A hierarchical splines-based $h$-adaptive isogeometric solver for all-electron Kohn--Sham equation

TL;DR

This work develops a novel $h$-adaptive isogeometric solver for all-electron Kohn–Sham equations using high-order hierarchical splines (HB/THB-splines) to enable local refinement near nuclei while preserving global regularity. The solver combines a self-consistent field loop with a residual-based a posteriori error indicator on a hierarchical mesh and solves the resulting generalized eigenproblem via LOBPCG with an elliptic preconditioner, yielding convergence that is largely independent of spline order. THB-splines and truncated hierarchical bases provide local refinement and sparse stiffness matrices, enabling accurate energies with only a few thousand degrees of freedom in challenging all-electron cases (e.g., CH$_4$ with ~6355 Dofs to reach 10$^{-3}$ Hartree/particle). The numerical experiments, spanning radial atoms to complex molecules (He, LiH, CH$_4$, benzene), validate accuracy, robustness, and efficiency of the approach and point to future work on scalable solvers and time-dependent KS. The framework holds promise for large-scale all-electron KS simulations where adaptive, high-order, smooth basis functions are advantageous.

Abstract

In this paper, a novel $h$-adaptive isogeometric solver utilizing high-order hierarchical splines is proposed to solve the all-electron Kohn--Sham equation. In virtue of the smooth nature of Kohn--Sham wavefunctions across the domain, except at the nuclear positions, high-order globally regular basis functions such as B-splines are well suited for achieving high accuracy. To further handle the singularities in the external potential at the nuclear positions, an $h$-adaptive framework based on the hierarchical splines is presented with a specially designed residual-type error indicator, allowing for different resolutions on the domain. The generalized eigenvalue problem raising from the discretized Kohn--Sham equation is effectively solved by the locally optimal block preconditioned conjugate gradient (LOBPCG) method with an elliptic preconditioner, and it is found that the eigensolver's convergence is independent of the spline basis order. A series of numerical experiments confirm the effectiveness of the $h$-adaptive framework, with a notable experiment that the numerical accuracy $10^{-3} \mathrm{~Hartree/particle}$ in the all-electron simulation of a methane molecule is achieved using only $6355$ degrees of freedom, demonstrating the competitiveness of our solver for the all-electron Kohn--Sham equation.

Figures (20)

• Figure 1: An example of cubic HB-spline with depth $3$. Refined B-spline basis (a1-a3) and Refined HB-spline basis (b1-b3) are marked by the black solid, blue dashed line, and red dashed-dotted line from level $0$ to level $2$, respectively.
• Figure 2: An example of cubic THB-spline with depth $3$. Refined HB-spline basis (b1-b3) and Refined THB-spline basis (c1-c3) are marked by the black solid, blue dashed line, and red dashed-dotted line from level $0$ to level $2$, respectively.
• Figure 3: Refined sliced mesh in the $X$-$Y$ plane for a helium atom ($p=3$).
• Figure 4: The convergence of LOBPCG in solving a helium atom on the uniform mesh with different orders. Left: without preconditioner. Right: with preconditioner.
• Figure 5: The residual of LOBPCG in solving a helium atom on the adaptive mesh ($p=3$). Left: without preconditioner. Right: with preconditioner.

Theorems & Definitions (4)

• Definition 1: Hierarchy of subdomains
• Definition 2: Hierarchical B-splines
• Definition 3: truncation
• Definition 4: Truncated Hierarchical B-splines