meta-GGA Functionals Research Articles

Performance of DFT for C60 Isomerization Energies: A Noticeable Exception to Jacob's Ladder.

The ability to accurately calculate relative energies of fullerenes is important in many areas of computational nanotechnology. Because of the large size of fullerenes, their relative energies cannot normally be calculated by means of high-level ab initio procedures, and therefore, density functional theory (DFT) represents a cost-effective alternative. In an extensive benchmark study, we calculate the electronic energies of eight C60 isomers by means of the high-level G4(MP2) composite procedure. G4(MP2) isomerization energies span a wide range between 307.5 and 1074.0 kJ mol-1. We use these benchmark data to assess the performance of DFT, double-hybrid DFT (DHDFT), and MP2-based ab initio methods. Surprisingly, functionals from the second and third rungs of Jacob's Ladder (i.e., GGA and meta-GGA functionals) significantly and systematically outperform hybrid and hybrid-meta-GGA functionals, which occupy higher rungs of Jacob's Ladder. In addition, DHDFT functionals do not offer a substantial improvement over meta-GGA functionals, with respect to isomerization energies. Overall, the best performing functionals with mean absolute deviations (MADs) below 15.0 kJ mol-1 are (MADs given in parentheses) the GGA N12 (14.7); meta-GGAs M06-L (10.6), M11-L (10.8), MN15-L (11.9), and TPSS-D3BJ (12.8); and the DHDFT functionals B2T-PLYP (9.3), mPW2-PLYP (9.8), B2K-PLYP (12.1), and B2GP-PLYP (12.3 kJ mol-1). In light of these results, we recommend the use of meta-GGA functionals for the calculation of fullerene isomerization energies. Finally, we show that inclusion of very small percentages of exact Hartree-Fock exchange (3-5%) slightly improves the performance of the GGA and meta-GGA functionals. However, their performance rapidly deteriorates with the inclusion of larger percentages of exact Hartree-Fock exchange.

Assessing the performance of the recent meta-GGA density functionals for describing the lattice constants, bulk moduli, and cohesive energies of alkali, alkaline-earth, and transition metals

The bulk properties such as lattice constants, bulk moduli, and cohesive energies of alkali, alkaline-earth, and transition metals are studied within the framework of the recently developed meta-GGA (meta-Generalized Gradient Approximation) level semilocal exchange-correlation functionals. To establish the applicability, broadness, and accuracy of meta-GGA functionals, we also put the results obtained using PBE (Perdew-Burke-Ernzerhof) [J. P. Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)] and PBE reparameterized for solid [J. P. Perdew et al., Phys. Rev. Lett. 100, 136406 (2008)] GGA functionals. The interesting feature of the present paper is that it measures the accuracy of the recently developed TM (Tao-Mo), TMTPSS [TM exchange with Tao-Perdew-Staroverov-Scuseria (TPSS)] [J. Tao and Y. Mo, Phys. Rev. Lett. 117, 073001 (2016)] correlation, and strongly constrained and appropriately normed [J. Sun et al., Phys. Rev. Lett. 115, 036402 (2015)] functionals to calculate the aforementioned properties. Not only that, we also include other (popular) meta-GGA functionals in order to have a closer look at the performance of the meta-GGA functionals too. The present systematic investigation shows that the TM functional is accurate in describing the lattice constants while for cohesive energies and bulk moduli, the PBE and modified TPSS perform better compared to others.

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

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

Deorbitalized meta-GGA exchange-correlation functionals in solids

A procedure for removing explicit orbital dependence from meta-generalized-gradient approximation (mGGA) exchange-correlation functionals by converting them into Laplacian-dependent functionals recently was developed by us and shown to be successful in molecules. It uses an approximate kinetic energy density functional (KEDF) parametrized to Kohn-Sham results (not experimental data) on a small training set. Here we present extensive validation calculations on periodic solids that demonstrate that the same deorbitalization with the same parametrization also is successful for those extended systems. Because of the number of stringent constraints used in its construction and its recent prominence, our focus is on the SCAN meta-GGA. Coded in \textsc{vasp}, the deorbitalized version, SCAN-L, can be as much as a factor of three faster than original SCAN, a potentially significant gain for large-scale ab initio molecular dynamics.

Time-dependent generalized Kohn–Sham theory

Generalized Kohn–Sham (GKS) theory extends the realm of density functional theory (DFT) by providing a rigorous basis for non-multiplicative potentials, the use of which is outside original Kohn–Sham theory. GKS theory is of increasing importance as it underlies commonly used approximations, notably (conventional or range-separated) hybrid functionals and meta-generalized-gradient-approximation (meta-GGA) functionals. While this approach is often extended in practice to time-dependent DFT (TDDFT), the theoretical foundation for this extension has been lacking, because the Runge–Gross theorem and the van Leeuwen theorem that serve as the basis of TDDFT have not been generalized to non-multiplicative potentials. Here, we provide the necessary generalization. Specifically, we show that with one simple but non-trivial additional caveat – upholding the continuity equation in the GKS electron gas – the Runge–Gross and van Leeuwen theorems apply to time-dependent GKS theory. We also discuss how this is manifested in common GKS-based approximations.

Theoretical Characterization by Density Functional Theory (DFT) of Delphinidin 3-O-Sambubioside and Its Esters Obtained by Chemical Lipophilization.

Anthocyanins are water-soluble phenolic pigments. However, their poor solubility in lipidic media limits their use. This hurdle can be overcome with the lipophilization of anthocyanins, which consists of adding an aliphatic chain to a hydrophilic compound, in order to increase its solubility in lipids. Still, the unspecific chemical lipophilization of anthocyanin-esters produces molecules with different properties from their precursors. In this work, experimental changes of anthocyanin-esters obtained by chemical lipophilization are investigated in silico aiming specifically at observing their molecular behavior and comparing it with their anthocyanin precursor. Thus, the analysis of delphinidin 3-O-sambubioside and its esters employing Density Functional Theory (DFT) methods, such as the hybrid functional B3LYP in combination with the 6-31++G(d,p) Pople basis set, provides the ground state properties, the local reactivity and the molecular orbitals (MOs) of these compounds. Excited states properties were analyzed by TD-DFT with the B3LYP functional, and the M06 and M06-2X meta-GGA functionals. Local reactivity calculations showed that the electrophilic site for all the anthocyanin-esters was the same as the one for the anthocyanin precursor, however the nucleophilic site changed depending localization of the esterification. TD-DFT results indicate that the place of esterification could change the electronic transitions and the MOs spatial distribution.

Assessment of DFT for endohedral complexes' dipole moment: PNO-LCCSD-F12 as a reference method.

We present a systematic evaluation of the performance of a wide range of exchange-correlation functionals and related dispersion correction schemes for the computation of dipole moments of endohedral complexes, formed through the encapsulation of an AB molecule (AB = LiF, HCl) inside carbon nanotubes (CNTs) of different diameter. The consistency and accuracy of (i) generalized gradient approximation, (ii) meta GGA, (iii) global hybrid, and (iv) range-separated hybrid density functionals are assessed. In total, 37 density functionals are tested. The results obtained using the highly accurate pair natural orbitals based explicitly correlated local coupled cluster singles doubles (PNO-LCCSD-F12) method of Werner and co-workers [Schwilk et al., J. Chem. Theory Comput., 2017, 13, 3650; Ma et al., J. Chem. Theory Comput., 2017, 13, 4871] with the aug-cc-pVTZ basis set serve as a reference. The static electric dipole moment is computed via the finite field response or, when possible, as the expectation value of the dipole operator. Among others, it is shown that functionals belonging to the class of range-separated hybrids, provide results closest to the coupled cluster reference data. In particular, the ωB97X as well as the M11 functional may be considered as a promising choice for computing electric properties of noncovalent endohedral complexes. On the other hand, the worst performance was found for the functionals which do not include the Hartree-Fock exchange. The analysis of both the coupled cluster and the DFT results indicates a strong coupling of dispersion and polarization that may also explain why lower level DFT methods, as well as Hartree-Fock and MP2, cannot yield dipole moments beyond a qualitative agreement with the higher order reference data. Interestingly, the much smaller and less systematically constructed basis sets of Pople of moderate size provide results of accuracy at least comparable with the extended Dunning's aug-cc-pVTZ basis set.

Calculation of Magnetic Shielding Constants with meta-GGA Functionals Employing the Multipole-Accelerated Resolution of the Identity: Implementation and Assessment of Accuracy and Efficiency.

We present a highly efficient implementation for density functional calculations of chemical shielding constants. It employs the multipole-accelerated resolution of the identity for the calculation of the Coulomb part, which complements the usage of low order scaling routines for the evaluation of the exchange-correlation part and the Hartree-Fock exchange part. Introduced errors for shifts of chemical shielding constants of H, C, F, and P are evaluated for respective test sets of molecules and are related to the accuracy of shifts obtained with hybrid and nonhybrid functionals of the generalized gradient approximation type as well as for meta-GGA functionals themselves. Efficiency is demonstrated for α-d-glucose chains with more than 2500 atoms on a single CPU as well as with an OpenMP parallelized version.

Spin states of Mn(III) meso-tetraphenylporphyrin chloride assessed by density functional methods.

The present work assessed several exchange-correlation functionals (including GGA, meta-GGA and hybrid functionals), in combination with a variety of basis sets and effective core potentials (ECP) for their ability to predict the ground spin state of Mn(III) meso-tetraphenylporphyrin chloride complex, labeled Mn(III)TPPCl, for which experimental data support the quintet high spin state. Geometry optimization of Mn(III)TPPCl was performed for three possible spin states (singlet state, LS; triplet state, IS; and quintet state, HS) at the TPSSh level using the LANL2DZ ECP for Mn and the 6-311G(d) basis set for C, N, Cl and H. Afterwards, single-point energy calculations were conducted by applying 18 exchange-correlation functionals (BLYP, B3LYP, PW91, BPW91, BP86, OLYP, OPBE, OPW91, O3LYP, PBE0, PBEh1PBE, HSEH1PBE, TPSS, TPSSh, M06L, M06, M062X and M06HF). The influence of the basis set for the metal center was assessed using a smaller group of functionals and varying between the Pople basis set 6-31G(d), its newer formulation m6-31G(d) and the larger Def2-QZVP basis set. All functionals in combination with Pople basis sets predict the quintet state as the ground spin state. In addition, the BLYP, BP86, BPW91, PW91, PBEh1PBE, TPSS and TPSSh functionals predicted the IS lying at most ~60kJmol-1 above the HS, which agrees with the reference data. Results including Def2-QZVP basis set were inconsistent, since only BLYP and B3LYP predict HS as the ground spin state. The recommended methodology for the treatment of such systems seems to be exchange-correlations functionals with few or none Hartree-Fock exchange and modest size basis sets. Graphical Abstract MnTPPCl molecule and the energy ordering of its spin states assessed by 18 functionals.

Accuracy of ab initio electron correlation and electron densities in vanadium dioxide

Diffusion quantum Monte Carlo results are used as a reference to analyze properties related to phase stability and magnetism in vanadium dioxide computed with various formulations of density functional theory. We introduce metrics related to energetics, electron densities and spin densities that give us insight on both local and global variations in the antiferromagnetic M1 and R phases. Importantly, these metrics can address contributions arising from the challenging description of the $3d$ orbital physics in this material. We observe that the best description of energetics between the structural phases does not correspond to the best accuracy in the charge density, which is consistent with observations made recently by Medvedev et al. [Science 355, 371 (2017)] in the context of isolated atoms. However, we do find evidence that an accurate spin density connects to correct energetic ordering of different magnetic states in ${\mathrm{VO}}_{2}$, although local, semilocal, and meta-GGA functionals tend to erroneously favor demagnetization of the vanadium sites. The recently developed SCAN functional stands out as remaining nearly balanced in terms of magnetization across the M1-R transition and correctly predicting the ground state crystal structure. In addition to ranking current density functionals, our reference energies and densities serve as important benchmarks for future functional development. With our reference data, the accuracy of both the energy and the electron density can be monitored simultaneously, which is useful for functional development. So far, this kind of detailed high accuracy reference data for correlated materials has been absent from the literature.

Exchange-correlation energies of atoms from efficient density functionals: influence of the electron density

The primary goal of Kohn–Sham density functional theory is to evaluate the exchange-correlation contribution to electronic properties. However, the accuracy of a density functional can be affected by the electron density. Here we apply the nonempirical Tao–Mo (TM) semilocal functional to study the influence of the electron density on the exchange and correlation energies of atoms and ions, and compare the results with the commonly used nonempirical semilocal functionals local spin-density approximation (LSDA), Perdew–Burke–Ernzerhof (PBE), Tao–Perdew–Staroverov–Scuseria (TPSS), and hybrid functional PBE0. We find that the spin-restricted Hartree–Fock density yields the exchange and correlation energies in good agreement with the Optimized Effective Potential method, particularly for spherical atoms and ions. However, the errors of these semilocal and hybrid functionals become larger for self-consistent densities. We further find that the quality of the electron density have greater effect on the exchange-correlation energies of kinetic energy density-dependent meta-GGA functionals TPSS and TM than on those of the LSDA and GGA, and therefore, should have greater influence on the performance of meta-GGA functionals. Finally, we show that the influence of the density quality on PBE0 is slightly reduced, compared to that of PBE, due to the exact mixing.

Density Functional Theory Calculation of Band Gap of Iron (II) disulfide and Tellurium

In this work, the band gap of Iron(II) disulfide and Tellurium were computed using the density functional theory with different meta-GGA functionals. The results showed that the meta-GGA functional M06L gave the closest value of the band gap as 0.96 eV compared to the experimental value of 0.95 eV for Iron(II) disulfide while the meta-GGA functional BJ06 gave the closest value of the band gap as 0.339 eV compared to the experimental value of 0.335 eV for Tellurium. This study showed that the meta-GGA functionals M06L and BJ06 can effectively predict the band gap of Iron(II)disulfide and Tellurium. In general, the use of meta-GGA functionals can be extended to compute the band gap of other semiconductors.

Mn(acac)2 and Mn(acac)3 complexes, a theoretical modeling of their L2,3-edges X-ray absorption spectra

The DFT/ROCIS method has been used to model the Mn(acac)2 (I) and Mn(acac)3 (II) L2,3-edges absorption spectra. Non-hybrid (BP86 and PBE), hybrid (B3P, PBE0 and B3LYP) and hybrid meta-GGA (M06) functionals have been considered to test their efficiency in reproducing experimental features. Structural parameters of both I and II are scarcely affected by the functional selection and the same holds as far as the IL2,3-edges absorption spectrum is concerned. At variance to that, the use of the hybrid meta-GGA functional is decisive for a detailed assignment of the IIL2,3-edges spectral features. The role played by the structural arrangement about the central Mn ion appears less important than the Mn oxidation number in determining the overall spectrum appearance and the spectral features’ excitation energies (EEs). DFT/ROCIS results indicate that, both in I and II, the lower EE side of the L3 intensity distribution mainly includes states having the ground state spin multiplicity (S=5/2 in I and S=2 in II), whereas states with different spin multiplicity contribute, both in I and II, to the L3 higher EE side. The occurrence of states involving metal to ligand charge transfer transitions and implying low lying empty ligand-based orbitals on the I/IIL3 higher EE sides has been further confirmed herein.

Exploring Systematic Discrepancies in DFT Calculations of Chlorine Nuclear Quadrupole Couplings.

Previous studies have revealed significant discrepancies between density functional theory (DFT)-calculated and experimental nuclear quadrupolar coupling constants (CQ) for chlorine atoms, particularly in ionic solids. Various aspects of the computations are systematically investigated here, including the choice of the DFT functional, basis set convergence, and geometry optimization protocol. The effects of fast (fs) time-scale dynamics are probed using molecular dynamics (MD) and nuclear quantum effects (NQEs) are considered using path-integral MD calculations. It is shown that the functional choice is the most important factor related to improving the accuracy of the quadrupolar coupling calculations, and that functionals beyond the generalized gradient approximation (GGA) level, such as hybrid and meta-GGA functionals, are required for good correlations with experiment. The influence of molecular dynamics and NQEs is less important than the functional choice in the studied systems. A method which involves scaling the calculated quadrupolar coupling constant is proposed here; its application leads to good agreement with experimental data.

Two-Component Noncollinear Time-Dependent Spin Density Functional Theory for Excited State Calculations.

We present a linear response formalism for the description of the electronic excitations of a noncollinear reference defined via Kohn-Sham spin density functional methods. A set of auxiliary variables, defined using the density and noncollinear magnetization density vector, allows the generalization of spin density functional kernels commonly used in collinear DFT to noncollinear cases, including local density, GGA, meta-GGA and hybrid functionals. Working equations and derivations of functional second derivatives with respect to the noncollinear density, required in the linear response noncollinear TDDFT formalism, are presented in this work. This formalism takes all components of the spin magnetization into account independent of the type of reference state (open or closed shell). As a result, the method introduced here is able to afford a nonzero local xc torque on the spin magnetization while still satisfying the zero-torque theorem globally. The formalism is applied to a few test cases using the variational exact-two-component reference including spin-orbit coupling to illustrate the capabilities of the method.

Exchange functionals based on finite uniform electron gases.

We show how one can construct a simple exchange functional by extending the well-know local-density approximation (LDA) to finite uniform electron gases. This new generalized local-density approximation functional uses only two quantities: the electron density ρ and the curvature of the Fermi hole α. This alternative "rung 2" functional can be easily coupled with generalized-gradient approximation (GGA) functionals to form a new family of "rung 3" meta-GGA (MGGA) functionals that we have named factorizable MGGAs. Comparisons are made with various LDA, GGA, and MGGA functionals for atoms and molecules.

Laplacian-dependent models of the kinetic energy density: Applications in subsystem density functional theory with meta-generalized gradient approximation functionals.

The development of semilocal models for the kinetic energy density (KED) is an important topic in density functional theory (DFT). This is especially true for subsystem DFT, where these models are necessary to construct the required non-additive embedding contributions. In particular, these models can also be efficiently employed to replace the exact KED in meta-Generalized Gradient Approximation (meta-GGA) exchange-correlation functionals allowing to extend the subsystem DFT applicability to the meta-GGA level of theory. Here, we present a two-dimensional scan of semilocal KED models as linear functionals of the reduced gradient and of the reduced Laplacian, for atoms and weakly bound molecular systems. We find that several models can perform well but in any case the Laplacian contribution is extremely important to model the local features of the KED. Indeed a simple model constructed as the sum of Thomas-Fermi KED and 1/6 of the Laplacian of the density yields the best accuracy for atoms and weakly bound molecular systems. These KED models are tested within subsystem DFT with various meta-GGA exchange-correlation functionals for non-bonded systems, showing a good accuracy of the method.

Ab initio molecular dynamics simulations of liquid water using high quality meta-GGA functionals.

We have used ab initio molecular dynamics (AIMD) to characterize water properties using two meta-generalized gradient approximation (meta-GGA) functionals, M06-L-D3 and B97M-rV, and compared their performance against a standard GGA corrected for dispersion, revPBE-D3, at ambient conditions (298 K, and 1 g cm-3 or 1 atm). Simulations of the equilibrium density, radial distribution functions, self-diffusivity, the infrared spectrum, liquid dipole moments, and characterizations of the hydrogen bond network show that all three functionals have overcome the problem of the early AIMD simulations that erroneously found ambient water to be highly structured, but they differ substantially among themselves in agreement with experiment on this range of water properties. We show directly using water cluster data up through the pentamer that revPBE-D3 benefits from a cancellation of its intrinsic functional error by running classical trajectories, whereas the meta-GGA functionals are demonstrably more accurate and would require the simulation of nuclear quantum effects to realize better agreement with all cluster and condensed phase properties.

Density functional theory calculation of band gap of Iron (II) disulfide and Tellurium

In this work, the band gap of Iron(II)disulfide and Tellurium were computed using the density functional theory with different meta-GGA functionals. The results showed that the meta-GGA functional M06L gave the closest value of the band gap as 0.96ev compared to the experimental value of 0.95ev for Iron(II)disulfide while the meta-GGA functional BJ06 gave the closest value of the band gap as 0.339ev compared to the experimental value of 0.335ev for Tellurium. This study showed that the meta-GGA functionals M06L and BJ06 can effectively predict the band gap of Iron(II)disulfide and Tellurium. In general the use of meta-GGA functionals can be extended to compute the band gap of other semiconductors.

In search of the best DFT functional for dealing with organic anionic species.

Quantum chemical computational methods are thought to have problems in dealing with unstable organic anions. This work assesses the ability of different Density Functional Theory (DFT) functionals to reproduce the electron affinity and reduction potential of organic compounds. The performance of 23 DFT functionals was evaluated by computing the negative electron affinities (from 0 eV to -3.0 eV) and reduction potentials in acetonitrile (from 0 to -2.7 V). In general, most of the hybrid GGA functionals work fine in the prediction of electron affinities, BPW91, B3PW91 and M06 being the best in each class of functionals (pure, hybrid and meta-GGA functionals, respectively). On the other hand, the ab initio post-Hartree-Fock methods, MP2 and coupled-cluster (CCSD(T)), as well as the double hybrid functionals, B2PLYP and mPW2PLYP, usually fail. For compounds with EAs lower than -1.75 eV, a method for stabilizing the anion, based on solvation with the IEFPCM model, was employed. In this case, BPW91, PBE0 and M06-HF could be the recommended option for the pure, hybrid and meta-GGA functionals, respectively. The situation improves for the evaluation and prediction of redox potentials. In this case the performance of the DFT functionals is better, in part because the solvent assists in the stabilization of the anions. Nevertheless, there is a systematic bias in the calculation of absolute redox potentials, which could be corrected by using a redox partner that helps by the cancellation of errors. In this case, the hybrid and meta-GGA functionals B3PW91, PBE0, TPSSh and M06 are also among the best for computing redox potentials with a mean absolute deviation (MAD) lower than 0.13 V.