Xinming Qin

The Kohn–Sham density functional theory (DFT) is a fundamental tool for investigating the electronic structures of molecules and solids. For periodic systems, the singularity problem does not arise in pure DFT calculations. However, when DFT is combined with the Hartree–Fock method and other wave function methods to improve accuracy, the long-range nature of the Coulomb potential 1/r leads to cumbersome interactions and introduces a 1/G2 singularity at the Γ point in reciprocal space. Handling these divergence singularities is crucial for enhancing both the accuracy and efficiency of calculations in periodic systems, particularly in various computational methods such as hybrid functionals, random phase approximation (RPA), second-order Møller–Plesset perturbation theory, GW approximation, and time-dependent density functional theory. In this study, we conduct detailed comprehensive calculations and comparisons of various approaches, including spherical truncation, Wigner–Seitz truncation, screened Coulomb potentials, auxiliary functions, and the probe charge method, to address this singularity problem in plane-wave calculations within periodic boundary conditions. In the context of plane-wave-based RPA and GW calculations, we pioneeringly implement the Wigner–Seitz truncation scheme and systematically investigate its convergence and accuracy through both supercell expansion and k-point sampling. Our results reveal optimal correction strategies tailored to different systems and computational methods, providing valuable guidance for future developments in the field.

The second-order Møller-Plesset perturbation (MP2) theory is a post-Hartree-Fock method widely used to describe weak correlation energies in solids and molecules, but its high computational cost scales as O(N⁵). Herein, we present an accurate and efficient implementation of MP2 within the plane-wave (PW) basis set for both periodic and molecular systems, which incorporates the interpolative separable density fitting (ISDF) decomposition and the Laplace transformation (LT) of the energy denominator. These innovations avoid the direct construction of electron repulsion integrals (ERIs) and reduce the computational complexity of MP2 from O(N⁵) to O(N⁴). The key idea for reducing the scaling is to exploit the numerical redundancy of occupied-virtual molecular orbital pairs on the real-space grid in the plane-wave basis set, which enables ERIs to be factorized into lower-rank quantities. This leads to further cost reductions in both the direct and exchange terms of the MP2 correlation energy. For a bulk silicon system consisting of 128 atoms, the LT-ISDF-MP2 method demonstrates a 13.5-fold speedup in total computation time compared to the standard approach. Using this plane-wave LT-ISDF-MP2 method, we simulate the π-π stacking interaction in the 1,3-butadiene dimer, successfully capturing the dispersion interaction and reproducing the self-assembled configuration.

Density functional theory (DFT) calculations of hybrid functionals have traditionally been limited to small systems containing hundreds of atoms due to substantial computational constraints. In this work, we introduce an interface between DeepH, a machine learning-based Hamiltonian approach, and HONPAS, a density functional theory software package. By leveraging DeepH's ability to bypass self-consistent field (SCF) iterations, DFT calculations in HONPAS become significantly more efficient, including computationally intensive hybrid functional calculations. This combined approach is particularly advantageous for twisted van der Waals systems, as demonstrated through examples of twisted bilayer graphene and twisted bilayer MoS₂. The substantial reduction in computation time for the HSE06 functional suggests that our method effectively addresses the efficiency-accuracy trade-off in DFT calculations, making high-accuracy calculations feasible for large systems containing more than ten thousand atoms.

Born-Oppenheimer molecular dynamics (BOMD) simulations are of great interest for the dynamic properties of molecular and solid systems. However, BOMD simulations necessitate not only an extensive period of dynamical evolution but also costly self-consistent-field (SCF) electronic structure calculations, especially for hybrid functional-based BOMD (H-BOMD) simulations within plane-wave basis sets. Here, we propose an improved always stable predictor-corrector (ASPC) method for the wave function extrapolation to accelerate the plane-wave H-BOMD simulations, named projected ASPC (PASPC), yielding a wave function closer to the actual solution space and efficiently reducing the number of SCF iterations at each MD step. We investigated the convergence properties of different extrapolation schemes for molecular and solid systems. Numerical results demonstrate that plane-wave H-BOMD simulations can be significantly faster than conventional cases by combining the accelerated algorithms with the PASPC method. The energy drift is also evaluated, showing that PASPC produces energy drift with smaller oscillations and can simulate a larger time step for systems containing heavy atoms, demonstrating the accuracy of the extrapolation schemes. Furthermore, H-BOMD simulations showcase more accurate power and infrared spectra of silicon dioxide and liquid water that are comparable to those of experimental measurements.

Hybrid functionals that incorporate exact Hartree-Fock exchange (HFX) into density functional theory (DFT) are crucial for accurately predicting the electronic structures of extended systems in condensed-matter physics and materials science. Although the exact exchange contributes only a small fraction of the total energy, HFX calculations in hybrid functionals demand significant computational resources. Here, we introduce dual-grid and mixed-precision techniques, based on two low-rank approximations, adaptively compressed exchange (ACE) and interpolative separable density fitting (ISDF) methods, to significantly improve the computational efficiency of plane-wave hybrid functional calculations in the software package PWDFT (plane wave density functional theory). The dual-grid method introduces a smaller cutoff energy, thereby reducing the number of Fourier and real-space grids involved in the HFX calculations. The mixed-precision method reduces the computational cost by shifting from double precision to single precision during the HFX construction while maintaining double precision for other DFT processes. Numerical results demonstrate that both the dual-grid and mixed-precision techniques can accelerate plane-wave hybrid functional calculations several times, with an acceptable trade-off in accuracy compared to the original low-rank approximations. Furthermore, by utilizing a hybrid MPI and OpenMP parallel implementation, we successfully perform large-scale plane-wave hybrid functional calculations with up to 8,000 silicon atoms using 16,000 CPU cores.

Linear-scaling Math 1 density functional theory (DFT) represents a significant advancement in the field of computational materials science, especially for simulations of large systems where traditional cubic-scaling methods become computationally prohibitive. The core operation in Math 2 methods is sparse general matrix-matrix multiplication (SpGEMM), which is the major performance bottleneck. To enhance the computational efficiency of SpGEMM, it is crucial to consider the inherent sparse pattern of these matrices. Targeting block-sparse matrices with moderate block sizes and regular block shapes, we have developed a distributed block-sparse matrix-matrix multiplication (DB-SpGEMM) algorithm for large-scale DFT calculations. Through deep optimizations in distributed matrix storage, computational task decomposition, asynchronous task scheduling, and load balancing, we have implemented a linear-scaling method based on this algorithm within the discontinuous Galerkin density functional theory (DGDFT). On the new Sunway supercomputer, our approach achieves a 8 ∼ 10x speedup compared to the original version on monolayer phosphorene systems, and demonstrates superior scalability.