Current density in exchange-correlation functionals: Application to atomic states

Spin Currents via the Gauge Principle for Meta-Generalized Gradient Exchange-Correlation Functionals

The prominence of density functional theory in the field of electronic structure computation stems from its ability to usefully balance accuracy and computational effort. At the base of this ability is a functional of the electron density: the exchange-correlation energy. This functional satisfies known exact conditions that guide the derivation of approximations. The strongly constrained and appropriately normed (SCAN) approximation stands out as a successful, modern, example. In this Letter, we demonstrate how the SU(2) gauge invariance of the exchange-correlation functional in spin current density functional theory allows us to add an explicit dependence on spin currents in the SCAN functional (here called JSCAN)-and similar meta-generalized-gradient functional approximations-solely invoking first principles. In passing, a spin-current dependent generalization of the electron localization function (here called JELF) is also derived. The extended forms are implemented in a developer's version of the crystal23 program. Applications on molecules and materials confirm the practical relevance of the extensions.

Comment on: "Toward Accurate Two-Photon Absorption Spectrum Simulations: Exploring the Landscape beyond the Generalized Gradient Approximation"

A recent benchmark study of two-photon absorption (2PA) strengths using meta-generalized gradient approximation (MGGA) exchange-correlation functionals by Ahmadzadeh, Li, Rinkevicius, Norman, and Zaleśny (ALRNZ24) [ J. Phys. Chem. Lett. 2024, 15, 969] misrepresents the state of the field in this area. Not only was an assessment of 2PA strengths for the exact same benchmark published previously; ALRNZ24 also uses a gauge-variant form of MGGA response theory which produces erratic behavior for certain benchmark systems. Applications of MGGAs to optical and magnetic response properties should use a gauge-invariant extension of MGGA functionals such as paramagnetic current-dependent MGGAs.

Paramagnetic Nuclear Magnetic Resonance Shifts for Triplet Systems and Beyond with Modern Relativistic Density Functional Methods

An efficient framework for the calculation of paramagnetic NMR (pNMR) shifts within exact two-component (X2C) theory and (current-dependent) density functional theory (DFT) up to the class of local hybrid functionals (LHFs) is presented. Generally, pNMR shifts for systems with more than one unpaired electron depend on the orbital shielding contribution and a temperature-dependent term. The latter includes zero-field splitting (ZFS), hyperfine coupling (HFC), and the g-tensor. For consistency, we calculate these three tensors at the same level of theory, i.e., using scalar-relativistic X2C augmented with spin-orbit perturbation theory. Results for pNMR chemical shifts of transition-metal complexes reveal that this X2C-DFT framework can yield good results for both the shifts and the individual tensor contributions of metallocenes and related systems, especially if the HFC constant is large. For small HFC constants, the relative error is often large, and sometimes the sign may be off. 4d and 5d complexes with more complicated structures demonstrate the limitations of a fully DFT-based approach. Additionally, a Co-based complex with a very large ZFS and pronounced multireference character is not well described. Here, a hybrid DFT-multireference framework is necessary for accurate results. Our results show that X2C is sufficient to describe relativistic effects and computationally cheaper than a fully relativistic approach. Thus, it allows use of large basis sets for converged HFCs. Overall, current-dependent meta-generalized gradient approximations and LHFs show some potential; however, the currently available functionals leave a lot to be desired, and the predictive power is limited.

Avoiding spin contamination and spatial symmetry breaking by exact-exchange-only optimized-effective-potential methods within the symmetrized Kohn-Sham framework

For open-shell atoms and molecules, Kohn-Sham (KS) methods typically resort to spin-polarized approaches that exhibit spin-contamination and often break spatial symmetries. As a result, the KS Hamiltonian operator and the KS orbitals do not exhibit the space and spin symmetry of the physical electron system. The KS formalism can be symmetrized in a rigorous way only in real space, only in spin space, or both in real and spin space. Within such symmetrized KS frameworks, we present exact-exchange-only optimized-effective-potential (OEP) methods that are free of spin contamination and/or spatial symmetry breaking. The effect of symmetrizations on the total energy and its parts and on the exchange potential is analyzed. The presented exact-exchange-only OEP methods may serve as a starting point for high-level symmetrized KS methods based, e.g., on the adiabatic-connection fluctuation-dissipation theorem.

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.

Exact two-component theory becoming an efficient tool for NMR shieldings and shifts with spin-orbit coupling

We present a gauge-origin invariant exact two-component (X2C) approach within a modern density functional framework, supporting meta-generalized gradient approximations such as TPSS and range-separated hybrid functionals such as CAM-B3LYP. The complete exchange-correlation kernel is applied, including the direct contribution of the field-dependent basis functions and the reorthonormalization contribution from the perturbed overlap matrix. Additionally, the finite nucleus model is available for the electron-nucleus potential and the vector potential throughout. Efficiency is ensured by the diagonal local approximation to the unitary decoupling transformation in X2C as well as the (multipole-accelerated) resolution of the identity approximation for the Coulomb term (MARI-J, RI-J) and the seminumerical exchange approximation. Errors introduced by these approximations are assessed and found to be clearly negligible. The applicability of our implementation to large-scale calculations is demonstrated for a tin pincer-type system as well as low-valent tin and lead complexes. Here, the calculation of the Sn nuclear magnetic resonance shifts for the pincer-type ligand with about 2400 basis functions requires less than 1 h for hybrid density functionals. Further, the impact of spin-orbit coupling on the nucleus-independent chemical shifts and the corresponding ring currents of all-metal aromatic systems is studied.

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.

Meta-generalized gradient approximations in time dependent generalized Kohn-Sham theory: Importance of the current density correction

We revisit the use of Meta-Generalized Gradient Approximations (mGGAs) in time-dependent density functional theory, reviewing conceptual questions and solving the generalized Kohn-Sham equations by real-time propagation. After discussing the technical aspects of using mGGAs in combination with pseudopotentials and comparing real-space and basis set results, we focus on investigating the importance of the current-density based gauge invariance correction. For the two modern mGGAs that we investigate in this work, TASK and r2SCAN, we observe that for some systems, the current density correction leads to negligible changes, but for others, it changes excitation energies by up to 40% and more than 0.8 eV. In the cases that we study, the agreement with the reference data is improved by the current density correction.

Gauge-Invariant Excited-State Linear and Quadratic Response Properties within the Meta-Generalized Gradient Approximation

Gauge invariance is a fundamental symmetry connected to charge conservation and is widely accepted as indispensable for any electronic structure method. Hence, the gauge variance of the time-dependent kinetic energy density τ used in many meta-generalized gradient approximations (MGGAs) to the exchange-correlation (XC) functional presents a major obstacle for applying MGGAs within time-dependent density functional theory (TDDFT). Replacing τ by the gauge-invariant generalized kinetic energy density τ̂ significantly improves the accuracy of various functionals for vertical excitation energies [R. Grotjahn, F. Furche, and M. Kaupp. J. Chem. Phys. 2022, 157, 111102]. However, the dependence of the resulting current-MGGAs (cMGGAs) on the paramagnetic current density gives rise to new exchange-correlation kernels and hyper-kernels ignored in previous implementations of quadratic and higher-order response properties. Here we report the first implementation of cMGGAs and hybrid cMGGAs for excited-state gradients and dipole moments, as well as an extension to quadratic response properties including dynamic hyperpolarizabilities and two-photon absorption cross sections. In the first comprehensive benchmark study of MGGAs and cMGGAs for two-photon absorption cross sections, the M06-2X functional is found to be superior to the GGA hybrid PBE0. Additionally, two case studies from the literature for the practical prediction of nonlinear optical properties are revisited and potential advantages of hybrid (c)MGGAs compared to hybrid GGAs are discussed. The effect of restoring gauge invariance varies depending on the employed MGGA functional, the type of excitation, and the property under investigation: While some individual excited-state equilibrium structures are significantly affected, on average, these changes result in marginal improvements when compared against high-level reference data. Although the gauge-variant MGGA quadratic response properties are generally close to their gauge-invariant counterparts, the resulting errors are not bounded and significantly exceed typical method errors in some of the cases studied. Despite the limited effects seen in benchmark studies, gauge-invariant implementations of cMGGAs for excited-state properties are desirable from a fundamental perspective, entail little additional computational cost, and are necessary for response properties consistent with cMGGA linear response calculations such as excitation energies.

Practical Post-Kohn-Sham Methods for Time-Reversal Symmetry Breaking References

The applicability of reduced scaling algorithms based on auxiliary subspace methods for the correlation energy from the random phase approximation (RPA) as well as the correlation part of the self-energy obtained from the GW method is outlined for time-reversal symmetry breaking Kohn-Sham (KS) references. The updated algorithms allow for an efficient evaluation of RPA energies and GW quasiparticle energies for molecular systems with KS references that break time-reversal symmetry. The latter occur, for example, in magnetic fields. Furthermore, KS references for relativistic open-shell molecules also break time-reversal symmetry due to the single determinant ansatz used. Errors of the updated reduced-scaling algorithms are shown to be negligible compared to reference implementations, while the overall computational scaling is reduced by 2 orders of magnitude. Ionization energies obtained from the GW approximation are shown to be robust even for the electronically complicated group of trivalent lanthanoid ions. Starting from GW quasiparticle energies, it is subsequently shown that light-matter interactions of these systems can be calculated using the Bethe-Salpeter equation (BSE). Using the combined GW-BSE method, the absorption and emission spectra of a molecular europium(III) complex can be obtained including spin-orbit coupling.

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.

Density-functional theory vs density-functional fits: The best of both

In a recent paper [A. D. Becke, J. Chem. Phys. 156, 214101 (2022)], we compared two Kohn-Sham density functionals based on physical modeling and theory with the best density-functional power series fits in the literature. With only a handful of physically motivated pre-factors, our functionals matched, and even slightly exceeded, the performance of the best power-series functionals on the general main group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) chemical database of Goerigk et al. [Phys. Chem. Chem. Phys. 19, 32184 (2017)]. This begs the question: how much can their performance be improved by adding power-series terms of our own? We address this question in the present work. First, we describe a series expansion variable that we believe contains more local physics than any other variable considered to date. Then we undertake modest, one-dimensional fits to the GMTKN55 data with our theory-based functional corrected by power-series exchange and dynamical correlation terms. We settle on 12 power-series terms (plus six parent terms) and achieve the lowest GMTKN55 "WTMAD2" error yet reported, by a substantial margin, for a hybrid Kohn-Sham density functional. The new functional is called "B22plus."

On the inhibition of cytochrome P450 3A4 by structurally diversified flavonoids

Cytochrome P450 3A4 (CYP3A4) is the most versatile enzyme involved in drug metabolism. The time-dependent inhibition of CYP3A4 by acacetin, apigenin, chrysin, and pinocembrin was experimentally detected, but not entirely elaborated so far. Thus, a two-level QM/MM (Quantum Mechanics/Molecular Mechanics) model is developed to yield insights into the receptor-flavonoid recognition at the molecular scale. Active site residues and the flavonoid are modelled using SCC-DFTB-D (QM level), while the rest of the complex is treated using AMBER force field (MM level). QM/MM binding free energies are well correlated with experimental data, indicating the largest inhibitory effect of chrysin on enzyme activity at a submicromolar concentration. Consequently, quercetin (QUE) and flavopiridol (FLP) are observed as representative examples of structurally different flavonoids. The inhibition parameters for QUE and FLP are evaluated using the well-calibrated QM/MM strategy, thereby aiding to quantitatively conceive the functional behavior of the whole family of flavonoids. A kinetic threshold for further assessment of the drug-drug interactions underlying the time-dependent inhibition of CYP3A4 by flavonoids is explored.Communicated by Ramaswamy H. Sarma.

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.

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].

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.

Density-functional theory vs density-functional fits

Kohn–Sham density-functional theory (DFT), the predominant framework for electronic structure computations in chemistry today, has undergone considerable evolution in the past few decades. The earliest DFT approximations were based on uniform electron gas models completely free of empirical parameters. Tremendous improvements were made by incorporating density gradients and a small number of parameters, typically one or two, obtained from fits to atomic data. Incorporation of exact exchange and fitting to molecular data, such as experimental heats of formation, allowed even further improvements. This, however, opened a Pandora’s Box of fitting possibilities, given the limitless choices of chemical reactions that can be fit. The result is a recent explosion of DFT approximations empirically fit to hundreds, or thousands, of chemical reference data. These fitted density functionals may contain several dozen empirical parameters. What has been lost in this fitting trend is physical modeling based on theory. In this work, we present a density functional comprising our best efforts to model exchange–correlation in DFT using good theory. We compare its performance to that of heavily fit density functionals using the GMTKN55 chemical reference data of Goerigk and co-workers [Phys. Chem. Chem. Phys. 19, 32184 (2017)]. Our density-functional theory, using only a handful of physically motivated pre-factors, competes with the best heavily fit Kohn–Sham functionals in the literature.

Linear Response of Current-Dependent Density Functional Approximations in Magnetic Fields

This Letter outlines the steps and derivations that are necessary to apply density functional approximations that depend on the current and kinetic energy density rigorously within the framework of linear-response methods, including adiabatic time-dependent current density functional theory. This includes systems with a non-zero current density in the ground state. The necessary exchange-correlation kernel for these density functional approximations is derived, and the matrix elements are given explicitly. Due to the gauge variance of the kinetic energy density in an external magnetic field, having access to the proper current-dependent exchange-correlation kernel is necessary to recover gauge invariance for excited states. As a proof of principle application, the excited states of two small molecules in strong external magnetic fields are calculated using linear-response time-dependent current density functional theory. Finally, the implications of the derived current density-dependent exchange-correlation kernel for systems with strong spin-orbit coupling are discussed.

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.

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).

Paramagnetic NMR Shielding Tensors and Ring Currents: Efficient Implementation and Application to Heavy Element Compounds

We present an efficient implementation of paramagnetic NMR shielding tensors and shifts in a nonrelativistic and scalar-relativistic density functional theory framework. For the latter, we make use of the scalar exact two-component Hamiltonian in its local approximation, and generally we apply the well established (multipole-accelerated) resolution of the identity approximation and the seminumerical exchange approximation. The perturbed density matrix of a paramagnetic NMR shielding calculation is further used to study the magnetically induced current density and ring currents of open-shell systems as illustrated for [U@Bi₁₂]^3-. [U@Bi₁₂]^3- features delocalized highest occupied molecular orbitals and sustains a net diatropic ring current of ca. 18 nA/T through the Bi₁₂ torus similar to the all-metal aromatic heavy-element cluster [Th@Bi₁₂]^4-.

Method for Calculating Excited Electronic States Using Density Functionals and Direct Orbital Optimization with Real Space Grid or Plane-Wave Basis Set

A direct orbital optimization method is presented for density functional calculations of excited electronic states using either a real space grid or a plane-wave basis set. The method is variational, provides atomic forces in the excited states, and can be applied to Kohn-Sham (KS) functionals as well as orbital-density-dependent (ODD) functionals including explicit self-interaction correction. The implementation for KS functionals involves two nested loops: (1) An inner loop for finding a stationary point in a subspace spanned by the occupied and a few virtual orbitals corresponding to the excited state; (2) an outer loop for minimizing the energy in a tangential direction in the space of the orbitals. For ODD functionals, a third loop is used to find the unitary transformation that minimizes the energy functional among occupied orbitals only. Combined with the maximum overlap method, the algorithm converges in challenging cases where conventional self-consistent field algorithms tend to fail. The benchmark tests presented include two charge-transfer excitations in nitrobenzene and an excitation of CO to degenerate π* orbitals where the importance of complex orbitals is illustrated. The application of this method to several metal-to-ligand charge-transfer and metal-centered excited states of an Fe^II photosensitizer complex is described, and the results are compared to reported experimental estimates. This method is also used to study the effect of the Perdew-Zunger self-interaction correction on valence and Rydberg excited states of several molecules, both singlet and triplet states, and the performance compared to semilocal and hybrid functionals.

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.

Optimizing Molecular Geometries in Strong Magnetic Fields

An efficient implementation of geometrical derivatives at the Hartree-Fock (HF) and current-density functional theory (CDFT) levels is presented for the study of molecular structure in strong magnetic fields. The required integral derivatives are constructed using a hybrid McMurchie-Davidson and Rys quadrature approach, which combines the amenability of the former to the evaluation of derivative integrals with the efficiency of the latter for basis sets with high angular momentum. In addition to its application to evaluating derivatives of four-center integrals, this approach is also applied to gradients using the resolution-of-the-identity approximation, enabling efficient optimization of molecular structure for many-electron systems under a strong magnetic field. The CDFT contributions have been implemented for a wide range of density functionals up to and including the meta-GGA level with current-density dependent contributions and (range-separated) hybrids for the first time. Illustrative applications are presented to the OH and benzene molecules, revealing the rich and complex chemistry induced by the presence of an external magnetic field. Challenges for geometry optimization in strong fields are highlighted, along with the requirement for careful analysis of the resulting electronic structure at each stationary point. The importance of correlation effects is examined by comparison of results at the HF and CDFT levels. The present implementation of molecular gradients at the CDFT level provides a cost-effective approach to the study of molecular structure under strong magnetic fields, opening up many new possibilities for the study of chemistry in this regime.

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.

Perspective: Fifty years of density-functional theory in chemical physics

Since its formal inception in 1964-1965, Kohn-Sham density-functional theory (KS-DFT) has become the most popular electronic structure method in computational physics and chemistry. Its popularity stems from its beautifully simple conceptual framework and computational elegance. The rise of KS-DFT in chemical physics began in earnest in the mid 1980s, when crucial developments in its exchange-correlation term gave the theory predictive power competitive with well-developed wave-function methods. Today KS-DFT finds itself under increasing pressure to deliver higher and higher accuracy and to adapt to ever more challenging problems. If we are not mindful, however, these pressures may submerge the theory in the wave-function sea. KS-DFT might be lost. I am hopeful the Kohn-Sham philosophical, theoretical, and computational framework can be preserved. This Perspective outlines the history, basic concepts, and present status of KS-DFT in chemical physics, and offers suggestions for its future development.

The importance of current contributions to shielding constants in density-functional theory

The sources of error in the calculation of nuclear-magnetic-resonance shielding constants determined by density-functional theory are examined.

Unrestricted prescriptions for open-shell singlet diradicals: using economical ab initio and density functional theory to calculate singlet-triplet gaps and bond dissociation curves

Here we present and test several computational prescriptions for calculating singlet-triplet (ST) gap energies and bond dissociation curves for open-shell singlet diradicals using economical unrestricted single reference type calculations. For ST gap energies from Slipchenko and Krylov's atom and molecule test set (C, O, Si, NH, NF, OH(+), O(2), CH(2), and NH(2)(+)) spin unrestricted Hartree-Fock and MP2 energies result in errors greater than 15 kcal/mol. However, spin-projected (SP) Hartree-Fock theory in combination with spin-component-scaled (SCS) or scaled-opposite-spin (SOS) second-order perturbation theory gives ST gap energies with a mean unsigned error (MUE) of less than 2 kcal/mol. Density functionals generally give poor results for unrestricted energies and only the ωB97X-D, the M06, and the M06-2X functionals provide reasonable accuracy after spin-projection with MUE values of 4.7, 4.3, and 3.0 kcal/mol, respectively, with the 6-311++G(2d,2p) basis set. We also present a new one parameter hybrid density functional, diradical-1 (DR-1), based on Adamo and Barone's modified PW exchange functional with the PW91 correlation functional. This DR-1 method gives a mean error (ME) of 0.0 kcal/mol and a MUE value of 1.3 kcal/mol for ST gap energies. As another test of unrestricted methods the bond dissociation curves for methane (CH(4)) and hydrofluoric acid (H-F) were calculated with the M06-2X, DR-1, and ωB97X-D density functionals. All three of these functionals give reasonable results for the methane C-H bond but result in errors greater than 50 kcal/mol for the H-F bond dissociation. Spin-projection is found to significantly degrade bond dissociation curves past ~2.2 Å. Although unrestricted Hartree-Fock theory provides a very poor description of H-F bond dissociation, unrestricted SCS-MP2 and SOS-MP2 methods give accurate results.

Tests of an exact-exchange-based density-functional theory on transition-metal complexes

We have compiled a benchmark set of mean ligand-removal enthalpies for 32 transition-metal complexes of relevance in organometallic and catalysis chemistry. Our recent exact-exchange-based density-functional model, DF07 ( J. Chem. Phys. 2007, 127 (12), 124108 ), is assessed on this benchmark set along with other representative GGA, meta-GGA, and hybrid functionals. DF07 performs remarkably well, despite its exact-exchange foundation, indicating that it properly describes nondynamical correlation in transition-metal–ligand bonds.

What differs on the enzymatic acetylation mechanisms for arylamines and arylhydrazines substrates? A theoretical study

The acetylation mechanisms of several selected typical substrates from experiments, including arylamines and arylhydrazines, are investigated with the density functional theory in this paper. The results indicate that all the transition states are characterized by a four-membered ring structure, and hydralazine (HDZ) is the most potent substrate. The bioactivity for all the compounds is increased in a sequence of PABA ≈ 4-AS < 4-MA < 5-AS ≈ INH < HDZ. The conjunction effect and the delocalization of the lone pairs of N atom play a key role in the reaction. All the results are consistent with the experimental data.