Meta-generalized Gradient Approximation Research Articles

Current density functional framework for spin-orbit coupling: Extension to periodic systems.

Spin-orbit coupling induces a current density in the ground state, which consequently requires a generalization for meta-generalized gradient approximations. That is, the exchange-correlation energy has to be constructed as an explicit functional of the current density, and a generalized kinetic energy density has to be formed to satisfy theoretical constraints. Herein, we generalize our previously presented formalism of spin-orbit current density functional theory [Holzer et al., J. Chem. Phys. 157, 204102 (2022)] to non-magnetic and magnetic periodic systems of arbitrary dimension. In addition to the ground-state exchange-correlation potential, analytical derivatives such as geometry gradients and stress tensors are implemented. The importance of the current density is assessed for band gaps, lattice constants, magnetic transitions, and Rashba splittings. In the latter, the impact of the current density may be larger than the deviation between different density functional approximations.

Semiconducting Characteristic of Antiferromagnetic Al4X3Mn (X = P, As and Sb) Compounds with Ab Initio Simulation Methods

This research reports the electronic characteristics of ternary aluminium-based Al4X3Mn (X=P, As and Sb) compounds for the most stable magnetic order which is A-type antiferromagnetic. The related systems are comforming 215 space number with P-43m space group which is simple cubic crystal structure. The computations in this research have been done within the framework of Density Functional Theory. The calculations utilized Perdew-Burke-Ernzerhof type correlation functionals within the meta-generalized gradient approximation. For considered four different type magnetic orders, the visualized volume-energy plots and the calculated formation energy values imply that the magnetic nature of these compositions is A-type antiferromagnetic. Besides, the investigated electronic natures in the detected stable magnetic phase of these systems are semiconductor since the band gaps were observed in their electronic band structures and density of states (Eg = 0.36 eV for Al4P3Mn, Eg = 0.33 eV for Al4As3Mn, and Eg = 0.18 eV for Al4Sb3Mn).

The rise and fall of stretched bond errors: Extending the analysis of Perdew-Zunger self-interaction corrections of reaction barrier heights beyond the LSDA.

Incorporating self-interaction corrections (SIC) significantly improves chemical reaction barrier height predictions made using density functional theory methods. We present a detailed orbital-by-orbital analysis of these corrections for three semi-local density functional approximations (DFAs) situated on the three lowest rungs of Jacob's ladder of approximations. The analysis is based on Fermi-Löwdin Orbital Self-Interaction Correction (FLOSIC) calculations performed at several steps along the reaction pathway from the reactants (R) to the transition state (TS) to the products (P) for four representative reactions selected from the BH76 benchmark set. For all three functionals, the major contribution to self-interaction corrections of the barrier heights can be traced to stretched bond orbitals that develop near the TS configuration. The magnitude of the ratio of the self-exchange-correlation energy to the self-Hartree energy (XC/H) for a given orbital is introduced as an indicator of one-electron self-interaction error. XC/H = 1.0 implies that an orbital's self-exchange-correlation energy exactly cancels its self-Hartree energy and that the orbital, therefore, makes no contribution to the SIC in the FLOSIC scheme. For the practical DFAs studied here, XC/H spans a range of values. The largest values are obtained for stretched or strongly lobed orbitals. We show that significant differences in XC/H for corresponding orbitals in the R, TS, and P configurations can be used to identify the major contributors to the SIC of barrier heights and reaction energies. Based on such comparisons, we suggest that barrier height predictions made using the strongly constrained and appropriately normed meta-generalized gradient approximation may have attained the best accuracy possible for a semi-local functional using the Perdew-Zunger SIC approach.

Performance of Density Functionals for Excited-State Properties of Isolated Chromophores and Exciplexes: Emission Spectra, Solvatochromic Shifts, and Charge-Transfer Character.

This study assesses the performance of various meta-generalized gradient approximation (meta-GGA), global hybrid, and range-separated hybrid (RSH) density functionals in capturing the excited-state properties of organic chromophores and their excited-state complexes (exciplexes). Motivated by their uses in solar energy harvesting and photoredox CO2 reduction, we use oligo-(p-phenylenes) and their excited-state complexes with triethylamine as model systems. We focus on the fluorescence properties of these systems, specifically emission energies. We also consider solvatochromic shifts and wave function characteristics. The latter is described by using reduced quantities such as natural transition orbitals (NTOs) and exciton descriptors. The functionals are benchmarked against the experimental fluorescence spectra and the equation-of-motion coupled-cluster method with single and double excitations. Both in isolated chromophores and in exciplexes, meta-GGA functionals drastically underestimate the emission energies and exhibit significant exciton delocalization and anticorrelation between electron and hole motion. The performance of global hybrid functionals is strongly dependent on the percentage of exact exchange. Our study identifies RSH GGAs as the best-performing functionals, with ωPBE demonstrating the best agreement with experimental results. RSH meta-GGAs often overestimate emission energies in exciplexes and yield larger hole NTOs. Their performance can be improved by optimally tuning the range-separation parameter.

Towards the "Eldorado" of pKa Determination: A Reliable and Rapid DFT Model.

The selection of a "perfect tool" for the theoretical determination of acid-base dissociation constants (Ka) is still puzzling. Recently, we developed a user-friendly model exploiting CAM-B3LYP for determining pKa with impressive reliability. Herein, a new challenge is faced, examining a panel of functionals belonging to different rungs of the "Jacob's ladder" organization, which classifies functionals according to their level of theory. Specifically, meta-generalized gradient approximations (GGAs), hybrid-GGAs, and the more complex range-separated hybrid (RSH)-GGAs were investigated in predicting the pKa of differently substituted carboxylic acids. Therefore, CAM-B3LYP, WB97XD, B3PW91, PBE1PBE, PBEPBE and TPSSTPSS were used, with 6-311G+(d,p) as the basis set and the solvation model based on density (SMD). CAM-B3LYP showed the lowest mean absolute error value (MAE = 0.23) with relatively high processing time. PBE1PBE and B3PW91 provided satisfactory predictions (MAE = 0.34 and 0.38, respectively) with moderate computational time cost, while PBEPBE, TPSSTPSS and WB97XD led to unreliable results (MAE > 1). These findings validate the reliability of our model in predicting carboxylic acids pKa, with MAE well below 0.5 units, using a simplistic theoretical level and a low-cost computational approach.

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

In this Letter, we present a pioneering analysis of the density functional approximations (DFAs) beyond the generalized gradient approximation (GGA) for predicting two-photon absorption (2PA) strengths of a set of push-pull π-conjugated molecules. In more detail, we have employed a variety of meta-generalized gradient approximation (meta-GGA) functionals, including SCAN, MN15, and M06-2X, to assess their accuracy in describing the 2PA properties of a chosen set of 48 organic molecules. Analytic quadratic response theory is employed for these functionals, and their performance is compared against the previously studied DFAs and reference data obtained at the coupled-cluster CC2 level combined with the resolution-of-identity approximation (RI-CC2). A detailed analysis of the meta-GGA functional performance is provided, demonstrating that they improve upon their predecessors in capturing the key electronic features of the π-conjugated two-photon absorbers. In particular, the Minnesota functional MN15 shows very promising results as it delivers pleasingly accurate chemical rankings for two-photon transition strengths and excited-state dipole moments.

Self-interaction corrected SCAN functional for molecules and solids in the numeric atom-center orbital framework.

Semilocal density-functional approximations (DFAs), including the state-of-the-art SCAN functional, are plagued by the self-interaction error (SIE). While this error is explicitly defined only for one-electron systems, it has inspired the self-interaction correction method proposed by Perdew and Zunger (PZ-SIC), which has shown promise in mitigating the many-electron SIE. However, the PZ-SIC method is known for its significant numerical instability. In this study, we introduce a novel constraint that facilitates self-consistent localization of the SIC orbitals in the spirit of Edmiston-Ruedenberg orbitals [Rev. Mod. Phys. 35, 457 (1963)]. Our practical implementation within the all-electron numeric atom-centered orbitals code FHI-aims guarantees efficient and stable convergence of the self-consistent PZ-SIC equations for both molecules and solids. We further demonstrate that our PZ-SIC approach effectively mitigates the SIE in the meta-generalized gradient approximation SCAN functional, significantly improving the accuracy for ionization potentials, charge-transfer energies, and bandgaps for a diverse selection of molecules and solids. However, our PZ-SIC method does have its limitations. It cannot improve the already accurate SCAN results for properties such as cohesive energies, lattice constants, and bulk modulus in our test sets. This highlights the need for new-generation DFAs with more comprehensive applicability.

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.

Origin of the success of mGGAs for bandgaps.

One of the well-known limitations of Kohn-Sham density functional theory is the tendency to strongly underestimate bandgaps. Meta-generalized gradient approximations (mGGAs), which include the kinetic energy density in the functional form, have been shown to significantly alleviate this deficiency. In this study, we explore the mechanisms responsible for this improvement from the angle of the underlying local densities. We find that the highest occupied and lowest unoccupied states are distinct in the space of the underlying descriptors. The gap opening is compared to a simple scaling of the local density approximation, and two mechanisms responsible for opening the mGGA gaps are identified. First of all, the relatively large negative derivative of the functional form with respect to reduced kinetic energy tends to elevate the lowest unoccupied state. Second, the curvature of functional, which ensures that it is bounded, tends to lower the highest occupied state. Remarkably, these two mechanisms are found to be transferable over a large and diverse database of compounds.

Exact exchange-like electric response from a meta-generalized gradient approximation: A semilocal realization of ultranonlocality

We review the concept of ultranonlocality in density functional theory and the relation between ultranonlocality, the derivative discontinuity of the exchange energy, and the static electric response in extended molecular systems. We present the construction of a new meta-generalized gradient approximation for exchange that captures the ultranonlocal response to a static electric field in very close correspondence to exact exchange, yet at a fraction of its computational cost. This functional, in particular, also captures the dependence of the response on the system size. The static electric polarizabilities of hydrogen chains and oligo-acetylene molecules calculated with this meta-GGA are quantitatively close to the ones obtained with exact exchange. The chances and challenges associated with the construction of meta-GGAs that are intended to combine a substantial derivative discontinuity and ultranonlocality with an accurate description of electronic binding are discussed.

Dispersion-corrected r2SCAN based double-hybrid functionals

The regularized and restored semi-local meta-generalized gradient approximation (meta-GGA) exchange–correlation functional r2SCAN [Furness et al., J. Phys. Chem. Lett. 11, 8208–8215 (2020)] is used to create adiabatic-connection-derived global double-hybrid functionals employing spin-opposite-scaled MP2. The 0-DH, CIDH, QIDH, and 0–2 type double-hybrid functionals are assessed as a starting point for further modification. Variants with 50% and 69% Hartree–Fock exchange (HFX) are empirically optimized (Pr2SCAN50 and Pr2SCAN69), and the effect of MP2-regularization (κPr2SCAN50) and range-separated HFX (ωPr2SCAN50) is evaluated. All optimized functionals are combined with the state-of-the-art London dispersion corrections D4 and NL. The resulting functionals are assessed comprehensively for their performance on main-group and metal-organic thermochemistry on 90 different benchmark sets containing 25 800 data points. These include the extensive GMTKN55 database, additional sets for main-group chemistry, and multiple sets for transition-metal complexes, including the ROST61, the MOR41, and the MOBH35 sets. As the main target of this study is the development of a broadly applicable, robust functional with low empiricism, special focus is put on variants with moderate amounts of HFX (50%), which are compared to the so far successful PWPB95-D4 (50% HFX, 20% MP2 correlation) functional. The overall best variant, ωPr2SCAN50-D4, performs well on main-group and metal-organic thermochemistry, followed by Pr2SCAN69-D4 that offers a slight edge for metal-organic thermochemistry and by the low HFX global double-hybrid Pr2SCAN50-D4 that performs robustly across all tested sets. All four optimized functionals, Pr2SCAN69-D4, Pr2SCAN50-D4, κPr2SCAN50-D4, and ωPr2SCAN50-D4, outperform the PWPB95-D4 functional.

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.

Enhancing strength and ductility of CrCoNi medium-entropy alloy through the 9R phase and doping

The 9R phase model based on experimental orientation relationship are constructed, the mechanical moduli of the face-centered cubic (FCC) and 9R phases in CrCoNi medium-entropy alloy (MEA) are investigated by first-principles calculations with the meta-generalized gradient approximation (meta-GGA) functional. This functional accurately describes CrCoNi MEA, captures localized electronic states, and overcomes the challenge of DFT + U parameters in special quasi-random structure (SQS) based transition metal atom models. Results indicate that the 9R phase exhibits higher mechanical moduli than the FCC phase, with a 12% increase in bulk modulus and a 7% increase in Poisson's ratio. Consequently, the strength and toughness of CrCoNi MEA are significantly enhanced. Doping Mo, Ta, and W in the MEA structures improves strength by promoting the formation of the 9R phase compared to the doped single FCC phase. Calculations of the work of adhesion and fracture toughness for the 9R/FCC interface reveal improved interface properties with Mo and W doping. These enhancements are attributed to the hybridization of electronic orbitals in the doped CrCoNi MEA, resulting in changes in bond types between neighboring atoms and subsequent modifications of mechanical properties in the 9R phase. Comparative computational values strongly support an alloy design strategy involving Mo doping and the construction of the 9R phase to achieve superior mechanical properties in MEA.

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.

A potential candidate material for quantum anomalous Hall effect: Heterostructures of ferromagnetic insulator and graphene

Discovery of novel topological materials have motivated tremendous research owing to their peculiar energy bands and potential fundamental and practical prospects. In contrast to conventional nonmagnetic topological insulators (TIs), new type of magnetic TIs, such as Cr2Ge2Te6, revealed a quantum Hall effect (QHE) at zero magnetic field. The phenomenon is called anomalous QHE. In the magnetic TIs showing QAHE, the interaction of spontaneous magnetism with conducting surface states opens an insulating band gap at the Dirac points by breaking the time-reversal symmetry. In contrast to inducing the phenomenon of QHE through spontaneous magnetism by chemical doping, the alternative suitable option is the heterostructure engineering of 2D Dirac materials with semiconductor magnetic insulators. In this study, we first time investigate the phenomenon of QAHE in semiconducting magnetic insulator Cr2Ge2Te6 by proposing its bulk and monolayer heterostructure with graphene by the first-principles calculations. We employed density functional theory to compute electronic properties of Cr2Ge2Te6 and its heterostructure with graphene. The exchange-correlation in view of meta-generalized gradient approximations with tight binding method reveal the results for bulk Cr2Ge2Te6 are close to the experimental values. The maximum achieved energy gap in the heterostructure is up to 5 meV.

Artificial Neural Network-Based Density Functional Approach for Adiabatic Energy Differences in Transition Metal Complexes.

During the past decades, approximate Kohn-Sham density functional theory schemes have garnered many successes in computational chemistry and physics, yet the performance in the prediction of spin state energetics is often unsatisfactory. By means of a machine learning approach, an enhanced exchange and correlation functional is developed to describe adiabatic energy differences in transition metal complexes. The functional is based on the computationally efficient revision of the regularized, strongly constrained, and appropriately normed functional and improved by an artificial neural network correction trained over a small data set of electronic densities, atomization energies, and/or spin state energetics. The training process, performed using a bioinspired nongradient-based approach adapted for this work from the particle swarm optimization, is analyzed and discussed extensively. The resulting machine learned meta-generalized gradient approximation functional is shown to outperform most known density functionals in the prediction of adiabatic energy differences for a diverse set of transition metal complexes with varying local coordinations and metal choices.

Assessment of the nonempirical r2SCAN-QIDH double-hybrid density functional against large and diverse datasets.

We update the Quadratic Integrand Double-Hybrid (QIDH) model [J. Chem. Phys. 141, 031101 (2014)] by incorporating the nonempirical restored-regularized Strongly Constrained and Appropriately Normed (r2SCAN) meta-generalized gradient approximation exchange-correlation functional, thus devising a robust density functional approximation free of any empirical parameter and incorporating all the constraints so far known for the exchange-correlation kernel. We assessed the new r2SCAN-QIDH expression on the GMTKN55 database and further extend its application to various types of non-covalent interactions (e.g., S66 × 8, O24 × 5). The assessment done shows that the model becomes very competitive in accuracy with respect to parent exchange-correlation functionals of any type, but without relying on any fitted parameter or numerical training.

Semilocal Meta-GGA Exchange-Correlation Approximation from Adiabatic Connection Formalism: Extent and Limitations.

The incorporation of a strong-interaction regime within the approximate semilocal exchange-correlation functionals still remains a very challenging task for density functional theory. One of the promising attempts in this direction is the recently proposed adiabatic connection semilocal correlation (ACSC) approach [Constantin, L. A.; Phys. Rev. B 2019, 99, 085117] allowing one to construct the correlation energy functionals by interpolation of the high and low-density limits for the given semilocal approximation. The current study extends the ACSC method to the meta-generalized gradient approximations (meta-GGA) level of theory, providing some new insights in this context. As an example, we construct the correlation energy functional on the basis of the high- and low-density limits of the Tao-Perdew-Staroverov-Scuseria (TPSS) functional. Arose in this way, the TPSS-ACSC functional is one-electron self-interaction free and accurate for the strictly correlated and quasi-two-dimensional regimes. Based on simple examples, we show the advantages and disadvantages of ACSC semilocal functionals and provide some new guidelines for future developments in this context.

Meta-generalized gradient approximation time-dependent density functional theory study of electron trapping in Hf- and Zr-doped lutetium oxide: influencing the oxygen vacancy.

This work analyzes the effects of a nearby Hf or Zr dopant on the electron density trapped at an oxygen vacancy site. The two metals are among the dopants used to achieve thermoluminescence and energy storage in phosphors based on cubic lutetium oxide (c-Lu2O3). The presence of oxygen vacancies is anticipated in those phosphors. If the dopant is located outside the immediate surroundings of the vacancy site, the resulting optical trap depth is similar to that of the isolated oxygen vacancies (1.6-1.7 eV versus 1.7 eV). If the dopant is one of the four metal cations surrounding the vacancy site, the corresponding trap depth is 2.0-2.1 eV. Using time-dependent density-functional theory calculations, it was found that the excitation of the vacancy-trapped electrons can take two forms: a local excited state at the vacancy site can be formed, or an electron transfer to Hf might occur. With charge compensation in mind, several structures with three defects were analyzed: the dopant cation, the vacancy and an interstitial oxygen (Hf/Zr plus a Frenkel pair). These last two systems with the dopant in a +4 oxidation state and a single electron trapped at the vacancy site correspond to zero total charge, while another electron can be trapped. The vacancy site is expected to trap the electron, not the dopant. The composite defects of the dopant and Frenkel pair are thus considered the most likely electron traps in cubic Lu2O3:Hf and cubic Lu2O3:Zr.

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 0.8eV. In the cases that we study, the agreement with the reference data is improved by the current density correction.