A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies

Density Functional Theory under the Bubbles and Cube Numerical Framework

Density functional theory within the Kohn-Sham density functional theory (KS-DFT) ansatz has been implemented into our bubbles and cube real-space molecular electronic structure framework, where functions containing steep cusps in the vicinity of the nuclei are expanded in atom-centered one-dimensional (1D) numerical grids multiplied with spherical harmonics (bubbles). The remainder, i.e., the cube, which is the cusp-free and smooth difference between the atomic one-center contributions and the exact molecular function, is represented on a three-dimensional (3D) equidistant grid by using a tractable number of grid points. The implementation of the methods is demonstrated by performing 3D numerical KS-DFT calculations on light atoms and small molecules. The accuracy is assessed by comparing the obtained energies with the best available reference energies.

The Elephant in the Room of Density Functional Theory Calculations

Using multiwavelets, we have obtained total energies and corresponding atomization energies for the GGA-PBE and hybrid-PBE0 density functionals for a test set of 211 molecules with an unprecedented and guaranteed μHartree accuracy. These quasi-exact references allow us to quantify the accuracy of standard all-electron basis sets that are believed to be highly accurate for molecules, such as Gaussian-type orbitals (GTOs), all-electron numeric atom-centered orbitals (NAOs), and full-potential augmented plane wave (APW) methods. We show that NAOs are able to achieve the so-called chemical accuracy (1 kcal/mol) for the typical basis set sizes used in applications, for both total and atomization energies. For GTOs, a triple-ζ quality basis has mean errors of ∼10 kcal/mol in total energies, while chemical accuracy is almost reached for a quintuple-ζ basis. Due to systematic error cancellations, atomization energy errors are reduced by almost an order of magnitude, placing chemical accuracy within reach also for medium to large GTO bases, albeit with significant outliers. In order to check the accuracy of the computed densities, we have also investigated the dipole moments, where in general only the largest NAO and GTO bases are able to yield errors below 0.01 D. The observed errors are similar across the different functionals considered here.

Optimization of numerical orbitals using the Helmholtz kernel

We present an integration scheme for optimizing the orbitals in numerical electronic structure calculations on general molecules. The orbital optimization is performed by integrating the Helmholtz kernel in the double bubble and cube basis, where bubbles represent the steep part of the functions in the vicinity of the nuclei, whereas the remaining cube part is expanded on an equidistant three-dimensional grid. The bubbles' part is treated by using one-center expansions of the Helmholtz kernel in spherical harmonics multiplied with modified spherical Bessel functions of the first and second kinds. The angular part of the bubble functions can be integrated analytically, whereas the radial part is integrated numerically. The cube part is integrated using a similar method as we previously implemented for numerically integrating two-electron potentials. The behavior of the integrand of the auxiliary dimension introduced by the integral transformation of the Helmholtz kernel has also been investigated. The correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations on H₂, H₂O, and CO. The obtained energies are compared with reference values in the literature showing that an accuracy of 10^-4 to 10^-7 E_h can be obtained with our approach.

A Generalized Grid-Based Fast Multipole Method for Integrating Helmholtz Kernels

A grid-based fast multipole method (GB-FMM) for optimizing three-dimensional (3D) numerical molecular orbitals in the bubbles and cube double basis has been developed and implemented. The present GB-FMM method is a generalization of our recently published GB-FMM approach for numerically calculating electrostatic potentials and two-electron interaction energies. The orbital optimization is performed by integrating the Helmholtz kernel in the double basis. The steep part of the functions in the vicinity of the nuclei is represented by one-center bubbles functions, whereas the remaining cube part is expanded on an equidistant 3D grid. The integration of the bubbles part is treated by using one-center expansions of the Helmholtz kernel in spherical harmonics multiplied with modified spherical Bessel functions of the first and second kind, analogously to the numerical inward and outward integration approach for calculating two-electron interaction potentials in atomic structure calculations. The expressions and algorithms for massively parallel calculations on general purpose graphics processing units (GPGPU) are described. The accuracy and the correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations (HF-SCF) on H₂, H₂O, and CO. Our calculations show that an accuracy of 10^-4 to 10^-7 E_h can be reached in HF-SCF calculations on general molecules.

Efficient Computation of the Hartree-Fock Exchange in Real-Space with Projection Operators

We describe an efficient projection-based real-space implementation of the nonlocal single-determinant exchange operator. Through a matrix representation of the projected operator, we show that this scheme works equally well for both occupied and virtual states. Our scheme reaches a speedup of 2 orders of magnitude and has no significant loss of accuracy compared to an implementation of the full nonlocal single-determinant exchange operator. We find excellent agreement upon comparing Hartree-Fock eigenvalues, dipoles, and polarizabilities of selected molecules calculated using our method to values in the literature. To illustrate the efficiency of this scheme we perform calculations on systems with up to 240 carbon atoms.

Asymptotic behavior and interpretation of virtual states: The effects of confinement and of basis sets

A real-space high order finite difference method is used to analyze the effect of spherical domain size on the Hartree-Fock (and density functional theory) virtual eigenstates. We show the domain size dependence of both positive and negative virtual eigenvalues of the Hartree-Fock equations for small molecules. We demonstrate that positive states behave like a particle in spherical well and show how they approach zero. For the negative eigenstates, we show that large domains are needed to get the correct eigenvalues. We compare our results to those of Gaussian basis sets and draw some conclusions for real-space, basis-sets, and plane-waves calculations.

The grid-based fast multipole method – a massively parallel numerical scheme for calculating two-electron interaction energies

A grid-based fast multipole method has been developed for calculating two-electron interaction energies for non-overlapping charge densities.

Real-space numerical grid methods in quantum chemistry

This themed issue reports on recent progress in the fast developing field of real-space numerical grid methods in quantum chemistry.

An auxilliary grid method for the calculation of electrostatic terms in density functional theory on a real-space grid

The calculation of electrostatic integrals is performed using domain decomposition and auxiliary non-uniform grids for density functional theory real-space implementation.