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

Abstract

Numerical oscillation of the total energy can be observed when the Kohn- Sham 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, 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 both pseudopotential and all-electron calculations, including a stabilized 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.

Modeling and Computation of Nano-Optics

“…Therefore the vector potential, as the conjugate variable of the magnetic field, will also be removed from the Hamiltonian, which reduces TD-CDFT to TD-DFT. In this section, we summarize our work on the adaptive Finite Element Methods (FEM) [7,[9][10][11][12] and spectral methods based on Frozen Gaussian Beams [13] for TD-DFT. We first introduce some notations.…”

“…according to the tree structure, each pair of elements from two different meshes would have a belonging-to relation, allowing the efficient interpolation of the solution between two different meshes, which is very important in a dynamical simulation. Inspired by the work done by Verfürth in [46], the following residual type elementwise error indicator in solving the Kohn-Sham equation with the finite element methods is proposed [11,12]:…”

“…The translational and rotational variance of the total energy are common issues which affect the reliability of the simulations with real space methods such as Finite Difference Methods and Finite Element Methods. As illustrated in [12], the adaptive meshes can also help to alleviate this problem. To demonstrate the effectiveness of the proposed adaptive method, results on the all-electron calculation of the ground state of a benzene molecule are shown in Fig.…”

“…Although the semiclassical approach reduces the computational cost that would otherwise be tremendous in QED, a time dependent many body Schrödinger equation is still involved, for which numerical solutions are prohibitively expensive in many practical situations. A recent effort by the authors [6][7][8][9][10][11][12] is to adopt the Time Dependent Current Density Functional Theory (TD-CDFT) [14] to further simplify the semiclassical model and its computation. In the Density Functional Theory (DFT), a one-to-one correspondence (up to an arbitrary constant) between the external potential and the ground state electron density has been proved in the seminal work of Hohenberg and Kohn [15].…”

“…More recently, we apply the method to study light driven nano devices by further incorporating Ehrenfest molecular dynamics [8]. Secondly, aiming at nonlinear problems involving strong attosecond (10 −18 s) laser pulses and metallic nano particles, we have developed several adaptive techniques for the time dependent Density Functional Theory [9][10][11][12], based on a posteriori error estimates of Finite Element Methods for the time dependent Kohn-Sham equation, and more recently a novel spectral method [13] in the semiclassical regime using a Fourier integral operator commonly known as the Frozen Gaussian approximation (FGA) ansatz.…”