A parallel eigensolver for real-space pseudopotential density functional theory calculations

First-principles electronic structure calculations are a popular tool for understanding and predicting properties of materials. Among such methods, the combination of real-space density functional theory and pseudopotentials to solve the Kohn–Sham equation has several advantages. Real-space methods, such as finite differences and finite elements, avoid the global communication needed in fast Fourier transformation and offer better scalability for large calculations on hundreds or thousands of compute nodes. Besides, finite-difference methods with a uniform real-space grid are easy to implement, e.g., the convergence of a Kohn–Sham solution is controlled by a single parameter – the grid spacing.

One promising algorithm for solving the Kohn–Sham eigenvalue problem in real space is the Chebyshev-filtered subspace iteration method (CheFSI). Within this algorithm, the charge density is constructed without regard to a solution for individual eigenvalues. However, for large systems CheFSI may suffer from super-linear scaling operations such as orthonormalization and the Rayleigh–Ritz procedure.

In the dissertation I will present two improvements in CheFSI to enhance its scalability and accelerate calculation. The first one is a hybrid method that combines a spectrum slicing method and CheFSI. The spectrum slicing method divides a Kohn–Sham eigenvalue problem into subproblems, wherein each subproblem can be solved in parallel using CheFSI. We will show that, by the simulations of confined systems with thousands of atoms, this hybrid method can be faster and possesses better scalability than CheFSI.

The second improvement is a grid partitioning method based on space-filling curves. Space-filling curves based grid partitioning improves the efficiency of the sparse matrix–vector multiplication, which is the key component of CheFSI. We will show that, by computations of confined systems with 50,000 atoms or 200,000 electrons, this method effectively reduces the communication overhead and improves the utilization of the vector processing capabilities provided by most modern parallel computers.

Along with the improvements, I will also present three applications. One is the study of the evolution of density of states of silicon nanocrystals from small ones to their bulk limit. The simulations can hardly be performed without the improvement in sparse matrix–vector multiplication enhanced by space-filling curves based grid partitioning. The other two applications are the studies of proton transfer in liquid water and the adsorption of water on titanium dioxide surfaces.