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.

Investigation of Isolated IrF5 -, IrF6 - Anions and M[IrF6] (M=Na, K, Rb, Cs) Ion Pairs by Matrix-Isolation Spectroscopy and Relativistic Quantum-Chemical Calculations.

The molecular IrF5 -, IrF6 - anions and M[IrF6] (M=Na, K, Rb, Cs) ion pairs were prepared by co-deposition of laser-ablated alkali metal fluorides MF with IrF6 and isolated in solid neon or argon matrices under cryogenic conditions. The free anions were obtained as well by co-deposition of IrF6 with laser-ablated metals (Ir or Pt) as electron sources. The products were characterized in a combined analysis of matrix IR spectroscopy and electronic structure calculations using two-component quasi-relativistic DFT methods accounting for spin-orbit coupling (SOC) effects as well as multi-reference configuration-interaction (MRCI) approaches with SOC. Inclusion of SOC is crucial in the prediction of spectra and properties of IrF6 - and its alkali-metal ion pairs. The observed IR bands and the computations show that the IrF6 - anion adopts an Oh structure in a nondegenerate ground state stabilized by SOC effects, and not a distorted D4h structure in a triplet ground state as suggested by scalar-relativistic calculations. The corresponding "closed-shell" M[IrF6] ion pairs with C3v symmetry are stabilized by coordination of an alkali metal ion to three F atoms, and their structural change in the series from M=Na to Cs was proven spectroscopically. There is no evidence for the formation of IrF7, IrF7 - or M[IrF7] (M=Na, K, Rb, Cs) ion pairs in our experiments.

Controlling the Activation at NiII-CO2 2- Moieties through Lewis Acid Interactions in the Second Coordination Sphere.

Nickel complexes with a two-electron reduced CO2 ligand (CO2 2-, "carbonite") are investigated with regard to the influence alkali metal (AM) ions have as Lewis acids on the activation of the CO2 entity. For this purpose complexes with NiII(CO2)AM (AM=Li, Na, K) moieties were accessed via deprotonation of nickel-formate compounds with (AM)N(iPr)2. It was found that not only the nature of the AM ions in vicinity to CO2 affect the activation, but also the number and the ligation of a given AM. To this end the effects of added (AM)N(R)2, THF, open and closed polyethers as well as cryptands were systematically studied. In 14 cases the products were characterized by X-ray diffraction and correlations with the situation in solution were made. The more the AM ions get detached from the carbonite ligand, the lower is the degree of aggregation. At the same time the extent of CO2 activation is decreased as indicated by the structural and spectroscopic analysis and reactivity studies. Accompanying DFT studies showed that the coordinating AM Lewis acidic fragment withdraws only a small amount of charge from the carbonite moiety, but it also affects the internal charge equilibration between the LtBuNi and carbonite moieties.

Structure and Photophysics of N-Tolanyl-phenochalcogenazines and their Radical Cations.

The study focuses on the structural and photophysical characteristics of neutral and oxidized forms of N-tolanyl-phenochalcogenazines PZX-tolan with X=O, S, Se, and Te. X-ray crystal structure analyses show a pseudo-equatorial (pe) structure of the tolan substituent in the O, S, and Se dyads, while the Te dyad possesses a pseudo-axial (pa) structure. DFT calculations suggest the pe structure for O and S, and the pa structure for Se and Te as stable forms. Steady-state and femtosecond-time resolved optical spectroscopy in toluene solution indicate that the O and S dyads emit from a CT state, whereas the Se and Te dyads emit from a tolan-localized state. The T1 state is tolan-localized in all cases, showing phosphorescence at 77 K. The heavy atom effect of chalcogens induces intersystem crossing from S1 to Tx, resulting in a decreasing S1 lifetime from 2.1 ns to 0.42 ps. The T1 states possess potential for singlet oxygen sensitization with a high quantum yield (ca. 40 %) for the O, S, and Se dyads. Radical cations exhibit spin density primarily localized at the heterocycle. EPR measurements and quasirelativistic DFT calculations reveal a very strong g-tensor anisotropy, supporting the pe structure for the S and Se derivatives.

Spectroscopic Identification of Trifluorosilylphosphinidene and Isomeric Phosphasilene and Silicon Trifluorophosphine Complex.

The perfluorinated silylphosphinidene, F3SiP, in the triplet ground state is generated by the reaction of laser-ablated silicon atoms with PF3 in solid neon and argon matrices. The reactions proceed with the initial formation of a silicon trifluorophosphine complex, F3PSi, in the triplet ground state, and a more stable inserted phosphasilene, FPSiF2, in the singlet ground state upon deposition. The trifluorosilylphosphinidene was formed through F-migration reactions of FPSiF2 and F3PSi following a two-state mechanism under irradiation with visible light (λ = 470 nm) and full arc light (λ > 220 nm), respectively. High-level quantum-chemical methods support the identification of F3PSi, FPSiF2, and F3SiP by matrix-isolation IR spectroscopy.

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.

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.

Hydrogen Bonding in Platinum Indolylphosphine Polyfluorido and Fluorido Complexes.

The reaction of the Pt complexes cis-[Pt(CH3 )(Ar){Ph2 P(Ind)}2 ] (Ind=2-(3-methyl)indolyl, Ar=4-tBuC6 H4 (1 a), 4-CH3 C6 H4 (1 b), Ph (1 c), 4-FC6 H4 (1 d), 4-ClC6 H4 (1 e), 4-CF3 C6 H4 (1 f)) with HF afforded the polyfluorido complexes trans-[Pt(F(HF)2 )(Ar){Ph2 P(Ind)}2 ] 2 a-f, which can be converted into the fluoride derivatives trans-[Pt(F)(Ar){Ph2 P(Ind)}2 ] (3 a-f) by treatment with CsF. The compounds 2 a-f and 3 a-f were characterised thoroughly by multinuclear NMR spectroscopy. The data reveal hydrogen bonding of the fluorido ligand with HF molecules and the indolylphosphine ligand. Polyfluorido complexes 2 a-f show larger |1 J(F,Pt)|, but lower 1 J(H,F) coupling constants when compared to the fluorido complexes 3 a-f. Decreasing 1 J(P,Pt) coupling constants in 2 a-f and 3 a-f suggest a cis influence of the aryl ligands in the following order: 4-tBuC6 H4 (a) ≈4-CH3 C6 H4 (b)

A 2,2-Difluoroimidazolidine Derivative for Deoxyfluorination Reactions: Mechanistic Insights by Experimental and Computational Studies.

A N-heterocyclic deoxyfluorinating agent SIMesF2 was synthesized by nucleophilic fluorination of N,N-1,3-dimesityl-2-chloroimidazolidinium chloride (3) at room temperature. SIMesF2 was applied to deoxyfluorinate carboxylic acids and alcohols and convert benzaldehyde into difluorotoluene. Mechanistic studies by NMR spectroscopy suggest reaction pathways of the carboxylic acid to acyl fluoride via outer-sphere fluorinations at an imidazolidinium ion by polyfluoride. DFT studies give further insight by exploring mechanistic details which distinguish the fluorination of aldehydes from that of carboxylic acids. Furthermore, a consecutive reaction sequence for the oxidation of an aldehyde followed by in situ fluorination of the generated carboxylic acid was developed.

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.

Strong-correlation density functionals made simple.

Recent work on incorporating strong-correlation (sc) corrections into the scLH22t local hybrid functional [A. Wodynski and M. Kaupp, J. Chem. Theory Comput. 18, 6111-6123 (2022)] used a hybrid procedure, applying a strong-correlation factor derived from the reverse Becke-Roussel machinery of the KP16/B13 and B13 functionals to the nonlocal correlation term of a local hybrid functional. Here, we show that adiabatic-connection factors for strong-correlation-corrected local hybrids (scLHs) can be constructed in a simplified way based on a comparison of semi-local and exact exchange-energy densities only, without recourse to exchange-hole normalization. The simplified procedure is based on a comparative analysis of Becke's B05 real-space treatment of nondynamical correlation and that in LHs, and it allows us to use, in principle, any semi-local exchange-energy density in the variable used to construct local adiabatic connections. The derivation of competitive scLHs is demonstrated based on either a modified Becke-Roussel or a simpler Perdew-Burke-Ernzerhof (PBE) energy density, leading to the scLH23t-mBR and scLH23t-tPBE functionals, which both exhibit low fractional spin errors while retaining good performance for weakly correlated situations. We also report preliminary attempts toward more detailed modeling of the local adiabatic connection, allowing a reduction of unphysical local maxima in spin-restricted bond-dissociation energy curves (scLH23t-mBR-P form). The simplified derivations of sc-factors reported here provide a basis for future constructions and straightforward implementation of exchange-correlation functionals that escape the zero-sum game between low self-interaction and static-correlation errors.

The Use of Bridging Ligand Substituents to Bias the Population of Localized and Delocalized Mixed-Valence Conformers in Solution.

The electronic structure and associated spectroscopic properties of ligand-bridged, bimetallic 'mixed-valence' complexes of the general form {M}(µ-B){M+ } are dictated by the electronic couplings, and hence orbital overlaps, between the metal centers mediated by the bridge. In the case of complexes such as [{Cp*(dppe)Ru}(µ-C≡CC6 H4 C≡C){Ru(dppe)Cp*}]+ , the low barrier to rotation of the half-sandwich metal fragments and the arylene bridge around the acetylene moieties results in population of many energy minima across the conformational energy landscape. Since orbital overlap is also sensitive to the particular mutual orientations of the metal fragment(s) and arylene bridge through a Karplus-like relationship, the different members of the population range exemplify electronic structures ranging from strongly localized (weakly coupled Robin-Day Class II) to completely delocalized (Robin-Day Class III). Here, we use electronic structure calculations with the hybrid density functional BLYP35-D3 and a continuum solvent model in combination with UV-vis-NIR and IR spectroelectrochemical studies to show that the conformational population in complexes [{Cp*(dppe)Ru}(µ-C≡CArC≡C){Ru(dppe)Cp*]+ , and hence the dominant electronic structure, can be biased through the steric and electronic properties of the diethynylarylene (Ar) moiety (Ar=1,4-C6 H4 , 1,4-C6 F4 , 1,4-C6 H2 -2,5-Me2 , 1,4-C6 H2 -2,5-(CF3 )2 , 1,4-C6 H2 -2,5-i Pr2 ).

Synthesis of Intramolecular P/Al-Based Frustrated Lewis Pairs via Aluminum-Tin-Exchange and their Reactivity toward CO2.

Frustrated Lewis pairs (FLPs) composed of acidic alane and basic phosphane functions, separated by a xanthene linker, can be prepared through the corresponding Me3 Sn derivative and methyl aluminum compounds with elimination of Me4 Sn. This way MeClAl-, Cl2 Al- and (C6 F5 )2 Al- moieties could be introduced and the resulting FLPs are stabilized by a further equivalent of the alane precursors. In contact with the FLPs CO2 is bound via the C atom at the phosphane functions and the two O atoms at the Al centers. The residues at the latter determine the binding strength. Hence, in case of MeClAl CO2 capture occurs at higher pressure and under ambient conditions CO2 is released again, while for Cl2 Al and (C6 F5 )2 Al CO2 binding becomes irreversible. The results of DFT calculations rationalize these findings by the high thermodynamic stabilization in case of more electronegative residues, which concomitantly lead to higher barriers, and in case of (C6 F5 )2 Al further stabilization is achieved through a low reorganization energy.

Investigation of Molecular Iridium Fluorides IrFn (n=1-6): A Combined Matrix-Isolation and Quantum-Chemical Study.

The photo-initiated defluorination of iridium hexafluoride (IrF6 ) was investigated in neon and argon matrices at 6 K, and their photoproducts are characterized by IR and UV-vis spectroscopies as well as quantum-chemical calculations. The primary photoproducts obtained after irradiation with λ=365 nm are iridium pentafluoride (IrF5 ) and iridium trifluoride (IrF3 ), while longer irradiation of the same matrix with λ=278 nm produced iridium tetrafluoride (IrF4 ) and iridium difluoride (IrF2 ) by Ir-F bond cleavage or F2 elimination. In addition, IrF5 can be reversed to IrF6 by adding a F atom when exposed to blue-light (λ=470 nm) irradiation. Laser irradiation (λ=266 nm) of IrF4 also generated IrF6 , IrF5 , IrF3 and IrF2 . Alternatively, molecular binary iridium fluorides IrFn (n=1-6) were produced by co-deposition of laser-ablated iridium atoms with elemental fluorine in excess neon and argon matrices under cryogenic conditions. Computational studies up to scalar relativistic CCSD(T)/triple-ζ level and two-component quasirelativistic DFT computations including spin-orbit coupling effects supported the formation of these products and provided detailed insights into their molecular structures by their characteristic Ir-F stretching bands. Compared to the Jahn-Teller effect, the influence of spin-orbit coupling dominates in IrF5 , leading to a triplet ground state with C4v symmetry, which was spectroscopically detected in solid argon and neon matrices.

Competition for Hydride Between Silicon and Boron: Synthesis and Characterization of a Hydroborane-Stabilized Silylium Ion.

Potent main-group Lewis acids are capable of activating element-hydrogen bonds. To probe the rivalry for hydride between silylium- and borenium-ion centers, a neutral precursor with the hydrosilane and hydroborane units in close proximity on a naphthalene-1,8-diyl platform was designed. Abstraction of one hydride leads to a hydroborane-stabilized silylium ion rather than a hydrosilane-coordinated borenium ion paired with [B(C6 F5 )4 ]- or [HCB11 Cl11 ]- as counteranions. Characterization by multinuclear NMR spectroscopy and X-ray diffraction supported by DFT calculations reveals a cationic, unsymmetrical open three-center, two-electron (3c2e) Si-H-B linkage.

Synthesis, Reactivity, and Bonding of Gold(I) Fluorido-Phosphine Complexes.

Gold(I) fluorido complexes with phosphine ligands have been synthesized from their respective iodido precursors. The bonding situation in comparison between complexes bearing phosphines and N-heterocyclic carbenes (NHCs) was explored quantum-chemically, obtaining similar results for both. Calculations of the 19F NMR chemical shifts match the experimental values well, including the approximately 40 ppm low-field shifts for the phosphine complexes compared to the NHC complexes, in spite of similar negative charges on fluorine. The reactivity of the highly water-sensitive gold(I) fluorido complexes was studied, resulting in substitution at the metal using trimethylsilyl reagents. The compounds studied were characterized using NMR as well as X-ray diffraction methods.

Relativistic Heavy-Neighbor-Atom Effects on NMR Shifts: Concepts and Trends Across the Periodic Table.

Chemical shifts present crucial information about an NMR spectrum. They show the influence of the chemical environment on the nuclei being probed. Relativistic effects caused by the presence of an atom of a heavy element in a compound can appreciably, even drastically, alter the NMR shifts of the nearby nuclei. A fundamental understanding of such relativistic effects on NMR shifts is important in many branches of chemical and physical science. This review provides a comprehensive overview of the tools, concepts, and periodic trends pertaining to the shielding effects by a neighboring heavy atom in diamagnetic systems, with particular emphasis on the "spin-orbit heavy-atom effect on the light-atom" NMR shift (SO-HALA effect). The analyses and tools described in this review provide guidelines to help NMR spectroscopists and computational chemists estimate the ranges of the NMR shifts for an unknown compound, identify intermediates in catalytic and other processes, analyze conformational aspects and intermolecular interactions, and predict trends in series of compounds throughout the Periodic Table. The present review provides a current snapshot of this important subfield of NMR spectroscopy and a basis and framework for including future findings in the field.

Excited states and spin-orbit coupling in chalcogen substituted perylene diimides and their radical anions.

While spin-orbit coupling does not play a decisive role in the photophysics of unsubstituted perylene diimides (PDI), this changes dramatically when two phenylselenyl or phenyltelluryl substituents were attached to the PDI bay positions. In the series of PhO-, PhS-, PhSe-, and PhTe-substituted PDIs we observed strongly decreasing fluorescence quantum yield as a consequence of strongly increasing intersystem crossing (ISC) rate, measured by transient absorption spectroscopy with fs- and ns-time resolution as well as by broadband fluorescence upconversion. Time-dependent density functional calculations suggest increasing spin-orbit coupling due to the internal heavy-atom effect as the reason for fast ISC. In case of the selenium PDI derivative we found significant singlet oxygen sensitization via the PDI triplet state. The corresponding radical anions of the chalcogen substituted PDIs were also prepared and investigated by optical and EPR spectroscopy. Here, the increasing SOC results in an increase of the g-tensor anisotropy, and of the isotropic g-value in solution, albeit quasirelativistic density functional calculations show only a relatively small fraction of the spin density to be located on the chalcogen atom.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Reaction Entropies in Solution from Analytical Three-Dimensional Reference Interaction Site Model Derivatives with Application to Redox and Spin-Crossover Processes.

An analytical approach to compute the excess entropy of solvation at constant pressure in three-dimensional reference interaction site model (3D-RISM) calculations is presented. It includes the changes in the macroscopic dielectric constant of the solvent upon variation of temperature and density. The approach is exact within the framework of force-field descriptions of the solute and gives reasonable results for self-consistently determined electrostatics as used in the 3D-RISM-self-consistent field approach, particularly for entropy differences. The new method is applied to simple examples of reaction entropies of iron complexes in aqueous solution, for which simple gas-phase calculations and many other approaches give unreliable estimates. For both redox half-reactions and spin-crossover processes, (semi)quantitative agreement with experimental reaction entropies can be achieved out of the box.