An Unconditionally Energy-Stable and Orthonormality-Preserving Iterative Scheme for the Kohn-Sham Gradient Flow Based Model

Abstract

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature.

Figures (9)

• Figure 5.1: The evolution of the computed total energy (in Hartree) with the time step (left) and the $L^2$ norm of the numerical solution (right) for the electronic structure of a helium atom (Example \ref{['Ex.1']}), demonstrating that our scheme preserves normalization exactly while being strictly energy stable even with large time steps.
• Figure 5.2: The nonuniform mesh used for domain discretization with a total number of degree of freedoms 5400 (left) and the contour plot (right) for the predicted electronic structure of a helium atom (Example \ref{['Ex.1']}).
• Figure 5.3: Profiles of the computed electron density function in a linear scale (left) and in a log scale (right) for the $X Y$ cross-section of the electronic structure of a helium atom (Example \ref{['Ex.1']})
• Figure 5.4: The evolution of the computed total energy (in Hartree) with the time step (left) and the $L^2$ norm of the first wave function and inner product of the two wave functions (right) for the electronic structure of a lithium hydride molecule (Example \ref{['Ex.2']}), demonstrating that our scheme preserves normalization and orthogonality exactly while being strictly energy stable even with large time steps.
• Figure 5.5: The nonuniform mesh used for domain discretization with a total number of degree of freedoms 6909 (left) and the 3D contour plot (right) for the predicted electronic structure of a lithium hydride molecule (Example \ref{['Ex.2']}).

Theorems & Definitions (15)

• Proposition 2.1
• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.2
• proof
• Corollary 4.1