Towards Translational Invariance of Total En- ergy with Finite Element Methods for Kohn-Sham Equation

Numerical oscillation of the total energy can be observed when the KohnSham equation is solved by real-space methods to simulate the translational move of an electronic system. Effectively remove or reduce the unphysical oscillation is crucial not only for the optimization of the geometry of the electronic structure such as molecules, but also for the study of molecular dynamics. In this paper, we study such unphysical oscillation based on the numerical framework in [G. Bao, G. H. Hu, and D. Liu, An h-adaptive finite element solver for the calculations of the electronic structures, Journal of Computational Physics, Volume 231, Issue 14, Pages 4967-4979, 2012], and deliver some numerical methods to constrain such unphysical effect for pseudopotential and all-electron calculations, respectively, including a stablized cubature strategy for Hamiltonian operator, and an a posteriori error estimator of the finite element methods for Kohn-Sham equation. The numerical results demonstrate the effectiveness of our method on restraining unphysical oscillation of the total energies. Corresponding author. Email addresses: [email protected] (G. Bao), [email protected] (G. H. Hu), [email protected] (D. Liu) AMS subject classifications: 52B10, 65D18, 68U05, 68U07