On the achievement of high fidelity and scalability for large‐scale diagonalizations in grid‐based DFT simulations

Recent advance in high performance computing (HPC) resources has opened the possibility to expand the scope of density functional theory (DFT) simulations toward large and complex molecular systems. This work proposes a numerically robust method that enables scalable diagonalizations of large DFT Hamiltonian matrices, particularly with thousands of computing CPUs (cores) that are usual these days in terms of sizes of HPC resources. The well‐known Lanczos method is extensively refactorized to overcome its weakness for evaluation of multiple degenerate eigenpairs that is the substance of DFT simulations, where a multilevel parallelization is adopted for scalable simulations in as many cores as possible. With solid benchmark tests for realistic molecular systems, the fidelity of our method are validated against the locally optimal block preconditioned conjugated gradient (LOBPCG) method that is widely used to simulate electronic structures. Our method may waste computing resources for simulations of molecules whose degeneracy cannot be reasonably estimated. But, compared to LOBPCG method, it is fairly excellent in perspectives of both speed and scalability, and particularly has remarkably less (< 10%) sensitivity of performance to the random nature of initial basis vectors. As a promising candidate for solving electronic structures of highly degenerate systems, the proposed method can make a meaningful contribution to migrating DFT simulations toward extremely large computing environments that normally have more than several tens of thousands of computing cores. A new algorithm is proposed for the degenerate eigenvalue problems. Computing resources are split into multiple groups, each performing a single Lanczos iteration with an initial vector orthogonal with the ones of different resource groups. If different groups find identical eigenvalues, their degeneracy is examined by checking the orthogonality of the corresponding eigenvectors. The practicality of the proposed algorithm is demonstrated against diagonalizations of large‐scale DFT Hamiltonian matrices