Meta-generalized Gradient Approximation Research Articles

Adsorption on Fe-MOF-74 for C1–C3 Hydrocarbon Separation

An increase in demand for energy efficient processes for the separation of a mixture of hydrocarbons drives the need for understanding metal–organic frameworks (MOFs) that can provide better noncryogenic alternatives for the fractionation of hydrocarbon mixtures. Here we study the structure and properties of a metal–organic framework, Fe-MOF-74, and its effectiveness to separate C1–C3 hydrocarbon mixtures. The binding of these hydrocarbons to the open metal sites of Fe-MOF-74 has been investigated using a meta-generalized gradient approximation density functional, M06-L, which has previously been validated for systems containing transition metals. For interpretive purposes, charge model 5 (CM5) is used to determine the partial atomic charges on the metal cations and the oxygen atoms of the ligands surrounding these metal centers. Our computations show preferential binding to the metal center of Fe-MOF-74 of unsaturated hydrocarbons over saturated ones in agreement with experimental results, and the calculated binding energies are in semiquantitative agreement with experiment. The results are analyzed in terms of various factors contributing to the binding, including structural distortion, electrostatics, damped dispersion, charge transfer, back bonding, and ligand field effects on the d orbitals. The CM5 charges are not sensitive to small differences in structure.

Isoorbital indicators for current density functional theory

Exchange-correlation density functionals at the meta-generalised gradient approximation level typically use the Kohn–Sham orbitals as an ingredient to create the isoorbital indicator , where is the von Weizsäcker kinetic energy density and is the kinetic energy density of the Kohn–Sham orbitals. When a magnetic field is included in the treatment, the Kohn–Sham orbitals vary with the vector-potential gauge, as does and implicitly the exchange-correlation energy. However, it is a known result in current density functional theory that the exact exchange-correlation energy is gauge independent. Three different gauge-independent replacements for have been proposed by other authors. Only one of these is, however, known to stay between 0 and 1. We modify one of the others, the Maximoff–Scuseria isoorbital indicator, so that it also stays between 0 and 1. We investigate the ability of the isoorbital indicators evaluated in a system with density and paramagnetic current density to reproduce the value of in a system having no paramagnetic current, but the same density . We provide the first example known to us where the Tao–Perdew isoorbital indicator goes outside of the interval from 0 to 1.

Ice phases under ambient and high pressure: Insights from density functional theory

Water is common and plays a crucial role in biological, chemical, and physical processes, but its crystalline or ice state has a complicated structure. In this work, we study the lattice mismatch challenge for ice nucleation on silver iodide, the sublimation energy for different ice phases, and the structural phase-transition pressures of ice, with various density functionals. Our calculations show that the recently developed meta-generalized gradient approximation made simple (MGGA_MS) yields a lattice mismatch (3$%$) of hexagonal ice (ice Ih) with \ensuremath{\beta}-AgI in good agreement with experiment (2$%$), significantly better than the Perdew-Burke-Ernzerhof (PBE) GGA mismatch (6$%$). MGGA_MS is a computationally efficient semilocal functional that incorporates intermediate-range van der Waals (vdW) interaction, which, overall, performs well for ice and may be expected to improve upon PBE for liquid water. While MGGA_MS predicts the most realistic volumes and volume changes in the phase transitions of ice Ih to trigonal ice (ice II) and tetragonal ice (ice VIII), a more accurate description of some other properties of the higher-pressure phases (ice II and ice VIII) is provided by some functionals that include long-range vdW corrections (e.g., revised Tao-Perdew-Staroverov-Scuseria+vdW for sublimation energy and optB88-vdW for transition pressure).

A Systematic Search for Structures, Stabilities, Electronic and Magnetic Properties of Silicon Doped Silver Clusters: Comparison with Pure Silver Clusters

The geometric structures, stabilities, electronic and magnetic properties of silicon doped silver clusters AgnSi (n = 1 - 9) have been systematically investigated by using meta-generalized gradient approximation (meta-GGA) exchange correlation Tao-Perdew-Staroverov-Scuseria (TPSS) functional. Due to the sp3 hybridization, the lowest energy structures of doped clusters favour the threedimensional structure. The silicon atom prefers to be located at the surface of the host silver clusters. The isomers that correspond to high coordination numbers of the Si-Ag bonds are found to be more stable. By analyzing the relative stabilities, the results show that the quadrangular bipyramid Ag4Si structure is the most stable geometry for the AgnSi clusters. Meanwhile, the fragmentation energies, second-order difference of energies, difference of highest occupied and lowest unoccupied molecular orbital (HOMO-LUMO gaps), and total magnetic moments exhibit pronounced even-odd alternations. The largest hardness difference (2:24 eV) exists between the clusters Ag4Si and Ag5, which illustrates that the corresponding Ag4Si cluster has dramatically enhanced chemical stability.

Which density functional is close to CCSD accuracy to describe geometry and interaction energy of small noncovalent dimers? A benchmark study using Gaussian09

A benchmark study on all possible density functional theory (DFT) methods in Gaussian09 is done to locate functionals that agree well with CCSD/aug-cc-pVTZ geometry and Ave-CCSD(T)/(Q-T) interaction energy (Eint) for small non-covalently interacting molecular dimers in "dispersion-dominated" (class 1), "dipole-induced dipole" (class 2), and "dipole-dipole" (class 3) classes. A DFT method is recommended acceptable if the geometry showed close agreement to CCSD result (RMSD < 0.045) and Eint was within 80-120% accuracy. Among 382 tested functionals, 1-46% gave good geometry, 13-44% gave good Eint , while 1-33% satisfied geometry and energy criteria. Further screening to locate the best performing functionals for all the three classes was made by counting the acceptable values of energy and geometry given by each functionals. The meta-generalized gradient approximation (GGA) functional M06L was the best performer with total 14 hits; seven acceptable energies and seven acceptable geometries. This was the only functional "recommended" for at least two dimers in each class. The functionals M05, B2PLYPD, B971, mPW2PLYPD, PBEB95, and CAM-B3LYP gave 11 hits while PBEhB95, PW91B95, Wb97x, BRxVP86, BRxP86, HSE2PBE, HSEh1PBE, PBE1PBE, PBEh1PBE, and PW91TPSS gave 10 hits. Among these, M05, B971, mPW2PLYPD, Wb97x, and PW91TPSS were among the "recommended" list of at least one dimer from each class. Long-range correction (LC) of Hirao and coworkers to exchange-correlation functionals showed massive improvement in geometry and Eint . The best performing LC-functionals were LC-G96KCIS and LC-PKZBPKZB. Our results predict that M06L is the most trustworthy DFT method in Gaussian09 to study small non-covalently interacting systems.

Studies on structures and electron affinities of the simplest alkyl dithio radicals and their anions with gaussian-3 theory and density functional theory

The equilibrium geometries and electron affinities of the R-SS/R-SS(-)(R=CH₃, C₂H₅, n-C₃H7, i-C₃H₇, n-C₄H₉, t-C₄H₉, n-C₅H₁₁) species have been studied using the higher level of the Gaussian-3(G3) theory and 21 carefully calibrated pure and hybrid density functionals (five generalized gradient approximation (GGA) methods, seven hybrid GGAs, three meta GGA methods, and six hybrid meta GGAs) in conjunction with diffuse function augmented double-ζ plus polarization (DZP++) basis sets. The geometries are fully optimized with each method and discussed. The reliable adiabatic electron affinity has been presented by means of the high level of G3 technique. With the DZP++ DFT method, three measures of neutral/anion energy differences reported in this work are the adiabatic electron affinity, the vertical electron affinity, and the vertical detachment energy. The adiabatic electron affinities, obtained at the BP86, M05-2X, B3LYP, M06, B98, M06-2X, mPW1PW91, HCTH, B97-1, M05, PBE1PBE, and VSXC methods, are in agreement with the G3 results. These methods perform better for EA prediction and are considered to be reliable.

Semilocal and hybrid meta-generalized gradient approximations based on the understanding of the kinetic-energy-density dependence

We present a global hybrid meta-generalized gradient approximation (meta-GGA) with three empirical parameters, as well as its underlying semilocal meta-GGA and a meta-GGA with only one empirical parameter. All of them are based on the new meta-GGA resulting from the understanding of kinetic-energy-density dependence [J. Sun, B. Xiao, and A. Ruzsinszky, J. Chem. Phys. 137, 051101 (2012)]. The obtained functionals show robust performances on the considered molecular systems for the properties of heats of formation, barrier heights, and noncovalent interactions. The pair-wise additive dispersion corrections to the functionals are also presented.

Structural and Electronic Properties of Stable AunIr2 (n = 1 – 7) Clusters: Comparison with Pure Gold Clusters

The geometrical structures, relative stabilities, electronic and magnetic properties of AunIr2 (n = 1 - 7) clusters have been systematically investigated by using meta-generalized gradient approximation (meta-GGA) Tao-Perdew-Staroverov-Scuseria (TPSS) functional in comparison with pure gold clusters. The optimized geometries show that the two doping iridium atoms can affect the structure of the host cluster. Compared with the pure Aun+2 clusters, the lowest energy AunIr2 (n=1 - 7) clusters favour higher spin multiplicity except for Au7Ir2. Furthermore, the calculated binding energies, fragmentation energies, second-order difference energies, and the highest occupied-lowest unoccupied energy gaps indicate that the stability of AunIr2 is enhanced. Natural population analysis reveals that the charges transfer from the Aun frames to the iridium atoms for Au3;4;6;7Ir2 clusters. In addition, charges and magnetic moments of 6s, 5d, and 6p states for the iridium atoms in AunIr2 (n = 1 - 7) clusters are also analyzed and compared.

A first-principles-based correlation functional for harmonious connection of short-range correlation and long-range dispersion

We present a physically motivated correlation functional belonging to the meta-generalized gradient approximation (meta-GGA) rung, which can be supplemented with long-range dispersion corrections without introducing double-counting of correlation contributions. The functional is derived by the method of constraint satisfaction, starting from an analytical expression for a real-space spin-resolved correlation hole. The model contains a position-dependent function that controls the range of the interelectronic correlations described by the semilocal functional. With minimal empiricism, this function may be adjusted so that the correlation model blends with a specific dispersion correction describing long-range contributions. For a preliminary assessment, our functional has been combined with an atom-pairwise dispersion correction and full Hartree-Fock (HF)-like exchange. Despite the HF-exchange approximation, its predictions compare favorably with reference interaction energies in an extensive set of non-covalently bound dimers.

Harnessing the meta-generalized gradient approximation for time-dependent density functional theory

Density functionals within the meta-generalized gradient approximation (MGGA) are widely used for ground-state electronic structure calculations. However, the gauge variance of the kinetic energy density τ confounds applications of MGGAs to time-dependent systems, excited states, magnetic properties, and states with strong spin-orbit coupling. Becke and Tao used the paramagnetic current density to construct a gauge invariant generalized kinetic energy density τ. We show that τ(W)≤τ, where τ(W) is the von Weizsäcker kinetic energy density of a one-electron system. Thus, replacing τ by τ leads to current-dependent MGGAs (cMGGAs) that are not only gauge invariant but also restore the accuracy of MGGAs in iso-orbital regions for time-dependent and current-carrying states. The current dependence of cMGGAs produces a vector exchange-correlation (XC) potential in the time-dependent adiabatic Kohn-Sham (KS) equations. While MGGA response properties of current-free ground states become manifestly gauge-variant to second order, linear response properties are affected by a new XC kernel appearing in the cMGGA magnetic orbital rotation Hessian. This kernel reflects the first-order coupling of KS orbitals due to changes in the paramagnetic current density and has apparently been ignored in previous MGGA response implementations. Inclusion of the current dependence increases total computation times by less than 50%. Benchmark applications to 109 adiabatic excitation energies using the Tao-Perdew-Staroverov-Scuseria (TPSS) MGGA and its hybrid version TPSSh show that cMGGA excitation energies are slightly lower than the MGGA ones on average, but exhibit fewer outliers. Similarly, the optical rotations of 13 small organic molecules show a small but systematic improvement upon inclusion of the magnetic XC kernel. We conclude that cMGGAs should replace MGGAs in all applications involving time-dependent or current-carrying states.

Improved CO Adsorption Energies, Site Preferences, and Surface Formation Energies from a Meta-Generalized Gradient Approximation Exchange-Correlation Functional, M06-L.

A notorious failing of approximate exchange-correlation functionals when applied to problems involving catalysis has been the inability of most local functionals to predict the correct adsorption site for CO on metal surfaces or to simultaneously predict accurate surface formation energies and adsorption energies for transition metals. By adding the kinetic energy density τ to the density functional, the revTPSS density functional was shown recently to achieve a balanced description of surface energies and adsorption energies. Here, we show that the older M06-L density functional, also containing τ, provides improved surface formation energies and CO adsorption energies over revTPSS for five transition metals and correctly predicted the on-top/hollow site adsorption preferences for four of the five metals, which was not achieved by most other local functionals. Because M06-L was entirely designed on the basis of atomic and molecular energies, its very good performance is a confirmation of the reasonableness of its functional form. Two GGA functionals with an expansion in the reduced gradient that is correct through second order, namely, SOGGA and SOGGA11, were also tested and found to produce the best surface formation energies of all tested GGA functionals, although they significantly overestimate the adsorption energies.

Graphene on metal surfaces and its hydrogen adsorption: A meta-GGA functional study

The interaction of graphene with various metal surfaces is investigated using density functional theory and the meta-generalized gradient approximation (MGGA) M06-L functional. We demonstrate that this method is of comparable accuracy to the random-phase approximation (RPA). With M06-L we study large systems inaccessible to RPA with H adsorbed on graphene on a selected strongly (Ni) and a selected weakly (Pt) interacting substrate. Very stable graphane-like clusters, where every other C atom binds to a H atom above and every other to a metal atom below, are found on both substrates. Such graphane-like clusters have been proposed to be responsible for opening a band gap in graphene. On Ni we find that the binding energies of the H clusters are almost constant with the cluster size, whereas on Pt the binding energies increase with the cluster size. Comparing the Perdew-Burke-Ernzerhof and M06-L functionals we demonstrate the importance of accounting for dispersive interactions.

Communication: Effect of the orbital-overlap dependence in the meta generalized gradient approximation

We study for the first time the effect of the dependence of meta generalized gradient approximation (MGGA) for the exchange-correlation energy on its input, the kinetic energy density, through the dimensionless inhomogeneity parameter, α, that characterizes the extent of orbital overlap. This leads to a simple MGGA exchange functional, which interpolates between the single-orbital regime, where α = 0, and the slowly varying density regime, where α ≈ 1, and then extrapolates to α → ∞. When combined with a variant of the Perdew-Burke-Ernzerhof GGA correlation, the resulting MGGA performs equally well for atoms, molecules, surfaces, and solids.

Semilocal dynamical correlation with increased localization

Using a localization technique for the correlation energy density and a linear response constraint, we construct a localized semilocal model for dynamical correlation. The model is incorporated into a meta-generalized gradient approximation functional which is very accurate for jellium surfaces, Hooke's atom at all regimes, and works well both in conjunction with a semilocal exchange and together with local and nonlocal exact exchange.

A meta-GGA Made Free of the Order of Limits Anomaly.

We have improved the revised Tao-Perdew-Staroverov-Scuseria (revTPSS) meta-generalized gradient approximation (GGA) in order to remove the order of limits anomaly in its exchange energy. The revTPSS meta-GGA recovers the second-order gradient expansion for a wide range of densities and therefore provides excellent atomization energies and lattice constants. For other properties of materials, however, even the revTPSS does not give the desired accuracy. The revTPSS does not perform as well as expected for the energy differences between different geometries for the same molecular formula and for the related nonbarrier height chemical reaction energies. The same order of limits problem might lead to inaccurate energy differences between different crystal structures and to inaccurate cohesive energies of insulating solids. Here we show a possible way to remove the order of limits anomaly with a weighted difference of the revTPSS exchange between the slowly varying and iso-orbitals (one- or two-electron) limits. We show that the new regularized (regTPSS) gives atomization energies comparable to revTPSS and preserves the accurate lattice constants as well. For other properties, the regTPSS gives at least the same performance as the revTPSS or TPSS meta-GGAs.

A comparative study on geometries, stabilities, and electronic properties between bimetallic AgnX (X = Au, Cu; n=1−8) and pure silver clusters

Using the meta-generalized gradient approximation (meta-GGA) exchange correlation TPSS functional, the geometric structures, the relative stabilities, and the electronic properties of bimetallic AgnX (X=Au, Cu; n=1−8) clusters are systematically investigated and compared with those of pure silver clusters. The optimized structures show that the transition point from preferentially planar to three-dimensional structure occurs at n = 6 for the AgnAu clusters, and at n = 5 for AgnCu clusters. For different-sized AgnX clusters, one X (X=Au or Cu) atom substituted Agn+1 structure is a dominant growth pattern. The calculated fragmentation energies, second-order differences in energies, and the highest occupied molecular orbital—lowest unoccupied molecular orbital (HOMO-LUMO) energy gaps show interesting odd-even oscillation behaviours, indicating that Ag2,4,6,8 and Ag1,3,5,7X (X=Au, Cu) clusters keep high stabilities in comparison with their neighbouring clusters. The natural population analysis reveals that the charges transfer from the Agn host to the impurity atom except for the Ag2Cu cluster. Moreover, vertical ionization potential (VIP), vertical electronic affinity (VEA), and chemical hardness (η) are discussed and compared in depth. The same odd—even oscillations are found for the VIP and η of the AgnX (X=Au, Cu; n=1−8) clusters.

Benchmarking the performance of time-dependent density functional methods

The performance of 24 density functionals, including 14 meta-generalized gradient approximation (mGGA) functionals, is assessed for the calculation of vertical excitation energies against an experimental benchmark set comprising 14 small- to medium-sized compounds with 101 total excited states. The experimental benchmark set consists of singlet, triplet, valence, and Rydberg excited states. The global-hybrid (GH) version of the Perdew-Burke-Ernzerhoff GGA density functional (PBE0) is found to offer the best overall performance with a mean absolute error (MAE) of 0.28 eV. The GH-mGGA Minnesota 2006 density functional with 54% Hartree-Fock exchange (M06-2X) gives a lower MAE of 0.26 eV, but this functional encounters some convergence problems in the ground state. The local density approximation functional consisting of the Slater exchange and Volk-Wilk-Nusair correlation functional (SVWN) outperformed all non-GH GGAs tested. The best pure density functional performance is obtained with the local version of the Minnesota 2006 mGGA density functional (M06-L) with an MAE of 0.41 eV.

Adsorption of successive layers of H2 molecules on a model copper surface: performances of second- to fifth-rung exchange-correlation functionals

The interaction of H2 molecules with a Cu(100) metallic surface has been investigated by DFT approaches using a (H2) n Cu13 cluster model. Nine exchange-correlation functionals, belonging to the generalized gradient approximations (GGA), meta GGA (mGGA), hybrid Kohn–Sham/Hartree–Fock models, either based on GGAs or mGGAs, range-separated hybrids, and double-hybrid families, have been tested on the chemisorption and physisorption processes involving one or two H2 layers. The addition of an empirical correction for dispersion has also been tested for some of these functionals. The calculated energies and structural parameters were compared to sophisticated multi reference configuration interaction including Davidson’s correction for quadruple excitations (MRCI + Q). Our results show that among the nine considered exchange-correlation functionals, none can accurately reproduce all processes involved in the successive layers adsorption. Although an hybrid based on a mGGA such as M06-2X can quantitatively describe both the physisorption step and dissociation barriers involved in the adsorption of the first H2, it fails to reproduce its chemisorption. On the other hand, significant discrepancies with the reference post-HF data are obtained for the description of the second layer interaction, no matter which functional is considered, outlining the need of improvement and/or development of exchange-correlation functionals suitable for complex systems such as H2/H2Cu.

Analysis of the performance of DFT functionals in the study of light emission by oxyluciferin analogs

AbstractFirefly bioluminescence has attracted much attention from the research community due to its most intriguing multicolor bioluminescence. Computational studies have been fundamental to the clarification of this phenomenon. However, no clear demonstration of the accuracy of the theoretical methods used in this field of research has been provided. In this work, we have used luciferin and dehydroluciferin in water as a model for the calculation of the emission wavelength of oxyluciferin and analogs. This model was used in a density functional theory (DFT) benchmarking in which 24 functionals were tested. A non‐negligible “case to case” dependence was observed. However in general, the better classes of the DFT functionals are generalized gradient approximation (GGA) and meta GGA. The generalized gradient exchange functional TPSSVWN5, the hybrid‐meta GGA functional mPWKCIS, and the hybrid GGA functional MPWLYP1M also present good results. For the contrary, the popular hybrid GGA functionals present a worse performance. © 2012 Wiley Periodicals, Inc.

Assessment of long-range corrected density functionals for dipole polarizability calculations of MX (M = Y-Cd; X = F, Cl, Br, and I) molecules

We present an extension of our previously published density functionals benchmarking for the static dipole polarizability calculations of 4d transition metal monohalides MX (M = Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd; X = F, Cl, Br, and I) [M. Alipour and A. Mohajeri, Mol. Phys. 109 (2011), 1439]. Based on a statistical analysis, the behaviors of five long-range corrected density functionals, CAM-B3LYP, LC-ωPBE, ωB97, ωB97X and ωB97X-D have been assessed. The high accuracy CCSD(T) methodology is used to benchmark the effect of long-range corrections in polarizability calculations. Among tested functionals, the LC-ωPBE functional is found to outperform others. Moreover, we have compared the performance of long-range corrected functionals with 42 previously tested functionals of various types including local spin density approximation (LSDA), generalized gradient approximation (GGA), meta GGA (MGGA), hybrid GGA (HGGA), and hybrid meta GGA (HMGGA). Our results show that there is a large variation in performance of various functionals in each type for calculations of the dipole polarizabilities. It has been indicated that long-range corrected functionals do not necessarily outperform older functionals, at least for the mean static dipole polarizability studied here. In particular, our extensive analysis shows that the functionals of B98, mPWIPW91, TPSSIKCIS, PBEIKCIS and B3PW91 are the best of all 47 tested density functionals. The results of this investigation can be used as a guidance to propose a new exchange-correlation functional for calculating the optical properties such as polarizability for considered molecules and also for larger transition metal systems.