Universal method for computation of electrostatic potentials

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.

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.

Performance of heterogeneous computing with graphics processing unit and many integrated core for hartree potential calculations on a numerical grid

We investigated the performance of heterogeneous computing with graphics processing units (GPUs) and many integrated core (MIC) with 20 CPU cores (20×CPU). As a practical example toward large scale electronic structure calculations using grid-based methods, we evaluated the Hartree potentials of silver nanoparticles with various sizes (3.1, 3.7, 4.9, 6.1, and 6.9 nm) via a direct integral method supported by the sinc basis set. The so-called work stealing scheduler was used for efficient heterogeneous computing via the balanced dynamic distribution of workloads between all processors on a given architecture without any prior information on their individual performances. 20×CPU + 1GPU was up to ∼1.5 and ∼3.1 times faster than 1GPU and 20×CPU, respectively. 20×CPU + 2GPU was ∼4.3 times faster than 20×CPU. The performance enhancement by CPU + MIC was considerably lower than expected because of the large initialization overhead of MIC, although its theoretical performance is similar with that of CPU + GPU. © 2016 Wiley Periodicals, Inc.

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.

The direct approach to gravitation and electrostatics method for periodic systems

The direct approach to gravitation and electrostatics (DAGE) algorithm is an accurate, efficient, and flexible method for calculating electrostatic potentials. In this paper, we show that the algorithm can be easily extended to consider systems with many different kinds of periodicities, such as crystal lattices, surfaces, or wires. The accuracy and performance are nearly the same for periodic and aperiodic systems. The electrostatic potential for semiperiodic systems, namely defects in crystal lattices, can be obtained by combining periodic and aperiodic calculations. The method has been applied to an ionic model system mimicking NaCl, and to a corresponding covalent model system.

Tensor product approximation with optimal rank in quantum chemistry

Tensor product decompositions with optimal separation rank provide an interesting alternative to traditional Gaussian-type basis functions in electronic structure calculations. We discuss various applications for a new compression algorithm, based on the Newton method, which provides for a given tensor the optimal tensor product or so-called best separable approximation for fixed Kronecker rank. In combination with a stable quadrature scheme for the Coulomb interaction, tensor product formats enable an efficient evaluation of Coulomb integrals. This is demonstrated by means of best separable approximations for the electron density and Hartree potential of small molecules, where individual components of the tensor product can be efficiently represented in a wavelet basis. We present a fairly detailed numerical analysis, which provides the basis for further improvements of this novel approach. Our results suggest a broad range of applications within density fitting schemes, which have been recently successfully applied in quantum chemistry.

Parallel implementation of a direct method for calculating electrostatic potentials

The authors present a method for calculating the electrostatic potential directly in a straightforward manner. While traditional methods for calculating the electrostatic potential usually involve solving the Poisson equation iteratively, the authors obtain the electrostatic interaction potential by performing direct numerical integration of the Coulomb-law expression using finite-element functions defined on a grid. The singularity of the Coulomb operator is circumvented by an integral transformation and the resulting auxiliary integral is obtained using Gaussian quadrature. The three-dimensional finite-element basis is constructed as a tensor (outer) product of one-dimensional functions, yielding a partial factorization of the expressions. The resulting algorithm has, without using any prescreening or other computational tricks, a formal computational scaling of Omicron(N4/3), where N is the size of the grid. The authors show here how to implement the method for efficiently running on parallel computers. The matrix multiplications of the innermost loops are completely independent, yielding a parallel algorithm with the computational costs scaling practically linearly with the number of processors.