System-Specific Separable Basis Based on Tucker Decomposition: Application to Density Functional Calculations

Abstract

For fast density functional calculations, a suitable basis that can accurately represent the orbitals within a reasonable number of dimensions is essential. Here, we propose a new type of basis constructed from Tucker decomposition of a finite-difference (FD) Hamiltonian matrix, which is intended to reflect the system information implied in the Hamiltonian matrix and satisfies orthonormality and separability conditions. By introducing the system-specific separable basis, the computation time for FD density functional calculations for seven two- and three-dimensional periodic systems was reduced by a factor of 2–71 times, while the errors in both the atomization energy per atom and the band gap were limited to less than 0.1 eV. The accuracy and speed of the density functional calculations with the proposed basis can be systematically controlled by adjusting the rank size of Tucker decomposition.

Dynamic Precision Approach for Accelerating Large-Scale Eigenvalue Solvers in Electronic Structure Calculations on Graphics Processing Units

“…We assume that H is a large-scale matrix, but only a limited number of the lowest eigenvalues and eigenvectors needs to be computed. Many iterative algorithms have been proposed to solve this problem efficiently. ,,, In this work, we used the LOBPCG method , to solve it, which is widely adopted in various simulation codes. ,,, …”

“…Electronic structure calculations were performed using the GOSPEL package, which is a grid-based KS solver written in Python. For all calculations, we used the PBE exchange-correlation functional and the norm-conserving pseudopotentials. − An equidistant grid with a grid spacing of 0.2 Å was used.…”

“…35,36,39,40 In this work, we used the LOBPCG method 35,36 to solve it, which is widely adopted in various simulation codes. 7,16,41,42 In LOBPCG, the eigenvalues and eigenvectors are obtained by minimizing residual vectors iteratively using the Rayleigh− Ritz procedure. The residual vector at the jth iteration is given by…”

Neural network-based pseudopotential: development of a transferable local pseudopotential

“…The pseudopotential approach is essential for efficient quantum simulations of materials by effectively removing core electrons. [1][2][3][4][5][6][7][8][9][10][11] Most pseudopotentials have a non-local form originating from their orbital-dependence. 12,13 This incurs additional computational processes and complicated implementation tasks.…”

“…In eqn (11), the l-dependence is still present. If all components of the loss function go to zero, V l (r) becomes the same for all l. However, since this is not the case in practice, we eventually need to choose which l is used for the LPP.…”