VAMPyR-A high-level Python library for mathematical operations in a multiwavelet representation.

Wavelets and multiwavelets have lately been adopted in quantum chemistry to overcome challenges presented by the two main families of basis sets: Gaussian atomic orbitals and plane waves. In addition to their numerical advantages (high precision, locality, fast algorithms for operator application, linear scaling with respect to system size, to mention a few), they provide a framework that narrows the gap between the theoretical formalism of the fundamental equations and the practical implementation in a working code. This realization led us to the development of the Python library called VAMPyR (Very Accurate Multiresolution Python Routines). VAMPyR encodes the binding to a C++ library for multiwavelet calculations (algebra and integral and differential operator application) and exposes the required functionality to write a simple Python code to solve, among others, the Hartree-Fock equations, the generalized Poisson equation, the Dirac equation, and the time-dependent Schrödinger equation up to any predefined precision. In this study, we will outline the main features of multiresolution analysis using multiwavelets and we will describe the design of the code. A few illustrative examples will show the code capabilities and its interoperability with other software platforms.

Scalar Relativistic Effects with Multiwavelets: Implementation and Benchmark.

The importance of relativistic effects in quantum chemistry is widely recognized, not only for heavier elements but throughout the periodic table. At the same time, relativistic effects are strongest in the nuclear region, where the description of electrons through a linear combination of atomic orbitals becomes more challenging. Furthermore, the choice of basis sets for heavier elements is limited compared with lighter elements where precise basis sets are available. Thanks to the framework of multiresolution analysis, multiwavelets provide an appealing alternative to overcoming this challenge: they lead to robust error control and adaptive algorithms that automatically refine the basis set description until the desired precision is reached. This allows one to achieve a proper description of the nuclear region. In this work, we extended the multiwavelet-based code MRChem to the scalar zero-order regular approximation framework. We validated our implementation by comparing the total energies for a small set of elements and molecules. To confirm the validity of our implementation, we compared both against a radial numerical code for atoms and the plane-wave-based code EXCITING.

Quantifying Intramolecular Basis Set Superposition Errors.

We show that medium-sized Gaussian basis sets lead to significant intramolecular basis set superposition errors at Hartree-Fock and density functional levels of theory, with artificial stabilization of compact over extended conformations for a 186 atom deca-peptide. Errors of ∼80 and ∼10 kJ/mol are observed, with polarized double zeta and polarized triple zeta quality basis sets, respectively. Two different procedures for taking the basis set superposition error into account are tested. While both reduce the error, it appears that polarized quadruple zeta basis sets are required to reduce the error below a few kJ/mol. Alternatively, the basis set superposition error can be eliminated using multiresolution methods based on Multiwavelets.

MRChem Multiresolution Analysis Code for Molecular Electronic Structure Calculations: Performance and Scaling Properties.

MRChem is a code for molecular electronic structure calculations, based on a multiwavelet adaptive basis representation. We provide a description of our implementation strategy and several benchmark calculations. Systems comprising more than a thousand orbitals are investigated at the Hartree-Fock level of theory, with an emphasis on scaling properties. With our design, terms that formally scale quadratically with the system size in effect have a better scaling because of the implicit screening introduced by the inherent adaptivity of the method: all operations are performed to the requested precision, which serves the dual purpose of minimizing the computational cost and controlling the final error precisely. Comparisons with traditional Gaussian-type orbitals-based software show that MRChem can be competitive with respect to performance.

Multiwavelets applied to metal-ligand interactions: Energies free from basis set errors.

Transition metal-catalyzed reactions invariably include steps where ligands associate or dissociate. In order to obtain reliable energies for such reactions, sufficiently large basis sets need to be employed. In this paper, we have used high-precision multiwavelet calculations to compute the metal-ligand association energies for 27 transition metal complexes with common ligands, such as H2, CO, olefins, and solvent molecules. By comparing our multiwavelet results to a variety of frequently used Gaussian-type basis sets, we show that counterpoise corrections, which are widely employed to correct for basis set superposition errors, often lead to underbinding. Additionally, counterpoise corrections are difficult to employ when the association step also involves a chemical transformation. Multiwavelets, which can be conveniently applied to all types of reactions, provide a promising alternative for computing electronic interaction energies free from any basis set errors.

Static Polarizabilities at the Basis Set Limit: A Benchmark of 124 Species.

Benchmarking molecular properties with Gaussian-type orbital (GTO) basis sets can be challenging, because one has to assume that the computed property is at the complete basis set (CBS) limit, without a robust measure of the error. Multiwavelet (MW) bases can be systematically improved with a controllable error, which eliminates the need for such assumptions. In this work, we have used MWs within Kohn-Sham density functional theory to compute static polarizabilities for a set of 92 closed-shell and 32 open-shell species. The results are compared to recent benchmark calculations employing the GTO-type aug-pc4 basis set. We observe discrepancies between GTO and MW results for several species, with open-shell systems showing the largest deviations. Based on linear response calculations, we show that these discrepancies originate from artifacts caused by the field strength and that several polarizabilies from a previous study were contaminated by higher order responses (hyperpolarizabilities). Based on our MW benchmark results, we can affirm that aug-pc4 is able to provide results close to the CBS limit, as long as finite difference effects can be controlled. However, we suggest that a better approach is to use MWs, which are able to yield precise finite difference polarizabilities even with small field strengths.

Modeled deposition of nitrogen and sulfur in Europe estimated by 14 air quality model systems: evaluation, effects of changes in emissions and implications for habitat protection.

The evaluation and intercomparison of air quality models is key to reducing model errors and uncertainty. The projects AQMEII3 and EURODELTA-Trends, in the framework of the Task Force on Hemispheric Transport of Air Pollutants and the Task Force on Measurements and Modelling, respectively (both task forces under the UNECE Convention on the Long Range Transport of Air Pollution, LTRAP), have brought together various regional air quality models to analyze their performance in terms of air concentrations and wet deposition, as well as to address other specific objectives. This paper jointly examines the results from both project communities by intercomparing and evaluating the deposition estimates of reduced and oxidized nitrogen (N) and sulfur (S) in Europe simulated by 14 air quality model systems for the year 2010. An accurate estimate of deposition is key to an accurate simulation of atmospheric concentrations. In addition, deposition fluxes are increasingly being used to estimate ecological impacts. It is therefore important to know by how much model results differ and how well they agree with observed values, at least when comparison with observations is possible, such as in the case of wet deposition. This study reveals a large variability between the wet deposition estimates of the models, with some performing acceptably (according to previously defined criteria) and others underestimating wet deposition rates. For dry deposition, there are also considerable differences between the model estimates. An ensemble of the models with the best performance for N wet deposition was made and used to explore the implications of N deposition in the conservation of protected European habitats. Exceedances of empirical critical loads were calculated for the most common habitats at a resolution of 100 × 100 m2 within the Natura 2000 network, and the habitats with the largest areas showing exceedances are determined. Moreover, simulations with reduced emissions in selected source areas indicated a fairly linear relationship between reductions in emissions and changes in the deposition rates of N and S. An approximate 20 % reduction in N and S deposition in Europe is found when emissions at a global scale are reduced by the same amount. European emissions are by far the main contributor to deposition in Europe, whereas the reduction in deposition due to a decrease in emissions in North America is very small and confined to the western part of the domain. Reductions in European emissions led to substantial decreases in the protected habitat areas with critical load exceedances (halving the exceeded area for certain habitats), whereas no change was found, on average, when reducing North American emissions in terms of average values per habitat.