Density functional theory calculations of the lowest energy quintet and triplet states of model hemes: role of functional, basis set, and zero-point energy corrections

We investigated the effect of several computational variables, including the choice of the basis set, application of symmetry constraints, and zero-point energy (ZPE) corrections, on the structural parameters and predicted ground electronic state of model 5-coordinate hemes (iron(II) porphines axially coordinated by a single imidazole or 2-methylimidazole). We studied the performance of B3LYP and B3PW91 with eight Pople-style basis sets (up to 6-311+G*) and B97-1, OLYP, and TPSS functionals with 6-31G and 6-31G* basis sets. Only hybrid functionals B3LYP, B3PW91, and B97-1 reproduced the quintet ground state of the model hemes. With a given functional, the choice of the basis set caused up to 2.7 kcal/mol variation of the quintet-triplet electronic energy gap (DeltaEel), in several cases, resulting in the inversion of the sign of DeltaEel. Single-point energy calculations with triple-zeta basis sets of the Pople (up to 6-311G++(2d,2p)), Ahlrichs (TZVP and TZVPP), and Dunning (cc-pVTZ) families showed the same trend. The zero-point energy of the quintet state was approximately 1 kcal/mol lower than that of the triplet, and accounting for ZPE corrections was crucial for establishing the ground state if the electronic energy of the triplet state was approximately 1 kcal/mol less than that of the quintet. Within a given model chemistry, effects of symmetry constraints and of a "tense" structure of the iron porphine fragment coordinated to 2-methylimidazole on DeltaEel were limited to 0.3 kcal/mol. For both model hemes the best agreement with crystallographic structural data was achieved with small 6-31G and 6-31G* basis sets. Deviation of the computed frequency of the Fe-Im stretching mode from the experimental value with the basis set decreased in the order: nonaugmented basis sets, basis sets with polarization functions, and basis sets with polarization and diffuse functions. Contraction of Pople-style basis sets (double-zeta or triple-zeta) affected the results insignificantly for iron(II) porphyrin coordinated with imidazole. Poor performance of a "locally dense" basis set with a large number of basis functions on the Fe center was observed in calculation of quintet-triplet gaps. Our results lead to a series of suggestions for density functional theory calculations of quintet-triplet energy gaps in ferrohemes with a single axial imidazole; these suggestions are potentially applicable for other transition-metal complexes.     

Avoiding pitfalls of a theoretical approach: the harmonic oscillator measure of aromaticity index from quantum chemistry calculations

The concept of the harmonic oscillator measure of aromaticity (HOMA) is based on comparing the geometrical parameters of a studied molecule with the parameters for an ideal aromatic system derived from bond lengths of the reference molecules. Nowadays, HOMA is routinely computed combining the geometries from quantum chemistry calculations with the experimentally based parameterization. Thus, obtained values of HOMA, however, are bound to suffer from inaccuracies of the theoretical methods and strongly depend on computational details. This could be avoided by obtaining both the input geometries and the parameters with the same theoretical method, but efficiency of the error compensation achieved in this way has not yet been probed. In our work, we have prepared a benchmark set of HOMA values for 25 cyclic hydrocarbons, based on the all core CCSD(T)/cc-pCVQ(T)Z geometries, and used it to investigate the impact of different choices of the exchange–correlation functionals and basis sets on HOMA, calculated against the experimentally based (HOMA^EP) or the consistently calculated (HOMA^CCP) parameters. We show that using HOMA^EP leads to large and unsystematic errors, and strong sensitivity to the choice of XC functional, basis set, and the experimental data for the reference geometry. This sensitivity is largely, although not completely attenuated in the consistent approach. We recommend the most suitable functionals for calculating HOMA in both approaches (HOMA^EP and HOMA^CCP), and provide the HOMA parameters for 25 studied exchange–correlation functionals and two popular basis sets.     

Theoretical investigation of substituted anthraquinone dyes

We have investigated with computational chemistry techniques the visible spectra of substituted anthraquinones. A wide panel of theoretical methods has been used, with various basis sets and density functional theory (DFT) functionals, in order to assess a level of theory that would lead to converged excitation energies. It turns out that the hybrid Becke-Lee-Yang-Parr and Perdew-Burke-Erzenrhof functionals with the 6-31G (d,p) atomic basis set provide reliable lambda(max) when the solvent effects are included in the model. Combining the results of both DFT schemes allows the prediction of lambda(max) with a standard deviation limited to 13 nm.     

Basis set and method dependence in quantum theory of atoms in molecules calculations for covalent bonds

The influence of various small- and medium-size basis sets used in Hartree-Fock (HF) and density functional theory (DFT)/B3LYP calculations on results of quantum theory of atoms in molecules based (QTAIM-based) analysis of bond parameters is investigated for several single, double, and triple covalent bonds. It is shown that, in general, HF and DFT/B3LYP methods give very similar QTAIM results with respect to the basis set. The smallest 6-31G basis set and DZ-quality basis sets of Dunning type lead to poor results in comparison to those obtained by the most reliable aug-cc-pVTZ. On the contrary, 6-311++G(2df,2pd) and in a somewhat lesser extent 6-311++G(3df,3pd) basis sets give satisfactory values of QTAIM parameters. It is also demonstrated that QTAIM calculations may be sensitive for the method and basis set in the case of multiple and more polarized bonds.     

How to compute isomerization energies of organic molecules with quantum chemical methods

The reaction energies for 34 typical organic isomerizations including oxygen and nitrogen heteroatoms are investigated with modern quantum chemical methods that have the perspective of also being applicable to large systems. The experimental reaction enthalpies are corrected for vibrational and thermal effects, and the thus derived "experimental" reaction energies are compared to corresponding theoretical data. A series of standard AO basis sets in combination with second-order perturbation theory (MP2, SCS-MP2), conventional density functionals (e.g., PBE, TPSS, B3-LYP, MPW1K, BMK), and new perturbative functionals (B2-PLYP, mPW2-PLYP) are tested. In three cases, obvious errors of the experimental values could be detected, and accurate coupled-cluster [CCSD(T)] reference values have been used instead. It is found that only triple-zeta quality AO basis sets provide results close enough to the basis set limit and that sets like the popular 6-31G(d) should be avoided in accurate work. Augmentation of small basis sets with diffuse functions has a notable effect in B3-LYP calculations that is attributed to intramolecular basis set superposition error and covers basic deficiencies of the functional. The new methods based on perturbation theory (SCS-MP2, X2-PLYP) are found to be clearly superior to many other approaches; that is, they provide mean absolute deviations of less than 1.2 kcal mol-1 and only a few (<10%) outliers. The best performance in the group of conventional functionals is found for the highly parametrized BMK hybrid meta-GGA. Contrary to accepted opinion, hybrid density functionals offer no real advantage over simple GGAs. For reasonably large AO basis sets, results of poor quality are obtained with the popular B3-LYP functional that cannot be recommended for thermochemical applications in organic chemistry. The results of this study are complementary to often used benchmarks based on atomization energies and should guide chemists in their search for accurate and efficient computational thermochemistry methods.     

A Systemic DFT Study on Several 5<Emphasis Type="Italic">d</Emphasis>-Electron Element Dimers: Hf<Subscript>2</Subscript>, Ta<Subscript>2</Subscript>, Re<Subscript>2</Subscript>, W<Subscript>2</Subscript>, and Hg<Subscript>2</Subscript>

Eleven kinds of density functionals in conjunction with three different basis sets are employed to investigate the homonuclear 5d-electron dimers: Hf₂, Ta₂, Re₂, W₂ and Hg₂. The computed bond lengths, vibrational frequencies and dissociation energies of these molecules are used to compare with available experimental data to find the appropriate combination of functional and basis set. The different functionals and basis sets favor different ground electronic state for Hf₂ and Re₂ molecules, indicating that these two dimers are sensitive to the functionals used. The molecular properties of Hg₂ dimer depend strongly on both functionals and basis sets used. It is found that the BP86 and PBEPBE functionals are generally successful in describing the 5d-electron dimers. For the ground states of these dimers, the bonding patterns are determined by natural bond orbital (NBO) analysis. Natural electron configurations show that the 6s and 5d orbitals in the bonding atoms hybrid with each other for the studied dimers except for Hg₂.     

What is the best DFT functional for vibronic calculations? A comparison of the calculated vibronic structure of the S1-S0 transition of phenylacetylene with cavity ringdown band intensities

The sensitivity of vibronic calculations to electronic structure methods and basis sets is explored and compared to accurate relative intensities of the vibrational bands of phenylacetylene in the S(1)(A(1)B(2)) ← S(0)(X(1)A(1)) transition. To provide a better measure of vibrational band intensities, the spectrum was recorded by cavity ringdown absorption spectroscopy up to energies of 2000 cm(-1) above the band origin in a slit jet sample. The sample rotational temperature was estimated to be about 30 K, but the vibrational temperature was higher, permitting the assignment of many vibrational hot bands. The vibronic structure of the electronic transition was simulated using a combination of time-dependent density functional theory (TD-DFT) electronic structure codes, Franck-Condon integral calculations, and a second-order vibronic model developed previously [Johnson, P. M.; Xu, H. F.; Sears, T. J. J. Chem. Phys. 2006, 125, 164331]. The density functional theory (DFT) functionals B3LYP, CAM-B3LYP, and LC-BLYP were explored. The long-range-corrected functionals, CAM-B3LYP and LC-BLYP, produced better values for the equilibrium geometry transition moment, but overemphasized the vibronic coupling for some normal modes, while B3LYP provided better-balanced vibronic coupling but a poor equilibrium transition moment. Enlarging the basis set made very little difference. The cavity ringdown measurements show that earlier intensities derived from resonance-enhanced multiphoton ionization (REMPI) spectra have relative intensity errors.     

First‐principle modelling of forsterite surface properties: Accuracy of methods and basis sets

The seven main crystal surfaces of forsterite (Mg₂SiO₄) were modeled using various Gaussian‐type basis sets, and several formulations for the exchange‐correlation functional within the density functional theory (DFT). The recently developed pob‐TZVP basis set provides the best results for all properties that are strongly dependent on the accuracy of the wavefunction. Convergence on the structure and on the basis set superposition error‐corrected surface energy can be reached also with poorer basis sets. The effect of adopting different DFT functionals was assessed. All functionals give the same stability order for the various surfaces. Surfaces do not exhibit any major structural differences when optimized with different functionals, except for higher energy orientations where major rearrangements occur around the Mg sites at the surface or subsurface. When dispersions are not accounted for, all functionals provide similar surface energies. The inclusion of empirical dispersions raises the energy of all surfaces by a nearly systematic value proportional to the scaling factor s of the dispersion formulation. An estimation for the surface energy is provided through adopting C₆ coefficients that are more suitable than the standard ones to describe O? O interactions in minerals. A 2 × 2 supercell of the most stable surface (010) was optimized. No surface reconstruction was observed. The resulting structure and surface energy show no difference with respect to those obtained when using the primitive cell. This result validates the (010) surface model here adopted, that will serve as a reference for future studies on adsorption and reactivity of water and carbon dioxide at this interface. © 2015 Wiley Periodicals, Inc.     

Structures and excitation energies of Zn–tetraarylporphyrin analogues: A theoretical study

The photophysical properties of β-substituted Zn–tetraarylporphyrin (ZnTAP) analogues used as dyes in dye-sensitized solar cells were studied using density functional theory (DFT). Singlet-excitation energy calculations of ZnTAP analogues were performed using time-dependent DFT with B3LYP, B3PW91, PBE0 exchange–correlation functionals at 6-31G(d) and 6-31+G(d) basis sets using B3LYP/6-31G(d) geometries. The PBE0 functional at 6-31+G(d) basis set provided a better correlation with the experimental data for both B- and Q-bands. The inclusion of solvation effect in the calculations provided a good agreement in terms of B:Q_ave ratio of the oscillator strengths for both analogues with the experimental values. Analogue 2 has a higher and a more balanced charge-carrier transport rates than analogue 1. In general, the addition of an electron-donating group in the meso-substituent (analogue 2) resulted in a narrower band gap, higher oscillator strength, a more red-shifted absorption spectra, and better charge-transfer characteristics than analogue 1.     

Is the uniform electron gas limit important for small Ag clusters? Assessment of different density functionals for Ag(n) (n < or = 4)

Twenty-three density functional theory (DFT) methods, including the second- and the third-generation functionals, are tested in conjunction with two basis sets (LANL2DZ and SDD) for studying the properties of neutral and ionic silver clusters. We find that DFT methods incorporating the uniform electron gas limit in the correlation functional, namely, those with Perdew's correlation functionals (PW91, PBE, P86, and TPSS), Becke's B95, and the Van Voorhis-Scuseria functional VSXC, generally perform better than the other group of functionals, e.g., those incorporating the LYP correlation functional and variations of the B97 functional. Strikingly, these two groups of functionals can produce qualitatively different results for the Ag3 and Ag4 clusters. The energetic properties and vibrational frequencies of Ag(n) are also evaluated by the different functionals. The present study shows that the choice of DFT methods for heavy metals may be critical. It is found that the exact-exchange-incorporated PBE functional (PBE1PBE) is among the best for predicting the range of properties.     

Sensitivity of the properties of ruthenium "blue dimer" to method, basis set, and continuum model

The ruthenium "blue dimer" [(bpy)(2)Ru(III)OH(2)](2)O(4+) is best known as the first well-defined molecular catalyst for water oxidation. It has been subject to numerous computational studies primarily employing density functional theory. However, those studies have been limited in the functionals, basis sets, and continuum models employed. The controversy in the calculated electronic structure and the reaction energetics of this catalyst highlights the necessity of benchmark calculations that explore the role of density functionals, basis sets, and continuum models upon the essential features of blue-dimer reactivity. In this paper, we report Kohn-Sham complete basis set (KS-CBS) limit extrapolations of the electronic structure of "blue dimer" using GGA (BPW91 and BP86), hybrid-GGA (B3LYP), and meta-GGA (M06-L) density functionals. The dependence of solvation free energy corrections on the different cavity types (UFF, UA0, UAHF, UAKS, Bondi, and Pauling) within polarizable and conductor-like polarizable continuum model has also been investigated. The most common basis sets of double-zeta quality are shown to yield results close to the KS-CBS limit; however, large variations are observed in the reaction energetics as a function of density functional and continuum cavity model employed.     

Performance of the density matrix functional theory in the quantum theory of atoms in molecules

The generalization to arbitrary molecular geometries of the energetic partitioning provided by the atomic virial theorem of the quantum theory of atoms in molecules (QTAIM) leads to an exact and chemically intuitive energy partitioning scheme, the interacting quantum atoms (IQA) approach, that depends on the availability of second-order reduced density matrices (2-RDMs). This work explores the performance of this approach in particular and of the QTAIM in general with approximate 2-RDMs obtained from the density matrix functional theory (DMFT), which rests on the natural expansion (natural orbitals and their corresponding occupation numbers) of the first-order reduced density matrix (1-RDM). A number of these functionals have been implemented in the promolden code and used to perform QTAIM and IQA analyses on several representative molecules and model chemical reactions. Total energies, covalent intra- and interbasin exchange-correlation interactions, as well as localization and delocalization indices have been determined with these functionals from 1-RDMs obtained at different levels of theory. Results are compared to the values computed from the exact 2-RDMs, whenever possible.     

Scrutinizing “Invisible” astatine: A challenge for modern density functionals

The main‐group 6p elements did not receive much attention in the development of recent density functionals. In many cases it is still difficult to choose among the modern ones a relevant functional for various applications. Here, we illustrate the case of astatine species (At, Z = 85) and we report the first, and quite complete, benchmark study on several properties concerning such species. Insights on geometries, transition energies and thermodynamic properties of a set of 19 astatine species, for which reference experimental or theoretical data has been reported, are obtained with relativistic (two‐component) density functional theory calculations. An extensive set of widely used functionals is employed. The hybrid meta‐generalized gradient approximation (meta‐GGA) PW6B95 functional is overall the best choice. It is worth noting that the range‐separated HSE06 functional as well as the old and very popular B3LYP and PBE0 hybrid‐GGAs appear to perform quite well too. Moreover, we found that astatine chemistry in solution can accurately be predicted using implicit solvent models, provided that specific parameters are used to build At cavities. © 2016 Wiley Periodicals, Inc.     

Consistent theoretical description of 1,3-dipolar cycloaddition reactions

The cycloaddition reactions of 18 1,3-dipolar molecules to ethylene and acetylene have been reinvestigated by quantum chemical methods that are based on a second-order perturbation treatment of electron correlation. It is found that SCS-MP2 and the new perturbative B2-PLYP density functional provide accurate reaction barriers and outperform MP2 as well as standard density functionals such as B3-LYP. The new second-order based methods have the additional advantage that they perform better with increasing quality of the one-particle space, as is desired for a good quantum chemical method. The errors for the reaction enthalpies are in general larger than for the barriers when compared to CBS-QB3 literature values, which is related to strong changes in the electronic structures, but the deviations are again smaller than with MP2 or B3-LYP and are also more systematic. The results of a detailed basis set study suggest that properly polarized triple-zeta AO basis sets represent a good compromise between accuracy and computational speed. The combination of very inaccurate density functionals with small (double-zeta) basis sets, which yields good results for the initial part of the reactions due to error compensation, is not recommended.     

Ab initio and density functional theory reinvestigation of gas-phase sulfuric acid monohydrate and ammonium hydrogen sulfate

We have calculated the thermochemical parameters for the reactions H(2)SO(4) + H(2)O <--> H(2)SO(4).H(2)O and H(2)SO(4) + NH(3) <--> H(2)SO(4).NH(3) using the B3LYP and PW91 functionals, MP2 perturbation theory and four different basis sets. Different methods and basis sets yield very different results with respect to, for example, the reaction free energies. A large part, but not all, of these differences are caused by basis set superposition error (BSSE), which is on the order of 1-3 kcal mol(-1) for most method/basis set combinations used in previous studies. Complete basis set extrapolation (CBS) calculations using the cc-pV(X+d)Z and aug-cc-pV(X+d)Z basis sets (with X = D, T, Q) at the B3LYP level indicate that if BSSE errors of less than 0.2 kcal mol(-1) are desired in uncorrected calculations, basis sets of at least aug-cc-pV(T+d)Z quality should be used. The use of additional augmented basis functions is also shown to be important, as the BSSE error is significant for the nonaugmented basis sets even at the quadruple-zeta level. The effect of anharmonic corrections to the zero-point energies and thermal contributions to the free energy are shown to be around 0.4 kcal mol(-1) for the H(2)SO(4).H(2)O cluster at 298 K. Single-point CCSD(T) calculations for the H(2)SO(4).H(2)O cluster also indicate that B3LYP and MP2 calculations reproduce the CCSD(T) energies well, whereas the PW91 results are significantly overbinding. However, basis-set limit extrapolations at the CCSD(T) level indicate that the B3LYP binding energies are too low by ca. 1-2 kcal/mol. This probably explains the difference of about 2 kcal mol(-1) for the free energy of the H(2)SO(4) + H(2)O <--> H(2)SO(4).H(2)O reaction between the counterpoise-corrected B3LYP calculations with large basis sets and the diffusion-based experimental values of S. M. Ball, D. R. Hanson, F. L Eisele and P. H. McMurry (J. Phys. Chem. A. 2000, 104, 1715). Topological analysis of the electronic charge density based on the quantum theory of atoms in molecules (QTAIM) shows that different method/basis set combinations lead to qualitatively different bonding patterns for the H(2)SO(4).NH(3) cluster. Using QTAIM analysis, we have also defined a proton transfer degree parameter which may be useful in further studies.     

Proper gaussian basis sets for density functional studies of water dimers and trimers

The accuracy of the Perdew-Burke-Ernzerhof and Tao-Perdew-Staroverov-Scuseria density functionals for describing noncovalent interaction energies in small water clusters is studied by testing 11 basis sets on a reduced test set proposed by Dahlke and Truhlar (J. Phys. Chem. B 2005, 109, 15677). We have also tested variants of the Perdew-Burke-Ernzerhof functional and the Becke98 hybrid functional. While moderate basis sets give converged density functional theory results for covalent dissociation energies, this is not true for noncovalent interaction energies. Our results show that density functionals give converged interaction energies with aug-cc-pVTZ and aug-cc-pVQZ basis sets. Gradual simplification of the basis set introduces an increasing overbinding effect. The best agreement with the high-level result was obtained by the Perdew-Burke-Ernzerhof functional at the basis set limit. The converged Tao-Perdew-Staroverov-Scuseria interaction energies show a systematic underbinding effect that can be compensated by a somewhat systematic overbinding basis set effect of smaller basis sets such as 6-31+G(d,2p). The inclusion of the diffuse functions in the oxygen basis set is very important, while the inclusion of the f functions practically does not influence the results. Improvement can be obtained by adding more hydrogen p functions to the 6-31+G basis set.     

Localized Orbitals vs. Pseudopotential-Plane Waves Basis Sets: Performances and Accuracy for Molecular Magnetic Systems

 Density functional theory, in combination with a) a careful choice of the exchange-correlation part of the total energy and b) localized basis sets for the electronic orbitals, has become the method of choice for calculating the exchange-couplings in magnetic molecular complexes. Orbital expansion on plane waves can be seen as an alternative basis set especially suited to allow optimization of newly synthesized materials of unknown geometries. However, little is known on the predictive power of this scheme to yield quantitative values for exchange coupling constants J as small as a few hundredths of eV (50–300 cm⁻¹). We have used density functional theory and a plane waves basis set to calculate the exchange couplings J of three homodinuclear Cu-based molecular complexes with experimental values ranging from +40 cm⁻¹ to −300 cm⁻¹. The plane waves basis set proves as accurate as the localized basis set, thereby suggesting that this approach can be reliably employed to predict and rationalize the magnetic properties of molecular-based materials. Corresponding author. E-mail: Carlo.Massobrio@ipcms.u-strasbg.fr Received August 5, 2002; accepted August 9, 2002     

High-level ab initio versus DFT calculations on (H2O2)2 and H2O2–H2O complexes as prototypes of multiple hydrogen bond systems

The performance of B-LYP, B-P86, B3-LYP, B3-P86, and B3-PW91 density functionals to describe multiple hydrogen bond systems was studied. For this purpose we have chosen the dimers of hydrogen peroxide and the hydrogen peroxide–water complexes. The geometries and vibrational frequencies obtained with a 6-311+G(d,p) basis set were compared with those obtained at the MP2 level using the same basis set expansion. The corresponding dimerization energies were obtained using a 6-311+G(3df,2p) basis set and compared with those obtained using the G2(MP2) theory. Red shiftings of the OH donor stretching frequencies were predicted by all approaches investigated; however, in all cases, the DFT values were sizably larger than the MP2 ones. Similarly, the blue shifting of the torsion of the hydrogen peroxide subunit was larger when evaluated at the DFT level. All functionals reproduced the G2(MP2) relative stabilities of the different local minima quite well. With the exception of the B-LYP and B3-PW91 approaches, all functionals yielded binding energies which deviated from the G2(MP2) values by less than 0.5 kcal/mol, provided that G2-type basis sets were used and that the corresponding BSSE corrections were included. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1124–1135     

Prediction of the reduction potential in transition‐metal containing complexes: How expensive? For what accuracy?

Accurate computationally derived reduction potentials are important for catalyst design. In this contribution, relatively inexpensive density functional theory methods are evaluated for computing reduction potentials of a wide variety of organic, inorganic, and organometallic complexes. Astonishingly, SCRF single points on B3LYP optimized geometries with a reasonably small basis set/ECP combination works quite well‐‐B3LYP with the BS1 [modified‐LANL2DZ basis set/ECP (effective core potential) for metals, LANL2DZ(d,p) basis set/LANL2DZ ECP for heavy nonmetals (Si, P, S, Cl, and Br), and 6‐31G(d') for other elements (H, C, N, O, and F)] and implicit PCM solvation models, SMD (solvation model based on density) or IEFPCM (integral equation formalism polarizable continuum model with Bondi atomic radii and α = 1.1 reaction field correction factor). The IEFPCM‐Bondi‐B3LYP/BS1 methodology was found to be one of the least expensive and most accurate protocols, among six different density functionals tested (BP86, PBEPBE, B3LYP, B3P86, PBE0, and M06) with thirteen different basis sets (Pople split‐valence basis sets, correlation consistent basis sets, or Los Alamos National Laboratory ECP/basis sets) and four solvation models (SMD, IEFPCM, IPCM, and CPCM). The MAD (mean absolute deviation) values of SCRF‐B3LYP/BS1 of 49 studied species were 0.263 V for SMD and 0.233 V for IEFPCM‐Bondi; and the linear correlations had respectable R ² values (R ² = 0.94 for SMD and R ² = 0.93 for IEFPCM‐Bondi). These methodologies demonstrate relatively reliable, convenient, and time‐saving functional/basis set/solvation model combinations in computing the reduction potentials of transition metal complexes with moderate accuracy. © 2017 Wiley Periodicals, Inc.