Hannah R. Leverentz

Perspectives on Basis Sets Beautiful: Seasonal Plantings of Diffuse Basis Functions

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.

Efficient diffuse basis sets: Cc-pVxZ+ and maug-cc-pVxZ

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.

Energetics of Atmospherically Implicated Clusters Made of Sulfuric Acid, Ammonia, and Dimethyl Amine

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.

Evaluation of the Electrostatically Embedded Many-Body Expansion and the Electrostatically Embedded Many-Body Expansion of the Correlation Energy by Application to Low-Lying Water Hexamers

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.

Electrostatically Embedded Many-Body Approximation for Systems of Water, Ammonia, and Sulfuric Acid and the Dependence of Its Performance on Embedding Charges

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.

Water 26-mers Drawn from Bulk Simulations: Benchmark Binding Energies for Unprecedentedly Large Water Clusters and Assessment of the Electrostatically Embedded Three-Body and Pairwise Additive Approximations

It is important to test methods for simulating water, but small water clusters for which benchmarks are available are not very representative of the bulk. Here we present benchmark calculations, in particular CCSD(T) calculations at the complete basis set limit, for water 26-mers drawn from Monte Carlo simulations of bulk water. These clusters are large enough that each water molecule participates in 2.5 hydrogen bonds on average. The electrostatically embedded three-body approximation with CCSD(T) embedded dimers and trimers reproduces the relative binding energies of eight clusters with a mean unsigned error (MUE, kcal per mole of water molecules) of only 0.009 and 0.015 kcal for relative and absolute binding energies, respectively. Using only embedded dimers (electrostatically embedded pairwise approximation) raises these MUEs to 0.038 and 0.070 kcal, and computing the energies with the M11 exchange-correlation functional, which is very economical, yields errors of only 0.029 and 0.042 kcal.

Assessing the Accuracy of Density Functional and Semiempirical Wave Function Methods for Water Nanoparticles: Comparing Binding and Relative Energies of (H2O)16 and (H2O)17 to CCSD(T) Results.

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.

Water 16-mers and hexamers: assessment of the three-body and electrostatically embedded many-body approximations of the correlation energy or the nonlocal energy as ways to include cooperative effects.

This work presents a new fragment method, the electrostatically embedded many-body expansion of the nonlocal energy (EE-MB-NE), and shows that it, along with the previously proposed electrostatically embedded many-body expansion of the correlation energy (EE-MB-CE), produces accurate results for large systems at the level of CCSD(T) coupled cluster theory. We primarily study water 16-mers, but we also test the EE-MB-CE method on water hexamers. We analyze the distributions of two-body and three-body terms to show why the many-body expansion of the electrostatically embedded correlation energy converges faster than the many-body expansion of the entire electrostatically embedded interaction potential. The average magnitude of the dimer contributions to the pairwise additive (PA) term of the correlation energy (which neglects cooperative effects) is only one-half of that of the average dimer contribution to the PA term of the expansion of the total energy; this explains why the mean unsigned error (MUE) of the EE-PA-CE approximation is only one-half of that of the EE-PA approximation. Similarly, the average magnitude of the trimer contributions to the three-body (3B) term of the EE-3B-CE approximation is only one-fourth of that of the EE-3B approximation, and the MUE of the EE-3B-CE approximation is one-fourth that of the EE-3B approximation. Finally, we test the efficacy of two- and three-body density functional corrections. One such density functional correction method, the new EE-PA-NE method, with the OLYP or the OHLYP density functional (where the OHLYP functional is the OptX exchange functional combined with the LYP correlation functional multiplied by 0.5), has the best performance-to-price ratio of any method whose computational cost scales as the third power of the number of monomers and is competitive in accuracy in the tests presented here with even the electrostatically embedded three-body approximation.

Assessment of New Meta and Hybrid Meta Density Functionals for Predicting the Geometry and Binding Energy of a Challenging System : The Dimer of H2S and Benzene

Noncovalent interactions of a hydrogen bond donor with an aromatic pi system present a challenge for density functional theory, and most density functionals do not perform well for this kind of interaction. Here we test seven recent density functionals from our research group, along with the popular B3LYP functional, for the dimer of H 2S with benzene. The functionals considered include the four new meta and hybrid meta density functionals of the M06 suite, three slightly older hybrid meta functionals, and the B3LYP hybrid functional, and they were tested for their abilities to predict the dissociation energies of three conformations of the H 2S-benzene dimer and to reproduce the key geometric parameters of the equilibrium conformation of this dimer. All of the functionals tested except B3LYP correctly predict which of the three conformations of the dimer is the most stable. The functionals that are best able to reproduce the geometry of the equilibrium conformation of the dimer with a polarized triple-zeta basis set are M06-L, PWB6K, and MPWB1K, each having a mean unsigned relative error across the two experimentally verifiable geometric parameters of only 8%. The success of M06-L is very encouraging because it is a local functional, which reduces the cost for large simulations. The M05-2X functional yields the most accurate binding energy of a conformation of the dimer for which a binding energy calculated at the CCSD(T) level of theory is available; M05-2X gives a binding energy for the system with a difference of merely 0.02 kcal/mol from that obtained by the CCSD(T) calculation. The M06 functional performs well in both categories by yielding a good representation of the geometry of the equilibrium structure and by giving a binding energy that is only 0.19 kcal/mol different from that calculated by CCSD(T). We conclude that the new generation of density functionals should be useful for a variety of problems in biochemistry and materials where aromatic functional groups can serve as hydrogen bond acceptors.

Quantum Mechanical Fragment Methods Based on Partitioning Atoms or Partitioning Coordinates

Conspectus The development of more efficient and more accurate ways to represent reactive potential energy surfaces is a requirement for extending the simulation of large systems to more complex systems, longer-time dynamical processes, and more complete statistical mechanical sampling. One way to treat large systems is by direct dynamics fragment methods. Another way is by fitting system-specific analytic potential energy functions with methods adapted to large systems. Here we consider both approaches. First we consider three fragment methods that allow a given monomer to appear in more than one fragment. The first two approaches are the electrostatically embedded many-body (EE-MB) expansion and the electrostatically embedded many-body expansion of the correlation energy (EE-MB-CE), which we have shown to yield quite accurate results even when one restricts the calculations to include only electrostatically embedded dimers. The third fragment method is the electrostatically embedded molecular tailoring approach (EE-MTA), which is more flexible than EE-MB and EE-MB-CE. We show that electrostatic embedding greatly improves the accuracy of these approaches compared with the original unembedded approaches. Quantum mechanical fragment methods share with combined quantum mechanical/molecular mechanical (QM/MM) methods the need to treat a quantum mechanical fragment in the presence of the rest of the system, which is especially challenging for those parts of the rest of the system that are close to the boundary of the quantum mechanical fragment. This is a delicate matter even for fragments that are not covalently bonded to the rest of the system, but it becomes even more difficult when the boundary of the quantum mechanical fragment cuts a bond. We have developed a suite of methods for more realistically treating interactions across such boundaries. These methods include redistributing and balancing the external partial atomic charges and the use of tuned fluorine atoms for capping dangling bonds, and we have shown that they can greatly improve the accuracy. Finally we present a new approach that goes beyond QM/MM by combining the convenience of molecular mechanics with the accuracy of fitting a potential function to electronic structure calculations on a specific system. To make the latter practical for systems with a large number of degrees of freedom, we developed a method to interpolate between local internal-coordinate fits to the potential energy. A key issue for the application to large systems is that rather than assigning the atoms or monomers to fragments, we assign the internal coordinates to reaction, secondary, and tertiary sets. Thus, we make a partition in coordinate space rather than atom space. Fits to the local dependence of the potential energy on tertiary coordinates are arrayed along a preselected reaction coordinate at a sequence of geometries called anchor points; the potential energy function is called an anchor points reactive potential. Electrostatically embedded fragment methods and the anchor points reactive potential, because they are based on treating an entire system by quantum mechanical electronic structure methods but are affordable for large and complex systems, have the potential to open new areas for accurate simulations where combined QM/MM methods are inadequate.