Hybrid Basis Set Research Articles

Accelerating computations of organometallic reaction energies through hybrid basis sets

A straightforward hybrid basis set approach to compute organometallic reaction energies is described to reduce computational times by up to 10-fold while maintaining the accuracy of DFT results.

Solution of Volume Integral Equation Using the SWG-Edge Hybrid Basis Functions for Inhomogeneous Dielectric Objects With Multiboundary

A hybrid discretization scheme for solution of volume integral equation (VIE) by method of moments (MoM) for electromagnetic scattering from dielectric objects is proposed in this article. The Schaubert-Wilton-Glisson and edge (SWG-Edge) hybrid basis functions are used in this discretization scheme. According to the divergence-free condition of electric displacement vector, a kind of edge basis functions defined in elements including boundary faces which separate a dielectric object from the background is derived. As a result, we get a SWG-Edge hybrid basis set. Details for the calculation of the corresponding matrix elements for the edge basis and testing functions are presented. Numerical results show the validity and accuracy of the hybrid discretization scheme. Finally, the proposed method is used for efficient solution of VIE for inhomogeneous dielectric objects with multiboundary. It is shown that for multiboundary problems, the number of unknowns of the hybrid basis is only about 71% of the traditional SWG basis. This means that the memory for solution of VIE by the traditional SWG basis functions can be reduced by half. Therefore, the SWG-edge hybrid basis is much more efficient than the traditional SWG basis for solution of VIE with multiboundary problems.

Investigating the exceptional adducts of alkoxides with Ru(II)-arene cations in alkyl alcohol solution using electrospray ionization mass spectrometry.

Exploring the formation mechanism of the exceptional adducts of alkoxides with Ru(II)-arene cations in alkyl alcohol solution using electrospray ionization mass spectrometry (ESI-MS) is crucial for further understanding the physicochemical properties of Ru(II)-arene complexes in solution. All mass spectra were collected with an AB SCIEX TripleTOF 5600+ mass spectrometer in positive mode. Theoretical calculations were carried out using the density functional theory method at the B3LYP level with a hybrid basis set consisting of 6-31G(d,p) and LanL2DZ in the Gaussian 03 program. When ruthenated [15 ]paracyclophanes (Ru-[15 ]PCPs) and Ru(II)-arene dimers were dissolved in alkyl alcohol solvents, the adducts of alkoxides with Ru(II)-arene cations were observed under positive ion mode ESI-MS, which resulted from the coordination of alkyl alcohol molecules with the Ru(II)-arene cations followed by the deprotonation of O-H bonds of the coordinated alcohols. Furthermore, the number of alkoxides binding to Ru-[15 ]PCPs was regulated by the number of ruthenium atoms. Attributed to good solubility and small steric hindrance, the signal intensity of the adducts of methoxides with Ru(II)-arene cations was the strongest among the three alkyl alcohols used in this study. The characteristic adducts of alkoxides with Ru(II)-arene cations were pervasively present in positive ion mode ESI-MS of nine Ru(II)-arene complexes dissolved in alkyl alcohol solvents. Taking into consideration the solubility and signal response, methanol is the most suitable solvent for the ESI-MS experiments with Ru(II)-arene complexes among the solvents studied, where almost only the diagnostic adducts of methoxides with Ru(II)-arene cations are present.

Электронно-возбужденные состояния в модельных комплексах кластеров благородных металлов с углеродными наноточками

Theoretical calculations of excited states in complexes of gold and silver three-atom nanoclusters with carbon quantum nanodots have been performed using the M062X functional and the def2SVP {H}/def2TZVP/def2TZVPP{Ag, Au} hybrid basis set. The subsequent calculation of the excited states was performed in the approximation of the time-dependent density functional theory implemented in Gaussian09. The chromophore centers of the points were modeled by heterocyclic molecules of isoquino-diazaanthracene and benzopyrano-naphthydrine. The clusters were attached to the points by means of ethyl mercaptan and a methoxyethane bridge of various lengths. Energy transfer channels are considered depending on the mutual arrangement of energy levels of clusters and heterocycles.

Experimental and Quantum Chemical Investigation on Piperazinium Hexachloro Stannous Trihydrate Single Crystal for Second Harmonic Generation Applications

A metal–organic single crystal of piperazinium hexachloro stannous trihydrate properties are explored through a combined form of experimental and density functional theory. The single crystal of piperazinium hexachloro stannous trihydrate was grown by a solvent evaporation method. All the theoretical calculations have been carried out using a hybrid basis set, B3LYP/6-31G(d,p)/LANL2DZ, with effective core potential. The structural and lattice parameters are confirmed from x-ray diffraction analysis. The functional group vibrational assignments have been theoretically calculated and are well correlated with experimental spectra. Optical studies revealed that the material has 60% transmission in the entire visible region, and a lower cut-off wavelength of about 271 nm where the possible electronic transition is n–σ*. The initial thermal decomposition is started at 60°C, and further decomposition stages have been investigated by thermal analysis. The intermolecular charge transfer, the HOMO–LUMO energy gap and chemical reactivity descriptors are calculated from the frontier molecular orbital analysis. The H···Cl/Cl···H (60%) intermolecular hydrogen bond interactions have the predominant role in determining molecular properties and crystal packing arrangements. The distinct atom-to-atom intermolecular interactions are interpreted through fingerprint plots. The second harmonic efficiency of a titled crystal is 1.8 times greater than that of typical KDP material. The dipole moment, polarizability and first-order hyperpolarizability are 23.89 D, − 2.29 × 10−23 esu and 3.26 × 10−30 esu, respectively, have been calculated from density functional theory. The obtained results show that a piperazinium hexachloro stannous trihydrate crystal could be accepted as a good candidate for nonlinear optical applications.

Single electron redox via an ion‐neutral complex in the fragmentation of protonated benzoylferrocenes

Ferrocene derivatives have become very popular molecules for biological applications. Although considerable experimental and theoretical calculation studies have demonstrated that ferrocene derivatives are easily oxidized during electrospray ionization (ESI), the details of the single electron redox reaction for protonated benzoylferrocenes in collision-induced dissociation (CID) mass spectrometry (MS) has not been obtained. Characterizing this mechanism is useful for further understanding the properties of ferrocene-containing biomaterials. All CID MS experiments were carried out using ESI ion trap mass spectrometry in positive ion mode. In addition, the accurate mass of fragments was measured on a ESI quadrupole time-of-flight (QTOF) mass spectrometer in positive ion mode. Theoretical calculations were carried out by the density functional theory (DFT) method with a hybrid basis set consisting of 6-31G (d) and ECP LanL2DZ in the Gaussian 03 program. In the fragmentation of protonated benzoylferrocenes, the characterized ferrocinium cation was observed, which was proposed from the cleavage of the bond between the ipso-carbon atom and the carbonyl carbon followed by a single electron redox in [substituted benzoyl/ferrocene] complexes. Moreover, when the complex contained an oxidant (substituted benzoyl cation) with higher electron affinity, the single electron redox reaction was more efficient. A correlation was established between the intensities of the two competitive product ions and the electron affinities of substituted benzoyl cations. The single electron redox reaction by the [substituted benzoyl/ferrocene] complexes was proposed by CID MS and theoretical calculations, which provided potential evidence to further understand the reversible reduction characteristics of ferrocene-containing biomaterials.

Performance of numerical atom‐centered basis sets in the ground‐state correlated calculations of noncovalent interactions: Water and methane dimer cases

Numerical atom‐centered basis sets (orbitals) (NAO) are known for their compactness and rapid convergence in the Hartree–Fock and density‐functional theory (DFT) molecular electronic‐structure calculations. To date, not much is known about the performance of the numerical sets against the well‐studied Gaussian‐type bases in correlated calculations. In this study, one instance of NAO [Blum et al., The Fritz Haber Institute ab initio Molecular Simulations Package (FHI‐aims), 2009] was thoroughly examined in comparison to the correlation‐consistent basis sets in the ground‐state correlated calculations on the hydrogen‐bonded water and dispersion‐dominated methane dimers. It was shown that these NAO demonstrate improved, comparing to the unaugmented correlation‐consistent based, convergence of interaction energies in correlated calculations. However, the present version of NAO constructed in the DFT calculations on covalently‐bound diatomics exhibits enormous basis‐set superposition error (BSSE)—even with the largest bases. Moreover, these basis sets are essentially unable to capture diffuse character of the wave function, necessary for example, for the complete convergence of correlated interaction energies of the weakly‐bound complexes. The problem is usually treated by addition of the external Gaussian diffuse functions to the NAO part, what indeed allows to obtain accurate results. However, the operation increases BSSE with the resulting hybrid basis sets even further and breaks down the initial concept of NAO (i.e., improved compactness) due to the significant increase in their size. These findings clearly point at the need in the alternative strategies for the construction of sufficiently‐delocalized and BSSE‐balanced purely‐numerical bases adapted for correlated calculations, possible ones were outlined here. For comparison with the considered NAOs, a complementary study on the convergence properties of the correlation‐consistent basis sets, with a special emphasis on BSSE, was also performed. Some of its conclusions may represent independent interest. © 2013 Wiley Periodicals, Inc.

The structural and bonding evolution in cysteine–gold cluster complexes

The bonding characteristics in cysteine-gold cluster complexes represented by thiolate (Au(n)·Cys(S) (n = 1, 3, 5, 7)) and thiol (Au(n)·Cys(SH) (n = 2, 4, 6, 8)) is investigated by density functional theory with 6-31G(d,p) and Lanl2DZ hybrid basis sets. The complexes exhibit very different bonding characteristic between these two forms. In the Au(n)·Cys(S) complexes, the charge transfers from gold clusters to sulfur atoms. The number of S-Au bonds in the Au(n)·Cys(S) complexes evolves from one to two when n is greater than three. For n equals three, i.e. Au(3)·Cys(S), its ground state only has one S-Au bond. While the only S-Au bond in Au(1)·Cys(S) is mainly covalent, the nature of the S-Au bond in other thiolates is featured with the combination of covalent and donor-acceptor interactions. In particular, one stable isomer of Au(3)·Cys(S) with two S-Au bonds, which is 2 kcal mol(-1) higher in energy than the corresponding ground state, consists of one covalent and one donor-acceptor S-Au bond explicitly. Moreover, the localized three center two electron bonds are formed within the Au clusters, which facilitates the formation of the two S-Au bonds in Au(5)·Cys(S) and Au(7)·Cys(S) complexes. In the Au(n)·Cys(SH) complexes, the donor-acceptor interaction prevails in the Au-SH bond by transferring lone pair electrons from the sulfur atom to the adjacent gold atom. Interestingly, the orbital with much more 6s-component in Au(4)·Cys(SH) enhances the donor-acceptor bonding character, thus yields the strongest bonding among all the Au(n)·Cys(SH) complexes studied in this paper. In general, the bonding strength between gold clusters and cysteine is positively correlated with the S-Au overlap-weighted bond order, but negatively correlated with the S-Au bond length. Lastly, the covalent and donor-acceptor S-Au bond strength is computed to be 48 and 18 kcal mol(-1), respectively.

Accurate and efficient calculation of excited vibrational states from quartic potential energy surfaces

Vibrational anharmonicity and resonances frequently complicate assignment of vibrational spectra. In order to analyse such spectra, these effects can be calculated from ab initio quartic potential energy surfaces (PESs) using second-order vibrational theory with resonances (VPT2 + K). This study compares the accuracies of using the cc-pVTZ basis set, the aug-cc-pVQZ basis set, and a hybrid approach that uses the cc-pVTZ basis set for the equilibrium geometry and quadratic force constants and the aug-cc-pVQZ basis set for the cubic and quartic force constants. Quartic PESs are computed using these basis sets for H2O, H2CO, HFCO, SCCl2, and their deuterated analogs with the CCSD(T) method. The computed PESs are assessed by comparing experimentally determined and theoretically calculated spectroscopic constants. The average absolute difference (⟨|error|⟩) between theoretical and experimental zero-point energy-corrected harmonic frequencies () decreases by 54.7% when the hybrid approach is used instead of the cc-pVTZ basis, but decreases by only an additional 2.3% when the aug-cc-pVQZ basis is used. The computed PESs are also assessed by comparing predicted and observed vibrational energy states. The weighted average root-mean-square (RMS) difference between predicted and observed vibrational energy levels decreases by 42.3% when the hybrid approach is used instead of cc-pVTZ, but decreases by only an additional 4.0% when aug-cc-pVQZ is used. These results demonstrate that calculations performed using the hybrid basis set approach, which have a substantially lower computational cost, are comparable in accuracy to those performed using the aug-cc-pVQZ basis set.

Multiple hydrogen-bonding interactions between macrocyclic triurea and F−, Cl−, Br−, I− and NO3−: a theoretical investigation

The binding behaviors of the 27-membered macrocyclic triurea 1 towards the five anions, F(-), Cl(-), Br(-), I(-) and NO(3)(-), through multiple hydrogen-bonding interactions, were investigated at the B3LYP/6-311++G(d,p)//B3LYP/6-31(1)++G(d,p) (6-31(1)++G(d,p) is a hybrid basis set; for more details see computational methods) level. Three binding modes (I, II, and III) were found for all the five anions in the gas phase, and seven structural parameters have been used to describe these binding modes. Binding mode I and II have similar binding geometries and their coordination number of anions is six. Binding mode III exhibits completely different binding characteristics and the coordination number is three except for NO(3)(-). Our calculation revealed that the binding strength of binding modes follows the trend, mode II > I > III, with the exception of F(-) complex. The binding affinity of anions in the gas phase goes in this order: F(-) > Cl(-) > NO(3)(-) > Br(-) > I(-). The changes in the binding affinity for all 15 urea-anion complexes under the influence of solvent environment were examined using the IEF-PCM continuum solvation model. Although the binding affinities are weakened substantially because of solvent effect, these drastic changes do not affect the affinity order in the gas phase. The experimentally observed affinity strength in chloroform, Cl(-) > NO(3)(-) > Br(-), was confirmed by this work. Moreover, we found a high correlation between ΔE(bind) (1) and ΔE(In) (1,3-dimethylurea) for all three binding modes, implying that the affinity strength of 1 to these five anions is determined mainly by the proton-accepting ability of anions, not by steric effect.

DFT model investigations on copper(II) chloride complexes: Halogen bonding of pyridyl bromine of CuCl 2(NH 3)(NC 5H 4Br-3) with several metal ligands

A set of theoretical model systems, investigated by the DFT method with hybrid basis sets, was utilized to mimic and exploit the halogen bond found in the neutral metal–organic coordination, CuCl 2(NC 5H 4Br-3) 2. The optimized calculations indicate that, while coordinating to metal ions, the end-bromine of organic halide subunit exhibits better directionality and affinity to electron density donors. Halogen bonding energies are substantial and range from −2.5 to −37.5 kcal/mol. These results reveal the importance of metal-influenced halogen bonding in directing supramolecular arrangements. To further study the nature of the halogen bond, the analyses of NBO and AIM were carried out. The conclusions show a considerable extent of charge transfer, complicated orbital interactions, and distinct bond critical points between interacting atoms upon halogen bonding.

Assessment of the CCSD and CCSD(T) Coupled-Cluster Methods in Calculating Heats of Formation for Zn Complexes

Heats of formation were calculated using coupled-cluster methods for a series of zinc complexes. The calculated values were evaluated against previously conducted computational studies using density functional methods as well as experimental values. Heats of formation for nine neutral ZnX(n) complexes [X = -Zn, -H, -O, -F2, -S, -Cl, -Cl2, -CH3, (-CH3)2] were determined at the CCSD and CCSD(T) levels using the 6-31G** and TZVP basis sets as well as the LANL2DZ-6-31G** (LACVP**) and LANL2DZ-TZVP hybrid basis sets. The CCSD(T)/6-31G** level of theory was found to predict the heat of formation for the nonalkyl Zn complexes most accurately. The alkyl Zn species were problematic in that none of the methods that were tested accurately predicted the heat of formation for these complexes. In instances where experimental geometric parameters were available, these were most accurately predicted by the CCSD/6-31G** level of theory; going to CCSD(T) did not improve agreement with the experimental values. Coupled-cluster methods did not offer a systemic improvement over DFT calculations for a given functional/basis set combination. With the exceptions of ZnH and ZnF2, there are multiple density functionals that outperform coupled-cluster calculations with the 6-31G** basis set.

A TDMA Hybrid SQUID Multiplexer

We have developed a multiplexed read-out for transition-edge sensors (TES) based on a hybrid time- and frequency-domain basis set, similar to that used in time-division multiple-access (TDMA) mobile phones. The hybrid basis set uses bandwidth more efficiently than microwave frequency-division SQUID multiplexing, making it possible to multiplex more detectors in each output line. The high open-loop bandwidth provided by our SQUID TDMA system also makes it possible to multiplex large arrays of fast, high dynamic range detectors such as fast x-ray calorimeters. In this approach, we embed the second-stage SQUID amplifier of our standard time-division multiplexer in an impedance matching circuit coupled to a broadband cryogenic high electron mobility transistor (HEMT) in a microwave reflectometer configuration. The input signals are flux coupled into the first-stage SQUID amplifiers whose signals are time-division multiplexed into the second-stage SQUID. At room temperature, the signal from the HEMT is mixed down to dc for analysis and further signal processing.

Modeling Metal Cation−Phosphate Interactions in Nucleic Acids in the Gas Phase via Alkali Metal Cation−Triethyl Phosphate Complexes

Threshold collision-induced dissociation techniques are employed to determine the bond dissociation energies (BDEs) of complexes of alkali metal cations, Na+, K+, Rb+, and Cs+, to triethyl phosphate (TEP). The primary and lowest energy dissociation pathway in all cases is the endothermic loss of the neutral TEP ligand. Theoretical electronic structure calculations at the B3LYP/6-311+G(2d,2p)//B3LYP/6-31G* level of theory are used to determine the structures, molecular parameters, and theoretical estimates for the BDEs of these complexes. For the complexes to Rb+ and Cs+, theoretical calculations were performed using hybrid basis sets in which the effective core potentials and valence basis sets of Hay and Wadt were used to describe the alkali metal cation, while the standard basis sets were used for all other atoms. The agreement between theory and experiment is excellent for the complexes to Na+ and K+ and is somewhat less satisfactory for the complexes to the heavier alkali metal cations, Rb+ and Cs+, where effective core potentials were used to describe the cation. The trends in the binding energies are examined. The binding of alkali metal cations to triethyl phosphate is compared with that to trimethylphosphate.

Structures and bonding of the complexes [M(η5-E5)] and [M(η5-E5)2] (M = Sc, Y, E = N, P): a DFT study

The mono- and bisligands complexes formed by rare earth cation (Sc, Y) with pentazolato anion or cyclo-P5− anion, the all-nitrogen counterparts of metallocenes or the all-phosphorus counterparts of metallocenes, have been studied at hybrid basis sets level with the DFT method. The geometric parameters, binding energy and the charge distributions of these complexes were characterized. And their stable orders were obtained. Monoligand complexes incline to become bisligands complexes due to their destabilization. The charge transferring between ligand and metallic cation has affinity with the stability of complex. The possibility of forming stable [M(η5-E5)2] (M = Sc, Y, E = N, P) complexes is predicted and they are viable synthetic targets theoretically, especially Sc(η5-N5)2.

Estimation procedures and error analysis for inferring the total plutonium (Pu) produced by a graphite-moderated reactor

Graphite isotope ratio method (GIRM) is a technique that uses measurements and computer models to estimate total plutonium (Pu) production in a graphite-moderated reactor. First, isotopic ratios of trace elements in extracted graphite samples from the target reactor are measured. Then, computer models of the reactor relate those ratios to Pu production. Because Pu is controlled under non-proliferation agreements, an estimate of total Pu production is often required, and a declaration of total Pu might need to be verified through GIRM. In some cases, reactor information (such as core dimensions, coolant details, and operating history) are so well documented that computer models can predict total Pu production without the need for measurements. However, in most cases, reactor information is imperfectly known, so a measurement and model-based method such as GIRM is essential. Here, we focus on GIRM's estimation procedure and its associated uncertainty. We illustrate a simulation strategy for a specific reactor that estimates GIRM's uncertainty and determines which inputs contribute most to GIRM's uncertainty, including inputs to the computer models. These models include a “local” code that relates isotopic ratios to the local Pu production, and a “global” code that predicts the Pu production shape over the entire reactor. This predicted shape is included with other 3D basis functions to provide a “hybrid basis set” that is used to fit the local Pu production estimates. The fitted shape can then be integrated over the entire reactor to estimate total Pu production. This GIRM evaluation provides a good example of several techniques of uncertainty analysis and introduces new reasons to fit a function using basis functions in the evaluation of the impact of uncertainty in the true 3D shape.

Simple and accurate method to evaluate tunneling splitting in polyatomic molecules.

A practical and accurate semiclassical method for calculating the tunneling splitting of the ground state in polyatomic molecules is presented based on a recent version of the instanton theory [J. Chem. Phys. 115, 6881 (2001)]. The method uses ab initio quantum chemical data for the potential energy surface without any concomitant extrapolation and requires only a small number of ab initio data points to get convergence even for large molecules. This enables one to use an advanced level of electronic structure theory and achieve a high accuracy of the result. The method is applied to the 9-atomic malonaldehyde molecule by making use of the potential energy surface at the level of CCSD(T) with the hybrid basis set of aug-cc-pVTZ (for oxygen atoms and the transferred hydrogen atom) and cc-pVTZ (for other atoms).

Quantum chemical calculation of vibrational spectra of large molecules--Raman and IR spectra for Buckminsterfullerene.

In this work we demonstrate how different modern quantum chemical methods can be efficiently combined and applied for the calculation of the vibrational modes and spectra of large molecules. We are aiming at harmonic force fields, and infrared as well as Raman intensities within the double harmonic approximation, because consideration of higher order terms is only feasible for small molecules. In particular, density functional methods have evolved to a powerful quantum chemical tool for the determination of the electronic structure of molecules in the last decade. Underlying theoretical concepts for the calculation of intensities are reviewed, emphasizing necessary approximations and formal aspects of the introduced quantities, which are often not explicated in detail in elementary treatments of this topic. It is shown how complex quantum chemistry program packages can be interfaced to new programs in order to calculate IR and Raman spectra. The advantages of numerical differentiation of analytical gradients, dipole moments, and static, as well as dynamic polarizabilities, are pointed out. We carefully investigate the influence of the basis set size on polarizabilities and their spatial derivatives. This leads us to the construction of a hybrid basis set, which is equally well suited for the calculation of vibrational frequencies and Raman intensities. The efficiency is demonstrated for the highly symmetric C(60), for which we present the first all-electron density functional calculation of its Raman spectrum.

First-principles calculation of coincidence Doppler broadening of positron annihilation radiation

We report a first-principles method for calculating the momentum density of positron-electron pairs in materials, which can be accurately measured, in a wide momentum range, by means of coincidence Doppler broadening (CDB) of positron annihilation radiation. The calculation is based on the two-component density-functional theory within the local-density approximation. The electron and positron wave functions are calculated by means of the full-potential linearized-augmented-plane-wave method with use of semicore orbitals and of the pure plane-wave method, respectively. This hybrid basis set accurately determines the wave functions of core and valence electrons and is free from any shape or symmetry assumption for the positron wave function. The method is applied to two typical systems. i.e., Al and graphite having isotropic and anisotropic positron densities, respectively. The calculations agree well with experiments over the entire measurable momentum region. Especially, the calculations well reproduce the anisotropic high-momentum CDB tails of graphite, which originate from the quasi-two-dimensionally distributed positron. This reproduction suggests that the present method is applicable for a variety of materials.

Comprehensiveab initio quantum mechanical and molecular orbital (MO) analysis of cisplatin: Structure, bonding, charge density, and vibrational frequencies

We carried out an extensive series of ab initio quantum mechanical (QM) calculations using pure effective core potential (ECP) and hybrid Hartree–Fock (HF)/ECP basis sets, including various electron correlation treatments up to the MP4 level on cis-diamminedichloroplatinum(II), cisplatin, an important anticancer drug. The optimized geometric parameters and vibrational frequencies of cisplatin were compared with the experimental values, and the effects of varying basis sets and correlation (MP level) treatments on the geometric parameters and vibrational frequencies were analyzed. We also present a detailed description of the bonding in cisplatin using qualitative MO analysis to characterize the key intramolecular interactions in cisplatin. The calculated molecular electrostatic potential (MEP), and the electrostatic potential (ESP) charges for cisplatin were used to identify regions of electrophilic and nucleophilic character. The charge density and the Laplacian of charge density of cisplatin were calculated to determine its bonding relationships. Using basis set performance, we identified two hybrid basis sets (HF/6-311G* and MP2/6-311G*) as basis sets of choice for studying cisplatin's molecular properties. Furthermore, we recommend a hybrid ECP/HF approach with electron correlation for the most accurate physicochemical and electronic description of cisplatin and related compounds. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 365–382, 1999