M06-L Functionals Research Articles

Generalized Born Solvation Model SM12.

We present a new self-consistent reaction-field implicit solvation model that employs the generalized Born approximation for the bulk electrostatic contribution to the free energy of solvation. The new solvation model (SM) is called SM12 (where ″12″ stands for 2012), and it is available with two sets of parameters, SM12CM5 and SM12ESP. The SM12CM5 parametrization is based on CM5 partial atomic charges, and the SM12ESP parametrization is based on charges derived from a quantum-mechanically calculated electrostatic potential (ESP) (in particular, we consider ChElPG and Merz-Kollman-Singh charges). The model was parametrized over 10 combinations of theoretical levels including the 6-31G(d) and MG3S basis sets and the B3LYP, mPW1PW, M06-L, M06, and M06-2X density functionals against 2979 reference experimental data. The reference data include 2503 solvation free energies and 144 transfer free energies of neutral solutes composed of H, C, N, O, F, Si, P, S, Cl, Br, and I in water and in 90 organic solvents as well as 332 solvation free energies of singly charged anions and cations in acetonitrile, dimethyl sulfoxide, methanol, and water. The advantages of the new solvation model over our previous generalized Born model (SM8) and all other previous generalized Born solvation models are (i) like the SMD model based on electron density distributions, it may be applied with a single set of parameters with arbitrary extended basis sets, whereas the SM8 model involves CM4 or CM4M charges that become unstable for extended basis sets, (ii) it is parametrized against a more diverse training sets than any previous solvation model, and (iii) it is defined for the entire periodic table.

ASSESSMENT OF THE PERFORMANCE OF THE M05-CLASS AND M06-CLASS FUNCTIONALS FOR THE STRUCTURE AND GEOMETRY OF THE HYDROGEN-BONDED AND HALOGEN-BONDED COMPLEXES

The M05-class (M05 and M05-2X) and M06-class (M06, M06-2X, M06-HF, and M06-L) functionals, developed by Zhao and Truhlar, have shown better performance than popular older DFT functionals in obtaining accurate binding energies of noncovalent complexes. However, the reliability of these functionals for the structure and geometry of noncovalent systems was seldom assessed. Here, using the MP2/aug-cc-pVTZ values as a benchmark, we assessed the performance of the M05-class and M06-class functionals for the structure and geometry of the hydrogen-bonded and halogen-bonded complexes. The results clearly show that the M05, M06 and M06L functionals totally fail to predict the structure of the hydrogen-bonded complex formed between glycine and carbonic acid whereas the M05-2X, M06-2X, M06-HF, and even B3LYP succeed. For the geometries of a series of halogen-bonded complexes, it is found that the M05-2X functional performs slightly better than the M06-2X and M06-HF functionals and much better than the M05, M06 and M06-L functionals on average. Based on these tests, we concluded that the M05, M06 and M06-L functionals are not good for the study of the structure and geometry of the hydrogen-bonded and halogen-bonded complexes and the density functional M05-2X is the best choice. In addition, we have also assessed the integration grid errors arising from the numerical integration of these functionals for the structure and geometry of the hydrogen-bonded and halogen-bonded complexes.

Theoretical Investigation of the Intrinsic Mechanical Properties of Single- and Double-Layer Graphene

In this study, the Young’s and flexural moduli of single- and double-layer graphene have been theoretically investigated using periodic boundary condition (PBC) density functional theory (DFT) with the PBE, HSE06H, and M06L functionals in conjunction with the 6-31G* and the 3-21G basis sets. The unit-cell size and shape dependence as well as the directional dependencies of the mechanical properties have also been investigated. Additionally, the calculated stretching and flexural strain–stress curves are reported. Finally, finite-element simulations have been performed so as to find a homogeneous equivalent isotropic transverse material for single-layer graphene, in order to reproduce mechanical behavior in both tensile and bending sollicitations.

Comparing the impact of different supporting ligands on copper(I)–ethylene interactions

The nature of copper(I)–ethylene bonding in a series of [Cu(L)(C2H2)]n+ complexes (where L=1,4,7-trithiacyclononane, 2,2′-bipyridine, N,N,N′,N′-tetramethylethylenediamine, 1,4,7-trimethyl-1,4,7-triazacyclononane, hydrotris(3,5-dimethyl-1-pyrazolyl)borate, the bis(2,6-dimethylphenyl)-substituted ß-diketiminate, and the iminophosphanamide (tBu)2P(NSiMe3)2) has been studied using density functional theory methods. The B3LYP, M06-L, and BP86 functionals performed similarly in reproducing important geometric parameters from experimental structures. Bonding interactions between the [Cu(L)]n+ and ethylene fragments were investigated using energy decomposition analysis (ALMO–EDA), molecular orbital and fragment orbital interaction analysis, and natural bond orbital (NBO) analysis. NBO and EDA calculations both predict an increase in [Cu(L)]n+→ethylene(π∗) charge donation and stabilization with an increase in the qualitatively-predicted electron donating ability of the supporting ligand, and a lesser amount of ethylene(π)→Cu charge donation, the magnitude of which is not strongly correlated to ligand donor ability. Molecular orbital analysis reveals an increased mixing of the empty ethylene π∗-orbital with filled orbitals as the qualitative ligand donating ability increases, consistent with enhanced Cu→ethylene(π∗) π-backbonding and the NBO and EDA results.

An assessment of pure, hybrid, meta, and hybrid‐meta GGA density functional theory methods for open‐shell systems: The case of the nonheme iron enzyme 8R–LOX

The performance of a range density functional theory functionals combined in a quantum mechanical (QM)/molecular mechanical (MM) approach was investigated in their ability to reliably provide geometries, electronic distributions, and relative energies of a multicentered open-shell mechanistic intermediate in the mechanism 8R-Lipoxygenase. With the use of large QM/MM active site chemical models, the smallest average differences in geometries between the catalytically relevant quartet and sextet complexes were obtained with the B3LYP(*) functional. Moreover, in the case of the relative energies between (4) II and (6) II, the use of the B3LYP(*) functional provided a difference of 0.0 kcal mol(-1). However, B3LYP(±) and B3LYP also predicted differences in energies of less than 1 kcal mol(-1). In the case of describing the electronic distribution (i.e., spin density), the B3LYP(*), B3LYP, or M06-L functionals appeared to be the most suitable. Overall, the results obtained suggest that for systems with multiple centers having unpaired electrons, the B3LYP(*) appears most well rounded to provide reliable geometries, electronic structures, and relative 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.

Methane Activation in Gold Cation-Exchanged Zeolites: A DFT Study

Activation of methane has attracted a great deal of interest in laboratory chemical synthesis and in large-scale industrial processes. We performed density functional theory studies to investigate the C–H bond breaking of methane on Au+ and Au2+ ions in vacuum and inside different types of zeolites. The density functional M06-L and the 6-31G(d,p) basis set were employed as this level of theory had already been shown to be reasonably accurate and affordable for transition metal systems. We investigated four industrially important catalysts, ZSM-5, FAU, FER, and MCM-22, each with a particular framework topology, with respect to their performance for methane activation. The bicoordinated character of the cationic site in the ZSM-5 structure provides a higher activity than the FAU structure with a 3-fold coordination of its cationic site. The activation energy of the reaction catalyzed by Au-ZSM-5 is lower than the one with the bare Au+ cation (13.2 vs 21.3 kcal/mol) because of the structural constraint imposed by the zeolite that leads to an earlier transition state with a high charge difference of the C–H atoms where the bond is broken. It is also found that the activity of Aun+ decreases already with n = 2, due to the shared positive charge. For the zeolites with large pores, Au-MCM-22 provides a higher activity due to the spacious framework of this particular type of zeolite is perfect for stabilizing the transition state structure but not the corresponding adsorption complex. The small and medium pore-sized zeolites, Au-FER and Au-ZSM-5 stabilize both the adsorption complex and the transition states, thus causing the activation energy to remain the same.

Interaction between alkyl radicals and single wall carbon nanotubes

The addition of primary, secondary, and tertiary alkyl radicals to single wall carbon nanotubes (SWCNTs) was studied by means of dispersion corrected density functional theory. The PBE, B97-D, M06-L, and M06-2X functionals were used. Consideration of Van der Waals interactions is essential to obtain accurate addition energies. In effect, the enthalpy changes at 298 K, for the addition of methyl, ethyl, isopropyl, and tert-butyl radicals onto a (5,5) SWCNT are: -25.7, -25.1, -22.4, and -16.6 kcal/mol, at the M06-2X level, respectively, whereas at PBE/6-31G* level they are significantly lower: -25.0, -19.0, -16.7, and -5.0 kcal/mol respectively. Although the binding energies are small, the attached alkyl radicals are expected to be stable because of the large desorption barriers. The importance of nonbonded interactions was more noticeable as we moved from primary to tertiary alkyl radicals. Indeed, for the tert-butyl radical, physisorption onto the (11,0) SWCNT is preferred rather than chemisorption. The bond dissociation energies determined for alkyl radicals and SWCNT follow the trend suggested by the consideration of radical stabilization energies. However, they are in disagreement with some degrees of functionalization observed in recent experiments. This discrepancy would stem from the fact that for some HiPco nanotubes, nonbonded interactions with alkyl radicals are stronger than covalent bonds.

Improving the Chemical Reactivity of Single-Wall Carbon Nanotubes with Lithium Doping

The effect of lithium doping on the reactivity of single-wall carbon nanotubes (SWCNTs) was studied by means of first principle calculations. Two prototype reactions were considered, the 1,3 dipolar cycloaddition and the attachment of free radicals. The results obtained employing the PBE, M06L, and M06-2X functionals indicated that lithium significantly increases the reaction energies. Although this effect is slightly increased upon diameter enlargement, it decreases when the distance between the functional group and lithium is augmented. We observed a stronger enhancement of reactivity for semiconducting SWCNTs as compared with the metallic ones. Regarding the endohedral and exohedral lithium dopings, both increase the binding energies, although for some functional groups the latter doping is far less effective because exohedral lithium can remove the fluorine atoms or hydroxyl or thiol groups from the tube walls. Therefore, endohedral lithium is expected to be more useful to enhance the reactions performed onto SWCNTs, whereas in some cases exohedral lithium may have a negative effect depending on the affinity that lithium has with the functional groups attached to the SWCNTs. If the lithium–functional group interaction is stronger than the binding energy between the nanotube and the functional group, the perfect sp2 framework of the nanotube will be restored. All of the functionals employed gave the same results from a qualitative standpoint, but important quantitative differences were observed in the magnitude of the reactivity enhancement.

Influence of Electron Doping on the Hydrogenation of Fullerene C60: A Theoretical Investigation

The influence of electron attachment on the stability of the mono- and dihydrogenated buckminsterfullerene C(60) was studied using density functional theory and semiempirical molecular orbital techniques. We have also assessed the reliability of computationally accessible methods that are important for investigating the reactivity of graphenic species and surfaces in general. The B3LYP and M06L functionals with the 6-311+G(d,p) basis set and MNDO/c are found to be the best methods for describing the electron affinities of C(60) and C(60)H(2) . It is shown that simple frontier molecular orbital analyses at both the AM1 and B3LYP/6-31G(d) levels are useful for predicting the most favourable position of protonation of C(60)H(-) , that is, formation of the kinetically controlled product 1,9-dihydro[60]fullerene, which is also the thermodynamically controlled product, in agreement with experimental and previous theoretical studies. We have shown that reduction of exo- and endo-C(60)H makes them more stable in contrast to the reduction of the exo,exo-1,9-C(60)H(2) , reduced forms of which decompose more readily, in agreement with experimental electrochemical studies. However, most other dihydro[60]fullerenes are stabilized by reduction and the regioselectivity of addition is predicted to decrease as the less stable isomers are stabilized more by the addition of electrons than the two most stable ones (1,9 and 1,7).

Performance of Density Functional Theory and Møller-Plesset Second-Order Perturbation Theory for Structural Parameters in Complexes of Ru.

We assess the performance of density functional theory (DFT) and Møller-Plesset second-order perturbation theory (MP2) for predicting structural parameters in Ru complexes, in particular, a Ru(IV) allyl dicationic complex with the formula [Ru(η(5)-Cp*)(η(3)-CH2CHCHC6H5)(NCCH3)2](2+) and the molecules RuO4 and Ru(C2O4)2(H2O)2(-), where Cp* denotes C5Me5 and Me denotes methyl. The density functionals studied are B3LYP, B3PW91, M05, M06, M06-L, MOHLYP, MPW3LYP, PBE0, PW6B95, SOGGA, τHCTHhyb, ωB97X, and ωB97X-D, in combination with three different basis sets, namely, LANL2DZ, def2-SVP, and def2-TZVP. The theoretically computed Ru-C distances corresponding to the phenylallyl complex are especially well predicted by the SOGGA (pure DFT) and ωB97X-D (DFT plus an empirical molecular mechanics term) methods. This contrasts with an article in this Journal [ Calhorda , M. J. , Pregosin , P. S. , and Veiros , L. F. J. Chem. Theory Comput. 2007 , 3 , 665 - 670 ] in which it was found that DFT cannot account for these Ru-C distances. Averaging over four Ru-C distances in the allyl complex and three unique Ru-O distances in RuO4 and Ru(C2O4)2(H2O)2(-), the SOGGA and ωB97X-D methods have both a smaller mean unsigned error than MP2 and the same maximum error. The M06, PW6B95, PBE0, M06-L, and ωB97X density functionals also have a smaller or the same mean unsigned error as MP2.

When noncovalent interactions are stronger than covalent bonds: Bilayer graphene doped with second row atoms, aluminum, silicon, phosphorus and sulfur

Herein, we investigate bilayer graphene doped with second-row atoms using the LDA, M06-L, and VDW-DF functionals. For 2–3at.% or lower Al and P dopings the structure with a short interlayer distance and without a covalent bond between the heteroatoms is more stable than that with a covalent bond and longer interlayer separation. However, for Si the linked structure is more stable. Doped bilayer-graphene is prone to the attachment of more heteroatoms, as reflected by the large adsorption energies, which decrease in the following order Al>Si>P. We show that it is feasible to tune the electronic properties, and the interlayer-interaction energy varying the type or amount of second row atoms.

Origin of the Unusual Ultraviolet Absorption of Arsenicin A

This paper presents a combined experimental and theoretical study of the electronic spectrum of the natural adamantane-type polyarsenical Arsenicin A. Experiments reveal that this molecule strongly absorbs UV light in the absence of an obvious chromophore. The observed absorbance is supported by the time-dependent density functional (TD-DFT) calculations with B3LYP, M06-L, and M06-2X functionals combined with the 6-311+G(3df,2pd) basis set, as well as by symmetry-adapted cluster/configuration interaction (SAC-CI) theory. The theoretical investigations reveal that the absorption is facilitated by through-space and through-bond interactions, between the lone pairs on the arsenic and oxygen atoms and the σ-bonding framework of the molecule, that destabilize occupied and stabilize unoccupied molecular orbitals.

CO and NO complexes of Fe(II) and Co(II) porphyrins

Density functional theory (DFT) calculations on iron(II) (FeP) and cobalt(II) (CoP) porphyrins with nitric oxide (NO) and carbon monoxide (CO) as axial ligands have been performed using BLYP, B3LYP, OLYP, B3PW91, and M06L functionals with double- and triple-ξ basis sets. Optimized geometries and binding energies were found to depend very strongly on the functional and basis set used. MP2 fails completely to describe geometries and binding energies of FeP–CO, CoP–CO, and CoP–NO complexes, but performs relatively well for the FeP–NO complex. The calculated binding energies range from −4.5 to −19.6 kcal mol−1 for CoP–CO, +4.0 to −29.1 kcal mol−1 for FeP–CO, +7.6 to −41.1 kcal mol−1 for FeP–NO, and −2.2 to −36.7 kcal mol−1 for CoP–NO. B3PW91/6–311 + G(d,p) and OLYP/6–311+G(d,p) perform better than the other functional/basis set combinations in reproducing the available experimental data. A complete active space self-consistent field/complete active space with second-order perturbation theory (CASSCF/CASPT2) analysis of the CO and NO interactions with FeP and CoP can be interpreted either in terms of σ-donation/π-back-donation mechanism (FeP–CO and CoP–CO) or σ bond formation between the orbital of metal and the π* orbital of NO.

Performance of CASPT2 and DFT for Relative Spin-State Energetics of Heme Models

The accuracy of the relative spin-state energetics of three small Fe(II) or Fe(III) heme models from multiconfigurational perturbation theory (CASPT2) and density functional theory with selected functionals (including the recently developed M06 and M06-L functionals) was assessed by comparing with recently available coupled cluster results. While the CASPT2 calculations of spin-state energetics were found to be very accurate for the studied Fe(III) complexes (including FeP(SH), a model of the active site of cytochrome P450 in its resting state), there is a strong indication of a systematic error (around 5 kcal/mol) in favor of the high-spin state for the studied Fe(II) complexes (including FeP(Im), a model of the active site of myoglobin). A larger overstabilization of the high-spin states was observed for the M06 and M06-L functionals, up to 22 and 11 kcal/mol, respectively. None of the tested density functionals consistently provides a better accuracy than CASPT2 for all model complexes.

Accurate Computed Enthalpies of Spin Crossover in Iron and Cobalt Complexes

Despite their importance in many chemical processes, the relative energies of spin states of transition metal complexes have so far been haunted by large computational errors. By the use of six functionals, B3LYP, BP86, TPSS, TPSSh, M06, and M06L, this work studies nine complexes (seven with iron and two with cobalt) for which experimental enthalpies of spin crossover are available. It is shown that such enthalpies can be used as quantitative benchmarks of a functional's ability to balance electron correlation in both the involved states. TPSSh achieves an unprecedented mean absolute error of approximately 11 kJ/mol in spin transition energies, with the local functional M06L a distant second (25 kJ/mol). Other tested functionals give mean absolute errors of 40 kJ/mol or more. This work confirms earlier suggestions that 10% exact exchange is near-optimal for describing the electron correlation effects of first-row transition metal systems. Furthermore, it is shown that given an experimental structure of an iron complex, TPSSh can predict the electronic state corresponding to that experimental structure. We recommend this functional as current state-of-the-art for studying spin crossover and relative energies of close-lying electronic configurations in first-row transition metal systems.

Validation study of the ability of density functionals to predict the planar-to-three-dimensional structural transition in anionic gold clusters

As gold clusters increase in size, the preferred structure changes from planar to three-dimensional and, for anionic clusters, Au(n)-, the two-dimensional(2D)-->three-dimensional (3D) transition is found experimentally to occur between n=11 and n=12. Most density functionals predict that planar structures are preferred up to higher n than is observed experimentally, an exception being the local spin density approximation. Here we test four relatively new functionals for this feature, in particular, M05, M06-L, M06, and SOGGA. We find that M06-L, M06, and SOGGA all predict the 2D-->3D transition at the correct value of n. Since the M06-L and M06 functionals have previously been shown to be reasonably accurate for transition metal bond energies, main group atomization energies, barrier heights, and noncovalent interaction energies, and, since they are here shown to perform well for the s-d excitation energy and ionization potential of Au atoms and for the size of Au(n)- clusters at which the 2D-->3D transition occurs, they are recommended for simulating processes catalyzed by gold clusters.

Evaluation of B3LYP, X3LYP, and M06-Class Density Functionals for Predicting the Binding Energies of Neutral, Protonated, and Deprotonated Water Clusters.

In this paper we assess the accuracy of the B3LYP, X3LYP, and newly developed M06-L, M06-2X, and M06 functionals to predict the binding energies of neutral and charged water clusters including (H2O)n, n = 2-8, 20), H3O(+)(H2O)n, n = 1-6, and OH(-)(H2O)n, n = 1-6. We also compare the predicted energies of two ion hydration and neutralization reactions on the basis of the calculated binding energies. In all cases, we use as benchmarks calculated binding energies of water clusters extrapolated to the complete basis set limit of the second-order Møller-Plesset perturbation theory with the effects of higher order correlation estimated at the coupled-cluster theory with single, double, and perturbative triple excitations in the aug-cc-pVDZ basis set. We rank the accuracy of the functionals on the basis of the mean unsigned error (MUE) between calculated benchmark and density functional theory energies. The corresponding MUE (kcal/mol) for each functional is listed in parentheses. We find that M06-L (0.73) and M06 (0.84) give the most accurate binding energies using very extended basis sets such as aug-cc-pV5Z. For more affordable basis sets, the best methods for predicting the binding energies of water clusters are M06-L/aug-cc-pVTZ (1.24), B3LYP/6-311++G(2d,2p) (1.29), and M06/aug-cc-PVTZ (1.33). M06-L/aug-cc-pVTZ also gives more accurate energies for the neutralization reactions (1.38), whereas B3LYP/6-311++G(2d,2p) gives more accurate energies for the ion hydration reactions (1.69).

Benchmark Energetic Data in a Model System for Grubbs II Metathesis Catalysis and Their Use for the Development, Assessment, and Validation of Electronic Structure Methods.

We present benchmark relative energetics in the catalytic cycle of a model system for Grubbs second-generation olefin metathesis catalysts. The benchmark data were determined by a composite approach based on CCSD(T) calculations, and they were used as a training set to develop a new spin-component-scaled MP2 method optimized for catalysis, which is called SCSC-MP2. The SCSC-MP2 method has improved performance for modeling Grubbs II olefin metathesis catalysts as compared to canonical MP2 or SCS-MP2. We also employed the benchmark data to test 17 WFT methods and 39 density functionals. Among the tested density functionals, M06 is the best performing functional. M06/TZQS gives an MUE of only 1.06 kcal/mol, and it is a much more affordable method than the SCSC-MP2 method or any other correlated WFT methods. The best performing meta-GGA is M06-L, and M06-L/DZQ gives an MUE of 1.77 kcal/mol. PBEh is the best performing hybrid GGA, with an MUE of 3.01 kcal/mol; however, it does not perform well for the larger, real Grubbs II catalyst. B3LYP and many other functionals containing the LYP correlation functional perform poorly, and B3LYP underestimates the stability of stationary points for the cis-pathway of the model system by a large margin. From the assessments, we recommend the M06, M06-L, and MPW1B95 functionals for modeling Grubbs II olefin metathesis catalysts. The local M06-L method is especially efficient for calculations on large systems.

Assessment of Density Functionals for Intramolecular Dispersion-Rich Interactions.

A range of density functional theory methods, including conventional hybrid and meta-hybrid functionals, a double-hybrid functional, and DFT-D (DFT augmented with an empirical dispersion term) were assessed for their ability to describe the three minima along the ϕGly rotational profile of one particular Tyr-Gly conformer. Previous work had shown that these minima are sensitive to intramolecular dispersion and basis set superposition error, the latter rendering MP2 calculations with small to medium-sized basis sets unsuitable for describing this molecule. Energy profiles for variation of the ϕGly torsion angle were compared to an estimated CCSD(T)/CBS reference profile. The hybrid functionals and the meta-hybrid PWB6K failed to predict all three minima; the meta-hybrid functionals M05-2X and M06-2X and the nonhybrid meta functional M06-L as well as the double-hybrid mPW2-PLYP and the B3LYP-D method did find all three minima but underestimated the relative stability of the two with rotated C-terminus. The best performance was delivered by the most elaborate density functional theory model employed: mPW2-PLYP-D. Only M06-2X and mPW2-PLYP-D predicted the correct order of stability of the three minima.