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

Gaussian-Approximated Poisson Preconditioner for Iterative Diagonalization in Real-Space Density Functional Theory

Various real-space methods optimized on massive parallel computers have been developed for efficient large-scale density functional theory (DFT) calculations of materials and biomolecules. The iterative diagonalization of the Hamiltonian matrix is a computational bottleneck in real-space DFT calculations. Despite the development of various iterative eigensolvers, the absence of efficient real-space preconditioners has hindered their overall efficiency. An efficient preconditioner must satisfy two conditions: appropriate acceleration of the convergence of the iterative process and inexpensive computation. This study proposed a Gaussian-approximated Poisson preconditioner (GAPP) that satisfied both conditions and was suitable for real-space methods. A low computational cost was realized through the Gaussian approximation of a Poisson Green's function. Fast convergence was achieved through the proper determination of Gaussian coefficients to fit the Coulomb energies. The performance of GAPP was evaluated for several molecular and extended systems, and it showed the highest efficiency among the existing preconditioners adopted in real-space codes.

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.