate computational cost. To predict how the potentials are affected by pH, we propose an improved version of the Alberty-Legendre transform that allows the construction of a more realistic Pourbaix diagram by taking into account how the protonation state changes with pH.

Linear-Scaling Open-Shell MP2 Approach: Algorithm, Benchmarks, and Large-Scale Applications

A linear-scaling local second-order Møller–Plesset (MP2) method is presented for high-spin open-shell molecules based on restricted open-shell (RO) reference functions. The open-shell local MP2 (LMP2) approach inherits the iteration- and redundancy-free formulation and the completely integral-direct, OpenMP-parallel, and memory and disk use economic algorithms of our closed-shell LMP2 implementation. By utilizing restricted local molecular orbitals for the demanding integral transformation step and by introducing a novel long-range spin-polarization approximation, the computational cost of RO-LMP2 approaches that of closed-shell LMP2. Extensive benchmarks were performed for reactions of radicals, ionization potentials, as well as spin-state splittings of carbenes and transition-metal complexes. Compared to the conventional MP2 reference for systems of up to 175 atoms, local errors of at most 0.1 kcal/mol were found, which are well below the intrinsic accuracy of MP2. RO-LMP2 computations are presented for challenging protein models of up to 601 atoms and 11 000 basis functions, which involve either spin states of a complexed iron ion or a highly delocalized singly occupied orbital. The corresponding runtimes of 9–15 h obtained with a single, many-core CPU demonstrate that MP2, as well as spin-scaled MP2 and double-hybrid density functional methods, become widely accessible for open-shell systems of unprecedented size and complexity.

Assessing challenging intra- and inter-molecular charge-transfer excitations energies with double-hybrid density functionals

We investigate the performance of a set of recently introduced range-separated double-hybrid functionals, namely ωB2-PLYP, ωB2GP-PLYP, RSX-0DH, and RSX-QIDH models for hard-to-calculate excitation energies. We compare with the parent (B2-PLYP, B2GP-PLYP, PBE0-DH, and PBE-QIDH) and other (DSD-PBEP86) double-hybrid models as well as with some of the most widely employed hybrid functionals (B3LYP, PBE0, M06-2X, and ωB97X). For this purpose, we select a number of medium-sized intra- and inter-molecular charge-transfer excitations, which are known to be challenging to calculate using time-dependent density-functional theory (TD-DFT) and for which accurate reference values are available. We assess whether the high accuracy shown by the newest double-hybrid models is also confirmed for those cases too. We find that asymptotically corrected double-hybrid models yield a superior performance, especially for the inter-molecular charge-transfer excitation energies, as compared to standard double-hybrid models. Overall, the PBE-QIDH and its corresponding range-separated RSX-QIDH functional are recommended for general-purpose TD-DFT applications, depending on whether long-range effects are expected to play a significant role.

Low-Order Scaling G0W0 by Pair Atomic Density Fitting

We derive a low-scaling G0W0 algorithm for molecules using pair atomic density fitting (PADF) and an imaginary time representation of the Green's function and describe its implementation in the Slater type orbital (STO)-based Amsterdam density functional (ADF) electronic structure code. We demonstrate the scalability of our algorithm on a series of water clusters with up to 432 atoms and 7776 basis functions and observe asymptotic quadratic scaling with realistic threshold qualities controlling distance effects and basis sets of triple-ζ (TZ) plus double polarization quality. Also owing to a very small prefactor, a G0W0 calculation for the largest of these clusters takes only 240 CPU hours with these settings. We assess the accuracy of our algorithm for HOMO and LUMO energies in the GW100 database. With errors of 0.24 eV for HOMO energies on the quadruple-ζ level, our implementation is less accurate than canonical all-electron implementations using the larger def2-QZVP GTO-type basis set. Apart from basis set errors, this is related to the well-known shortcomings of the GW space-time method using analytical continuation techniques as well as to numerical issues of the PADF approach of accurately representing diffuse atomic orbital (AO) products. We speculate that these difficulties might be overcome by using optimized auxiliary fit sets with more diffuse functions of higher angular momenta. Despite these shortcomings, for subsets of medium and large molecules from the GW5000 database, the error of our approach using basis sets of TZ and augmented double-ζ (DZ) quality is decreasing with system size. On the augmented DZ level, we reproduce canonical, complete basis set limit extrapolated reference values with an accuracy of 80 meV on average for a set of 20 large organic molecules. We anticipate our algorithm, in its current form, to be very useful in the study of single-particle properties of large organic systems such as chromophores and acceptor molecules.

Double hybrid DFT calculations with Slater type orbitals

On a comprehensive database with 1,644 datapoints, covering several aspects of main-group as well as of transition metal chemistry, we assess the performance of 60 density functional approximations (DFA), among them 36 double hybrids (DH). All calculations are performed using a Slater type orbital (STO) basis set of triple-ζ (TZ) quality and the highly efficient pair atomic resolution of the identity approach for the exchange- and Coulomb-term of the KS matrix (PARI-K and PARI-J, respectively) and for the evaluation of the MP2 energy correction (PARI-MP2). Employing the quadratic scaling SOS-AO-PARI-MP2 algorithm, DHs based on the spin-opposite-scaled (SOS) MP2 approximation are benchmarked against a database of large molecules. We evaluate the accuracy of STO/PARI calculations for B3LYP as well as for the DH B2GP-PLYP and show that the combined basis set and PARI-error is comparable to the one obtained using the well-known def2-TZVPP Gaussian-type basis set in conjunction with global density fitting. While quadruple-ζ (QZ) calculations are currently not feasible for PARI-MP2 due to numerical issues, we show that, on the TZ level, Jacob's ladder for classifying DFAs is reproduced. However, while the best DHs are more accurate than the best hybrids, the improvements are less pronounced than the ones commonly found on the QZ level. For conformers of organic molecules and noncovalent interactions where very high accuracy is required for qualitatively correct results, DHs provide only small improvements over hybrids, while they still excel in thermochemistry, kinetics, transition metal chemistry and the description of strained organic systems.