Multiconfigurational Correlation at DFT + U Cost: On-Site Electron-Electron Interactions Yield a Block-Localized Configuration Interaction Hamiltonian.

This work presents a first-principles wavefunction-in-DFT approach based on the Hubbard density functional theory (DFT) + U method. This approach begins with the standard DFT reference system of noninteracting electrons and introduces an electron-electron interaction projected onto DFT+U-type atomic states. The reference system's configuration interaction Hamiltonian is block-localized to these states and can be expressed in terms of state occupation numbers, state self-energies (which correspond to unscreened Hubbard U values), and the promotion energies of doubly excited Slater determinants. Simple approximations for the promotion energies provide multiconfigurational correlation energies without requiring explicit orbital localization/transform. Numerical results for fractionally occupied chromium atom, bonded chromium dimer, dissociating covalent bonds, and large active spaces show that the approach provides beyond-zero-sum accuracy at computational cost comparable to standard DFT+U.

A two-electron reducing reaction of CO2 to an oxalate anion: a theoretical study of delocalized (presolvated) electrons in Al(CH3)n(NH3)m, n = 0-2 and m = 1-6, clusters.

Presolvated electron possibility in three oxidation states of aluminum - Al(0), Al(I), and Al(II) - has been theoretically investigated for the Al + 6NH3, Al(CH3) + 5NH3, and Al(CH3)2 + 4NH3 reactions. It has been shown that the metal center adopts a tetrahedral shape for its most stable geometric structure, irrespective of the degree of Al oxidation states. Using different analysis techniques (highest occupied molecular orbital shapes, spin density distributions, and electron delocalization ranges), we showed that presolvated (delocalized) electrons are only formed in the Al(CH3)2(NH3)p coordination complexes when 2 ≤ p ≤ 4. It has also been evidenced that these delocalized electrons being powerful reducing agents allowed two CO2 molecules to be captured and form an oxalate ion in close contact with the [Al2(CH3)2(CH2)2(NH3)4]2+ dication core.

Modeling Multi-Step Organic Reactions: Can Density Functional Theory Deliver Misleading Chemistry?

Many organic reactions are characterized by a complex mechanism with a variety of transition states and intermediates of different chemical natures. Their correct and accurate theoretical characterization critically depends on the accuracy of the computational method used. In this work, we study a complex ambimodal cycloaddition with five transition states, two intermediates, and three products, and we ask whether density functional theory (DFT) can provide a correct description of this type of complex and multifaceted reaction. Our work fills a gap in that most systematic benchmarks of DFT for chemical reactions have considered much simpler reactions. Our results show that many density functionals not only lead to seriously large errors but also differ from one another in predicting whether the reaction is ambimodal. Only a few of the available functionals provide a balanced description of the complex and multifaceted reactions. The parameters varied in the tested functionals are the ingredients, the treatment of medium-range and nonlocal correlation energy, and the inclusion of Hartree-Fock exchange. These results show a clear need for more benchmarks on the mechanisms of large molecules in complex reactions.

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.

Comparing Density Functional Theory Metal-Ligand Bond Dissociation Enthalpies with Experimental Solution-Phase Enthalpies of Activation for Bond Dissociation.

The predictive ability of density functional theory is fundamental to its usefulness in chemical applications. Recent work has compared solution-phase enthalpies of activation for metal-ligand bond dissociation to enthalpies of reaction for bond dissociation, and the present work continues those comparisons for 43 density functional methods. The results for ligand dissociation enthalpies of 30 metal-ligand complexes tested in this work reveal significant inadequacies of some functionals as well as challenges from the dispersion corrections to some functionals. The analysis presented here demonstrates the excellent performance of a recent density functional, M11plus, which contains nonlocal rung-3.5 correlation. We also find a good agreement between theory and experiment for some functionals without empirical dispersion corrections such as M06, r2SCAN, M06-L, and revM11, as well as good performance for some functionals with added dispersion corrections such as ωB97X-D (which always has a correction) and BLYP, B3LYP, CAM-B3LYP, and PBE0 when the optional dispersion corrections are added.

Core-Projected Hybrids Fix Systematic Errors in Time-Dependent Density Functional Theory Predicted Core-Electron Excitations.

Linear response time-dependent density functional theory (TDDFT) is widely applied to valence, Rydberg, and charge-transfer excitations but, in its current form, makes large errors for core-electron excitations. This work demonstrates that the admixture of nonlocal exact exchange in atomic core regions significantly improves TDDFT-predicted core excitations. Exact exchange admixture is accomplished using projected hybrid density functional theory [ J. Chem. Theory Comput. 2023, 19, 837-847]. Scalar relativistic TDDFT calculations using core-projected B3LYP accurately model core excitations of second-period elements C-F and third-period elements Si-Cl, without sacrificing performance for the relative shifts of core excitation energies. Predicted K-edge X-ray near absorption edge structure (XANES) of a series of sulfur standards highlight the value of this approach. Core-projected hybrids appear to be a practical solution to TDDFT's limitations for core excitations, in the way that long-range-corrected hybrids are a practical solution to TDDFT's limitations for Rydberg and charge-transfer excitations.

Barrier Heights for Diels-Alder Transition States Leading to Pentacyclic Adducts: A Benchmark Study of Crowded, Strained Transition States of Large Molecules.

Theoretical characterization of reactions of complex molecules depends on providing consistent accuracy for the relative energies of intermediates and transition states. Here we employ the DLPNO-CCSD(T) method with core-valence correlation, large basis sets, and extrapolation to the CBS limit to provide benchmark values for Diels-Alder transition states leading to competitive strained pentacyclic adducts. We then used those benchmarks to test a diverse set of wave function and density functional methods for the absolute and relative barrier heights of these transition states. Our results show that only a few of the tested density functionals can predict the absolute barrier heights satisfactorily, although relative barrier heights are more accurate. The most accurate functionals tested are ωB97M-V, M11plus, ωB97X-V, PBE-D3(0), M11, and MN15 with MUDs from best estimates less than 3.0 kcal. These findings can guide selection of density functionals for future studies of crowded, strained transition states of large molecules.

Controlling Swing Rates in Macrocyclic Molecular Mortise Hinges.

Hinge motion is observed in macrocyclic, mortise-type molecular hinges using variable temperature NMR spectroscopy. The data is consistent with dynamic hinging from a folded-to-extended-to-folded enantiomeric state. Crystallographic and solution structures of the folded states are reported. Chemical shift predictions derived from crystallographic data corroborate fully revolute hinge motion. The rate of hinging is affected by steric congestion at the hinge axis. A macrocycle containing glycine, 1, hinges faster than one comprising aminoisobutyric acid, 2. The free energies of activation, ΔG≠ , for 1 and 2 were determined to be 13.3±0.3 kcal/mol and 16.3±0.3 kcal/mol, respectively. This barrier is largely independent of solvent across those surveyed (CD3 OD, CD3 CN, DMSO-d6 , pyridine-d5 , D2 O). Experiment and computation predict energy barriers that are consistent with disruption of an intramolecular network of hydrogen bonds. DFT calculations reveal a pathway for hinge motion.

Reimagining the Wave Function in Density Functional Theory: Exploring Strongly Correlated States in Pancake-Bonded Radical Dimers.

Pancake bonding between π-conjugated radicals challenges conventional electronic structure approximations, due to the presence of both dispersion (van der Waals) interactions and "strong" electron correlation. Here we use a reimagined wave function-in-density functional theory (DFT) approach to model pancake bonds. Our generalized self-interaction correction extends DFT's reference system of noninteracting electrons, by introducing electron-electron interactions within an active space. We show that a small variation on our previous derivation recovers a DFT-corrected complete active space method proposed by Pijeau and Hohenstein. Comparison of the two approaches shows that the latter provides reasonable dissociation curves for single bonds and pancake bonds, including excited states inaccessible to conventional linear response time-dependent DFT. The results motivate broader adoption of wavefunction-in-DFT approaches for modeling pancake bonds.

Computational and Experimental Evidence for Templated Macrocyclization: The Role of a Hydrogen Bond Network in the Quantitative Dimerization of 24-Atom Macrocycles.

In the absence of preorganization, macrocyclization reactions are often plagued by oligomeric and polymeric side products. Here, a network of hydrogen bonds was identified as the basis for quantitative yields of macrocycles derived from the dimerization of monomers. Oligomers and polymers were not observed. Macrocyclization, the result of the formation of two hydrazones, was hypothesized to proceed in two steps. After condensation to yield the monohydrazone, a network of hydrogen bonds formed to preorganize the terminal acetal and hydrazine groups for cyclization. Experimental evidence for preorganization derived from macrocycles and acyclic models. Solution NMR spectroscopy and single-crystal X-ray diffraction revealed that the macrocycles isolated from the cyclization reaction were protonated twice. These protons contributed to an intramolecular network of hydrogen bonds that engaged distant carbonyl groups to realize a long-range order. DFT calculations showed that this network of hydrogen bonds contributed 8.7 kcal/mol to stability. Acyclic models recapitulated this network in solution. Condensation of an acetal and a triazinyl hydrazine, which adopted a number of conformational isomers, yielded a hydrazone that adopted a favored rotamer conformation in solution. The critical hydrogen-bonded proton was also evident. DFT calculations of acyclic models showed that the rotamers were isoenergetic when deprotonated. Upon protonation, however, energies diverged with one low-energy rotamer adopting the conformation observed in the macrocycle. This conformation anchored the network of hydrogen bonds of the intermediate. Computation revealed that the hydrogen-bonded network in the acyclic intermediate contributed up to 14 kcal/mol of stability and preorganized the acetal and hydrazine for cyclization.

Projected Hybrid Density Functionals: Method and Application to Core Electron Ionization.

This work introduces a new class of hybrid density functional theory (DFT) approximations, which incorporate different fractions of nonlocal exact exchange in predefined states such as core atomic orbitals (AOs). These projected hybrid density functionals are related to range-separated hybrid functionals, which incorporate different fractions of nonlocal exchange at different electron-electron separations. This work derives projected hybrids using the Adiabatic Projection formalism. One projects the electron-electron interaction operator onto the chosen predefined states, introduces the projected operator into the noninteracting Kohn-Sham reference system, and employs a formally exact density functional to model the remaining electron-electron interactions. Projected hybrids, like range-separated hybrids, approximate the partially interacting reference system's ground-state wave function as a single Slater determinant. Projected hybrids are readily implemented into existing density functional codes, requiring only a projection of the one-particle density matrices and exchange operators entering existing routines. This work also presents an application to core electron ionization. Projecting onto core atomic orbitals allows us to introduce additional nonlocal exchange into atomic core regions. This reduces the impact of self-interaction error on computed core electron properties. Benchmark studies are reported for PBE0c70, a core-projected variant of the Perdew-Burke-Ernzerhof global hybrid PBE0, in which the fraction of nonlocal exchange is increased from 25% to 70% in atomic core regions. PBE0c70-predicted core orbital energies accurately recover nonrelativistic core-electron binding energies of second-period elements Li-Ne and third-period elements Na-Ar, without degrading the good performance of PBE0 for atomization energies and valence ionization potentials.

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.

Well-Tempered Metadynamics Simulations Predict the Structural and Dynamic Properties of a Chiral 24-Atom Macrocycle in Solution.

Inspired by therapeutic potential, the molecular engineering of macrocycles is garnering increased interest. Exercising control with design, however, is challenging due to the dynamic behavior that these molecules must demonstrate in order to be bioactive. Herein, the value of metadynamics simulations is demonstrated: the free-energy surfaces calculated reveal folded and flattened accessible conformations of a 24-atom macrocycle separated by barriers of ∼6 kT under experimentally relevant conditions. Simulations reveal that the dominant conformer is folded-an observation consistent with a solid-state structure determined by X-ray crystallography and a network of rOes established by 1H NMR. Simulations suggest that the macrocycle exists as a rapidly interconverting pair of enantiomeric, folded structures. Experimentally, 1H NMR shows a single species at room temperature. However, at lower temperature, the interconversion rate between these enantiomers becomes markedly slower, resulting in the decoalescence of enantiotopic methylene protons into diastereotopic, distinguishable resonances due to the persistence of conformational chirality. The emergence of conformational chirality provides critical experimental support for the simulations, revealing the dynamic nature of the scaffold-a trait deemed critical for oral bioactivity.

Systematically Improvable Generalization of Self-Interaction Corrected Density Functional Theory.

Perdew-Zunger self-interaction correction (PZSIC) reintroduces an exact constraint to approximate density functional theory (DFT). However, PZSIC can paradoxically degrade performance, and standard DFT approximations (with or without PZSIC) are not systematically improvable. We use the adiabatic projection formalism (Janesko, B. G. J. Chem. Phys. 2022, 156, 014111, https://doi.org/10.1063/5.0076144) to derive PZSIC in terms of a reference system experiencing only electron self-interaction. Generalization to a "self-and-some-others" interaction introduces correlation into the reference system, systematically bridging from PZSIC to exact wave function theory without the double counting of correlation. Minimal active spaces accurately treat nearly one electron, near-equilibrium, and strongly correlated model systems at modest computational expense.

Adiabatic projection: Bridging ab initio, density functional, semiempirical, and embedding approximations.

Modern electronic structure approximations routinely employ reference systems described by approximate Hamiltonians. This work introduces the adiabatic projection formalism for building formally exact corrections to such reference systems. Starting from the real Hamiltonian of a many-electron system, one constructs a reference system Hamiltonian by projecting the kinetic and electron-electron interaction operators onto "interesting" states. The reference system is corrected by density functionals for the difference between the projected and unprojected kinetic and electron-electron energies. These density functionals are constructed from adiabatic connections between the reference and real systems. The Hohenberg-Kohn theorems imply the existence of exact functionals, which can ensure that the reference system's ground-state energy and density match the real system. Adiabatic projection further generalizes Kohn-Sham density functional theory (DFT) and the generalized adiabatic connection [W. Yang, J. Chem. Phys. 109, 10107 (1998)] and recovers these methods for certain choices of projection operators. Other choices of projection operators offer new opportunities, including formally exact and systematically improvable analogues to wavefunction-in-DFT embedding, DFT+U, and semiempirical theories. Numerical results are presented for two representative choices: a projected exchange-correlation correction to small-basis-set coupled cluster theory and a projected kinetic energy density functional correcting basis set errors in DFT. The latter offers performance for dimerization energies approaching the Boys-Bernardi counterpoise correction while also correcting intramolecular basis set superposition errors.

Density functionals with full nonlocal exchange, nonlocal rung-3.5 correlation, and D3 dispersion: Combined accuracy for general main-group thermochemistry, kinetics, and noncovalent interactions.

We introduce the HF-R35-D3(BJ) functional combining full nonlocal exact (Hartree-Fock-like, HF) exchange, inexpensive rung-3.5 correlation constructed from nonlocal one-electron operators, and nonlocal D3 dispersion corrections. HF-R35-D3(BJ) is among the first full-exact-exchange functionals offering competitive accuracy for general main-group thermochemistry, kinetics, and noncovalent interactions. HF-R35-D3(BJ) gives weighted mean absolute deviation WTMAD-2 8.5 kcal/mol across the entire GMTKN55 dataset, outperforming most dispersion-corrected semilocal functionals and approaching the accuracy of dispersion-corrected global hybrids. This requires six fitted parameters, three each in the nonlocal correlation and dispersion corrections. Full nonlocal exchange appears to help give accurate binding energies and reasonable energy orderings for water hexamers. These results motivate continued exploration of inexpensive nonlocal correlation corrections to nonlocal exchange.

Replacing hybrid density functional theory: motivation and recent advances.

Density functional theory (DFT) is the most widely-used electronic structure approximation across chemistry, physics, and materials science. Every year, thousands of papers report hybrid DFT simulations of chemical structures, mechanisms, and spectra. Unfortunately, hybrid DFT's accuracy is ultimately limited by tradeoffs between over-delocalization and under-binding. This review summarizes these tradeoffs, and introduces six modern attempts to go beyond them while maintaining hybrid DFT's relatively low computational cost: DFT+U, self-interaction corrections, localized orbital scaling corrections, local hybrid functionals, real-space nondynamical correlation, and our rung-3.5 approach. The review concludes with practical suggestions for DFT users to identify and mitigate these tradeoffs' impact on their simulations.

Mechanistic Insights into Iron-Catalyzed C-H Bond Activation and C-C Coupling.

Iron-catalyzed C-C coupling reactions of pyrrole provide a unique alternative to the traditional Pd-catalyzed counterpart. However, many details regarding the actual mechanism remain unknown. A series of macrocyclic iron(III) complexes were used to evaluate specifics related to the role of O2, radicals, and µ-oxodiiron-complex participation in the catalytic cycle. It was determined that the mononuclear tetra-azamacrocyclic complex is a true catalyst and not a stoichiometric reagent, while more than one equivalent of a sacrificial oxidant is needed. Furthermore, the reaction does not proceed through an organic radical pathway. µ-Oxodiiron complexes are not involved in the main catalytic pathway, and the dimers are, in fact, off-cycle species that decrease catalytic efficiency.

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.

Calculation of magnetic properties with density functional approximations including rung 3.5 ingredients.

Density functional theory is widely used for modeling the magnetic properties of molecules, solids, and surfaces. Rung-3.5 ingredients, based on the expectation values of nonlocal one-electron operators, are new promising tools for the construction of exchange-correlation functional approximations. We present the formal extension of rung-3.5 ingredients to the calculation of magnetic properties. We add to the underlying nonlocal operators a dependence on the gauge of the magnetic field, and we derive the working equations for rung-3.5 expectation values in basis sets of gauge-including atomic orbitals. We demonstrate that the gauge corrections are significant. We conclude with an initial study of chemical shifts, optical rotatory dispersion, and Raman optical activity spectra predicted by M11plus, a range-separated hybrid meta functional incorporating nonlocal rung-3.5 correlation. M11plus proves to be reasonably accurate, further motivating the incorporation of nonlocal rung-3.5 ingredients in new density functional approximations.