SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Isolated clusters

As the first component of SPARC (Simulation Package for Ab-initio Real-space Calculations), we present an accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory (DFT) for isolated clusters. Specifically, utilizing a local reformulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local component of the force, we develop a framework using the finite-difference representation that enables the efficient evaluation of energies and atomic forces to within the desired accuracies in DFT. Through selected examples consisting of a variety of elements, we demonstrate that SPARC obtains exponential convergence in energy and forces with domain size; systematic convergence in the energy and forces with mesh-size to reference plane-wave result at comparably high rates; forces that are consistent with the energy, both free from any noticeable ‘egg-box’ effect; and accurate ground-state properties including equilibrium geometries and vibrational spectra. In addition, for systems consisting up to thousands of electrons, SPARC displays weak and strong parallel scaling behavior that is similar to well-established and optimized plane-wave implementations, but with a significantly reduced prefactor. Overall, SPARC represents an attractive alternative to plane-wave codes for practical DFT simulations of isolated clusters.作为SPARC（从头算实空间计算模拟软件包，Simulation Package for Ab-initio Real-space Calculations）的首个组成模块，我们提出了针对孤立团簇的密度泛函理论（Density Functional Theory, DFT）的精确高效有限差分格式与并行实现方案。具体而言，通过采用静电相互作用的局域重述、切比雪夫多项式滤波自洽场迭代，以及力的非局域分量重述，我们构建了基于有限差分表示的计算框架，可在DFT框架下以所需精度高效求解体系能量与原子力。通过选取涵盖多种元素的典型算例，我们验证了SPARC具备如下特性：其一，能量与原子力随计算域尺寸呈指数收敛；其二，能量与原子力随网格尺寸系统收敛至参考平面波结果，且收敛速率可观；其三，力与能量自洽一致，二者均无显著“蛋盒”效应；其四，可准确预测包括平衡几何构型与振动光谱在内的基态性质。此外，对于包含数千个电子的体系，SPARC展现出与成熟优化的平面波程序相近的强、弱并行扩展性，但前置因子显著更低。总体而言，针对孤立团簇的实际DFT模拟，SPARC是极具吸引力的平面波程序替代方案。