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

Benchmarking Correlation-Consistent Basis Sets for Frequency-Dependent Polarizabilities with Multiresolution Analysis

This paper presents the first converged frequency-dependent HF polarizability results for general molecules, on a set of 89 closed-shell atoms and molecules. The solver employs multiresolution analysis (MRA) in a multiwavelet basis to compute both ground and response states to a guaranteed precision, which are validated against independent numerical grid calculations on atoms and linear molecules. The MRA ground-state energies and response properties are used to evaluate results in correlation-consistent basis sets up to 5Z augmented with either single or double diffuse functions and core-polarization functions. Systematic trends are revealed through consideration of chemical composition as well as the use of machine learning to cluster convergence trends, the latter suggesting the possibility of learning and correcting basis-set error.

Basis Set Superposition Errors Are Partly Basis Set Imbalances

When calculating fragment interaction energies by electronic structure methods employing medium-sized atom-centered basis sets, it is often observed that the effect is systematically overestimated. The common interpretation is that the systematic error arises because the basis set for the complex is more complete than for the isolated fragments, and this is denoted basis set superposition errors. It has been observed, however, that the interaction energy in some cases is underestimated, which defies the interpretation in terms of basis set completeness, and instead suggests that the effect partly is due to basis set imbalance. The imbalance can be removed by explicit optimization of the basis sets for each structure, and it is shown that this to a significant extent reduces the systematic overestimation attributed to basis set superposition error.

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.

Nested gausslet basis sets

We introduce nested gausslet bases, an improvement on previous gausslet bases that can treat systems containing atoms with much larger atomic numbers. We also introduce pure Gaussian distorted gausslet bases, which allow the Hamiltonian integrals to be performed analytically, as well as hybrid bases in which the gausslets are combined with standard Gaussian-type bases. All these bases feature the diagonal approximation for the electron-electron interactions so that the Hamiltonian is completely defined by two Nb × Nb matrices, where Nb ≈ 104 is small enough to permit fast calculations at the Hartree-Fock level. In constructing these bases, we have gained new mathematical insight into the construction of one-dimensional diagonal bases. In particular, we have proved an important theorem relating four key basis set properties: completeness, orthogonality, zero-moment conditions, and diagonalization of the coordinate operator matrix. We test our basis sets on small systems with a focus on high accuracy, obtaining, for example, an accuracy of 2 × 10-5 Ha for the total Hartree-Fock energy of the neon atom in the complete basis set limit.

Reproducibility of density functional approximations: How new functionals should be reported

Density functional theory is the workhorse of chemistry and materials science, and novel density functional approximations are published every year. To become available in program packages, the novel density functional approximations (DFAs) need to be (re)implemented. However, according to our experience as developers of Libxc [Lehtola et al., SoftwareX 7, 1 (2018)], a constant problem in this task is verification due to the lack of reliable reference data. As we discuss in this work, this lack has led to several non-equivalent implementations of functionals such as Becke-Perdew 1986, Perdew-Wang 1991, Perdew-Burke-Ernzerhof, and Becke's three-parameter hybrid functional with Lee-Yang-Parr correlation across various program packages, yielding different total energies. Through careful verification, we have also found many issues with incorrect functional forms in recent DFAs. The goal of this work is to ensure the reproducibility of DFAs. DFAs must be verifiable in order to prevent the reappearance of the above-mentioned errors and incompatibilities. A common framework for verification and testing is, therefore, needed. We suggest several ways in which reference energies can be produced with free and open source software, either with non-self-consistent calculations with tabulated atomic densities or via self-consistent calculations with various program packages. The employed numerical parameters-especially the quadrature grid-need to be converged to guarantee a ≲0.1 μEh precision in the total energy, which is nowadays routinely achievable in fully numerical calculations. Moreover, as such sub-μEh level agreement can only be achieved when fully equivalent implementations of the DFA are used, the source code of the reference implementation should also be made available in any publication describing a new DFA.

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.

Cavity-Free Continuum Solvation: Implementation and Parametrization in a Multiwavelet Framework

We present a multiwavelet-based implementation of a quantum/classical polarizable continuum model. The solvent model uses a diffuse solute-solvent boundary and a position-dependent permittivity, lifting the sharp-boundary assumption underlying many existing continuum solvation models. We are able to include both surface and volume polarization effects in the quantum/classical coupling, with guaranteed precision, due to the adaptive refinement strategies of our multiwavelet implementation. The model can account for complex solvent environments and does not need a posteriori corrections for volume polarization effects. We validate our results against a sharp-boundary continuum model and find a very good correlation of the polarization energies computed for the Minnesota solvation database.

GA, Jensen SR, Eikås RDR, Frediani L. 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.