Local hybrid functionals: Theory, implementation, and performance of an emerging new tool in quantum chemistry and beyond

Chemically Accurate Singlet-Triplet Gaps of Arylcarbenes from Local Hybrid Density Functionals

Singlet-triplet (ST) gaps are key descriptors of carbenes, because their properties and reactivity are strongly spin-dependent. However, the theoretical prediction of ST gaps is challenging and generally thought to require elaborate correlated wave function methods or double-hybrid density functionals. By evaluating two recent test sets of arylcarbenes (AC12 and AC18), we show that local hybrid functionals based on the "common t" local mixing function (LMF) model achieve mean absolute errors below 1 kcal/mol at a computational cost only slightly higher than that of global hybrid functionals. An analysis of correlation contributions to the ST gaps suggests that the accuracy of the common t-LMF model is mainly due to an improved description of nondynamical correlation which, unlike exchange, is not additive in each spin-channel. Although spin-nonadditivity can be achieved using the local spin polarization alone, using the "common", i.e., spin-unresolved, iso-orbital indicator t for constructing the LMF is found to be critical for consistent accuracy in ST gaps of arylcarbenes. The results support the view of LHs as vehicles to improve the description of nondynamical correlation rather than sophisticated exchange mixing approaches.

BSE@GW Prediction of Charge Transfer Exciton in Molecular Complexes: Assessment of Self-Energy and Exchange-Correlation Dependence

The Bethe-Salpeter equation using the GW approximation to the self-energy (BSE@GW) is a computationally attractive method for studying electronic excitation from first principles within the many-body Green's function theory framework. We examine its dependence on the underlying exchange-correlation (XC) approximation as well as on the GW approximation for predicting the charge transfer exciton formation at representative type-II interfaces between molecular systems of tetrachloro-1,2-benzoquinone (TCBQ) and acene derivatives. For the XC approximation, we consider several widely used generalized gradient approximation (GGA) and hybrid GGA functionals. For the GW self-energy approximation, we examine the recently proposed renormalized singles approach by Yang and coauthors [J. Phys. Chem. Lett. 2019, 10 (3), 447-452; J. Chem. Theory Comput. 2022, 18, 7570-7585] in addition to other commonly employed approximated GW schemes. We demonstrate a reliable prediction of the charge transfer exciton within the BSE@GW level of theory.

Toward the Next Generation of Density Functionals: Escaping the Zero-Sum Game by Using the Exact-Exchange Energy Density

ConspectusKohn-Sham density functional theory (KS DFT) is arguably the most widely applied electronic-structure method with tens of thousands of publications each year in a wide variety of fields. Its importance and usefulness can thus hardly be overstated. The central quantity that determines the accuracy of KS DFT calculations is the exchange-correlation functional. Its exact form is unknown, or better "unknowable", and therefore the derivation of ever more accurate yet efficiently applicable approximate functionals is the "holy grail" in the field. In this context, the simultaneous minimization of so-called delocalization errors and static correlation errors is the greatest challenge that needs to be overcome as we move toward more accurate yet computationally efficient methods. In many cases, an improvement on one of these two aspects (also often termed fractional-charge and fractional-spin errors, respectively) generates a deterioration in the other one. Here we report on recent notable progress in escaping this so-called "zero-sum-game" by constructing new functionals based on the exact-exchange energy density. In particular, local hybrid and range-separated local hybrid functionals are discussed that incorporate additional terms that deal with static correlation as well as with delocalization errors. Taking hints from other coordinate-space models of nondynamical and strong electron correlations (the B13 and KP16/B13 models), position-dependent functions that cover these aspects in real space have been devised and incorporated into the local-mixing functions determining the position-dependence of exact-exchange admixture of local hybrids as well as into the treatment of range separation in range-separated local hybrids. While initial functionals followed closely the B13 and KP16/B13 frameworks, meanwhile simpler real-space functions based on ratios of semilocal and exact-exchange energy densities have been found, providing a basis for relatively simple and numerically convenient functionals. Notably, the correction terms can either increase or decrease exact-exchange admixture locally in real space (and in interelectronic-distance space), leading even to regions with negative admixture in cases of particularly strong static correlations. Efficient implementations into a fast computer code (Turbomole) using seminumerical integration techniques make such local hybrid and range-separated local hybrid functionals promising new tools for complicated composite systems in many research areas, where simultaneously small delocalization errors and static correlation errors are crucial. First real-world application examples of the new functionals are provided, including stretched bonds, symmetry-breaking and hyperfine coupling in open-shell transition-metal complexes, as well as a reduction of static correlation errors in the computation of nuclear shieldings and magnetizabilities. The newest versions of range-separated local hybrids (e.g., ωLH23tdE) retain the excellent frontier-orbital energies and correct asymptotic exchange-correlation potential of the underlying ωLH22t functional while improving substantially on strong-correlation cases. The form of these functionals can be further linked to the performance of the recent impactful deep-neural-network "black-box" functional DM21, which itself may be viewed as a range-separated local hybrid.

Implementation and First Evaluation of Strong-Correlation-Corrected Local Hybrid Functionals for the Calculation of NMR Shieldings and Shifts

Local hybrid functionals containing strong-correlation factors (scLHs) and range-separated local hybrids (RSLHs) have been integrated into an efficient coupled-perturbed Kohn-Sham implementation for the calculation of nuclear shielding constants. Several scLHs and the ωLH22t RSLH have then been evaluated for the first time for the extended NS372 benchmark set of main-group shieldings and shifts and the TM70 benchmark of 3d transition-metal shifts. The effects of the strong-correlation corrections have been analyzed with respect to the spatial distribution of the sc-factors, which locally diminish exact-exchange admixture at certain regions in a molecule. The scLH22t, scLH23t-mBR, and scLH23t-mBR-P functionals, which contain a "damped" strong-correlation factor to retain the excellent performance of the underlying LH20t functional for weakly correlated situations, tend to make smaller corrections to shieldings and shifts than the "undamped" scLH22ta functional. While the latter functional can also deteriorate agreement with the reference data in certain weakly correlated cases, it provides overall better performance, in particular for systems where static correlation is appreciable. This pertains only to a minority of systems in the NS372 main-group test set but to many more systems in the TM70 transition-metal test set, in particular for high-oxidation-state complexes, e.g., Cr(+VI) complexes and other systems with stretched bonds. Another undamped scLH, the simpler LDA-based scLH21ct-SVWN-m, also tends to provide significant improvements in many cases. The differences between the functionals and species can be rationalized on the basis of one-dimensional plots of the strong-correlation factors, augmented by isosurface plots of the fractional orbital density (FOD). Position-dependent exact-exchange admixture is thus shown to provide substantial flexibility in treating response properties like NMR shifts for both weakly and strongly correlated systems.

Spin-Symmetry Breaking and Hyperfine Couplings in Transition-Metal Complexes Revisited Using Density Functionals Based on the Exact-Exchange Energy Density

A small set of mononuclear manganese complexes evaluated previously for their Mn hyperfine couplings (HFCs) has been analyzed using density functionals based on the exact-exchange energy density─in particular, the spin symmetry breaking (SSB) found previously when using hybrid functionals. Employing various strong-correlation corrected local hybrids (scLHs) and strong-correlation corrected range-separated local hybrids (scRSLHs) with or without additional corrections to their local mixing functions (LMFs) to mitigate delocalization errors (DE), the SSB and the associated dipolar HFCs of [Mn(CN)4]2-, MnO3, [Mn(CN)4N]-, and [Mn(CN)5NO]2- (the latter with cluster embedding) have been examined. Both strong-correlation (sc)-correction and DE-correction terms help to diminish SSB and correct the dipolar HFCs. The DE corrections are more effective, and the effects of the sc corrections depend on their damping factors. Interestingly, the DE-corrections reduce valence-shell spin polarization (VSSP) and thus SSB by locally enhancing exact-exchange (EXX) admixture near the metal center and thereby diminishing spin-density delocalization onto the ligand atoms. In contrast, sc corrections diminish EXX admixture locally, mostly on specific ligand atoms. This then reduces VSSP and SSB as well. The performance of scLHs and scRSLHs for the isotropic Mn HFCs has also been analyzed, with particular attention to core-shell spin-polarization contributions. Further sc-corrected functionals, such as the KP16/B13 construction and the DM21 deep-neural-network functional, have been examined.

How well do one-electron self-interaction-correction methods perform for systems with fractional electrons?

Recently developed locally scaled self-interaction correction (LSIC) is a one-electron SIC method that, when used with a ratio of kinetic energy densities (zσ) as iso-orbital indicator, performs remarkably well for both thermochemical properties as well as for barrier heights overcoming the paradoxical behavior of the well-known Perdew-Zunger self-interaction correction (PZSIC) method. In this work, we examine how well the LSIC method performs for the delocalization error. Our results show that both LSIC and PZSIC methods correctly describe the dissociation of H2+ and He2+ but LSIC is overall more accurate than the PZSIC method. Likewise, in the case of the vertical ionization energy of an ensemble of isolated He atoms, the LSIC and PZSIC methods do not exhibit delocalization errors. For the fractional charges, both LSIC and PZSIC significantly reduce the deviation from linearity in the energy vs number of electrons curve, with PZSIC performing superior for C, Ne, and Ar atoms while for Kr they perform similarly. The LSIC performs well at the endpoints (integer occupations) while substantially reducing the deviation. The dissociation of LiF shows both LSIC and PZSIC dissociate into neutral Li and F but only LSIC exhibits charge transfer from Li+ to F- at the expected distance from the experimental data and accurate ab initio data. Overall, both the PZSIC and LSIC methods reduce the delocalization errors substantially.

Revisiting Gauge-Independent Kinetic Energy Densities in Meta-GGAs and Local Hybrid Calculations of Magnetizabilities

In a recent study [J. Chem. Theory Comput. 2021, 17, 1457-1468], some of us examined the accuracy of magnetizabilities calculated with density functionals representing the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA (mGGA), as well as global hybrid (GH) and range-separated (RS) hybrid functionals by assessment against accurate reference values obtained with coupled-cluster theory with singles, doubles, and perturbative triples [CCSD(T)]. Our study was later extended to local hybrid (LH) functionals by Holzer et al. [J. Chem. Theory Comput. 2021, 17, 2928-2947]; in this work, we examine a larger selection of LH functionals, also including range-separated LH (RSLH) functionals and strong-correlation LH (scLH) functionals. Holzer et al. also studied the importance of the physically correct handling of the magnetic gauge dependence of the kinetic energy density (τ) in mGGA calculations by comparing the Maximoff-Scuseria formulation of τ used in our aforementioned study to the more physical current-density extension derived by Dobson. In this work, we also revisit this comparison with a larger selection of mGGA functionals. We find that the newly tested LH, RSLH, and scLH functionals outperform all of the functionals considered in the previous studies. The various LH functionals afford the seven lowest mean absolute errors while also showing remarkably small standard deviations and mean errors. Most strikingly, the best two functionals are scLHs that also perform remarkably well in cases with significant multiconfigurational character, such as the ozone molecule, which is traditionally excluded from statistical error evaluations due to its large errors with common density functionals.

M11pz: A Nonlocal Meta Functional with Zero Hartree-Fock Exchange and with Broad Accuracy for Chemical Energies and Structures

The accuracy of Kohn-Sham density functional theory depends strongly on the approximation to the exchange-correlation functional. In this work, we present a new exchange-correlation functional called M11pz (M11 plus rung-3.5 terms with zero Hartree-Fock exchange) that is built on the M11plus functional with the goal of using its rung-3.5 terms without a Hartree-Fock exchange term, especially to improve the accuracy for strongly correlated systems. The M11pz functional is optimized with the same local and rung-3.5 ingredients that are used in M11plus but without any percentage of Hartree-Fock exchange. The performance of M11pz is compared with eight local functionals, and M11pz is found to be in top three when the errors or ranks are averaged over eight grouped and partially overlapping databases: AME418/22, atomic and molecular energies; MGBE172, main-group bond energies; TMBE40, transition-metal bond energies; SR309, single-reference systems; MR54, multireference systems; BH192, barrier heights; NC579, noncovalent interaction energies; and MS20, molecular structures. For calculations of band gaps of solids, M11pz is the second best of the nine tested functionals that have zero Hartree-Fock exchange.

Toward a correct treatment of core properties with local hybrid functionals

In local hybrid functionals (LHs), a local mixing function (LMF) determines the position-dependent exact-exchange admixture. We report new LHs that focus on an improvement of the LMF in the core region while retaining or partly improving upon the high accuracy in the valence region exhibited by the LH20t functional. The suggested new pt-LMFs are based on a Padé form and modify the previously used ratio between von Weizsäcker and Kohn-Sham local kinetic energies by different powers of the density to enable flexibly improved approximations to the correct high-density and iso-orbital limits relevant for the innermost core region. Using TDDFT calculations for a set of K-shell core excitations of second- and third-period systems including accurate state-of-the-art relativistic orbital corrections, the core part of the LMF is optimized, while the valence part is optimized as previously reported for test sets of atomization energies and reaction barriers (Haasler et al., J Chem Theory Comput 2020, 16, 5645). The LHs are completed by a calibration function that minimizes spurious nondynamical correlation effects caused by the gauge ambiguities of exchange-energy densities, as well as by B95c meta-GGA correlation. The resulting LH23pt functional relates to the previous LH20t functional but specifically improves upon the core region.

Zero-field splitting parameters within exact two-component theory and modern density functional theory using seminumerical integration

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin-dipole contribution and the response equations of the spin-orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn-Sham equations for the spin-orbit perturbation term. Second, the spin-orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin-orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin-orbit mean field Ansatz.

Range-Separated Local Hybrid Functionals with Small Fractional-Charge and Fractional-Spin Errors: Escaping the Zero-Sum Game of DFT Functionals

Extending recent developments on strong-correlation (sc) corrections to local hybrid functionals to the recent accurate ωLH22t range-separated local hybrid, a series of highly flexible strong-correlation-corrected range-separated local hybrids (scRSLHs) has been constructed and evaluated. This has required the position-dependent reduction of both short- and long-range exact-exchange admixtures in regions of space characterized by strong static correlations. Using damping procedures provides scRSLHs that retain largely the excellent performance of ωLH22t for weakly correlated situations and, in particular, for accurate quasiparticle energies of a wide variety of systems while reducing dramatically static-correlation errors, e.g., in stretched-bond situations. An additional correction to the local mixing function to reduce delocalization errors in abnormal open-shell situations provides further improvements in thermochemical and kinetic parameters, making scRSLH functionals such as ωLH23tdE or ωLH23tdP promising tools for complex molecular or condensed-phase systems, where low fractional-charge and fractional-spin errors are simultaneously important. The proposed rung 4 functionals thereby largely escape the usual zero-sum game between these two quantities and are expected to open new areas of accurate computations by Kohn-Sham DFT. At the same time, they require essentially no extra computational effort over the underlying ωLH22t functional, which means that their use is only moderately more demanding than that of global, local, or range-separated hybrid functionals.

Learning from the 4-(dimethylamino)benzonitrile twist: Two-parameter range-separated local hybrid functional with high accuracy for triplet and charge-transfer excitations

The recent ωLH22t range-separated local hybrid (RSLH) is shown to provide outstanding accuracy for the notorious benchmark problem of the two lowest excited-state potential energy curves for the amino group twist in 4-(dimethylamino)benzonitrile (DMABN). However, the design of ωLH22t as a general-purpose functional resulted in less convincing performance for triplet excitations, which is an important advantage of previous LHs. Furthermore, ωLH22t uses 8 empirical parameters to achieve broad accuracy. In this work, the RSLH ωLH23ct-sir is constructed with minimal empiricism by optimizing its local mixing function prefactor and range-separation parameter for only 8 excitation energies. ωLH23ct-sir maintains the excellent performance of ωLH22t for the DMABN twist and charge-transfer benchmarks but significantly improves the errors for triplet excitation energies (0.17 vs 0.24 eV). Additional test calculations for the AE6BH6 thermochemistry test set and large dipole moment and static polarizability test sets confirm that the focus on excitation energies in the optimization of ωLH23ct-sir has not caused any dramatic errors for ground-state properties. Although ωLH23ct-sir cannot replace ωLH22t as a general-purpose functional, it is preferable for problems requiring a universally good description of localized and charge-transfer excitations of both singlet and triplet multiplicity. Current limitations on the application of ωLH23ct-sir and other RSLHs to the study of singlet-triplet gaps of emitters for thermally activated delayed fluorescence are discussed. This work also includes the first systematic analysis of the influence of the local mixing function prefactor and the range-separation parameter in an RSLH on different types of excitations.

TURBOMOLE: Today and Tomorrow

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light-matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE's functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree-Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

Nonempirical Range-Separated Hybrid Functional with Spatially Dependent Screened Exchange

Electronic structure calculations based on density functional theory (DFT) have successfully predicted numerous ground-state properties of a variety of molecules and materials. However, exchange and correlation functionals currently used in the literature, including semilocal and hybrid functionals, are often inaccurate to describe the electronic properties of heterogeneous solids, especially systems composed of building blocks with large dielectric mismatch. Here, we present a dielectric-dependent range-separated hybrid functional, screened-exchange range-separated hybrid (SE-RSH), for the investigation of heterogeneous materials. We define a spatially dependent fraction of exact exchange inspired by the static Coulomb-hole and screened-exchange (COHSEX) approximation used in many-body perturbation theory, and we show that the proposed functional accurately predicts the electronic structure of several nonmetallic interfaces, three- and two-dimensional, pristine, and defective solids and nanoparticles.

Comparative Density Functional Theory Study of Magnetic Exchange Couplings in Dinuclear Transition-Metal Complexes

Multicenter transition-metal complexes (MCTMs) with magnetically interacting ions have been proposed as components for information-processing devices and storage units. For any practical application of MCTMs as magnetic units, it is crucial to characterize their magnetic behavior, and in particular, the isotropic magnetic exchange coupling, J, between its magnetic centers. Due to the large size of typical MCTMs, density functional theory is the only practical electronic structure method for evaluating the J coupling. Here, we assess the accuracy of different density functional approximations for predicting the magnetic couplings of eight dinuclear transition-metal complexes, including five dimanganese, two dicopper, and one divanadium with known reliable experimental J couplings spanning from ferromagnetic to strong antiferromagnetic. The density functionals considered include global hybrid functionals which mix semilocal density functional approximations and exact exchange with a fixed admixing parameter, six local hybrid functionals where the admixing parameters are extended to be spatially dependent, the SCAN and r2SCAN meta-generalized gradient approximations (GGAs), and two widely used GGAs. We found that global hybrids tested in this work have a tendency to over-correct the error in magnetic coupling parameters from the Perdew-Burke-Ernzerhof (PBE) GGA as seen for manganese complexes. The performance of local hybrid density functionals shows no improvement in terms of bias and is scattered without a clear trend, suggesting that more efforts are needed for the extension from global to local hybrid density functionals for this particular property. The SCAN and r2SCAN meta-GGAs are found to perform as well as benchmark global hybrids on most tested complexes. We further analyze the charge density redistribution of meta-GGAs as well as global and local hybrid density functionals with respect to that of PBE, in connection to the self-interaction error or delocalization error.

Reducing Exact Two-Component Theory for NMR Couplings to a One-Component Approach: Efficiency and Accuracy

The self-consistent and complex spin-orbit exact two-component (X2C) formalism for NMR spin-spin coupling constants [ J. Chem. Theory Comput. 17, 2021, 3874-3994] is reduced to a scalar one-component ansatz. This way, the first-order response term can be partitioned into the Fermi-contact (FC) and spin-dipole (SD) interactions as well as the paramagnetic spin-orbit (PSO) contribution. The FC+SD terms are real and symmetric, while the PSO term is purely imaginary and antisymmetric. The relativistic one-component approach is combined with a modern density functional treatment up to local hybrid functionals including the response of the current density. Computational demands are reduced by factors of 8-24 as shown for a large tin compound consisting of 137 atoms. Limitations of the current ansatz are critically assessed for Sn, Pb, Pd, and Pt compounds, i.e. the one-component treatment is not sufficient for tin compounds featuring a few heavy halogen atoms.

Full Implementation, Optimization, and Evaluation of a Range-Separated Local Hybrid Functional with Wide Accuracy for Ground and Excited States

We report the first full and efficient implementation of range-separated local hybrid functionals (RSLHs) into the TURBOMOLE program package. This enables the computation of ground-state energies and nuclear gradients as well as excitation energies. Regarding the computational effort, RSLHs scale like regular local hybrid functionals (LHs) with system or basis set size and increase timings by a factor of 2-3 in total. An advanced RSLH, ωLH22t, has been optimized for atomization energies and reaction barriers. It is an extension of the recent LH20t local hybrid and is based on short-range PBE and long-range HF exchange-energy densities, a pig2 calibration function to deal with the gauge ambiguity of exchange-energy densities, and reoptimized B95c correlation. ωLH22t has been evaluated for a wide range of ground-state and excited-state quantities. It further improves upon the already successful LH20t functional for the GMTKN55 main-group energetics test suite, and it outperforms any global hybrid while performing close to the top rung-4 functional, ωB97M-V, for these evaluations when augmented by D4 dispersion corrections. ωLH22t performs excellently for transition-metal reactivity and provides good balance between delocalization errors and left-right correlation for mixed-valence systems, with a somewhat larger bias toward localized states compared to LH20t. It approaches the accuracy of the best local hybrids to date for core, valence singlet and triplet, and Rydberg excitation energies while improving strikingly on intra- and intermolecular charge-transfer excitations, comparable to the most successful range-separated hybrids available.

Faster Exact Exchange for Solids via occ-RI-K: Application to Combinatorially Optimized Range-Separated Hybrid Functionals for Simple Solids with Pseudopotentials Near the Basis Set Limit

In this work, we developed and showcased the occ-RI-K algorithm to compute the exact exchange contribution in density functional calculations of solids near the basis set limit. Within the Gaussian planewave (GPW) density fitting, our algorithm achieves a 1-2 orders of magnitude speedup compared to conventional GPW algorithms. Since our algorithm is well suited for simulations with large basis sets, we applied it to 12 hybrid density functionals with pseudopotentials and a large uncontracted basis set to assess their performance on band gaps of 25 simple solids near the basis set limit. The largest calculation performed in this work involves 16 electrons and 350 basis functions in the unit cell utilizing a 6 × 6 × 6 k-mesh. With 20-27% exact exchange, global hybrid functionals (B3LYP, PBE0, revPBE0, B97-3, SCAN0) perform similarly with a root-mean-square deviation (RMSD) of 0.61-0.77 eV, while other global hybrid functionals such as M06-2X (2.02 eV) and MN15 (1.05 eV) show higher RMSD due to their increased fraction of exact exchange. A short-range hybrid functional, HSE achieves a similar RMSD (0.76 eV) but shows a notable underestimation of band gaps due to the complete lack of long-range exchange. We found that two combinatorially optimized range-separated hybrid functionals, ωB97X-rV (3.94 eV) and ωB97M-rV (3.40 eV), and the two other range-separated hybrid functionals, CAM-B3LYP (2.41 eV) and CAM-QTP01 (4.16 eV), significantly overestimate the band gap because of their high fraction of long-range exact exchange. Given the failure of ωB97X-rV and ωB97M-rV, we have yet to find a density functional that offers consistent performance for both molecules and solids. Our algorithm development and density functional assessment will serve as a stepping stone toward developing more accurate hybrid functionals and applying them to practical applications.

Unification of Perdew-Zunger self-interaction correction, DFT+U, and Rung 3.5 density functionals

This Communication presents a unified derivation of three different approximations used in density functional theory (DFT): the Perdew-Zunger self-interaction correction (PZSIC), the Hubbard correction DFT+U, and the Rung 3.5 density functionals. All three approximations can be derived by introducing electron self-interaction into the Kohn-Sham (KS) reference system of noninteracting electrons. The derivation uses the Adiabatic Projection formalism: one projects the electron-electron interaction operator onto certain states, introduces the projected operator into the reference system, and defines a density functional for the remainder. Projecting onto individual localized KS orbitals recovers our previous derivation of the PZSIC [B. G. Janesko, J. Phys. Chem. Lett. 13, 5698-5702 (2022)]. Projecting onto localized atom-centered orbitals recovers a variant of DFT+U. Projecting onto localized states at each point in space recovers Rung 3.5 approaches. New results include an "atomic state PZSIC" that does not require localizing the KS orbitals, a demonstration that typical Hubbard U parameters reproduce a scaled-down PZSIC, and a Rung 3.5 variant of DFT+U that does not require choosing atom-dependent states.

Local Hybrid Functional Applicable to Weakly and Strongly Correlated Systems

The recent idea (Wodyński, A.; Arbuznikov, A. V.; Kaupp M. J. Chem. Phys. 2021, 155, 144101) to augment local hybrid functionals by a strong-correlation (sc) factor obtained from the adiabatic connection in the spirit of the KP16 model has been extended and applied to generate the accurate sc-corrected local hybrid functional scLH22t. By damping small values of the ratio between nondynamical and dynamical correlation entering the correction factor, it has become possible to avoid double counting of nondynamical correlation for weakly correlated situations and thereby preserve the excellent accuracy of the underlying LH20t local hybrid for such cases almost perfectly. On the other hand, scLH22t improves substantially over LH20t in reducing fractional-spin errors (FSEs), in providing improved spin-restricted bond dissociation curves, and in treating some typical systems with multireference character. The obtained FSEs are similar to those of the KP16/B13 model and slightly larger than for B13, but performance for weakly correlated systems is better than for these two related methods, which are also difficult to use self-consistently. The recent DM21 functional based on the training of a deep neural network still performs somewhat better than scLH22t but allows no physical insights into the origins of reduced FSEs. Examination of local mixing functions (LMFs) for the corrected scLH22t and uncorrected LH20t functionals provides further insights: in weakly correlated situations, the LMF remains essentially unchanged. Strong-correlation effects manifest in a reduction of the LMF values in certain regions of space, even to the extent of producing negative LMF values. It is suggested that this is the mechanism by which also DM21, which may be viewed as a range-separated local hybrid, is able to reduce FSEs.

Importance of imposing gauge invariance in time-dependent density functional theory calculations with meta-generalized gradient approximations

It has been known for more than a decade that the gauge variance of the kinetic energy density τ leads to additional terms in the magnetic orbital rotation Hessian used in linear-response time-dependent density functional theory (TDDFT), affecting excitation energies obtained with τ-dependent exchange-correlation functionals. While previous investigations found that a correction scheme based on the paramagnetic current density has a small effect on benchmark results, we report more pronounced effects here, in particular, for the popular M06-2X functional and for some other meta-generalized gradient approximations (mGGAs). In the first part of this communication, this is shown by a reassessment of a set of five Ni(II) complexes for which a previous benchmark study that did not impose gauge invariance has found surprisingly large errors for excitation energies obtained with M06-2X. These errors are more than halved by restoring gauge invariance. The variable importance of imposing gauge invariance for different mGGA-based functionals can be rationalized by the derivative of the mGGA exchange energy integrand with respect to τ. In the second part, a large set of valence excitations in small main-group molecules is analyzed. For M06-2X, several selected n → π* and π→π_⊥ ^* excitations are heavily gauge-dependent with average changes of -0.17 and -0.28 eV, respectively, while π→π_‖ ^* excitations are marginally affected (-0.04 eV). Similar patterns, but of the opposite signs, are found for SCAN0. The results suggest that reevaluation of previous gauge variant TDDFT results based on M06-2X and other mGGA functionals is warranted.

Paramagnetic NMR Shielding Tensors Based on Scalar Exact Two-Component and Spin-Orbit Perturbation Theory

The temperature-dependent Fermi-contact and pseudocontact terms are important contributions to the paramagnetic NMR shielding tensor. Herein, we augment the scalar-relativistic (local) exact two-component (X2C) framework with spin-orbit perturbation theory including the screened nuclear spin-orbit correction for the EPR hyperfine coupling and g tensor to compute these temperature-dependent terms. The accuracy of this perturbative ansatz is assessed with the self-consistent spin-orbit two-component and four-component treatments serving as reference. This shows that the Fermi-contact and pseudocontact interaction is sufficiently described for paramagnetic NMR shifts; however, larger deviations are found for the EPR spectra and the principle components of the EPR properties of heavy elements. The impact of the perturbative treatment is further compared to that of the density functional approximation and the basis set. Large-scale calculations are routinely possible with the multipole-accelerated resolution of the identity approximation and the seminumerical exchange approximation, as shown for [CeTi₆O₃(OiPr)₉(salicylate)₆].

A Local Hybrid Exchange Functional Approximation from First Principles

Local hybrid functionals are a more flexible class of density functional approximations allowing for a position-dependent admixture of exact exchange. This additional flexibility, however, comes with a more involved mathematical form and a more complicated design. A common denominator for previously constructed local hybrid funtionals is usage of thermochemical benchmark data to construct these functionals. Herein, we design a local hybrid functional without relying on benchmark data. Instead, we construct it in a more ab initio manner, following the principles of modern meta-generalized gradient approximations and considering theoretical constrains. To achieve this, we make use of the density matrix expansion and a local mixing function based on an approximate correlation length. The accuracy of the developed density functional approximation is assessed for thermochemistry, excitation energies, polarizabilities, magnetizabilities, NMR spin-spincoupling constants, NMR shieldings and shifts, as well as EPR g-tensors and hyperfine coupling constants. Here, the new exchange functional shows a robust performance and is especially well suited for atomization energies, barrier heights, excitation energies, NMR coupling constants, and EPR properties, whereas it looses some ground for the NMR shifts.Therefore, the designed functional is a major step forwards for functionals that have been designed from first principles.

Communication: Impact of the current density on paramagnetic NMR properties

Meta-generalized gradient approximations (meta-GGAs) and local hybrid functionals generally depend on the kinetic energy density τ. For magnetic properties, this necessitates generalizations to ensure gauge invariance. In most implementations, τ is generalized by incorporating the external magnetic field. However, this introduces artifacts in the response of the density matrix and does not satisfy the iso-orbital constraint. Here, we extend previous approaches based on the current density to paramagnetic NMR shieldings and EPR g-tensors. The impact is assessed for main-group compounds and transition-metal complexes considering 25 density functional approximations. It is shown that the current density leads to substantial improvements-especially for the popular Minnesota and SCAN functional families. Thus, we strongly recommend to use the current density generalized τ in paramagnetic NMR and EPR calculations with meta-GGAs.

Hybrid functionals with local range separation: Accurate atomization energies and reaction barrier heights

Range-separated hybrid approximations to the exchange-correlation density functional mix exact and semi-local exchange in a position-dependent manner. In their conventional form, the range separation is controlled by a constant parameter. Turning this constant into a density functional leads to a locally space-dependent range-separation function and thus a more powerful and flexible range-separation approach. In this work, we explore the self-consistent implementation of a local range-separated hybrid, taking into account a one-electron self-interaction correction and the behavior under uniform density scaling. We discuss different forms of the local range-separation function that depend on the electron density, its gradient, and the kinetic energy density. For test sets of atomization energies, reaction barrier heights, and total energies of atoms, we demonstrate that our best model is a clear improvement over common global range-separated hybrid functionals and can compete with density functionals that contain multiple empirical parameters. Promising results for equilibrium bond lengths, harmonic vibrational frequencies, and vertical ionization potentials further underline the potential and flexibility of our approach.

A general justification for hybrid functionals in DFT by means of linear response theory

In the present work, resorting to linear response theory, we examine the plausibility of postulating Kohn-Sham (KS)-type equations which contain, by definition, an effective hybrid potential made up by some arbitrary mixture of local and non-local terms. In this way a general justification for the construction of hybrid functionals is provided without resorting to arguments based on the adiabatic connection, the generalized KS theory or the Levy's constrained search (or its variations). In particular, we examine the cases of single-hybrid functionals, derived from non-local exchange and of double-hybrid functionals, emerging from non-local second-order expressions obtained from the KS perturbation theory. A further generalization for higher-order hybrid functionals is also included.

Systematic Evaluation of Modern Density Functional Methods for the Computation of NMR Shifts of 3d Transition-Metal Nuclei

A wide range of density functionals from all rungs of Jacob's ladder have been evaluated systematically for a set of experimental 3d transition-metal NMR shifts of 70 complexes encompassing 12 × ⁴⁹Ti, 10 × ⁵¹V, 10 × ⁵³Cr, 11 × ⁵⁵Mn, 9 × ⁵⁷Fe, 9 × ⁵⁹Co, and 9 × ⁶¹Ni shift values, as well as a diverse range of electronic structure characteristics. The overall 39 functionals evaluated include one LDA, eight GGAs, seven meta-GGAs (including their current-density-functional─CDFT─versions), nine global hybrids, four range-separated hybrids, eight local hybrids, and two double hybrids, and we also include Hartree-Fock and MP2 calculations. While recent evaluations of the same functionals for a very large coupled-cluster-based benchmark of main-group shieldings and shifts achieved in some cases aggregate percentage mean absolute errors clearly below 2%, the best results for the present 3d-nuclei set are in the range between 4 and 5%. Strikingly, the overall best-performing functionals are the recently implemented CDFT versions of two meta-GGAs, namely cM06-L (4.0%) and cVSXC (4.3%), followed by cLH14t-calPBE (4.9%), B3LYP (5.0%), and cLH07t-SVWN (5.1%), i.e., the previously best-performing global hybrid and two local hybrids. A number of further functionals achieve aggregate deviations in the range 5-6%. Range-separated hybrids offer no particular advantage over global hybrids. Due to the overall poor performance of Hartree-Fock theory for all systems except the titanium complexes, MP2 and double-hybrid functionals are unsuitable for these 3d-nucleus shifts and provide large errors. Global hybrid functionals with larger EXX admixtures, such as BHLYP or M06-2X, also perform poorly, and some other highly parametrized global hybrids also are unsuitable. For many functionals depending on local kinetic energy τ, their CDFT variants perform much better than their "non-CDFT" versions. This holds notably also for the above-mentioned M06-L and VSXC, while the effect is small for τ-dependent local hybrids and can even be somewhat detrimental to the agreement with experiment for a few other cases. The separation between well-performing and more poorly performing functionals is mainly determined by their results for the most critical nuclei ⁵⁵Mn, ⁵⁷Fe, and ⁵⁹Co. Here either moderate exact-exchange admixtures or CDFT versions of meta-GGAs are beneficial for the accuracy. The overall deviations of the better-performing global or local hybrids are then typically dominated by the ⁵³Cr shifts, where triplet instabilities appear to disfavor exact-exchange admixture. Further detailed analyses help to pinpoint specific nuclei and specific types of complexes that are challenges for a given functional.

Accurate X-ray Absorption Spectra near L- and M-Edges from Relativistic Four-Component Damped Response Time-Dependent Density Functional Theory

The simulation of X-ray absorption spectra requires both scalar and spin–orbit (SO) relativistic effects to be taken into account, particularly near L- and M-edges where the SO splitting of core p and d orbitals dominates. Four-component Dirac–Coulomb Hamiltonian-based linear damped response time-dependent density functional theory (4c-DR-TDDFT) calculates spectra directly for a selected frequency region while including the relativistic effects variationally, making the method well suited for X-ray applications. In this work, we show that accurate X-ray absorption spectra near L_2,3- and M_4,5-edges of closed-shell transition metal and actinide compounds with different central atoms, ligands, and oxidation states can be obtained by means of 4c-DR-TDDFT. While the main absorption lines do not change noticeably with the basis set and geometry, the exchange–correlation functional has a strong influence with hybrid functionals performing the best. The energy shift compared to the experiment is shown to depend linearly on the amount of Hartee–Fock exchange with the optimal value being 60% for spectral regions above 1000 eV, providing relative errors below 0.2% and 2% for edge energies and SO splittings, respectively. Finally, the methodology calibrated in this work is used to reproduce the experimental L_2,3-edge X-ray absorption spectra of [RuCl₂(DMSO)₂(Im)₂] and [WCl₄(PMePh₂)₂], and resolve the broad bands into separated lines, allowing an interpretation based on ligand field theory and double point groups. These results support 4c-DR-TDDFT as a reliable method for calculating and analyzing X-ray absorption spectra of chemically interesting systems, advance the accuracy of state-of-the art relativistic DFT approaches, and provide a reference for benchmarking more approximate techniques.

The paper demonstrates that relativistic four-component TDDFT theory can reproduce and analyze experimental X-ray absorption spectra near L_2,3- and M_4,5-edges of transition metal and actinide compounds with different central atoms, ligands, and oxidation states. With variational inclusion of scalar and spin−orbit relativistic effects and hybrid functionals with an optimized amount of Hartee−Fock exchange (60%), it achieves relative errors below 0.2% and 2% for edge energies and spin−orbit (SO) splittings.

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).

Local hybrid functionals augmented by a strong-correlation model

The strong-correlation factor of the recent KP16/B13 exchange-correlation functional has been adapted and applied to the framework of local hybrid (LH) functionals. The expression identifiable as nondynamical (NDC) and dynamical (DC) correlations in LHs is modified by inserting a position-dependent KP16/B13-style strong-correlation factor q_AC(r) based on a local version of the adiabatic connection. Different ways of deriving this factor are evaluated for a simple one-parameter LH based on the local density approximation. While the direct derivation from the LH NDC term fails due to known deficiencies, hybrid approaches, where the factor is determined from the B13 NDC term as in KP16/B13 itself, provide remarkable improvements. In particular, a modified B13 NDC expression using Patra's exchange-hole curvature showed promising results. When applied to the simple LH as a first attempt, it reduces atomic fractional-spin errors and deficiencies of spin-restricted bond dissociation curves to a similar extent as the KP16/B13 functional itself while maintaining the good accuracy of the underlying LH for atomization energies and reaction barriers in weakly correlated situations. The performance of different NDC expressions in deriving strong-correlation corrections is analyzed, and areas for further improvements of strong-correlation corrected LHs and related approaches are identified. All the approaches evaluated in this work have been implemented self-consistently into a developers' version of the Turbomole program.

Assessment of hybrid functionals for singlet and triplet excitations: Why do some local hybrid functionals perform so well for triplet excitation energies?

The performance of various hybrid density functionals is assessed for 105 singlet and 105 corresponding triplet vertical excitation energies from the QUEST database. The overall lowest mean absolute error is obtained with the local hybrid (LH) functional LH12ct-SsirPW92 with individual errors of 0.11 eV (0.11 eV) for singlet (triplet) n → π* excitations and 0.29 eV (0.17 eV) for π → π* excitations. This is slightly better than with the overall best performing global hybrid M06-2X [n → π*: 0.13 eV (0.17 eV), π → π*: 0.30 eV (0.20 eV)], while most other global and range-separated hybrids and some LHs suffer from the "triplet problem" of time-dependent density functional theory. This is exemplified by correlating the errors for singlet and triplet excitations on a state-by-state basis. The excellent performance of LHs based on a common local mixing function, i.e., an LMF constructed from the spin-summed rather than the spin-resolved semilocal quantities, is systematically investigated by the introduction of a spin-channel interpolation scheme that allows us to continuously modulate the fraction of opposite-spin terms used in the LMF. The correlation of triplet and singlet errors is systematically improved for the n → π* excitations when larger fractions of the opposite-spin-channel are used in the LMF, whereas this effect is limited for the π → π* excitations. This strongly supports a previously made hypothesis that attributes the excellent performance of LHs based on a common LMF to cross-spin-channel nondynamical correlation terms.

Accurate Prediction of Band Structure of FeS2: A Hard Quest of Advanced First-Principles Approaches

The pyrite and marcasite polymorphs of FeS2 have attracted considerable interests for their potential applications in optoelectronic devices because of their appropriate electronic and optical properties. Controversies regarding their fundamental band gaps remain in both experimental and theoretical materials research of FeS2. In this work, we present a systematic theoretical investigation into the electronic band structures of the two polymorphs by using many-body perturbation theory with the GW approximation implemented in the full-potential linearized augmented plane waves (FP-LAPW) framework. By comparing the quasi-particle (QP) band structures computed with the conventional LAPW basis and the one extended by high-energy local orbitals (HLOs), denoted as LAPW + HLOs, we find that one-shot or partially self-consistent GW (G 0 W 0 and GW 0, respectively) on top of the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation with a converged LAPW + HLOs basis is able to remedy the artifact reported in the previous GW calculations, and leads to overall good agreement with experiment for the fundamental band gaps of the two polymorphs. Density of states calculated from G 0 W 0@PBE with the converged LAPW + HLOs basis agrees well with the energy distribution curves from photo-electron spectroscopy for pyrite. We have also investigated the performances of several hybrid functionals, which were previously shown to be able to predict band gaps of many insulating systems with accuracy close or comparable to GW. It is shown that the hybrid functionals considered in general fail badly to describe the band structures of FeS2 polymorphs. This work indicates that accurate prediction of electronic band structure of FeS2 poses a stringent test on state-of-the-art first-principles approaches, and the G 0 W 0 method based on semi-local approximation performs well for this difficult system if it is practiced with well-converged numerical accuracy.

Examination of How Well Long-Range-Corrected Density Functionals Satisfy the Ionization Energy Theorem

We calculated the vertical ionization energies (VIE) of 99 species in two ways to examine the accuracy of several long-range-corrected (LC) hybrid meta functionals in comparison with a gradient approximation (GA), global hybrids, and doubly hybrids. In the category of LC functionals, we examined both those with meta ingredients (i.e., that depend on the kinetic energy density) and those without them. The LC-hybrid meta functionals examined are M11, revM11, M11plus, and ωB97M-V. The reference data used to assess accuracy consist of 95 molecules and 4 atoms in the GW100 set. The two methods studied are the ΔSCF method (involving the difference of neutral and cation self-consistent field (SCF) energies) and the ionization energy theorem (involving the orbital energy of the highest occupied molecular orbital, HOMO). We calculated linear correlation coefficients (r²) and mean absolute deviations (MADs) between each approach and the reference VIE value from the CCSD(T)/def2-TZVPP level of theory. We compared the new LC-hybrid meta calculations to calculations with the 10 functionals in a previous VIE study by Brémond et al. and to the calculations with LC-BLYP (LC-Becke, Lee-Yang-Parr), CAM-B3LYP (Coulomb-attenuating-method Becke-3-parameter Lee-Yang-Parr), LC-ωHPBE, and ωB97X-D. The results show that Minnesota LC-hybrid meta functionals have the smallest mean absolute deviation of ionization energy theorem VIEs with the reference data; the LC-ωHPBE functional also does quite well in this test. This is very encouraging and indicates that LC-hybrid meta functionals would be the best starting points for the tuning strategy that has been shown to be a very good procedure for improving time-dependent density functional calculations, and it also helps explain the good success of LC-hybrid meta functionals for molecular excitation energies.

Reliable TDDFT Protocol Based on a Local Hybrid Functional for the Prediction of Vibronic Phosphorescence Spectra Applied to Tris(2,2'-bipyridine)-Metal Complexes

An efficient computational protocol for the prediction of vibrationally resolved phosphorescence spectra is developed and validated for five tris(2,2'-bipyridine)-metal complexes ([M(bpy)₃]ⁿ⁺, where M = Zn, Ru, Rh, Os, Ir). The outstanding feature of this protocol is the use of full linear-response time-dependent density functional theory (TDDFT) for the excited-state triplet calculation, i.e., the commonly seen strategies employing the Tamm-Dancoff approximation (TDA) or unrestricted density functional theory (DFT) calculations for the T₁ state are not needed. This is achieved by the use of a local hybrid functional (LH12ct-SsirPW92) that features a real-space dependent admixture of exact exchange governed by a local mixing function. The excellent performance of this LH for triplet excitation energies known from previous studies transfers to a remarkable mean absolute error of 0.06 eV for the phosphorescence 0-0 energies investigated herein, while the popular B3PW91 functional gives an error of 0.27 eV in TDDFT and 0.09 eV in unrestricted DFT calculations, respectively. The advantages of the local hybrid are particularly apparent for excited states with a mixed-valence character. The influence of spin-orbit coupling was found to be significant for [Os(bpy)₃]²⁺ red-shifting the 0-0 energy for phosphorescence by 0.17 eV, while the effect is negligible for the other complexes (<0.03 eV). The influence of the basis-set and integration-grid sizes is evaluated, and a computationally lighter protocol is validated that leads to drastic savings in computation time with negligible loss in accuracy.

NMR Indirect Spin-Spin Coupling Constants in a Modern Quasi-Relativistic Density Functional Framework

A quasi-relativistic implementation of NMR indirect spin-spin coupling constants is presented. The exact two-component (X2C) Hamiltonian and its diagonal local approximation to the unitary decoupling transformation (DLU) are utilized together with the (modified) screened nuclear spin-orbit approach. In a restricted kinetic balance, the finite nucleus model is available for both the scalar and vector potentials. The implementation supports density functionals up to the fourth rung of Jacob's ladder, i.e., (range-separated) hybrid and local hybrid functionals based on a seminumerical ansatz. We assess the quality of our quasi-relativistic X2C approach by comparison with "fully" relativistic four-component results for small main-group molecules and alkynyl compounds. The mean absolute error introduced by the DLU scheme is less than 0.05 × 10¹⁹ T J^-2 of the reduced coupling constant for the small main-group molecules and 0.5 Hz for the alkynyl compounds. Thus, the error is significantly smaller than finite nucleus size effects for heavy elements. The basis set convergence and the impact of different density functional approximations are further studied. We propose a simple scheme to develop segmented-contracted relativistic all-electron basis sets for NMR spin-spin couplings. Our implementation allows us to perform calculations of extended molecules with reasonable computational effort, which is illustrated for the ¹J(¹¹⁹Sn, ³¹P) coupling constant of a low-valent tin phosphinidenide complex. The corresponding results are in good agreement with the experimental findings.

GGA-Level Subsystem DFT Achieves Sub-kcal/mol Accuracy Intermolecular Interactions by Mimicking Nonlocal Functionals

The key feature of nonlocal kinetic energy functionals is their ability to reduce to the Thomas-Fermi functional in the regions of high density and to the von Weizsäcker functional in the region of low-density/high reduced density gradient. This behavior is crucial when these functionals are employed in subsystem DFT simulations to approximate the nonadditive kinetic energy. We propose a GGA nonadditive kinetic energy functional which mimics the good behavior of nonlocal functionals, retaining the computational complexity of typical semilocal functionals. Crucially, this functional depends on the inter-subsystem density overlap. The new functional reproduces Kohn-Sham DFT and benchmark CCSD(T) interaction energies of weakly interacting dimers in the S22-5 and S66 test sets with a mean absolute deviation well below 1 kcal/mol.

Assessing locally range-separated hybrid functionals from a gradient expansion of the exchange energy density

Locally range-separated hybrid (LRSH) functionals feature a real-space-dependent range separation function (RSF) instead of a system-independent range-separation parameter, which thus enables a more flexible admixture of exact exchange than conventional range-separated hybrid functionals. In particular, the development of suitable RSF models and exploring the capabilities of the LRSH approach, in general, are tasks that require further investigations and will be addressed in this work. We propose a non-empirical scheme based on a detailed scaling analysis with respect to a uniform coordinate scaling and on a short-range expansion of the range-separated exchange energy density to derive new RSF models from a gradient expansion of the exchange energy density. After optimizing a small set of empirical parameters introduced to enhance their flexibility, the resulting second- and fourth-order RSFs are evaluated with respect to atomic exchange energies, atomization energies, and transition barrier heights.

Assessing the Accuracy of Local Hybrid Density Functional Approximations for Molecular Response Properties

A comprehensive overview of the performance of local hybrid functionals for molecular properties like excited states, ionization potentials within the GW framework, polarizabilities, magnetizabilities, NMR chemical shifts, and NMR spin-spin coupling constants is presented. We apply the generalization of the kinetic energy, τ, with the paramagnetic current density to all magnetic properties and the excitation energies from time-dependent density functional theory. This restores gauge invariance for these properties. Different ansätze for local mixing functions such as the iso-orbital indicator, the correlation length, the Görling-Levy second-order limit, and the spin polarization are compared. For the latter, we propose a modified version of the corresponding hyper-generalized gradient approximation functional of Perdew, Staroverov, Tao, and Scuseria (PSTS) [Phys. Rev. A 2008, 78, 052513] to allow for a numerically stable evaluation of the exchange-correlation kernel and hyperkernel. The PSTS functional leads to a very consistent improvement compared to the related TPSSh functional. It is further shown that the "best" choice of the local mixing function depends on the studied property and molecular class. While functionals based on the iso-orbital indicator lead to rather accurate excitation energies and ionization energies, the results are less impressive for NMR properties, for which a considerable dependence on the considered molecular test set and nuclei is observed. Johnson's local hybrid functional based on the correlation length yields remarkable results for NMR shifts of compounds featuring heavy elements and also for the excitation energies of organic compounds.

Implementation and Validation of Local Hybrid Functionals with Calibrated Exchange-Energy Densities for Nuclear Shielding Constants

A recently reported coupled-perturbed Kohn-Sham implementation to compute nuclear shielding constants with gauge-including atomic orbitals and local hybrid functionals has been extended to cover higher derivatives of the density in the local mixing function (LMF) of the local hybrid as well as the calibration function (CF) needed to deal with the ambiguity of exchange-energy densities. This allowed the first evaluation of state-of-the-art local hybrids with "calibrated" exchange-energy densities for nuclear shieldings. Compared to previously evaluated simpler local hybrids without a CF, appreciable improvements are found for proton shieldings. Furthermore, the recent LH20t functional is still competitive with the outstanding performance of the uncalibrated LH12ct-SsirSVWN and LH12ct-SsifSVWN LHs for heavier nuclei, suggesting that LH20t is possibly the most robust choice of any rung-four functional for computing the nuclear shieldings of main-group nuclei so far. Interestingly, the presence of a CF in the functional significantly reduces the number of artifacts introduced by the widely used Maximoff-Scuseria framework to treat the local kinetic energy τ. The latter occurs in so-called t-LMFs used in many of the present local hybrids. In any case, the use of Dobson's current-density functional framework is also recommended with more advanced calibrated τ-dependent local hybrid functionals.

XABOOM: An X-ray Absorption Benchmark of Organic Molecules Based on Carbon, Nitrogen, and Oxygen 1s → π* Transitions

The performance of several standard and popular approaches for calculating X-ray absorption spectra at the carbon, nitrogen, and oxygen K-edges of 40 primarily organic molecules up to the size of guanine has been evaluated, focusing on the low-energy and intense 1s → π* transitions. Using results obtained with CVS-ADC(2)-x and fc-CVS-EOM-CCSD as benchmark references, we investigate the performance of CC2, ADC(2), ADC(3/2), and commonly adopted density functional theory (DFT)-based approaches. Here, focus is on precision rather than on accuracy of transition energies and intensities-in other words, we target relative energies and intensities and the spread thereof, rather than absolute values. The use of exchange-correlation functionals tailored for time-dependent DFT calculations of core excitations leads to error spreads similar to those seen for more standard functionals, despite yielding superior absolute energies. Long-range corrected functionals are shown to perform particularly well compared to our reference data, showing error spreads in energy and intensity of 0.2-0.3 eV and ∼10%, respectively, as compared to 0.3-0.6 eV and ∼20% for a typical pure hybrid. In comparing intensities, state mixing can complicate matters, and techniques to avoid this issue are discussed. Furthermore, the influence of basis sets in high-level ab initio calculations is investigated, showing that reasonably accurate results are obtained with the use of 6-311++G**. We name this benchmark suite as XABOOM (X-ray absorption benchmark of organic molecules) and provide molecular structures and ground-state self-consistent field energies and spectroscopic data. We believe that it provides a good assessment of electronic structure theory methods for calculating X-ray absorption spectra and will become useful for future developments in this field.

The Trip to the Density Functional Theory Zoo Continues: Making a Case for Time-Dependent Double Hybrids for Excited-State Problems

This account is written for general users of time-dependent density functional theory (TD-DFT) methods as well as chemists who are unfamiliar with the field. It includes a brief overview of conventional TD-DFT approaches and recommendations for applications to organic molecules based on our own experience. The main emphasis of this work, however, lies in providing the first in-depth review of time-dependent double-hybrid density functionals. They were first established in 2007 with very promising follow-up studies in the subsequent four years before developments or applications became scarce. The topic has regained more interest since 2017, and this account reviews those latest developments led by our group. These developments have shown unprecedented robustness for a variety of different types of electronic excitations when compared to more conventional TD-DFT methods. In particular, time-dependent double hybrids do not suffer from artificial ghost states and are able to reproduce exciton-coupled absorption spectra. Our latest methods include range separation and belong to the currently best TD-DFT methods for singlet-singlet excitations in organic molecules. While there is still room for improvement and further development in this space, we hope that this account encourages users to adjust their computational protocols to such new methods to provide more real-life testing and scenarios.

Composition- and Condition-Dependent Kinetics of Homogeneous Ester Hydrogenation by a Mn-Based Catalyst

The reaction medium and conditions are the key parameters defining the efficiency and performance of a homogeneous catalyst. In the state-of-the-art molecular descriptions of catalytic systems by density functional theory (DFT) calculations, the reaction medium is commonly reduced to an infinitely diluted ideal solution model. In this work, we carry out a detailed operando computational modeling analysis of the condition dependencies and nonideal solution effects on the mechanism and kinetics of a model ester hydrogenation reaction by a homogeneous Mn(I)-P,N catalyst. By combining DFT calculations, COSMO-RS solvent model, and the microkinetic modeling approach, the kinetic behavior of the multicomponent homogeneous catalyst system under realistic reaction conditions was investigated in detail. The effects of the reaction medium and its dynamic evolution in the course of the reaction were analyzed by comparing the results obtained for the model methyl acetate hydrogenation reaction in a THF solution and under solvent-free neat reaction conditions. The dynamic representations of the reaction medium give rise to strongly nonlinear effects in the kinetic models. The nonideal representation of the reaction medium results in pronounced condition dependencies of the computed energetics of the elementary reaction steps and the computed kinetic profiles but affects only slightly such experimentally accessible kinetic descriptors as the apparent activation energy and the degree of rate control.

Nonlocal rung-3.5 correlation from the density matrix expansion: Flat-plane condition, thermochemistry, and kinetics

The rung-3.5 approach to density functional theory constructs nonlocal approximate correlation from the expectation values of nonlocal one-electron operators. This offers an inexpensive solution to hybrid functionals' imbalance between exact nonlocal exchange and local approximate correlation. Our rung-3.5 correlation functionals also include a local complement to the nonlocal ingredient, analogous to the local exchange component of a hybrid functional. Here, we use the density matrix expansion (DME) to build rung-3.5 complements. We demonstrate how these provide a measure of local fractional occupancy and use them to approximate the flat-plane condition. We also use these complements in a three-parameter nonlocal correlation functional compatible with full nonlocal exchange. This functional approaches the accuracy of widely used hybrids for molecular thermochemistry and kinetics. The DME provides a foundation for practical, minimally empirical, nonlocal correlation functionals compatible with full nonlocal local exchange.

Toward a First-Principles Evaluation of Transport Mechanisms in Molecular Wires

Understanding charge transport through molecular wires is important for nanoscale electronics and biochemistry. Our goal is to establish a simple first-principles protocol for predicting the charge transport mechanism in such wires, in particular the crossover from coherent tunneling for short wires to incoherent hopping for longer wires. This protocol is based on a combination of density functional theory with a polarizable continuum model introduced by Kaupp et al. for mixed-valence molecules, which we had previously found to work well for length-dependent charge delocalization in such systems. We combine this protocol with a new charge delocalization measure tailored for molecular wires, and we show that it can predict the tunneling-to-hopping transition length with a maximum error of one subunit in five sets of molecular wires studied experimentally in molecular junctions at room temperature. This suggests that the protocol is also well suited for estimating the extent of hopping sites as relevant, for example, for the intermediate tunneling-hopping regime in DNA.