Real-space techniques for computing the electronic structure of nearly a million electrons*

By employing density functional theory and pseudopotentials, the electronic structure problem can be solved, although the computational cost associated with the Kohn–Sham equations remains a challenge. Here, we explore innovative approaches for the analysis of systems comprising over 100,000 atoms. Our approach centers on new computational algorithms and offers several advantages. For example, real-space formalisms, such as finite difference, circumvent the need for global communications based on FFT's. Our method exhibits excellent scalability, spanning hundreds or thousands of computer nodes. Also, finite-difference methods utilizing a uniform real-space grid facilitate straightforward implementations, where the grid spacing determines convergence. Employing a Chebyshev-filtered subspace iteration method, offers a promising technique for solving the Kohn–Sham eigenvalue problem [1–3]. Our computations involving confined systems, comprising over 200,000 atoms or 800,000 electrons, demonstrate the effectiveness of this method in reducing communication overhead and maximizing the utilization of vector processing capabilities offered by parallel computers [4,5]. Our quantitative analysis of the electronic structure shows how it approaches its bulk counterpart as a function of nanocluster size. The band gap is enlarged due to quantum confinement in nanoclusters, but decreases as the system size increases. Our work serves as a proof of concept for real-space approaches in efficiently parallelizing very large calculations.