Projectability-Disentangled Wannier Functions

• PDWFs are highly localized Wannier functions generated via atomic orbital projectability measures, enabling precise tight-binding models.
• The method employs a two-step disentanglement and localization process that automates state selection without manual energy windows.
• Enhanced by projector augmentation and extensions for spin–orbit coupling and magnetization, PDWFs achieve interpolation errors below 15 meV.

Projectability-Disentangled Wannier Functions (PDWFs) are a class of maximally localized Wannier functions generated via an automated protocol that leverages atomic-orbital projectability measures rather than manual inner/outer energy windows. The PDWF methodology constructs Wannier functions that are both highly localized and chemically intuitive, enabling robust and efficient tight-binding models for a wide variety of materials—including metals, insulators, systems with magnetization, and those with strong spin–orbit coupling. PDWFs have been established as a central tool in large-scale electronic-structure workflows and high-throughput computational materials science, offering numerically precise (meV-level) interpolations of band structures with minimal manual intervention (Qiao et al., 2023, Jiang et al., 9 Jul 2025).

1. Mathematical Foundations and Projectability Measures

The construction of PDWFs is based on the use of a set of “atomic-like” trial orbitals, typically chosen as pseudopotential pseudo-atomic orbitals (PAOs) or hydrogenic atomic orbitals (AOs). For each Bloch state $|\psi_{m\mathbf{k}}\rangle$ (band index $m$ and wavevector $\mathbf{k}$), the core quantity is the projectability:

$$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$

where $\{|g_n\rangle\}$ are the atomic-like projectors. This total projectability $p_{m\mathbf{k}}$ quantifies the overlap of $|\psi_{m\mathbf{k}}\rangle$ with the projector manifold. For a complete set, $p_{m\mathbf{k}} \to 1$; for free-electron-like states, $p_{m\mathbf{k}} \ll 1$.

The mathematical framework underlying PDWFs closely follows the Marzari–Vanderbilt localization paradigm. The functional minimized is the quadratic spread

$$\Omega = \sum_{n=1}^J \left[ \langle w_{n0} | \mathbf{r}^2 | w_{n0} \rangle - |\langle w_{n0} | \mathbf{r} | w_{n0} \rangle|^2 \right]$$

where the Wannier functions $m$0 are built from the Bloch manifold using semi-unitary matrices $m$1.

2. Selection Criteria and Disentanglement Algorithm

PDWF subspace selection is governed by projectability-based decision thresholds:

• If $m$2: discard the state from the Wannier subspace.
• If $m$3: freeze the state (include it as-is).
• If $m$4: include the state in the disentanglement optimization (“flexible”).

Typical default values are $m$5–$m$6 and $m$7, but these are tunable to target specified interpolation errors. The actual Wannier subspace is constructed by (i) including all frozen bands, (ii) excluding discarded bands, and (iii) variationally building a smooth subspace from the “flexible” bands by minimizing the gauge-invariant part of the spread $m$8.

This process fully replaces the conventional practice of using inner and outer energy windows in the Souza–Marzari elucidation of entangled manifolds. The two-step minimization, first of $m$9 (disentanglement) and then the remainder ($\mathbf{k}$0, for localization), yields compact and well-localized MLWFs (Qiao et al., 2023, Jiang et al., 9 Jul 2025).

3. Projector Manifold Augmentation and Automated Robustness

In a nontrivial fraction of cases (∼3–12% of materials, especially near $\mathbf{k}$1 eV above the Fermi level), the default set of PAOs may fail to fully span the relevant Hilbert space, leading to poor projectability and interpolation discontinuities. To address this, PDWFs incorporate an automated mechanism for extending the projector manifold by generating additional hydrogenic AOs:

• For each chemical element, minimal sets are defined (e.g., $\mathbf{k}$2 for Period 1; $\mathbf{k}$3 for Periods 2–3; $\mathbf{k}$4 for transition metals, etc.).
• Hydrogenic AOs are generated with radial parameter $\mathbf{k}$5 fitted to PAO shapes and orthogonalized via Löwdin and Gram–Schmidt orthonormalization.
• This augmentation continues iteratively until all target states are sufficiently projectable (i.e., $\mathbf{k}$6 at all relevant $\mathbf{k}$7).

This procedural step achieves pseudopotential-agnostic, fully automatic Wannierization, and guarantees >99% success without manual tuning across large and diverse materials sets (Jiang et al., 9 Jul 2025).

4. Spin–Orbit Coupling, Magnetization, and Advanced Protocols

The PDWF protocol has been extended to accommodate spin–orbit coupling (SOC) and arbitrary magnetization:

• For SOC, Bloch states are two-component spinors $\mathbf{k}$8, while projectors become $\mathbf{k}$9-dependent (total angular momentum eigenstates), typically derived from fully relativistic pseudopotentials (e.g., PseudoDojo or PAW from pslibrary).
• Magnetization is treated as collinear (separate spin-up and spin-down Wannierizations from spin-polarized DFT) or non-collinear (with SOC, via joint spinor analysis and using a dedicated MagneticStructureData plugin in AiiDA).
• Workflow modules (Quantum ESPRESSO, Wannier90, AiiDA-wannier90-workflows) accommodate these extensions, supporting parallel computation and robust automation.

PDWFs thus support accurate tight-binding models even for systems where exchange-correlation spin effects and relativistic interactions are pronounced (Jiang et al., 9 Jul 2025).

5. Workflow Implementation and Automation

PDWFs are implemented as a fully automated protocol combining:

• DFT (Quantum ESPRESSO with suitable pseudopotentials for either non-relativistic or relativistic cases).
• Projector generation scripts that augment PAOs with hydrogenic AOs as needed.
• Automated overlap computation, projectability analysis, subspace selection, and spread functional minimization in the Wannier90 code.
• Post-processing for tight-binding Hamiltonian construction and band-structure interpolation.
• Comprehensive workflow management by AiiDA infrastructure, enabling k-point pool parallelization and end-to-end reproducibility (Qiao et al., 2023, Jiang et al., 9 Jul 2025).

Typical workflow steps are:

1. DFT: obtain $$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$0, and projectors $$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$1.
2. Projector analysis and augmentation (if needed).
3. Compute all overlaps $$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$2 and $$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$3.
4. Apply PDWF selection and perform disentanglement/localization.
5. Construct Wannier functions and real-space Hamiltonian.

6. Performance Benchmarks and Comparative Analysis

Comprehensive benchmarks have established the superior accuracy and locality of PDWFs relative to other protocols:

Method	Mean $$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$4 (meV)	Max $$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$5 (meV)	MLWF Mean Spread (Å$$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$6)	Hamiltonian Decay $$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$7 (Å)	Interpolation Success (%)
SCDM	11.2	84	3.5	2.66	Not reported
PDWF	4.2	36.7	1.4	2.27	$$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$897
Robust PDWF (+hydrogenic, no SOC)	2.00	$$p_{mn\mathbf{k}} = |\langle g_n | \psi_{m\mathbf{k}} \rangle|^2, \quad p_{m\mathbf{k}} = \sum_{n} p_{mn\mathbf{k}}$$915	Not specified	Not specified	100
Standard PDWF (SOC, PseudoDojo)	10.14	Not specified	Not specified	Not specified	87.5
Robust PDWF (+hydrogenic, SOC)	$\{|g_n\rangle\}$02.1	$\{|g_n\rangle\}$115	Not specified	Not specified	100

Key metrics used include:

• Band-distance $\{|g_n\rangle\}$2 (root mean squared error up to $\{|g_n\rangle\}$3 eV),
• Wannier function quadratic spread $\{|g_n\rangle\}$4,
• Real-space Hamiltonian decay length $\{|g_n\rangle\}$5 via $\{|g_n\rangle\}$6,
• Wannier center positions relative to atoms.

In high-throughput tests across tens of thousands of materials, robust PDWF protocols (with hydrogenic projector augmentation) achieved $\{|g_n\rangle\}$7 meV and $\{|g_n\rangle\}$8100% success without manual intervention (Jiang et al., 9 Jul 2025, Qiao et al., 2023).

7. Applications, Case Studies, and Best Practices

PDWFs have been applied to diverse systems and phenomena:

• Graphene: Avoids free-electron contamination in $\{|g_n\rangle\}$9 bands via customized $p_{m\mathbf{k}}$0/$p_{m\mathbf{k}}$1, yielding $p_{m\mathbf{k}}$2, $p_{m\mathbf{k}}$3-like Wannier functions with meV-level interpolation.
• Silicon: Conduction bands with $p_{m\mathbf{k}}$4 character are accurately Wannierized via either d-type PAO inclusion or joint projectability/disentanglement (PD+ED).
• Metals (Cu, SrVO$p_{m\mathbf{k}}$5): Achieves atomic $p_{m\mathbf{k}}$6- and $p_{m\mathbf{k}}$7-like MLWFs with $p_{m\mathbf{k}}$85 meV interpolation errors; more localized and less k-point-intensive than SCDM Wannierization.
• High-throughput materials screening: Applied to 21,737 nonmagnetic materials, with >97% achieving $p_{m\mathbf{k}}$9 meV automatically (Qiao et al., 2023).

Best practices include use of high-quality PAOs as projectors, initializing with $|\psi_{m\mathbf{k}}\rangle$0/$|\psi_{m\mathbf{k}}\rangle$1 and a frozen energy window $|\psi_{m\mathbf{k}}\rangle$2, automating threshold refinement to minimize $|\psi_{m\mathbf{k}}\rangle$3, and validating output MLWF centers and spreads. For robust operation, the “projector generation” step for hydrogenic AOs is recommended wherever low projectability arises, enabling consistent operation across different pseudopotential sets and aberrant chemical environments (Jiang et al., 9 Jul 2025).

PDWFs have thus enabled automated, chemically meaningful, and highly accurate tight-binding models suitable for both detailed single-system analyses and large-scale materials informatics.