Spread-balanced Wannier functions - Robust and automatable orbital localization
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2021-09-27 |
| Journal | Physical review. B./Physical review. B |
| Authors | P. Fontana, Ask Hjorth Larsen, Thomas Olsen, Kristian S. Thygesen |
| Institutions | Technical University of Denmark |
| Citations | 8 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive SummaryâThis research introduces a highly robust and automatable scheme for generating Wannier Functions (WFs) with balanced spread distributions, addressing a major challenge in high-throughput computational materials science.
- Core Innovation (Ωvar Functional): A new localization functional (Ωvar) is introduced that minimizes the conventional quadratic spread while adding a penalty term proportional to the variance of the spread distribution.
- Enhanced Robustness: This method prevents the common issue of âsacrificingâ one or a few WFs, where a single orbital becomes highly delocalized to slightly improve the localization of the rest.
- Significant Smax Reduction: The maximum spread (Smax) of the least localized WF is consistently and significantly reduced. For a challenging monolayer material (WMO3Te8), Smax was reduced by 76% (from 21.5 A2 to 5.1 A2).
- Full Automation Protocol: A complete, automatable protocol is proposed for selecting the initial orbital guess (based on atomic valence configuration) and determining the optimal number of WFs (Nw) via a trial-and-error search minimizing Smax.
- Broad Applicability: The method is validated across diverse, complex systems, including NV centers in diamond, metal slabs with adsorbates, ferroelectrics (BaTiO3), and a set of 30 two-dimensional (2D) materials.
- Software Integration: All methodologies are implemented in Python as part of the open-source Atomic Simulation Environment (ASE), facilitating immediate use in automated workflows.
Technical Specifications
Section titled âTechnical SpecificationsâThe following table summarizes key performance metrics and parameters derived from the application of the Spread Balanced Wannier Function (Ωvar) method compared to the standard Maximally Localized Wannier Function (Ω) method.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Localization Functional | Ωvar | N/A | New functional minimizing spread variance. |
| NV Center Max Spread (Ω) | 6.8 ± 0.3 | A2 | Standard method, Nw = 127. |
| NV Center Max Spread (Ωvar) | 4.46 ± 0.04 | A2 | Spread Balanced method (34% reduction in Smax). |
| WMO3Te8 Max Spread (Ω) | 21.5 | A2 | Monolayer material, minimal Nw (64 WFs). |
| WMO3Te8 Max Spread (Ωvar) | 5.1 | A2 | Monolayer material, minimal Nw (76% reduction). |
| BaTiO3 Spontaneous Polarization | 45.4 | ”C/cm2 | Calculated using Ωvar WFs; matches Berry phase method. |
| Average Smax (30 2D Materials, Optimal Nw, Ω) | 2.9 ± 0.2 | A2 | Mean maximum spread across 30 materials. |
| Average Smax (30 2D Materials, Optimal Nw, Ωvar) | 2.5 ± 0.1 | A2 | Mean maximum spread across 30 materials (Improved robustness). |
| Energy Threshold (E0) | CBM (or EF) + 2 | eV | Used for selecting target bands for insulators (metals). |
| NV Center DFT Cutoff | 400 | eV | Plane wave basis set cutoff for NV center calculation. |
| NV Center Force Threshold | 0.05 | eV/A | Used during structure optimization (LBFGS algorithm). |
Key Methodologies
Section titled âKey MethodologiesâThe robust and automatable Wannierization scheme relies on a three-part protocol: initial guess generation, optimal Nw selection, and minimization of the variance-penalized functional.
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Initial Guess Generation (Automated):
- The initial guess for the WFs is automatically generated based on the valence configuration of the atoms in the unit cell.
- A set of Nw atomic orbitals (NAOs) is constructed:
- s-orbitals are placed at random positions within a radius of 1.5 A from the atoms.
- p- and d-orbitals are centered on the atoms.
- Robustness Measure: Five independent optimizations are run using different random seeds for the initial s-orbital positions to ensure the iterative algorithm avoids local minima.
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Optimal Number of WFs (Nw) Selection:
- The minimum required number of WFs (Nmin) is calculated based on the energy threshold (E0).
- Wannierization is performed for a range of Nw values (typically Nmin up to Nmin + 5).
- The optimal Nw (Nopt) is selected as the value that yields the lowest maximum spread (Smax) across the five independent runs.
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Spread Balanced Localization (Ωvar):
- The localization procedure maximizes the functional Ωvar: $$ \Omega_{var} = \Omega - w_{var} \text{Var} \left[ \sum_{\alpha=1}^{N_G} W_{\alpha} |Z_{\alpha,nn}|^2 \right] $$ (Note: The term $\sum W_{\alpha} |Z_{\alpha,nn}|^2$ is the inverse spread of the n-th WF.)
- The weight parameter $w_{var}$ is typically set equal to the number of WFs ($N_{w}$) for consistent performance.
- This functional explicitly penalizes large variations in the spread among individual WFs, forcing a more balanced localization.
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Underlying DFT Calculations:
- Density Functional Theory (DFT) calculations utilized the PBE exchange-correlation functional.
- The GPAW code was used for DFT calculations.
- Monkhorst-Pack k-point grids were used with a minimum density of 5 k-points per A-1.
Commercial Applications
Section titled âCommercial ApplicationsâThe development of robust and automatable Wannier function generation is critical for engineering applications requiring accurate, localized electronic structure models, particularly in high-throughput computational design.
| Industry/Field | Application/Benefit |
|---|---|
| High-Throughput Materials Screening | Enables fully automated calculation of electronic properties (band structure, polarization) for large material databases (e.g., 2D materials), accelerating the discovery of new functional materials. |
| Ferroelectric and Piezoelectric Devices | Accurate calculation of spontaneous polarization (demonstrated on BaTiO3) using the modern theory of polarization, essential for designing non-volatile memory and sensors. |
| Quantum Computing and Sensing | Reliable modeling of localized defects (like the NV center in diamond) where precise orbital localization is necessary to understand spin states and coherence properties for qubit design. |
| Electronic Transport Simulation | WFs provide the localized basis set required for advanced transport calculations, including electron-phonon matrix elements, Berry curvatures, and non-equilibrium Greenâs functions (NEGF) used in device modeling. |
| Surface and Interface Engineering | Robust localization in complex, non-periodic systems (metal slabs with adsorbates, defects) allows for microscopic analysis of how interfaces affect charge transfer and polarization. |
| Computational Software Development | The methods are implemented in the open-source ASE Python package, providing a robust, standardized tool for academic and industrial computational materials engineers. |
View Original Abstract
We introduce a new type of Wannier functions (WFs) obtained by minimizing the\nconventional spread functional with a penalty term proportional to the variance\nof the spread distribution. This modified Wannierisation scheme is less prone\nto produce ineffective solutions featuring one or several poorly localized\norbitals, making it well suited for complex systems or high-throughput\napplications. Furthermore, we propose an automatable protocol for selecting the\ninitial guess and determine the optimal number of bands (or equivalently WFs)\nfor the localization algorithm. The improved performance and robustness of the\napproach is demonstrated for a diverse set of test systems including the NV\ncenter in diamond, metal slabs with atomic adsorbates, spontaneous polarization\nof ferroelectrics and 30 inorganic monolayer materials comprising both metals\nand semiconductors. The methods are implemented in Python as part of the Atomic\nSimulation Environment (ASE).\n