Exploring the Convergence and Properties of Intrinsic Bond Orbitals in Solids

TL;DR

This paper develops intrinsic bond orbitals (IBOs) for solids by combining Pipek–Mezey localization with intrinsic atomic orbitals in a plane-wave framework, yielding a bond-centric, localized orbital set that is largely independent of the band gap. It casts the localization as a Riemannian optimization on the unitary group $U(n)$ and compares multiple solvers (BFGS, CG, SA, DIIS) implemented in Lucon.jl and in VASP, analyzing convergence, memory effects, and scalability to large supercells. Key findings include a robust link between orbital spread and local geometry (e.g., $\sigma_{\text{HF}} \approx 2.1 \alpha$ with $\alpha = d_{\textNN}/\#\textNN$), and substantial sparsity in the Fock exchange matrix that persists across materials and band gaps, enabling reduced-cost many-electron methods. Metal-oxide systems emerge as convergence bottlenecks, underscoring the need for improved initial guesses, preconditioning, or non-iterative approaches for reliable large-scale Wannier localization in solids.

Abstract

We present a study of the construction and spatial properties of localized Wannier orbitals in large supercells of insulating solids using plane waves as the underlying basis. The Pipek-Mezey (PM) functional in combination with intrinsic atomic orbitals (IAOs) as projectors is employed, resulting in so-called intrinsic bond orbitals (IBOs). Independent of the bonding type and band gap, a correlation between orbital spreads and geometric properties is observed. As a result, comparable sparsity patterns of the Hartree-Fock exchange matrix are found across all considered bulk 3D materials, exhibiting covalent bonds, polar covalent bonds, and ionic bonds. Recognizing the considerable computational effort required to construct localized Wannier orbitals for large periodic simulation cells, we address the performance and scaling of different solvers for the localization problem. This includes the Broyden-Fletcher-Goldfarb-Shanno (BFGS), Conjugate-Gradient (CG), Steepest Ascent (SA) as well as the Direct Inversion in the Iterative Subspace (DIIS) method. Each algorithm performs a Riemannian optimization under unitary matrix constraint, efficiently reaching the optimum in the "curved parameter space" on geodesics. We hereby complement the quantum chemistry and materials science literature with an introduction to this topic along with key references. The solvers have been implemented both within the Vienna Ab initio Simulation Package (VASP) and as a standalone open-source software package. Furthermore, we observe that the construction of Wannier orbitals for supercells of metal oxides presents a significant challenge, requiring approximately one order of magnitude more iteration steps than other systems studied.

Figures (11)

• Figure 1: Visual representation of ibo in a selection of materials. The top row shows an only with those sites of the periodic structure it connects, indicating the bond. The bottom row shows the conventional unit cell of the corresponding material. All pictures were made with VESTA Momma2011, using an isosurface level of 5.0 for the orbitals.
• Figure 2: Box-and-whisker plot of the number of required iterations against the number of memorized steps compared to for a graphene supercell (162 atoms) with flower defects.
• Figure 3: Box-and-whisker plot of the iterations against occupied orbitals $n$ for several systems and supercell sizes. The dashed lines are proportional to $\sqrt{n}$ and $\sqrt[4]{n}$ respectively, giving an idea of scaling.
• Figure 4: Convergence of with several memory size settings and , for a TiO2 supercell (72 atoms, 288 occupied orbitals), the unity matrix serves as starting point (i.e. starting from bloch orbitals).
• Figure 5: Convergence of with several memory size settings and , for a graphene supercell (162 atoms) with flower defects, the unity matrix serves as starting point (i.e. starting from bloch orbitals).