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.

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.

Nuclear Spin-Spin Couplings: Efficient Evaluation of Exact Exchange and Extension to Local Hybrid Functionals.

We present an efficient implementation for the computation of nuclear spin-spin coupling tensors within density functional theory into the TURBOMOLE software suite. Emphasis is put on methods to efficiently evaluate the Hartree-Fock exchange needed for hybrid functionals: resolution of the identity and seminumerical evaluation on a grid. Our algorithm allows for the selection of specific nuclei for the reduction of calculation times. Further, the accuracy of locally dense basis sets in the density functional theory framework is investigated. These features allow for the routine computation of coupling constants in systems comprising about 100 carbon atoms within less than one day on a single CPU and within a few hours when using the OpenMP variant. Based on seminumerical integration, the first implementation of local hybrid functionals for spin-spin couplings is reported. This has allowed a preliminary evaluation of position-dependent exact-exchange admixture in three local hybrid functionals for a set of 80 isotropic spin-spin couplings in 23 small main-group molecules against CC3 and MCSCF reference data. Two of the local hybrids (LH14t-calPBE and LH07t-SVWN) are the top performers in the overall statistical evaluation compared to several standard functionals (TPSS, TPSSh, B3LYP, PBE0, and BHLYP), in particular, as they do not exhibit notable outliers for specific coupling types.

Unusually Large Effects of Charge-assisted C-H⋠⋠⋠F Hydrogen Bonds to Anionic Fluorine in Organic Solvents: Computational Study of 19 F NMR Shifts versus Thermochemistry.

A comparison of computed 19 F NMR chemical shifts and experiment provides evidence for large specific solvent effects for fluoride-type anions interacting with the σ*(C-H) orbitals in organic solvents like MeCN or CH2 Cl2 . We show this for systems ranging from the fluoride ion and the bifluoride ion [FHF]- to polyhalogen anions [ClFx ]- . Discrepancies between computed and experimental shifts when using continuum solvent models like COSMO or force-field-based descriptions like the 3D-RISM-SCF model show specific orbital interactions that require a quantum-mechanical treatment of the solvent molecules. This is confirmed by orbital analyses of the shielding constants, while less negatively charged fluorine atoms (e. g., in [EF4 ]- ) do not require such quantum-mechanical treatments to achieve reasonable accuracy. The larger 19 F solvent shift of fluoride in MeCN compared to water is due to the larger coordination number in the former. These observations are due to unusually strong charge-assisted C-H⋅⋅⋅F- hydrogen bonds, which manifest beyond some threshold negative natural charge on fluorine of ca. < -0.6 e. The interactions are accompanied by sizable free energies of solvation, in the order F- ≫[FHF]- >[ClF2 ]- >[ClF4 ]- . COSMO-RS solvation free energies tend to moderately underestimate those from the micro-solvated cluster treatment. Red-shifted and intense vibrational C-H stretching bands, potentially accessible in bulk solution, are further spectroscopic finger prints.

Lessons from the Spin-Polarization/Spin-Contamination Dilemma of Transition-Metal Hyperfine Couplings for the Construction of Exchange-Correlation Functionals.

Hyperfine couplings (HFCs) of open-shell transition-metal centers are known to often depend crucially on core-shell spin polarization (CSSP). The latter is typically underestimated by semilocal density functionals, while admixture of exact exchange (EXX) in (global) hybrid functionals enhances CSSP. Unfortunately, a metal-ligand antibonding character of one or more of the singly occupied molecular orbitals of the complex will cause substantial valence-shell spin polarization (VSSP), which for global hybrids with higher EXX admixtures may lead to substantial spin contamination, thereby deteriorating the overall electronic structure and the dipolar couplings. In view of this known dilemma, we use a subset of 3d complexes from an earlier study (M. Munzarová, M. Kaupp J. Phys. Chem. A 1999, 103, 9966-9983) to examine systematically a wide range of exchange-correlation functionals for metal HFCs, including highly parametrized (meta-)GGAs, global, and range-separated hybrid functionals not yet available in earlier studies, as well as for the first time local hybrids with real-space position-dependent EXX admixture. Both CSSP and VSSP have been carefully analyzed in terms of their orbital contributions, both for cases dominated only by CSSP and for systems influenced crucially by VSSP and spin contamination. While some more parametrized meta-GGA functionals (τ-HCTH, VSXC, partially M06-L) provide surprisingly realistic CSSP, some others (MN12-L, MN15-L) and some global hybrids (M05, M06, partly MN15) exhibit dramatic shortcomings in describing the CSSP contributions. Local hybrid functionals provide a promising way of enhancing CSSP by high EXX admixture in the core region while avoiding excessive VSSP and thus spin contamination. These analyses provide important insights that may help to construct improved functionals for HFCs and related properties (e.g., contact NMR shifts).