Efficient Shift-and-Invert Preconditioning for Multi-GPU Accelerated Density Functional Calculations.

To accelerate the iterative diagonalization of electronic structure calculations, we propose a new inexact shift-and-invert (ISI) preconditioning method. The key idea is to improve shift values in the ISI preconditioning to be closer to the exact eigenvalues, leading to a significant boost in the convergence speed of the iterative diagonalization. Furthermore, we adopted a preconditioned conjugate gradient solver to rapidly evaluate an inversion process. Finally, we accelerated overall processes, including the proposed modification, with state-of-the-art graphical processing units (GPUs) and assessed its parallel efficiency with real-space density functional calculations of 1D, 2D, and 3D periodic systems. Our method attains both fast diagonalization convergence and high multi-GPU parallel efficiency. This is evident from the fact that single-point density functional calculations for hundreds of atom systems can be done in approximately 10 s using 8 GPUs. The proposed method can be generally applied to any electronic structure calculation methods involving large-scale diagonalizations.

Configuration interaction singles based on the real-space numerical grid method: Kohn-Sham versus Hartree-Fock orbitals.

We developed a program code of configuration interaction singles (CIS) based on a numerical grid method. We used Kohn-Sham (KS) as well as Hartree-Fock (HF) orbitals as a reference configuration and Lagrange-sinc functions as a basis set. Our calculations show that KS-CIS is more cost-effective and more accurate than HF-CIS. The former is due to the fact that the non-local HF exchange potential greatly reduces the sparsity of the Hamiltonian matrix in grid-based methods. The latter is because the energy gaps between KS occupied and virtual orbitals are already closer to vertical excitation energies and thus KS-CIS needs small corrections, whereas HF results in much larger energy gaps and more diffuse virtual orbitals. KS-CIS using the Lagrange-sinc basis set also shows a better or a similar accuracy to smaller orbital space compared to the standard HF-CIS using Gaussian basis sets. In particular, KS orbitals from an exact exchange potential by the Krieger-Li-Iafrate approximation lead to more accurate excitation energies than those from conventional (semi-) local exchange-correlation potentials.