We present a perspective on the use of diffuse basis functions for electronic structure calculations by density functional theory and wave function theory. We especially emphasize minimally augmented basis sets and calendar basis sets. We base our conclusions on our previous experience with commonly computed quantities, such as bond energies, barrier heights, electron affinities, noncovalent (van der Waals and hydrogen bond) interaction energies, and ionization potentials, on Stephens et al.'s results for optical rotation and on our own new calculations (presented here) of polarizabilities and of potential energy curves of van der Waals complexes. We emphasize the benefits of partial augmentation of the higher-zeta basis sets in preference to full augmentation at a lower ζ level. Benefits and limitations of the use of fully, partially, and minimally augmented basis sets are reviewed for different electronic structure methods and molecular properties. We have found that minimal augmentation is almost always enough for density functional theory (DFT) when applied to ionization potentials, electron affinities, atomization energies, barrier heights, and hydrogen-bond energies. For electric dipole polarizabilities, we find that augmentation beyond minimal has an average effect of 8% at the polarized triple-ζ level and 5% at the polarized quadruple-ζ level. The effects are larger for potential energy curves of van der Waals complexes. The effects are also larger for wave function theory (WFT). Even for WFT though, full augmentation is not needed for most purposes, and a level of augmentation between minimal and full is optimal for most problems. The calendar basis sets named after the months provide a convergent sequence of partially augmented basis sets that can be used for such calculations. The jun-cc-pV(T+d)Z basis set is very useful for MP2-F12 calculations of barrier heights and hydrogen bond strengths.

Water hexamers provide a critical testing ground for validating potential energy surface predictions because they contain structural motifs not present in smaller clusters. We tested the ability of 11 density functionals (four of which are local and seven of which are nonlocal) to accurately predict the relative energies of a series of low-lying water hexamers, relative to the CCSD(T)/aug′-cc-pVTZ level of theory, where CCSD(T) denotes coupled cluster theory with an interative treatment of single and double excitations and a quasiperturbative treatment of connected triple excitations. Five of the density functionals were tested with two different basis sets, making a total of 16 levels of density functional theory (DFT) tested. When single-point energy calculations are carried out on geometries obtained with second-order Møller-Plesset perturbation theory (MP2), only three density functionals, M06-L, M05-2X, and M06-2X, are able to correctly predict the relative energy ordering of the hexamers. These three functionals predict that the range of energies spanned by the six isomers is 3.2-5.6 kcal/mol, whereas the other eight functionals predict ranges of 1.0-2.4 kcal/ mol; the benchmark value for this range is 3.1 kcal/mol. When the hexamers are optimized at each level of theory, all methods are able to reproduce the MP2 geometries well for all isomers except the boat and bag isomers, and DFT optimization changes the energy ordering for seven of the 16 methods tested. The addition of zero-point energy changes the energy ordering for all of the density functionals studied except for M05-2X and M06-2X. The variation in relative energies predicted by the different methods highlights the necessity for exercising caution in the choice of density functionals used in future studies. Of the 11 density functionals tested, the most accurate results for energies were obtained with the PWB6K, MPWB1K, and M05-2X functionals.

We combine the diffuse basis functions from the 6-31+G basis set of Pople and co-workers with the correlation-consistent basis sets of Dunning and co-workers. In both wave function and density functional calculations, the resulting basis sets reduce the basis set superposition error almost as much as the augmented correlation-consistent basis sets, although they are much smaller. In addition, in density functional calculations the new basis sets, called cc-pVxZ+ where x = D, T, Q, ..., or x = D+d, T+d, Q+d, ..., give very similar energetic predictions to the much larger aug-cc-pVxZ basis sets. However, energetics calculated from correlated wave function calculations are more slowly convergent with respect to the addition of diffuse functions. We also examined basis sets with the same number and type of functions as the cc-pVxZ+ sets but using the diffuse exponents of the aug-cc-pVxZ basis sets and found very similar performance to cc-pVxZ+; these basis sets are called minimally augmented cc-pVxZ, which we abbreviate as maug-cc-pVxZ.

The formation of atmospheric aerosol particles through clustering of condensable vapors is an important contributor to the overall concentration of these atmospheric particles. However, the details of the nucleation process are not yet well understood and are difficult to probe by experimental means. Computational chemistry is a powerful tool for gaining insights about the nucleation mechanism. Here, we report accurate electronic structure calculations of the potential energies of small clusters made from sulfuric acid, ammonia, and dimethylamine. We also assess and validate the accuracy of less expensive methods that might be used for the calculation of the binding energies of larger clusters for atmospheric modeling. The PW6B95-D3 density-functional-plus-molecular-mechanics calculation with the MG3S basis set stands out as yielding excellent accuracy while still being affordable for very large clusters.

We have applied a many-body (MB) expansion, the electrostatically embedded many-body (EE-MB) approximation, and the electrostatically embedded many-body expansion of the correlation energy (EE-MB-CE), each at the two-body (MB = PA, where PA denotes pairwise additive) and three-body (MB = 3B) levels, to calculate total energies for a series of low-lying water hexamers using eight correlated levels of theory including second-order and fourth-order Møller-Plesset perturbation theory (MP2 and MP4) and coupled cluster theory with single, double, and quasipertubative triple excitations (CCSD(T)). Comparison of the expansion methods to energies obtained from full (i.e., unexpanded) calculations shows that the EE-3B-CE method is able to reproduce the full cluster energies to within 0.03 kcal/mol, on average. We have also found that the deviations of the results predicted by the expansion methods from those obtained with full calculations are nearly independent of the correlated level of theory used; this observation will allow validation of the many-body methods on large clusters at less expensive levels of theory (such as MP2) to be extrapolated to the CCSD(T) level of theory. Furthermore, we have been able to rationalize the accuracies of the MB, EE-MB, and EE-MB-CE methods for the six hexamers in terms of the specific many-body effects present in each cluster.

Addendum. We also present here some further calculations that do not correct an error in the original article but that provide further relevant information. In particular, we note that the article tested the new plus basis sets for ionization potentials, electron affinities, atomization energies, barrier heights, and basis set superposition errors. We then presented tests of another set of basis sets, called maug basis sets, obtained by truncating the aug basis sets to the same size as the plus basis sets. The maug basis sets were tested only for barrier heights and basis set superposition errors, and we found very similar performance to the plus basis sets. As an example of the differences in the basis sets, diffuse functions on O in maug-cc-pVTZ have exponential parameters of 0.07376 for s functions and 0.05974 for p; these exponential parameters are smaller than those in the plus basis set, where both parameters are 0.0845. The most difficult tests of the adequacy of a scheme for diffuse basis functions are provided by electron affinities. We have now tested maug-cc-pVxZ against cc-pVxZ+ with both x ) D and x ) T for electron affinities, and we found better performance with the maug basis sets for M06-2X (better on average) and CCSD(T) (always better), especially for systems containing oxygen atoms (and to a lesser extent for Si -and C -), but better performance (on average) with the plus basis set for B3LYP. However, in all 78 cases the anion energies are lower for the maug basis set than the corresponding plus one, so the improvement of the plus basis sets for B3LYP electron affinities is presumably due to cancellation of basis set error with a large error in the opposite direction from the functional itself. Table A1 gives two additional rows for the original Table 4 that show the mean unsigned errors in electron affinities with two maug basis sets. The conclusion is that anion energies and electron affinities are more sensitive than barrier heights and basis set superposition errors to the precise values of the diffuse exponents, and the maug basis sets are more accurate for such calculations, probably because the exponents were optimized for atomic anions. 1

The binding energies and relative conformational energies of five configurations of the water 16-mer are computed using 61 levels of density functional (DF) theory, 12 methods combining DF theory with molecular mechanics damped dispersion (DF-MM), seven semiempirical-wave function (SWF) methods, and five methods combining SWF theory with molecular mechanics damped dispersion (SWF-MM). The accuracies of the computed energies are assessed by comparing them to recent high-level ab initio results; this assessment is more relevant to bulk water than previous tests on small clusters because a 16-mer is large enough to have water molecules that participate in more than three hydrogen bonds. We find that for water 16-mer binding energies the best DF, DF-MM, SWF, and SWF-MM methods (and their mean unsigned errors in kcal/mol) are respectively M06-2X (1.6), ωB97X-D (2.3), SCC-DFTB-γ(h) (35.2), and PM3-D (3.2). We also mention the good performance of CAM-B3LYP (1.8), M05-2X (1.9), and TPSSLYP (3.0). In contrast, for relative energies of various water nanoparticle 16-mer structures, the best methods (and mean unsigned errors in kcal/mol), in the same order of classes of methods, are SOGGA11-X (0.3), ωB97X-D (0.2), PM6 (0.4), and PMOv1 (0.6). We also mention the good performance of LC-ωPBE-D3 (0.3) and ωB97X (0.4). When both relative and binding energies are taken into consideration, the best methods overall (out of the 85 tested) are M05-2X without molecular mechanics and ωB97X-D when molecular mechanics corrections are included; with considerably higher average errors and considerably lower cost, the best SWF or SWF-MM method is PMOv1. We use six of the best methods for binding energies of the water 16-mers to calculate the binding energies of water hexamers and water 17-mers to test whether these methods are also reliable for binding energy calculations on other types of water clusters.

This work tests the capability of the electrostatically embedded many-body (EE-MB) method to calculate accurate (relative to conventional calculations carried out at the same level of electronic structure theory and with the same basis set) binding energies of mixed clusters (as large as 9-mers) consisting of water, ammonia, sulfuric acid, and ammonium and bisulfate ions. This work also investigates the dependence of the accuracy of the EE-MB approximation on the type and origin of the charges used for electrostatically embedding these clusters. The conclusions reached are that for all of the clusters and sets of embedding charges studied in this work, the electrostatically embedded three-body (EE-3B) approximation is capable of consistently yielding relative errors of less than 1% and an average relative absolute error of only 0.3%, and that the performance of the EE-MB approximation does not depend strongly on the specific set of embedding charges used. The electrostatically embedded pairwise approximation has errors about an order of magnitude larger than EE-3B. This study also explores the question of why the accuracy of the EE-MB approximation shows such little dependence on the types of embedding charges employed.