An Implicit Solver for the Time-Dependent Kohn-Sham Equation

Abstract

The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation. However, the efficiency issue prevents the practical applications of those implicit methods. In this paper, an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation. The efficiency issue is partially resolved by three approaches, i.e., an h-adaptive mesh method is proposed to effectively restrain the size of the discretized problem, a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization, as well as the OpenMP based parallelization of the algorithm. The numerical convergence, the ability on preserving the physical properties, and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

“…In other words, the gradient flow based model is quite promising in ground state electronic structure calculations and deserves further investigation. For the sake of clarity, we would like to mention that the gradient flow based Kohn-Sham DFT model is different from the time dependent Kohn-Sham equation in [28,36,43].…”

A Linearized Structure-Preserving Numerical Scheme for a Gradient Flow Model of the Kohn-Sham Density Functional Theory

“…The solution of the Kohn-Sham equation [2,27] by self-consistent field (SCF) iteration is a popular approach to determine the ground state of electronic structure systems. It has been realized in various mature software, including DFT-FE † , VASP ‡ , and Quantum Espresso [14].…”