In this paper, we introduce a highly accurate and efficient numerical solver for the radial Kohn--Sham equation. The equation is discretized using a high-order finite element method, with its performance further improved by incorporating a parameter-free moving mesh technique. This approach greatly reduces the number of elements required to achieve the desired precision. In practice, the mesh redistribution involves no more than three steps, ensuring the algorithm remains computationally efficient. Remarkably, with a maximum of $13$ elements, we successfully reproduce the NIST database results for elements with atomic numbers ranging from $1$ to $92$.
In [Dai et al, Multi. Model. Simul., 2020], a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory, based on which a linearized method was developed in [Hu, et al, EAJAM, accepted] for further improving the numerical efficiency. In this paper, a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham model. Temporally, the convergence, the asymptotic stability, as well as the structure-preserving property of the linearized numerical scheme in the method is discussed followin…
We address the problem of bounding rigorously the errors in the numerical solution of the Kohn-Sham equations due to (i) the finiteness of the basis set, (ii) the convergence thresholds in iterative procedures, (iii) the propagation of rounding errors in floating-point arithmetic. In this contribution, we compute fully-guaranteed bounds on the solution of the non-self-consistent equations in the pseudopotential approximation in a plane-wave basis set. We demonstrate our methodology by providing band structure diagrams of silicon annotated with error bars indicating the combined error.
To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations. In this paper, we propose and analyze a class of iteration schemes for the discretized Kohn- Sham Density Functional Theory model, with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals exponentially without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly. In addition, we present a feasible and efficient approach to get suitable time step siz…
Numerical Analysis of Finite Dimensional Approximations of Kohn-Sham Models
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；Department of Physics, Fudan University（复旦大学物理系）
In this paper, we propose and analyze a gradient flow based Kohn-Sham density functional theory. First, we prove that the critical point of the gradient flow based model can be a local minimizer of the Kohn-Sham total energy. Then we apply a midpoint scheme to carry out the temporal discretization. It is shown that the critical point of the Kohn-Sham energy can be well-approximated by the scheme. In particular, based on the midpoint scheme, we design an orthogonality preserving iteration scheme to minimize the Kohn-Sham energy and show that the orthogonality preserving iteration scheme produce…
Based on the numerical method proposed in [G. Hu, X. Xie, F. Xu, J. Comput. Phys., 355 (2018), 436-449.] for Kohn-Sham equation, further improvement on the efficiency is obtained in this paper by i). designing a numerical method with the strategy of separately handling the nonlinear Hartree potential and exchange-correlation potential, and ii).parallelizing the algorithm in an eigenpairwise approach. The feasibility of two approaches are analyzed in detail, and the new algorithm is described completely. Compared with previous results, a significant improvement of numerical efficiency can be ob…
Numerical analysis of the planewave discretization of orbital-free and Kohn-Sham models Part I: The Thomas-Fermi-von Weizacker model
Université Paris-Est, CERMICS, Project-team Micmac, INRIA-Ecole des Ponts（巴黎东大学，CERMICS，Micmac项目组，INRIA-路桥学院）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）
Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
Université Paris-Est, CERMICS, Project-team Micmac, INRIA-Ecole des Ponts（巴黎东大学，CERMICS，Micmac项目组，INRIA-路桥学院）；UPMC Univ Paris 06, UMR 7598 LJLL（巴黎第六大学索邦大学，UMR 7598 LJK实验室）；CNRS, UMR 7598 LJLL（法国国家科学研究中心，UMR 7598 LJK实验室）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）
Adaptive Finite Element Approximations for Kohn-Sham Models
Technische Universität München（慕尼黑工业大学）；LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；Department of Physics, Fudan University（复旦大学物理系）
Moreau-Yosida-based Kohn-Sham Inversion for Periodic Systems
Oslo Metropolitan University（奥斯陆城市大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Ludwig-Maximilians-Universität München（慕尼黑大学）；Max Planck Institute for the Structure and Dynamics of Matter（马普物质结构动力学研究所）；University of Oslo（奥斯陆大学）
