Highly accurate first-principles benchmark data sets for the parametrization and validation of density functional and other approximate methods. Derivation of a robust, generally applicable, double-hybrid functional for thermochemistry and thermochemical kinetics

Supervised learning of a chemistry functional with damped dispersion

Kohn-Sham density functional theory is widely used in chemistry, but no functional can accurately predict the whole range of chemical properties, although recent progress by some doubly hybrid functionals comes close. Here, we optimized a singly hybrid functional called CF22D with higher across-the-board accuracy for chemistry than most of the existing non-doubly hybrid functionals by using a flexible functional form that combines a global hybrid meta-nonseparable gradient approximation that depends on density and occupied orbitals with a damped dispersion term that depends on geometry. We optimized this energy functional by using a large database and performance-triggered iterative supervised training. We combined several databases to create a very large, combined database whose use demonstrated the good performance of CF22D on barrier heights, isomerization energies, thermochemistry, noncovalent interactions, radical and nonradical chemistry, small and large systems, simple and complex systems and transition-metal chemistry.

Spin-State Splittings in 3d Transition-Metal Complexes Revisited: Benchmarking Approximate Methods for Adiabatic Spin-State Energy Differences in Fe(II) Complexes

The CASPT2+δMRCI composite approach reported in a companion paper has been extended and used to provide high-quality reference data for a series of adiabatic spin gaps (defined as ΔE = E_quintet - E_singlet) of [Fe^IIL₆]²⁺ complexes (L = CNH, CO, NCH, NH₃, H₂O), either at nonrelativistic level or including scalar relativistic effects. These highly accurate data have been used to evaluate the performance of various more approximate methods. Coupled-cluster theory with singles, doubles, and perturbative triples, CCSD(T), is found to agree well with the new reference data for Werner-type complexes but exhibits larger underestimates by up to 70 kJ/mol for the π-acceptor ligands, due to appreciable static correlation in the low-spin states of these systems. Widely used domain-based local CCSD(T) calculations, DLPNO-CCSD(T), are shown to depend very sensitively on the cutoff values used to construct the localized domains, and standard values are not sufficient. A large number of density functional approximations have been evaluated against the new reference data. The B2PLYP double hybrid gives the smallest deviations, but several functionals from different rungs of the usual ladder hierarchy give mean absolute deviations below 20 kJ/mol. This includes the B97-D semilocal functional, the PBE0* global hybrid with 15% exact-exchange admixture, as well as the local hybrids LH07s-SVWN and LH07t-SVWN. Several further functionals achieve mean absolute errors below 30 kJ/mol (M06L-D4, SSB-D, B97-1-D4, LC-ωPBE-D4, LH12ct-SsirPW92-D4, LH12ct-SsifPW92-D4, LH14t-calPBE-D4, LHJ-HFcal-D4, and several further double hybrids) and thereby also still overall outperform CCSD(T) or uncorrected CASPT2. While exact-exchange admixture is a crucial factor in favoring high-spin states, the present evaluations confirm that other aspects can be important as well. A number of the better-performing functionals underestimate the spin gaps for the π-acceptor ligands but overestimate them for L = NH₃, H₂O. In contrast to a previous suggestion, non-self-consistent density functional theory (DFT) computations on top of Hartree-Fock orbitals are not a promising path to produce accurate spin gaps in such complexes.

Barrier heights, reaction energies and bond dissociation energies for RH + HO2 reactions with coupled-cluster theory, density functional theory and diffusion quantum Monte Carlo methods

Hydrogen abstraction reactions by the HO₂ radical from hydrocarbon molecules are an important class of reactions in the autoignition of hydrocarbon fuels. Performance of DLPNO-CC and DFT methods using three hybrids and four double hybrids as well as FN-DMC with the single-Slater-Jastrow trial wavefunction on barrier heights and reaction energies of RH + HO₂ reactions as well as bond dissociation energies of the involved X-H molecules is evaluated by comparison with the highly accurate CCSD(T)-F12b/CBS results in this study. Our results show that the DLPNO-CCSD(T)-F12 method can achieve highly accurate barrier heights, reaction energies and X-H bond energies for RH + HO₂ reactions at a relatively low computational cost, and it is applicable to the H-abstraction reactions of larger molecules. Among all DFAs, MN15 and the employed double hybrids can achieve accurate barrier heights and reaction energies with MADs of less than or around 2 kJ mol^-1, but their error on X-H bond energies is more pronounced. Only DSD-BLYP and DSD-PBEB95 can provide X-H bond energies with MADs less than 4 kJ mol^-1. Considering dispersion correction in DFT calculations does not improve these barrier heights and reaction energies. The error of FN-DMC on barrier heights and reaction energies is slightly larger than that of MN15 and those of double hybrids, but it can achieve results within chemical accuracy for these reactions and the X-H bond energies.

Systematic Evaluation of Counterpoise Correction in Density Functional Theory

A widespread belief persists that the Boys-Bernardi function counterpoise (CP) procedure "overcorrects" supramolecular interaction energies for the effects of basis-set superposition error. To the extent that this is true for correlated wave function methods, it is usually an artifact of low-quality basis sets. The question has not been considered systematically in the context of density functional theory, however, where basis-set convergence is generally less problematic. We present a systematic assessment of the CP procedure for a representative set of functionals and basis sets, considering both benchmark data sets of small dimers and larger supramolecular complexes. The latter include layered composite polymers with ∼150 atoms and ligand-protein models with ∼300 atoms. Provided that CP correction is used, we find that intermolecular interaction energies of nearly complete-basis quality can be obtained using only double-ζ basis sets. This is less expensive as compared to triple-ζ basis sets without CP correction. CP-corrected interaction energies are less sensitive to the presence of diffuse basis functions as compared to uncorrected energies, which is important because diffuse functions are expensive and often numerically problematic for large systems. Our results upend the conventional wisdom that CP "overcorrects" for basis-set incompleteness. In small basis sets, CP correction is mandatory in order to demonstrate that the results do not rest on error cancellation.

Reduced-Scaling Double Hybrid Density Functional Theory with Rapid Basis Set Convergence through Localized Pair Natural Orbital F12

Following earlier work [Mehta, N.; Martin, J. M. L. J. Chem. Theory Comput.2022, 10.1021/acs.jctc.2c00426] that showed how the slow basis set convergence of the double hybrid density functional theory can be obviated by the use of F12 explicit correlation in the GLPT2 step (second order Görling-Levy perturbation theory), we demonstrate here for the very large and chemically diverse GMTKN55 benchmark suite that the CPU time scaling of this step can be reduced (asymptotically linearized) using the localized pair natural orbital (PNO-L) approximation at negligible cost in accuracy.

Explicitly Correlated Double-Hybrid DFT: A Comprehensive Analysis of the Basis Set Convergence on the GMTKN55 Database

Double-hybrid density functional theory (DHDFT) offers a pathway to accuracy approaching composite wavefunction approaches such as G4 theory. However, the Görling-Levy second-order perturbation theory (GLPT2) term causes them to partially inherit the slow ∝L-3 (with L the maximum angular momentum) basis set convergence of correlated wavefunction methods. This could potentially be remedied by introducing F12 explicit correlation: we investigate the basis set convergence of both DHDFT and DHDFT-F12 (where GLPT2 is replaced by GLPT2-F12) for the large and chemically diverse general main-group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) benchmark suite. The B2GP-PLYP-D3(BJ) and revDSD-PBEP86-D4 DHDFs are investigated as test cases, together with orbital basis sets as large as aug-cc-pV5Z and F12 basis sets as large as cc-pVQZ-F12. We show that F12 greatly accelerates basis set convergence of DHDFs, to the point that even the modest cc-pVDZ-F12 basis set is closer to the basis set limit than cc-pV(Q+d)Z or def2-QZVPPD in orbital-based approaches, and in fact comparable in quality to cc-pV(5+d)Z. Somewhat surprisingly, aug-cc-pVDZ-F12 is not required even for the anionic subsets. In conclusion, DHDF-F12/VDZ-F12 eliminates concerns about basis set convergence in both the development and applications of double-hybrid functionals. Mass storage and I/O bottlenecks for larger systems can be circumvented by localized pair natural orbital approximations, which also exhibit much gentler system size scaling.

Automatic generation of complementary auxiliary basis sets for explicitly correlated methods

Explicitly correlated calculations, aside from the orbital basis set, typically require three auxiliary basis sets: Coulomb-exchange fitting (JK), resolution of the identity MP2 (RI-MP2), and complementary auxiliary basis set (CABS). If unavailable for the orbital basis set and chemical elements of interest, the first two can be auto-generated on the fly using existing algorithms, but not the third. In this paper, we present a quite simple algorithm named autoCABS; a Python implementation under a free software license is offered at Github. For the cc-pVnZ-F12 (n = D,T,Q,5), the W4-08 thermochemical benchmark, and the HFREQ2014 set of harmonic frequencies, we demonstrate that autoCABS-generated CABS basis sets are comparable in quality to purpose-optimized OptRI basis sets from the literature, and that the quality difference becomes entirely negligible as n increases.

Benefits of Range-Separated Hybrid and Double-Hybrid Functionals for a Large and Diverse Data Set of Reaction Energies and Barrier Heights

To better understand the thermochemical kinetics and mechanism of a specific chemical reaction, an accurate estimation of barrier heights (forward and reverse) and reaction energies is vital. Because of the large size of reactants and transition state structures involved in real-life mechanistic studies (e.g., enzymatically catalyzed reactions), density functional theory remains the workhorse for such calculations. In this paper, we have assessed the performance of 91 density functionals for modeling the reaction energies and barrier heights on a large and chemically diverse data set (BH9) composed of 449 organic chemistry reactions. We have shown that range-separated hybrid functionals perform better than the global hybrids for BH9 barrier heights and reaction energies. Except for the PBE-based range-separated nonempirical double hybrids, range separation of the exchange term helps improve the performance for barrier heights and reaction energies. The 16-parameter Berkeley double hybrid, ωB97M(2), performs remarkably well for both properties. However, our minimally empirical range-separated double hybrid functionals offer marginally better accuracy than ωB97M(2) for BH9 barrier heights and reaction energies.

Molecular Structures of the Silicon Pyridine-2-(thi)olates Me3Si(pyX), Me2Si(pyX)2 and Ph2Si(pyX)2 (py = 2-Pyridyl, X = O, S), and Their Intra- and Intermolecular Ligand Exchange in Solution

A series of pyridine-2-olates (pyO) and pyridine-2-thiolates (pyS) of silicon was studied in solid state and in solution. The crystal structures of Me3Si(pyO) (1a), Me3Si(pyS) (1b), Me2Si(pyO)2 (2a), Me2Si(pyS)2 (2b), Ph2Si(pyO)2 (3a) and Ph2Si(pyS)2 (3b) were determined by X-ray diffraction. For that purpose, crystals of the (at room temperature) liquid compounds 1a and 1b were grown in a capillary on the diffractometer. Compounds 1a, 1b, 2a, 2b and 3a feature tetracoordinate silicon atoms in the solid state, whereas 3b gave rise to a series of four crystal structures in which the Si atoms of this compound are hexacoordinate. Two isomers (3b1 with all-cis arrangement of the C2N2S2 donor atoms in P, and 3b2 with trans S-Si-S axis in P21/n) formed individual crystal batches, which allowed for their individual 29Si NMR spectroscopic study in the solid state (the determination of their chemical shift anisotropy tensors). Furthermore, the structures of a less stable modification of 3b2 (in C2/c) as well as a toluene solvate 3b2×(toluene) (in P) were determined. In CDCl3, the equimolar solutions of the corresponding pairs of pyO and pyS compounds (2a/2b and 3a/3b) showed substituent scrambling with the formation of the products Me2Si(pyO)(pyS) (2c) and Ph2Si(pyO)(pyS) (3c), respectively, as minor components in the respective substituent exchange equilibrium.

Fullerenes Pose a Strain on Hybrid Density Functional Theory

The computational modeling of fullerenes plays a fundamental role in designing low-dimension carbon nanostructures. Nevertheless, the relative energies of fullerenes larger than C20 and C24 have not been comprehensively examined by means of highly accurate ab initio methods, for example, the CCSD(T) method. Here we report such an investigation for a diverse set of 29 C40 isomers. We calculate the energies of the C40 fullerenes using the G4(MP2) composite ab initio method, which approximates the CCSD(T) energy in conjunction with a triple-ζ-quality basis set (CCSD(T)/TZ). The CCSD(T)/TZ isomerization energies span 43.1-763.3 kJ mol-1. We find a linear correlation (R2 = 0.96) between the CCSD(T)/TZ isomerization energies and the fullerene pentagon signatures (P1 index), which reflect the strain associated with fused pentagon-pentagon rings. Using the reference CCSD(T)/TZ isomerization energies, we examine the relationship between the percentage of exact Hartree-Fock (HF) exchange in hybrid density functional theory (DFT) methods and the pentagon-pentagon strain energies. We find that the performance of hybrid DFT methods deteriorates with the pentagon-pentagon strain energy. This deterioration in performance becomes more pronounced with the inclusion of high amounts of HF exchange. For example, for B3LYP (20% HF exchange), the root-mean-square deviation (RMSD) relative to G4(MP2) increases from 8.9 kJ mol-1 for the low-strain isomers (P1 = 11) to 18.0 kJ mol-1 for the high-strain isomers (P1 > 13). However, for BH&HLYP (50% HF exchange) the RMSD increases from 23.0 (P1 = 11) to 113.2 (P1 > 13) kJ mol-1. A similar trend is observed for the M06/M06-2X pair of functionals. Namely, for M06 (27% HF exchange) the RMSD increases from 0.8 (P1 = 11) to 21.0 (P1 > 13) kJ mol-1, whereas for M06-2X (54% HF exchange) the RMSD increases from 16.7 (P1 = 11) to 77.7 (P1 > 13) kJ mol-1. Overall, we find that the strain associated with pentagon adjacency is an inherently challenging problem for hybrid DFT methods involving high amounts of HF exchange and that there is an inverse relationship between the optimal percentage of HF exchange and the pentagon-pentagon strain energy. For example, for BLYP the optimal percentages of HF exchange are 13% (P1 = 11), 10% (P1 = 12), 7.5% (P1 = 13), and 6% (P1 > 13).

MP2-F12 Basis Set Convergence near the Complete Basis Set Limit: Are h Functions Sufficient?

We have investigated the title question for the W4-08 thermochemical benchmark using l-saturated truncations of a large reference (REF) basis set, as well as for standard F12-optimized basis sets. With the REF basis set, the root-mean-square (RMS) contribution of i functions to the MP2-F12 total atomization energies (TAEs) is about 0.01 kcal/mol, the largest individual contributions being 0.04 kcal/mol for P₂ and P₄. However, even for these cases, basis set extrapolation from {g,h} basis sets adequately addresses the problem. Using basis sets insufficiently saturated in the spdfgh angular momenta may lead to exaggerated i function contributions. For extrapolation from spdfg and spdfgh basis sets, basis set convergence appears to be quite close to the theoretical asymptotic ∝ L^–7 behavior. We hence conclude that h functions are sufficient even for highly demanding F12 applications. With one-parameter extrapolation, spdf and spdfg basis sets are adequate, aug-cc-pV{T,Q}Z-F12 yielding a RMSD = 0.03 kcal/mol. A limited exploration of CCSD(F12*) and CCSD-F12b suggests our conclusions are applicable to higher-level F12 methods as well.

Analytic Gradients for the Long-Range-Corrected XYG3 Type of Doubly Hybrid Density Functionals: Theory, Implementation, and Assessment

An analytic gradient algorithm for the long-range-corrected (lrc-) XYG3 type of doubly hybrid functionals has been derived and implemented. Due to the introduction of a long-range second-order perturbation (lrPT2) as the correction, the Z-vector equations need to be modified, where the construction of a unique total Lagrangian of the method is required. Geometry optimizations using lrc-XYG3 and lrc-XYGJ-OS have then been applied to the A21 data set that consists of 21 noncovalently interacting systems with CCSD(T) reference data for structures, in order to testify the idea of introducing lrPT2 for a better description of intermolecular interactions. While lrc-XYG3 was observed to offer a better description in dispersion-dominant systems as compared to the XYG3 functional, lrc-XYGJ-OS stands out with an overall better balanced performance in reproducing intermolecular geometries.

Assessing Recent Time-Dependent Double-Hybrid Density Functionals on Doublet-Doublet Excitations

This work is the first thorough investigation of time-dependent double-hybrid density functionals (DHDFs) for the calculation of doublet-doublet excitation energies. It sheds light on the current state-of-the-art techniques in the field and clarifies if there is still room for future improvements. Overall, 29 hybrid functionals and DHDFs are investigated. We separately analyze the individual impacts of the Tamm-Dancoff approximation (TDA), range separation, and spin-component/opposite scaling (SCS/SOS) on 45 doublet-doublet excitations in 23 radicals before concluding with an overarching analysis that includes and excludes challenging excitations with double-excitation or multireference character. Our results show again that so-called "nonempirical" DHDFs are outperformed by semiempirical ones. While the best assessed functionals are DHDFs, some of the worst are also DHDFs and outperformed by all assessed hybrids. SCS/SOS is particularly beneficial for range-separated DHDFs. Spin-scaled, range-separated DHDFs paired with the TDA belong to the best tested methods here, and we particularly highlight SCS-ωB2GP-PLYP, SOS-ωB2PLYP, SOS-ωB2GP-PLYP, SOS-ωB88PP86, SOS-RSX-QIDH, and SOS-ωPBEPP86. When comparing our functional rankings with previous studies on singlet-singlet and singlet-triplet excitations, we recommend TDA-SOS-ωB88PP86 and TDA-SOS-ωPBEPP86 as robust methods for excitation energies in general until further improvements have been achieved that surpass the chemical accuracy threshold for challenging open-shell excitations without increasing the computational effort.

Assessment of XC functionals for the study of organic molecules with superhalogen substitution. A systematic comparison between DFT and CCSD(T)

A systematic density functional theory study, including 17 exchange-correlation functionals, was performed on 22 composite structures consisting of organic molecules, e.g., ethylene, ethane, and benzene, and superhalogen substitutions arising from [MgX₃]^- and [Mg₂X₅]^- (X = F, Cl). Range-separated hybrid functionals ωB97M-V, ωB97X-D3(BJ), ωB97XD, ωB97X, and CAM-B3LYP, as well as double-hybrid functionals B2PLYP and DSD-PBEP86-D3(BJ), are verified to provide reliable results with accuracy approaching that at the coupled-cluster single double triple [CCSD(T)] level. The basis set effect of density functional theory calculation is usually moderate, and triple-ξ quality, e.g., Def2-TZVP, is enough in most cases. In addition, the average values from HF and MP2 method, indicated as (MP2 + HF)/2, are also quite close to those of CCSD(T).

Noncovalently bound excited-state dimers: a perspective on current time-dependent density functional theory approaches applied to aromatic excimer models

Excimers are supramolecular systems whose binding strength is influenced by many factors that are ongoing challenges for computational methods, such as charge transfer, exciton coupling, and London dispersion interactions. Treating the various intricacies of excimer binding at an adequate level is expected to be particularly challenging for Time-Dependent Density Functional Theory (TD-DFT) methods. In addition to well-known limitations for some TD-DFT methods in the description of charge transfer or exciton coupling, the inherent London dispersion problem from ground-state DFT translates to TD-DFT. While techniques to appropriately treat dispersion in DFT are well-developed for electronic ground states, these dispersion corrections remain largely untested for excited states. Herein, we aim to shed light on current TD-DFT methods, including some of the newest developments. The binding of four model excimers is studied across nine density functionals with and without the application of additive dispersion corrections against a wave function reference of SCS-CC2/CBS(3,4) quality, which approximates select CCSDR(3)/CBS data adequately. To our knowledge, this is the first study that presents single-reference wave function dissociation curves at the complete basis set level for the assessed model systems. It is also the first time range-separated double-hybrid density functionals are applied to excimers. In fact, those functionals turn out to be the most promising for the description of excimer binding followed by global double hybrids. Range-separated and global hybrids-particularly with large fractions of Fock exchange-are outperformed by double hybrids and yield worse dissociation energies and inter-molecular equilibrium distances. The deviation between each assessed functional and reference increases with system size, most likely due to missing dispersion interactions. Additive dispersion corrections of the DFT-D3(BJ) and DFT-D4 types reduce the average errors for TD-DFT methods but do so inconsistently and therefore do not offer a black-box solution in their ground-state parametrised form. The lack of appropriate description of dispersion effects for TD-DFT methods is likely hindering the practical application of the herein identified more efficient methods. Dispersion corrections parametrised for excited states appear to be an important next step to improve the applicability of TD-DFT methods and we hope that our work assists with the future development of such corrections.

Tackling an accurate description of molecular reactivity with double-hybrid density functionals

In this Communication, we assess a panel of 18 double-hybrid density functionals for the modeling of the thermochemical and kinetic properties of an extended dataset of 449 organic chemistry reactions belonging to the BH9 database. We show that most of DHs provide a statistically robust performance to model barrier height and reaction energies in reaching the “chemical accuracy.” In particular, we show that nonempirical DHs, such as PBE0-DH and PBE-QIDH, or minimally parameterized alternatives, such as ωB2PLYP and B2K-PLYP, succeed to accurately model both properties in a balanced fashion. We demonstrate, however, that parameterized approaches, such as ωB97X-2 or DSD-like DHs, are more biased to only one of both properties.

Accurate Prediction of Nuclear Magnetic Resonance Parameters via the XYG3 Type of Doubly Hybrid Density Functionals

Nuclear magnetic resonance (NMR) spectroscopy is one of the most powerful and versatile tools in elucidating molecular structures. To eliminate ambiguities of experimental assignments, accurate calculations of NMR spectra are of great importance. Here, a method for theoretical evaluation of the NMR shielding constants by analytic derivatives using gauge including atomic orbitals (GIAO) has been implemented for the XYG3 type of doubly hybrid density functionals (xDH), namely, the GIAO-xDH method. Benchmark calculations on shielding constants and chemical shifts demonstrate the remarkable accuracy of the GIAO-xDH method, compared to the accurate CCSD(T) references. It is shown here that the XYGJ-OS functional is able to give a mean absolute deviation (MAD) of ∼3.0 ppm in the calculated shielding constants for ¹³C, ¹⁵N, ¹⁷O, ¹⁹F, while both XYGJ-OS and xDH-PBE0 functionals are able to provide a satisfactory estimation of chemical shifts with MADs of ∼0.03 and 1.0 ppm for ¹H and ¹³C, respectively. The basis set influence upon the method has been examined and a computational scheme considering both accuracy and efficiency has been proposed and tested to predict the experimental ¹³C chemical shifts of five medium-sized natural product molecules, yielding a MAD of ∼1.0 ppm, which demonstrates the practical feasibility of the GIAO-xDH method.

Do Double-Hybrid Functionals Benefit from Regularization in the PT2 Term? Observations from an Extensive Benchmark

We put to the test a recent suggestion [Shee, J., et al. J. Phys. Chem. Lett. 2021, 12 (50), 12084-12097] that MP2 regularization might improve the performance of double-hybrid density functionals. Using the very large and chemically diverse GMTKN55 benchmark, we find that κ-regularization is indeed beneficial at lower percentages of Hartree-Fock exchange, especially if spin-component scaling is not applied [such as in B2GP-PLYP or ωB97M(2)]. This benefit dwindles for DSD and DOD functionals and vanishes entirely in the ∼70% HF exchange region optimal for them.

Implementation and Optimization of the Embedded Cluster Reference Interaction Site Model with Atomic Charges

In this work, we implemented the embedded cluster reference interaction site model (EC-RISM) originally developed by Kloss, Heil, and Kast (J. Phys. Chem. B2008, 112, 4337–4343). This method combines quantum mechanical calculations with the 3D reference interaction site model (3D-RISM). Numerous options, such as buffer, grid space, basis set, charge model, water model, closure relation, and so forth, were investigated to find the best settings. Additionally, the small point charges, which are derived from the solvent distribution from the 3D-RISM solution to represent the solvent in the QM calculation, were neglected to reduce the overhead without the loss of accuracy. On the MNSOL[a], MNSOL, and FreeSolv databases, our implemented and optimized method provides solvation free energies in water with 5.70, 6.32, and 6.44 kJ/mol root-mean-square deviations, respectively, but with different settings, 5.22, 6.08, and 6.63 kJ/mol can also be achieved. Only solvent models containing fitting parameters, like COSMO-RS and EC-RISM with universal correction and directly used electrostatic potential, perform better than our EC-RISM implementation with atomic charges.

Structural Elucidation of an Atropisomeric Entcassiflavan-(4β→8)-Epicatechin Isolated from Dalbergia monetaria L.f. Based on NMR and ECD Calculations in Comparison to Experimental Data

A rare dihydoxyflavan-epicatechin proanthocyanidin, entcassiflavan-(4β→8)-epicatechin, was isolated from Dalbergia monetaria, a plant widely used by traditional people from the Amazon to treat urinary tract infections. The constitution and relative configuration of the compound were elucidated by HR-MS and detailed 1D- and 2D-NMR measurements. By comparing the experimental electronic circular dichroism (ECD) spectrum with the calculated ECD spectra of all 16 possible isomers, the absolute configuration, the interflavan linkage, and the atropisomers could be determined.

Optimal Tuning Perspective of Range-Separated Double Hybrid Functionals

We study the optimal tuning of the free parameters in range-separated double hybrid functionals, based on enforcing the exact conditions of piecewise linearity and spin constancy. We find that introducing the range separation in both the exchange and the correlation terms allows for the minimization of both fractional charge and fractional spin errors for singlet atoms. The optimal set of parameters is system specific, underlining the importance of the tuning procedure. We test the performance of the resulting optimally tuned functionals for the dissociation curves of diatomic molecules. We find that they recover the correct dissociation curve for the one-electron system, H2+, and improve the dissociation curves of many-electron molecules such as H2 and Li2, but they also yield a nonphysical maximum and only converge to the correct dissociation limit at very large distances.

Small-Basis Set Density-Functional Theory Methods Corrected with Atom-Centered Potentials

Density functional theory (DFT) is currently the most popular method for modeling noncovalent interactions and thermochemistry. The accurate calculation of noncovalent interaction energies, reaction energies, and barrier heights requires choosing an appropriate functional and, typically, a relatively large basis set. Deficiencies of the density-functional approximation and the use of a limited basis set are the leading sources of error in the calculation of noncovalent and thermochemical properties in molecular systems. In this article, we present three new DFT methods based on the BLYP, M06-2X, and CAM-B3LYP functionals in combination with the 6-31G* basis set and corrected with atom-centered potentials (ACPs). ACPs are one-electron potentials that have the same form as effective-core potentials, except they do not replace any electrons. The ACPs developed in this work are used to generate energy corrections to the underlying DFT/basis-set method such that the errors in predicted chemical properties are minimized while maintaining the low computational cost of the parent methods. ACPs were developed for the elements H, B, C, N, O, F, Si, P, S, and Cl. The ACP parameters were determined using an extensive training set of 118655 data points, mostly of complete basis set coupled-cluster level quality. The target molecular properties for the ACP-corrected methods include noncovalent interaction energies, molecular conformational energies, reaction energies, barrier heights, and bond separation energies. The ACPs were tested first on the training set and then on a validation set of 42567 additional data points. We show that the ACP-corrected methods can predict the target molecular properties with accuracy close to complete basis set wavefunction theory methods, but at a computational cost of double-ζ DFT methods. This makes the new BLYP/6-31G*-ACP, M06-2X/6-31G*-ACP, and CAM-B3LYP/6-31G*-ACP methods uniquely suited to the calculation of noncovalent, thermochemical, and kinetic properties in large molecular systems.

Protocol for Directing Nudged Elastic Band Calculations to the Minimum Energy Pathway: Nurturing Errant Calculations Back to Convergence

The combination of density functional theory (DFT) and the nudged elastic band (NEB) method offers a practical tool for the discovery of underlying reaction mechanisms related to the synthesis of functional materials. However, in practice, the lack of a standardized protocol for minimum energy pathway determination too often leads to an inefficient and computationally intensive design process. To that end, we define a verifiable DFT+NEB protocol for efficiently locating and confirming the transition state of a reaction. To test this assertion, we curate 226 unique reactions within 14 classes of reactions and investigate their performance in terms of the number of NEB iterations they require to locate the transition state and an estimate of the associated mean absolute error. Leveraging this protocol, we demonstrate its application for an initial set of parameters: number of frames, N_frames = 11; maximum step size, S_max = 0.04 Å; optimizer = LBFGS; and spring constant, k_spr = 0.1 eV/Ų. We report a convergence rate of 73% and find that a root-mean-square force (F_RMS) of 0.01 eV/Å provides a "rule of thumb" below which NEB simulations are likely to converge. Venturing beyond this baseline enquiry, we delineate the effect on performance of altering the number of frames, maximum step size, choice of optimizer and spring constant. We find improvements in performance with increasing N_frames and S_max, ostensibly approaching some asymptotic limit. We also see substantial improvement in efficiency with the LBFGS optimizer and a clear minimum in performance for the spring constant value of 0.1 eV/Ų. Finally, we provide five case studies that demonstrate typical convergence issues for NEB simulations and suggest methods to overcome them. Our results provide specific and transferable recommendations, offering a transparent and practical tool for beginner and expert researchers alike toward a more rational NEB simulation design.

EPR Spectroscopy of Cu(II) Complexes: Prediction of g-Tensors Using Double-Hybrid Density Functional Theory

Computational electron paramagnetic resonance (EPR) spectroscopy is an important field of applied quantum chemistry that contributes greatly to connecting spectroscopic observations with the fundamental description of electronic structure for open-shell molecules. However, not all EPR parameters can be predicted accurately and reliably for all chemical systems. Among transition metal ions, Cu(II) centers in inorganic chemistry and biology, and their associated EPR properties such as hyperfine coupling and g-tensors, pose exceptional difficulties for all levels of quantum chemistry. In the present work, we approach the problem of Cu(II) g-tensor calculations using double-hybrid density functional theory (DHDFT). Using a reference set of 18 structurally and spectroscopically characterized Cu(II) complexes, we evaluate a wide range of modern double-hybrid density functionals (DHDFs) that have not been applied previously to this problem. Our results suggest that the current generation of DHDFs consistently and systematically outperform other computational approaches. The B2GP-PLYP and PBE0-DH functionals are singled out as the best DHDFs on average for the prediction of Cu(II) g-tensors. The performance of the different functionals is discussed and suggestions are made for practical applications and future methodological developments.

Charge-Transfer Excitations within Density Functional Theory: How Accurate Are the Most Recommended Approaches?

The performance of the most recent density functionals is assessed for charge-transfer (CT) excitations using comprehensive intra- and intermolecular CT benchmark sets with high-quality reference values. For this comparison, the state-of-the-art range-separated (RS) and long-range-corrected (LC) double hybrid (DH) approaches are selected, and global DH and LC hybrid functionals are also inspected. The correct long-range behavior of the exchange-correlation (XC) energy is extensively studied, and various CT descriptors are compared as well. Our results show that the most robust performance is attained by RS-PBE-P86/SOS-ADC(2), as it is suitable to describe both types of CT excitations with outstanding accuracy. Furthermore, concerning the intramolecular transitions, unexpectedly excellent results are obtained for most of the global DHs, but their limitations are also demonstrated for bimolecular complexes. Despite the outstanding performance of the LC-DH methods for common intramolecular excitations, serious deficiencies are pointed out for intermolecular CT transitions, and the wrong long-range behavior of the XC energy is revealed. The application of LC hybrids to such transitions is not recommended in any respect.

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.

Accurate Spectral Properties within Double-Hybrid Density Functional Theory: A Spin-Scaled Range-Separated Second-Order Algebraic-Diagrammatic Construction-Based Approach

Our second-order algebraic-diagrammatic construction [ADC(2)]-based double-hybrid (DH) ansatz (J. Chem. Theory Comput. 2019, 15, 4440. DOI: 10.1021/acs.jctc.9b00391) is combined with range-separation techniques. In the present scheme, both the exchange and the correlation contributions are range-separated, while spin-scaling approaches are also applied. The new methods are thoroughly tested for the most popular benchmark sets including 250 singlet and 156 triplet excitations, as well as 80 oscillator strengths. It is demonstrated that the range separation for the correlation contributions is highly recommended for both the genuine and the ADC(2)-based DH approaches. Our results show that the latter scheme slightly but consistently outperforms the former one for single excitation dominated transitions. Furthermore, states with larger fractions of double excitations are assessed as well, and challenging charge-transfer excitations are also discussed, where the recently proposed spin-scaled long-range corrected DHs fail. The suggested iterative fourth-power scaling RS-PBE-P86/SOS-ADC(2) method, using only three adjustable parameters, provides the most robust and accurate excitation energies within the DH theory. In addition, the relative error of the oscillator strengths is reduced by 65% compared to the best genuine DH functionals.

Eliminating Delocalization Error to Improve Heterogeneous Catalysis Predictions with Molecular DFT + U

Approximate semilocal density functional theory (DFT) is known to underestimate surface formation energies yet paradoxically overbind adsorbates on catalytic transition-metal oxide surfaces due to delocalization error. The low-cost DFT + U approach only improves surface formation energies for early transition-metal oxides or adsorption energies for late transition-metal oxides. In this work, we demonstrate that this inefficacy arises due to the conventional usage of metal-centered atomic orbitals as projectors within DFT + U. We analyze electron density rearrangement during surface formation and O atom adsorption on rutile transition-metal oxides to highlight that a standard DFT + U correction fails to tune properties when the corresponding density rearrangement is highly delocalized across both metal and oxygen sites. To improve both surface properties simultaneously while retaining the simplicity of a single-site DFT + U correction, we systematically construct multi-atom-centered molecular-orbital-like projectors for DFT + U. We demonstrate this molecular DFT + U approach for tuning adsorption energies and surface formation energies of minimal two-dimensional models of representative early (i.e., TiO₂) and late (i.e., PtO₂) transition-metal oxides. Molecular DFT + U simultaneously corrects adsorption energies and surface formation energies of multilayer models of rutile TiO₂(110) and PtO₂(110) to resolve the paradoxical description of surface stability and surface reactivity of semilocal DFT.

Graphene-induced planarization of cyclooctatetraene derivatives

Stable equilibrium compounds containing a planar antiaromatic cyclooctatetraene (COT) ring are promising candidates for organic electronic devices such as organic semiconductor transistors. The planarization of COT by incorporation into rigid planar π-systems, as well as by oxidation or reduction has attracted considerable attention in recent years. Using dispersion-corrected density functional theory calculations, we explore an alternative approach of planarizing COT derivatives by adsorption onto graphene. We show that strong π-π stacking interactions between graphene and COT derivatives induce a planar structure with an antiaromatic central COT ring. In addition to being reversible, this strategy provides a novel approach for planarizing COT without the need for incorporation into a rigid structure, atomic substitution, oxidation, or reduction.

Accurate predictions of the electronic excited states of BODIPY based dye sensitizers using spin-component-scaled double-hybrid functionals: a TD-DFT benchmark study

The vertical excitation energies of 13 BODIPY based dye sensitizers are benchmarked by means of TD-DFT, using 36 functionals from different DFT rungs. Most TD-DFT results were found to overestimate the excitation energies, and show mean absolute error (MAE) values in the range 0.2-0.5 eV. The dispersion-corrected, spin-component-scaled, double-hybrid (DSD) functionals DSD-BLYP and DSD-PBEP86 were found to have the smallest MAE values of 0.083 eV and 0.106 eV, respectively, which is close to the range of average errors found in the more expensive coupled-cluster methods. Moreover, DSD-BLYP and DSD-PBEP86 functionals show excellent consistency and quality of results (standard deviation = 0.048 eV and 0.069 eV respectively). However, the range separated hybrid (RSH) and the range separated double hybrid (RSDH) functionals were found to provide the best predictability (linear determination coefficient R 2 > 0.97 eV).

Mechanism, Thermochemistry, and Kinetics of the Reversible Reactions: C2H3 + H2 ⇌ C2H4 + H ⇌ C2H5

High-level coupled cluster theory, in conjunction with Active Thermochemical Tables (ATcT) and E,J-resolved master equation calculations were used in a study of the title reactions, which play an important role...

S66x8 noncovalent interactions revisited: new benchmark and performance of composite localized coupled-cluster methods

The S66x8 noncovalent interactions benchmark has been re-evaluated at the “sterling silver” level. Against this, a selection of computationally more economical alternatives has been assayed, ranging from localized CC to double hybrids and SAPT(DFT).

Assessment of time-dependent density functionals for the electronic excitation energies of organic dyes used in DSSCs

The absorption spectra modeled as the vertical excitation energies of 13 dye sensitizers used in dye-sensitized solar cells (DSSCs) are benchmarked by means of time-dependent (TD)-DFT, using 36 functionals from different DFT rungs.

BH9, a New Comprehensive Benchmark Data Set for Barrier Heights and Reaction Energies: Assessment of Density Functional Approximations and Basis Set Incompleteness Potentials

The calculation of accurate reaction energies and barrier heights is essential in computational studies of reaction mechanisms and thermochemistry. To assess methods regarding their ability to predict these two properties, high-quality benchmark sets are required that comprise a reasonably large and diverse set of organic reactions. Due to the time-consuming nature of both locating transition states and computing accurate reference energies for reactions involving large molecules, previous benchmark sets have been limited in scope, the number of reactions considered, and the size of the reactant and product molecules. Recent advances in coupled-cluster theory, in particular local correlation methods like DLPNO-CCSD(T), now allow the calculation of reaction energies and barrier heights for relatively large systems. In this work, we present a comprehensive and diverse benchmark set of barrier heights and reaction energies based on DLPNO-CCSD(T)/CBS called BH9. BH9 comprises 449 chemical reactions belonging to nine types common in organic chemistry and biochemistry. We examine the accuracy of DLPNO-CCSD(T) vis-a-vis canonical CCSD(T) for a subset of BH9 and conclude that, although there is a penalty in using the DLPNO approximation, the reference data are accurate enough to serve as a benchmark for density functional theory (DFT) methods. We then present two applications of the BH9 set. First, we examine the performance of several density functional approximations commonly used in thermochemical and mechanistic studies. Second, we assess our basis set incompleteness potentials regarding their ability to mitigate basis set incompleteness errors. The number of data points, the diversity of the reactions considered, and the relatively large size of the reactant molecules make BH9 the most comprehensive thermochemical benchmark set to date and a useful tool for the development and assessment of computational methods.

Symmetry Dilemma of Doubly Hybrid Density Functionals for Equilibrium Molecular Property Calculations

In electronic structure theory, when an approximate wavefunction tends to artifactually break the symmetry of the exact Hamiltonian, the corresponding method is referred to as having a "symmetry dilemma" problem. Such types of artifacts were often reported when Hartree-Fock (HF) and the low-level post-HF methods were used, while the traditional Kohn-Sham density functional theory (KS-DFT) methods were usually found to be more resistant to this breakdown. In this work, we present a systematic study on the reliability of the doubly hybrid (DH) DFT methods for several violable cases. Almost all the commonly used B2PLYP-type (bDH) functionals are shown to have a severe "symmetry dilemma" problem and yield dramatically unreliable molecular properties, such as dipole moment, vibrational frequency, and static polarizability at the equilibrium geometry. A one-parameter bDH functional model study demonstrates that such a problem is a combined effect of the inappropriate portion of the HF exchange (over 50%) for the self-consistent field (SCF) calculation and the augmentation of the second-order perturbative contribution. It is remarkable that the XYG3-type (xDH) functionals show a good capability to resist the artifactual symmetry breaking and yield reliable molecular properties when the same critical cases are calculated. In the xDH, there are two functionals of different purposes, namely, the SCF functional and the energy functional, which have different amounts of the HF exchange and different portions of the correlation contributions. The success of the xDHs can be attributed to this flexibility in xDH construction to avoid using an improperly large portion of the HF exchange in the SCF functional. The insights gained in this work are of significance for the development of an improved DH functional.

Extended Benchmark Set of Main-Group Nuclear Shielding Constants and NMR Chemical Shifts and Its Use to Evaluate Modern DFT Methods

An extended theoretical benchmark set, NS372, for light main-group nuclear shieldings and NMR shifts has been constructed based on high-level GIAO-CCSD(T)/pcSseg-3//CCSD(T)/cc-pVQZ reference data. After removal of the large static-correlation cases O₃, F₃^-, and BH from the statistical evaluations for the ¹⁷O, ¹⁹F, and ¹¹B subsets, the benchmark comprises overall 372 shielding values in 117 molecules with a wide range of electronic-structure situations, containing 124 ¹H, 14 ¹¹B, 93 ¹³C, 43 ¹⁵N, 31 ¹⁷O, 47 ¹⁹F, 14 ³¹P, and 6 ³³S shielding constants. The CCSD(T)/pcSseg-3 data are shown to be close to the basis-set and method limit and thus provide an excellent benchmark to evaluate more approximate methods, such as density functional approaches. This dataset has been used to evaluate Hartree-Fock (HF) and MP2, and a wide range of exchange-correlation functionals from local density approximation (LDA) to generalized gradient approximations (GGAs) and meta-GGAs (focusing on their current-density functional implementations), as well as global hybrid, range-separated hybrid, local hybrid, and double-hybrid functionals. Starting with absolute shielding constants, the DSD-PBEP86 double hybrid is confirmed to provide the highest accuracy, with an aggregate relative mean absolute error (rel. MAE) of only 0.9%, followed by MP2 (1.1%). MP2 and double hybrids only show larger errors for a few systems with the largest static-correlation effects. The double-hybrid B2GP-PLYP, the two local hybrids cLH12ct-SsirPW92 and cLH12ct-SsifPW92, and the current-density functional meta-GGA cB97M-V follow closely behind (all 1.5%), as do some further functionals, cLH20t and cMN15-L (both 1.6%), as well as B2PLYP and KT3 (both 2.0%). Functionals on the lower rungs of the usual ladder offer the advantage of lower computational cost and access to larger molecules. Closer examination also reveals the best-performing methods for individual nuclei in the test set. Different ways of treating τ-dependent functionals are evaluated. When moving from absolute shielding constants to chemical shifts, some of the methods can benefit from systematic error compensation, and the overall error range somewhat narrows. Further methods now achieve the 2% threshold of relative MAEs, including functionals based on TPSS (TPSSh, cmPSTS).

Structural and Energetic Properties of Amino Acids and Peptides Benchmarked by Accurate Theoretical and Experimental Data

Structural, energetic, and spectroscopic data derived in this work aim at the setup of an "experimentally validated" database for amino acids and polypeptides conformers. First, the "cheap" composite scheme (ChS, CCSD(T)/(CBS+CV)^MP2) is tested for evaluation of conformational energies of all eight stable conformers of glycine, by comparing to the more accurate CCSD(T)/CBS+CV computations (Phys. Chem. Chem. Phys. 2013, 15, 10094-10111 and J Mol. Model. 2020, 26, 129). The recently proposed jun-ChS (J. Chem. Theory and Comput. 2020, 16, 988-1006), employing the jun-cc-pVnZ basis set family for CCSD(T) computations and CBS extrapolation, yields conformational energies accurate to 0.2 kJ·mol^-1, at reduced computational cost with respect to aug-ChS employing aug-cc-pVnZ basis sets. The jun-ChS composite scheme is further applied to derive conformational energies for three dipeptide analogues Ac-Gly-NH₂, Ac-Ala-NH₂, and Gly-Gly. Finally, dipeptide conformational energies and semiexperimental equilibrium rotational constants along with the CCSD(T)/(CBS+CV)^MP2 structural parameters (J. Phys. Chem. Lett. 2014, 5, 534-540) stand as the reference for benchmarking of selected density functional methodologies. The double-hybrid functionals B2-PLYP-D3(BJ) and DSD-PBEP86, perform best for structural and energetic characterization of all dipeptide analogues. From hybrid functionals CAM-B3LYP-D3(BJ) and ωB97X-D3(BJ) represent promising methods applicable for larger peptide-based systems for which computations with double-hybrid functionals are not feasible.

Elaborated thermochemical treatment of HF, CO, N2, and H2O: Insight into HEAT and its extensions

Empirical, highly accurate non-relativistic electronic total atomization energies (eTAEs) are established by combining experimental or computationally converged treatments of the nuclear motion and relativistic contributions with the total atomization energies of HF, CO, N₂, and H₂O obtained from the Active Thermochemical Tables. These eTAEs, which have estimated (2σ) uncertainties of less than 10 cm^-1 (0.12 kJ mol^-1), form the basis for an analysis of high-level ab initio quantum chemical calculations that aim at reproducing these eTAEs for the title molecules. The results are then employed to analyze the performance of the high-accuracy extrapolated ab initio thermochemistry, or High-Accuracy Extrapolated Ab Initio Thermochemistry (HEAT), family of theoretical methods. The method known as HEAT-345(Q), in particular, is found to benefit from fortuitous error cancellation between its treatment of the zero-point energy, extrapolation errors in the Hartree-Fock and coupled cluster contributions, neglect of post-(T) core-correlation, and the basis-set error involved in higher-level correlation corrections. In addition to shedding light on a longstanding curiosity of the HEAT protocol-where the cheapest HEAT-345(Q) performs comparably to the theoretically more complete HEAT-456QP procedure-this study lays the foundation for extended HEAT variants that offer substantial improvements in accuracy relative to the established approaches.

Do Double-Hybrid Exchange-Correlation Functionals Provide Accurate Chemical Shifts? A Benchmark Assessment for Proton NMR

A benchmark density functional theory (DFT) study of ¹H NMR chemical shifts for data sets comprising 200 chemical shifts, including complex natural products, has been carried out to assess the performance of DFT methods. Two new benchmark data sets, NMRH33 and NMRH148, have been established. The meta-GGA revTPSS performs remarkably well against the NMRH33 benchmark set (mean absolute deviation (MAD), 0.10 ppm; maximum deviation (max), 0.26 ppm) with the smallest MAD of all evaluated functionals. The best-performing double-hybrid density functional (DHDF), revDSD-BLYP (MAD, 0.16 ppm; max, 0.35 ppm), performs similarly to hybrid-GGA methods (e.g., mPW1PW91/6-311G(d) (MAD, 0.15 ppm; max, 0.36 ppm)), but at a considerably higher computational cost. The results indicate that currently available double-hybrid DFT methods offer no benefit over GGA (including hybrid and meta) functionals in the calculation of ¹H NMR chemical shifts.

Machine learning to tame divergent density functional approximations: a new path to consensus materials design principles

Virtual high-throughput screening (VHTS) with density functional theory (DFT) and machine-learning (ML)-acceleration is essential in rapid materials discovery. By necessity, efficient DFT-based workflows are carried out with a single density functional approximation (DFA). Nevertheless, properties evaluated with different DFAs can be expected to disagree for cases with challenging electronic structure (e.g., open-shell transition-metal complexes, TMCs) for which rapid screening is most needed and accurate benchmarks are often unavailable. To quantify the effect of DFA bias, we introduce an approach to rapidly obtain property predictions from 23 representative DFAs spanning multiple families, “rungs” (e.g., semi-local to double hybrid) and basis sets on over 2000 TMCs. Although computed property values (e.g., spin state splitting and frontier orbital gap) differ by DFA, high linear correlations persist across all DFAs. We train independent ML models for each DFA and observe convergent trends in feature importance, providing DFA-invariant, universal design rules. We devise a strategy to train artificial neural network (ANN) models informed by all 23 DFAs and use them to predict properties (e.g., spin-splitting energy) of over 187k TMCs. By requiring consensus of the ANN-predicted DFA properties, we improve correspondence of computational lead compounds with literature-mined, experimental compounds over the typically employed single-DFA approach.

Machine learning (ML)-based feature analysis reveals universal design rules regardless of density functional choices. Using the consensus among multiple functionals, we identify robust lead complexes in ML-accelerated chemical discovery.

Assessing Density Functional Theory for Chemically Relevant Open-Shell Transition Metal Reactions

Due to the principle lack of systematic improvement possibilities of density functional theory, careful assessment of the performance of density functional approximations (DFAs) on well-designed benchmark sets, for example, for reaction energies and barrier heights, is crucial. While main-group chemistry is well covered by several available sets, benchmark data for transition metal chemistry is sparse. This is especially the case for larger, chemically relevant molecules. Addressing this issue, we recently introduced the MOR41 benchmark which covers chemically relevant reactions of closed-shell complexes. In this work, we extend these efforts to single-reference open-shell systems and introduce the "reactions of open-shell single-reference transition metal complexes" (ROST61) benchmark set. ROST61 includes accurate coupled-cluster reference values for 61 reaction energies with a mean reaction energy of -42.8 kcal mol^-1. Complexes with 13-93 atoms covering 20 d-block elements are included, but due to the restriction to single-reference open-shell systems, important elements such as iron or platinum could not be taken into account, or only to a small extent. We assess the performance of 31 DFAs in combination with three London dispersion (LD) correction schemes. Further, DFT-based composite methods, MP2, and a few semiempirical quantum chemical methods are evaluated. Consistent with the results for the MOR41 closed-shell benchmark, we find that the ordering of DFAs according to Jacob's ladder is preserved and that adding an LD correction is crucial, clearly improving almost all tested methods. The recently introduced r²SCAN-3c composite method stands out with a remarkable mean absolute deviation (MAD) of only 2.9 kcal mol^-1, which is surpassed only by hybrid DFAs with low amounts of Fock exchange (e.g., 2.3 kcal mol^-1 for TPSS0-D4/def2-QZVPP) and double-hybrid (DH) DFAs but at a significantly higher computational cost. The lowest MAD of only 1.6 kcal mol^-1 is obtained with the DH DFA PWPB95-D4 in the def2-QZVPP basis set approaching the estimated accuracy of the reference method. Overall, the ROST61 set adds important reference data to a sparsely sampled but practically relevant area of chemistry. At this point, it provides valuable orientation for the application and development of new DFAs and electronic structure methods in general.

Chemical accuracy with σ-functionals for the Kohn-Sham correlation energy optimized for different input orbitals and eigenvalues

Recently, a new type of orbital-dependent functional for the Kohn-Sham (KS) correlation energy, σ-functionals, was introduced. Technically, σ-functionals are closely related to the well-known direct random phase approximation (dRPA). Within the dRPA, a function of the eigenvalues σ of the frequency-dependent KS response function is integrated over purely imaginary frequencies. In σ-functionals, this function is replaced by one that is optimized with respect to reference sets of atomization, reaction, transition state, and non-covalent interaction energies. The previously introduced σ-functional uses input orbitals and eigenvalues from KS calculations with the generalized gradient approximation (GGA) exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE). Here, σ-functionals using input orbitals and eigenvalues from the meta-GGA TPSS and the hybrid-functionals PBE0 and B3LYP are presented and tested. The number of reference sets taken into account in the optimization of the σ-functionals is larger than in the first PBE based σ-functional and includes sets with 3d-transition metal compounds. Therefore, also a reparameterized PBE based σ-functional is introduced. The σ-functionals based on PBE0 and B3LYP orbitals and eigenvalues reach chemical accuracy for main group chemistry. For the 10 966 reactions from the highly accurate W4-11RE reference set, the B3LYP based σ-functional exhibits a mean average deviation of 1.03 kcal/mol compared to 1.08 kcal/mol for the coupled cluster singles doubles perturbative triples method if the same valence quadruple zeta basis set is used. For 3d-transition metal chemistry, accuracies of about 2 kcal/mol are reached. The computational effort for the post-self-consistent evaluation of the σ-functional is lower than that of a preceding PBE0 or B3LYP calculation for typical systems.