Projectability-Disentangled Wannier Functions

• PDWFs are maximally-localized Wannier functions constructed by filtering Bloch states based on their projection onto selected atomic-like orbitals.
• The method uses high and low projectability thresholds to freeze, discard, or mix states, yielding compact tight-binding representations with lower interpolation errors.
• Extensions for magnetism and spin–orbit coupling, including automated projector manifold expansion, enhance the method’s applicability across diverse materials.

Projectability-disentangled Wannier functions (PDWFs) are maximally-localized Wannier functions (MLWFs) constructed from a Bloch subspace selected and disentangled according to the projectability of each Bloch state onto a chosen localized-orbital manifold, typically pseudo-atomic orbitals (PAOs) taken from pseudopotential projectors. In this formulation, states with high projectability are kept identically, states with negligible projectability are discarded, and states with intermediate projectability are mixed within a Souza–Marzari–Vanderbilt (SMV)-type disentanglement procedure. The resulting Wannier Hamiltonians are intended to span all occupied bands together with their “natural complement” in the unoccupied manifold, yielding compact tight-binding representations of optimized atomic orbitals in crystals (Qiao et al., 2023). Subsequent work extended the same logic to collinear and non-collinear magnetism and to spin–orbit-coupled systems, while introducing an automatic expansion of the projector manifold with additional hydrogenic atomic orbitals when PAOs alone are insufficient (Jiang et al., 9 Jul 2025).

1. Definition and conceptual basis

The defining object in PDWFs is the projectability of a Bloch state $|\psi_{m\mathbf{k}}\rangle$ onto a localized basis $\{|g_n\rangle\}$. The orbital-resolved projectability is

$$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$

and the total projectability is

$$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$

If the localized orbitals form a complete basis for the Bloch state, then $p_{m\mathbf{k}}=1$; if the Bloch state is free-electron-like and has negligible atomic character, then $p_{m\mathbf{k}}\approx 0$ (Qiao et al., 2023).

PDWFs replace the usual energy-only disentanglement logic by “projectability windows.” Two thresholds are introduced: a high threshold $p_\text{keep}$ and a low threshold $p_\text{drop}$. A Bloch state is discarded if $p_{m\mathbf{k}} < p_\text{drop}$, kept identically if $p_{m\mathbf{k}} \ge p_\text{keep}$, and included in the disentangling pool if $\{|g_n\rangle\}$0. The selection of frozen, discarded, and mixable states is therefore not based on energy alone, but on how well each state is represented in the chosen atomic-like basis (Qiao et al., 2023).

This construction targets a specific type of low-energy model. The occupied manifold is always included, and the unoccupied manifold is augmented only by conduction states that continue the same orbital characters, such as antibonding partners of valence orbitals. The paper characterizes this as the occupied bands and their “natural complement” for the empty states, producing a tight-binding picture of optimized atomic orbitals in crystals (Qiao et al., 2023).

A common misconception is that PDWFs define a new localization functional. They do not. The novelty lies in the choice of subspace. Once that subspace has been selected through projectability-guided disentanglement, the method proceeds with the standard MLWF machinery.

2. Formal construction and algorithmic realization

The starting point is a DFT calculation, implemented in the original work with Quantum ESPRESSO, providing Bloch eigenstates $\{|g_n\rangle\}$1 and eigenvalues $\{|g_n\rangle\}$2, together with a set of localized orbitals $\{|g_n\rangle\}$3 chosen as PAOs from the pseudopotential. Projection matrices are defined by

$\{|g_n\rangle\}$4

As in standard Wannier90, trial Bloch-like functions are formed as

$\{|g_n\rangle\}$5

and after Löwdin orthonormalization they define the initial gauge for the Wannier transformation

$\{|g_n\rangle\}$6

For isolated bands $\{|g_n\rangle\}$7 is square and unitary; for entangled manifolds it is rectangular and semi-unitary (Qiao et al., 2023).

The disentanglement step itself remains the SMV procedure. PDWFs redefine which states are available to that procedure. High-projectability states are frozen, low-projectability states are excluded, and only the intermediate states are mixed to construct a smooth $\{|g_n\rangle\}$8-dimensional subspace across the Brillouin zone. After this subspace has been obtained, the Marzari–Vanderbilt spread functional

$\{|g_n\rangle\}$9

is minimized without modification (Qiao et al., 2023).

Typical benchmark values are $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$0 and $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$1 or $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$2. In graphene, a representative choice is $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$3 and $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$4, which freezes high-projectability $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$5 and $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$6 states, discards free-electron bands with $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$7, and disentangles mixed-character states in between. In practice PD is often combined with a small frozen energy window, denoted PD+ED, to guarantee exact reproduction of bands within approximately $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$8 eV of $$p_{nm\mathbf{k}} = \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2,$$9 in metals or up to CBM $$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$0 eV in insulators when projectability alone is not sufficient (Qiao et al., 2023).

The real-space Hamiltonian in the Wannier basis is

$$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$1

with Bloch-space representation

$$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$2

Its Frobenius norm decays approximately as

$$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$3

where the fitted decay length $$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$4 quantifies the range of the effective tight-binding model (Qiao et al., 2023).

Implementation is centered on pw2wannier90.x and Wannier90. PAO projections are computed in $$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$5-space, which the authors describe as cheaper than the QRCP step used in SCDM. A “k-pool” parallelization is implemented in pw2wannier90.x. In automated AiiDA workflows, Wannierization is performed on regular $$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$6-point grids with spacing approximately $$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$7, and the workflow scans $$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$8 from $$p_{m\mathbf{k}} = \sum_n p_{nm\mathbf{k}} = \sum_n \left| \langle g_n | \psi_{m\mathbf{k}} \rangle \right|^2.$$9 down to $p_{m\mathbf{k}}=1$0 in steps of $p_{m\mathbf{k}}=1$1, with $p_{m\mathbf{k}}=1$2 or $p_{m\mathbf{k}}=1$3, to reach a target $p_{m\mathbf{k}}=1$4 such as $p_{m\mathbf{k}}=1$5 meV (Qiao et al., 2023).

3. Relation to energy disentanglement, SCDM, and closest-Wannier constructions

Standard SMV disentanglement uses an outer energy window and an inner frozen window. The same windows are applied at all $p_{m\mathbf{k}}=1$6, even though bands disperse. The original PDWF paper identifies the resulting pathologies directly: if the window is too narrow, antibonding partners can be missed; if it is too wide, free-electron bands can be included, leading to delocalized Wannier functions or “ghost” bands. PDWFs address this by making selection local in $p_{m\mathbf{k}}=1$7: atomic-like states are retained regardless of their energy, while free-electron-like states can be removed even when they lie in the same energy range (Qiao et al., 2023).

The most detailed quantitative comparison in the original PDWF work is against the selected-columns-of-the-density-matrix algorithm (SCDM) on a 200-material benchmark. SCDM is fully automated once the quasi-density-matrix filter parameters are fixed, but it remains basis-neutral and can include more free-electron-like components. PDWFs instead exploit the PAO manifold directly.

Quantity on 200 materials	PDWF	SCDM
Mean $p_{m\mathbf{k}}=1$8	$p_{m\mathbf{k}}=1$9 meV	$p_{m\mathbf{k}}\approx 0$0 meV
Mean $p_{m\mathbf{k}}\approx 0$1	$p_{m\mathbf{k}}\approx 0$2 meV	$p_{m\mathbf{k}}\approx 0$3 meV
Success rate for $p_{m\mathbf{k}}\approx 0$4 meV	96.5%	91.5%
Initial average spread $p_{m\mathbf{k}}\approx 0$5	$p_{m\mathbf{k}}\approx 0$6	$p_{m\mathbf{k}}\approx 0$7
Final average spread $p_{m\mathbf{k}}\approx 0$8	$p_{m\mathbf{k}}\approx 0$9	$p_\text{keep}$0
Mean Hamiltonian decay length $p_\text{keep}$1	$p_\text{keep}$2 Å	$p_\text{keep}$3 Å

These figures support the claim that PDWF Hamiltonians are both more accurate in interpolation up to $p_\text{keep}$4 eV and more local in real space. The paper also reports that PDWF Wannier centers are systematically closer to atoms, whereas SCDM more often yields bond-centered or floating Wannier functions (Qiao et al., 2023).

A related but distinct development is Ozaki’s closest Wannier functions (CWFs), which construct orthonormal Wannier functions closest to a chosen set of guiding localized orbitals by minimizing a Frobenius-norm distance functional. The optimum is obtained non-iteratively through the singular value decomposition and polar decomposition of a projection matrix $p_\text{keep}$5, and the singular values explicitly diagnose projectability onto the guiding set. The detailed synthesis in the supplied material argues that this makes CWFs a projectability-aware, local-in-$p_\text{keep}$6 alternative to spread-driven disentanglement, and that the singular values provide a natural projectability measure for PDWF-style filtering (Ozaki, 2023).

4. Benchmarks and representative materials

The original PDWF study benchmarked the method on the same 200-system test set used by Vitale et al., including metals and valence-plus-conduction descriptions of insulators. The band-distance metric was defined as

$p_\text{keep}$7

with a corresponding maximum metric

$p_\text{keep}$8

For $p_\text{keep}$9 eV, PDWFs achieved mean $p_\text{drop}$0 meV and mean $p_\text{drop}$1 meV. The same work then applied the automated workflow to 21,737 non-magnetic materials from Materials Cloud MC3D, obtaining mean $p_\text{drop}$2 meV, mean $p_\text{drop}$3 meV, and a fraction with $p_\text{drop}$4 meV of about 97.8% (21,259 / 21,737); among those “good” cases, the mean $p_\text{drop}$5 was approximately $p_\text{drop}$6 meV (Qiao et al., 2023).

Graphene is the canonical example of why projectability selection differs from energy-window selection. Its conduction manifold contains many free-electron bands intertwined with $p_\text{drop}$7 and $p_\text{drop}$8 states of strong C $p_\text{drop}$9 character. If an energy-only inner window is set too high, free-electron bands become frozen and delocalized Wannier functions result; if the outer window is too low, antibonding partners are lost. PDWFs instead drop the free-electron bands through low projectability and freeze the atomic-like conduction states even in the same energy region, producing $p_{m\mathbf{k}} < p_\text{drop}$0- and $p_{m\mathbf{k}} < p_\text{drop}$1-like Wannier functions that retain the orbital bands embedded inside the free-electron manifold (Qiao et al., 2023).

Silicon illustrates the method’s dependence on projector completeness. The conduction-band minimum along $p_{m\mathbf{k}} < p_\text{drop}$2–X has significant $p_{m\mathbf{k}} < p_\text{drop}$3 character. With only $p_{m\mathbf{k}} < p_\text{drop}$4 and $p_{m\mathbf{k}} < p_\text{drop}$5 PAOs, its projectability is approximately $p_{m\mathbf{k}} < p_\text{drop}$6, so pure PD with $p_{m\mathbf{k}} < p_\text{drop}$7 does not freeze the CBM and interpolation errors appear there. Two remedies are explicitly given. In PD+ED, states within CBM $p_{m\mathbf{k}} < p_\text{drop}$8 eV are frozen by energy regardless of projectability. Alternatively, a pseudopotential including $p_{m\mathbf{k}} < p_\text{drop}$9 PAOs is used; then the CBM projectability rises to approximately $p_{m\mathbf{k}} \ge p_\text{keep}$0 and pure PD becomes accurate (Qiao et al., 2023).

The paper also reports metallic examples such as Cu and SrVO$p_{m\mathbf{k}} \ge p_\text{keep}$1, where PD+ED yields very good interpolation around the Fermi level with a minimal orbital count, while the Wannier functions remain atomic-like, including localized $p_{m\mathbf{k}} \ge p_\text{keep}$2 states in SrVO$p_{m\mathbf{k}} \ge p_\text{keep}$3 (Qiao et al., 2023).

5. Magnetism, spin–orbit coupling, and projector-manifold expansion

A later extension generalized PDWFs to collinear magnetism, non-collinear magnetism, and spin–orbit coupling. In the SOC and non-collinear cases, the Kohn–Sham states are two-component spinors,

$p_{m\mathbf{k}} \ge p_\text{keep}$4

and the projector set must use $p_{m\mathbf{k}} \ge p_\text{keep}$5-dependent PAOs from fully relativistic pseudopotentials. The implementation modifies pw2wannier90.x to read and use $p_{m\mathbf{k}} \ge p_\text{keep}$6-dependent PAOs, to apply projectors to spinor plane-wave states, and to read external projector files for SOC as well. Collinear magnetism is treated as two independent scalar calculations, one for each spin channel, merged afterward into a block-diagonal tight-binding model (Jiang et al., 9 Jul 2025).

The central practical innovation of the extension is an “extended protocol” that automatically enlarges the projector manifold when the available PAOs do not adequately represent low-lying conduction states up to the target window, typically $p_{m\mathbf{k}} \ge p_\text{keep}$7 eV. The added projectors are hydrogenic atomic orbitals chosen by periodic-table rules. For periods $p_{m\mathbf{k}} \ge p_\text{keep}$8, the required set includes $p_{m\mathbf{k}} \ge p_\text{keep}$9 and $\{|g_n\rangle\}$00; for higher periods, alkali and alkaline earth elements require $\{|g_n\rangle\}$01 and $\{|g_n\rangle\}$02, transition metals require $\{|g_n\rangle\}$03, $\{|g_n\rangle\}$04, and $\{|g_n\rangle\}$05, and post-transition elements from the B group to noble gases require $\{|g_n\rangle\}$06 and $\{|g_n\rangle\}$07. Missing orbitals relative to a given pseudopotential library are filled using analytic hydrogenic radial functions, with $\{|g_n\rangle\}$08 parameters fitted either by enforcing orthogonality to existing PAOs of the same angular momentum or by reproducing OpenMX-like PAOs when no such inner shell exists (Jiang et al., 9 Jul 2025).

The final combined projector set is orthonormalized in two stages: Löwdin orthonormalization within the PAO subset and within the hydrogenic subset, followed by modified Gram–Schmidt that keeps the PAO subset fixed while orthogonalizing the hydrogenic orbitals against the PAOs and against each other. The purpose is to preserve the chemistry encoded in the pseudopotential PAOs while repairing incompleteness in the low-energy conduction manifold (Jiang et al., 9 Jul 2025).

The AlCo example makes the mechanism explicit. With only the PAOs present in the PseudoDojo pseudopotentials, the calculation gives $\{|g_n\rangle\}$09 meV, and projectability drops rapidly above $\{|g_n\rangle\}$10, reaching essentially zero for some states around $\{|g_n\rangle\}$11 eV at the R point. Adding a Co $\{|g_n\rangle\}$12 hydrogenic AO raises the minimum projectability near the target window to approximately $\{|g_n\rangle\}$13, removes discontinuities, reduces Co Wannier spreads from $\{|g_n\rangle\}$14–$\{|g_n\rangle\}$15 to $\{|g_n\rangle\}$16–$\{|g_n\rangle\}$17, and lowers the band distance to $\{|g_n\rangle\}$18 meV (Jiang et al., 9 Jul 2025).

The benchmark results are correspondingly strong. For 200 chemically diverse materials without SOC, the combination PDWF + PAOs + external hydrogenic AOs gave mean $\{|g_n\rangle\}$19 meV, median $\{|g_n\rangle\}$20 meV, and a 100% success rate, defined as $\{|g_n\rangle\}$21 meV for all 200 systems. For 200 SOC materials, the protocol reduced PseudoDojo results from 25/200 systems with $\{|g_n\rangle\}$22 meV, mean $\{|g_n\rangle\}$23 meV, median $\{|g_n\rangle\}$24 meV, to mean $\{|g_n\rangle\}$25 meV and median $\{|g_n\rangle\}$26 meV, again with all 200 systems at $\{|g_n\rangle\}$27 meV. A modified-pslibrary set showed the same pattern, improving from 7/200 failures, mean $\{|g_n\rangle\}$28 meV, median $\{|g_n\rangle\}$29 meV, to mean $\{|g_n\rangle\}$30 meV and median $\{|g_n\rangle\}$31 meV with 100% success. The 40 worst systems from the earlier 21,737-material run were also repaired: after adding hydrogenic AOs and turning on the guiding-center option in Wannier90, all 40 had $\{|g_n\rangle\}$32 meV. For 16 collinear magnetic systems, the mean and median $\{|g_n\rangle\}$33 were both below $\{|g_n\rangle\}$34 meV; for 16 non-collinear systems, both were below $\{|g_n\rangle\}$35 meV (Jiang et al., 9 Jul 2025).

6. Limitations, theoretical context, and broader significance

PDWFs remain dependent on the quality and completeness of the localized orbital manifold. The original study states this explicitly: if important states have low projectability with the chosen PAOs, one must either add more PAOs or supplement PD with ED. Silicon’s $\{|g_n\rangle\}$36-character CBM is the clearest example. The later SOC extension identifies projector insufficiency as the main residual failure mode in automated runs and responds by expanding the projector manifold automatically rather than altering the core PD logic (Qiao et al., 2023).

Within the broader theory of entangled-band Wannier functions, PDWFs should be understood as a subspace-selection strategy rather than a resolution of all localization questions. The variational formulation for generalized MLWFs with entangled band structures shows that standard disentanglement can be interpreted as a splitting method for a constrained optimization over both subspace and gauge, and that in free-electron models the maximally-localized functions for entangled bands may decay only algebraically rather than exponentially. The same work shows that super-algebraic decay can be obtained by smoothing the gauge, which suggests that localization asymptotics are controlled not only by subspace choice but also by regularity in $\{|g_n\rangle\}$37-space (Damle et al., 2018).

An adjacent theoretical boundary is provided by results on compactly supported Wannier functions. In one dimension, a subspace admits a compactly supported orthogonal basis if and only if its projector is strictly local; in higher dimensions, strictly local projectors guarantee compactly supported hybrid Wannier functions along any chosen axis, and NN-reducibility provides a sufficient condition for compactly supported orthogonal bases. The same analysis shows that strictly local projectors are topologically trivial in many cases. This suggests that exact compact support is not the generic target for PDWFs, especially for topologically nontrivial manifolds; rather, PDWFs are best viewed as a practical route to compact, chemically interpretable, and automatable Wannier Hamiltonians within the limitations imposed by entanglement and topology (Sathe et al., 2020).

The original PDWF paper itself points toward several generalizations. It suggests using projectability-guided selection for correlated subspaces, combining PD with symmetry-adapted Wannierization in strongly entangled or topological systems, extending projectability beyond PAOs to more complex localized orbitals such as molecular units or spin–orbit-entangled bases, and coupling PD with variational formulations that minimize total spread over both subspace and gauge (Qiao et al., 2023). A plausible implication is that PDWFs occupy an intermediate position between chemically guided Wannierization and fully variational projector-based methods: they retain the standard MLWF localization machinery, but replace trial-and-error subspace selection by an explicitly projectability-resolved disentanglement protocol.