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arxiv: 2201.07751 · v4 · submitted 2022-01-19 · ❄️ cond-mat.mtrl-sci

Wannier Functions Dually Localized in Space and Energy

Aaron Mahler , Jacob Z. Williams , Neil Qiang Su , Weitao Yang This is my paper

Pith reviewed 2026-05-24 12:48 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords Wannier functionsdual localizationvalence and conduction bandsfrontier orbitalssiliconethylenecopperdensity functional approximation
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The pith

Wannier functions localized in both space and energy can be constructed from the combined valence and conduction bands, producing frontier orbitals near the Fermi energy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends a prior method for dual localization of Wannier functions, which had been limited to occupied bands, to now include both valence and conduction bands in one manifold. By minimizing a weighted combination of spatial spread and energy spread, the resulting functions remain localized while carrying orbital energy information. In examples, these functions recover bonding and antibonding character in bulk silicon and molecular ethylene, plus d-orbital features in metallic copper. The construction supports orbital-dependent calculations that tie functions to specific energy windows and produces fractional occupations that can adjust density functional total energies.

Core claim

Departing from previous work limited to the occupied manifold, the dual localization procedure applied to the joint valence-plus-conduction Bloch states produces Wannier functions that are simultaneously localized in real space and in energy. Near the Fermi level these functions correspond to frontier bonding and antibonding orbitals in silicon and ethylene and to d-orbital character in copper, while naturally inducing fractional occupations suitable for corrections within density functional approximations.

What carries the argument

The dual localization functional that minimizes a weighted sum of spatial variance and energy variance over the combined valence and conduction manifold.

Load-bearing premise

A single unitary transformation of the Bloch states can be chosen to keep both spatial and energy spreads small across the full set of valence and conduction bands without losing chemically useful orbital character.

What would settle it

Visual inspection or projection of the resulting functions in bulk silicon onto known bonding and antibonding orbitals shows no resemblance to frontier states.

Figures

Figures reproduced from arXiv: 2201.07751 by Aaron Mahler, Jacob Z. Williams, Neil Qiang Su, Weitao Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. Degeneracy pattern of the average energy [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Spatial cost Ω, energy cost Ξ, and total cost [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. DLWF density isoplots for silicon (isovalue 0 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. DLWF density isoplots of copper at isovalue 0 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Disentangled band structure of silicon used for the isosurface plots in the main text. The solid [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Disentangled band structure of copper used for the isosurface plots in the main text. The solid [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
read the original abstract

The construction of Wannier functions from Bloch orbitals offers a unitary freedom that can be exploited to yield Wannier functions with advantageous properties. Minimizing the spatial variance is a well-known choice; another, previously proposed for Wannier functions constructed from the occupied Bloch manifold, minimizes a weighted sum of spatial and energy variance. Departing from all previous work, we extend dual localization to include both valence and conduction bands together. Near the Fermi energy, these dually localized Wannier functions yield frontier (bonding and antibonding) orbitals in bulk silicon and molecular ethylene, as well as $d$-orbital character in metallic copper. Because they are both localized and retain information about the orbital energy spectrum, dually localized Wannier functions are well suited to orbital-dependent methods that associate Wannier functions with specific energy ranges. They naturally induce fractional occupations, allowing for corrections to the DFA total energy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends dual localization of Wannier functions—previously limited to the occupied manifold—to the combined valence-plus-conduction manifold by minimizing a weighted sum of spatial and energy variances. Numerical examples in bulk silicon, molecular ethylene, and metallic copper are presented to show that the resulting functions correspond to frontier bonding/antibonding orbitals near the Fermi energy (and d-character in Cu), with suggested utility for orbital-dependent methods and fractional-occupation corrections to DFA total energies.

Significance. If the construction reliably produces functions that are simultaneously spatially localized and energetically concentrated near EF without additional post-selection, the method would supply a practical route to energy-resolved localized orbitals for use in orbital-dependent functionals. The numerical demonstrations in three chemically distinct systems constitute the primary evidence offered for this utility.

major comments (2)
  1. [functional definition and extension paragraph] The central claim—that the dually localized WFs automatically yield frontier orbitals near EF—rests on the minimization of the weighted spatial-plus-energy variance functional over the full valence+conduction manifold. No explicit term, projection, or post-selection that biases the mean energies toward the gap is stated in the functional definition; the energy-variance contribution can be satisfied by any spread of energies across the wide window. The manuscript should clarify, with the explicit functional and optimization details, why the stationary points concentrate near EF rather than distributing across the manifold (see the section introducing the dual-localization functional and the paragraph describing the extension to unoccupied bands).
  2. [results for Si and C2H4] Table or figure presenting the orbital characters (e.g., the silicon or ethylene results): the reported frontier character is asserted to follow directly from the minimization, yet the energy variances and mean energies of the obtained WFs are not compared against a control minimization that omits the energy term. Without this comparison it is difficult to isolate the contribution of the dual-localization term to the observed concentration near EF.
minor comments (2)
  1. Notation for the weight parameter balancing spatial versus energy variance should be introduced once and used consistently; its numerical value(s) used in the reported calculations should be stated explicitly.
  2. The abstract states that the functions 'naturally induce fractional occupations'; the precise mapping from the dually localized WFs to these occupations is not spelled out in the main text and would benefit from a short algorithmic outline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [functional definition and extension paragraph] The central claim—that the dually localized WFs automatically yield frontier orbitals near EF—rests on the minimization of the weighted spatial-plus-energy variance functional over the full valence+conduction manifold. No explicit term, projection, or post-selection that biases the mean energies toward the gap is stated in the functional definition; the energy-variance contribution can be satisfied by any spread of energies across the wide window. The manuscript should clarify, with the explicit functional and optimization details, why the stationary points concentrate near EF rather than distributing across the manifold (see the section introducing the dual-localization functional and the paragraph describing the extension to unoccupied bands).

    Authors: We agree that the manuscript would benefit from a more explicit statement of the functional and a clearer discussion of the stationary points. The dual-localization functional is defined as the sum over Wannier functions of a weighted combination of spatial variance and energy variance, minimized via a unitary transformation of the combined valence-plus-conduction Bloch manifold. The energy-variance term favors linear combinations drawn from narrow energy windows within the manifold. In the presence of a band gap, the spatial-variance term preferentially selects the bonding and antibonding combinations near the gap because they permit superior real-space localization compared with states deeper in the valence or conduction bands. We will revise the relevant section to state the functional explicitly, describe the numerical optimization procedure, and add a brief argument explaining the preference for the gap region. This revision will be made without introducing any additional bias term or post-selection. revision: yes

  2. Referee: [results for Si and C2H4] Table or figure presenting the orbital characters (e.g., the silicon or ethylene results): the reported frontier character is asserted to follow directly from the minimization, yet the energy variances and mean energies of the obtained WFs are not compared against a control minimization that omits the energy term. Without this comparison it is difficult to isolate the contribution of the dual-localization term to the observed concentration near EF.

    Authors: The referee is correct that a control calculation omitting the energy term would help isolate its contribution. In the revised manuscript we will add results obtained by minimizing only the spatial variance (standard Marzari-Vanderbilt functional) over the identical combined valence-plus-conduction manifold for silicon and ethylene. We will report the mean energies and energy variances of the resulting functions alongside those from the dual-localization procedure, thereby demonstrating the role of the energy term in concentrating the Wannier functions near the Fermi energy. The new comparison will be presented in an updated table or figure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a standard variational extension

full rationale

The paper describes extending a known weighted spatial-plus-energy variance minimization (previously applied only to occupied bands) to the combined valence-plus-conduction manifold. The reported frontier orbital characters near the Fermi level are presented as numerical outcomes of this minimization, not as quantities defined into the functional or recovered by construction. No load-bearing premise reduces to a self-citation chain, fitted input renamed as prediction, or ansatz smuggled via prior work; the central construction remains an independent application of unitary freedom whose results can be checked against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Ledger inferred strictly from abstract text; full paper may contain additional parameters or assumptions not visible here.

free parameters (1)
  • weight parameter balancing spatial versus energy variance
    Abstract refers to minimization of a weighted sum; the relative weight is a tunable choice not fixed by prior literature.
axioms (1)
  • domain assumption Unitary freedom exists in the Bloch-to-Wannier transformation and can be used to minimize a joint spatial-energy variance functional
    This is the explicit premise stated in the abstract for constructing the dually localized functions.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Correcting Delocalization Error in Materials with Localized Orbitals and Linear-Response Screening

    cond-mat.mtrl-sci 2024-06 unverdicted novelty 6.0

    lrLOSC corrects delocalization error in DFT for materials, predicting fundamental gaps of eleven materials to within 0.22 eV while providing a nonzero total-energy correction.

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