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arxiv: 2606.19471 · v1 · pith:TC5RY7BGnew · submitted 2026-06-17 · 🧮 math.NA · cond-mat.mtrl-sci· cs.NA· math.FA· physics.chem-ph

Moreau-Yosida-based Kohn-Sham Inversion for Periodic Systems

Vebj{\o}rn H. Bakkestuen , Michael F. Herbst , Vegard Falm{\aa}r , Markus Penz , Andre Laestadius This is my paper

Pith reviewed 2026-06-26 19:55 UTC · model grok-4.3

classification 🧮 math.NA cond-mat.mtrl-scics.NAmath.FAphysics.chem-ph
keywords Moreau-Yosida regularizationKohn-Sham inversiondensity-potential inversionperiodic systemsdensity functional theoryproximal mappingSobolev spaceexchange-correlation potential
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The pith

Moreau-Yosida regularization recovers the Kohn-Sham exchange-correlation potential for periodic systems via a limiting procedure.

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

This paper develops a density-potential inversion framework for periodic systems in density functional theory by working inside a periodic homogeneous Sobolev space and applying Moreau-Yosida regularization. It proves lower semicontinuity of the non-interacting kinetic-energy functional to support a limiting procedure that extracts the exchange-correlation potential. The proximal mapping supplies both the theoretical backbone and the practical algorithmic step for the inversion. Numerical experiments on the Kohn-Sham and Gross-Pitaevskii equations illustrate convergence behavior and practical properties. Readers gain a concrete route from computed densities back to potentials under periodic boundary conditions typical of solid-state calculations.

Core claim

The framework recovers the exchange-correlation potential of Kohn-Sham theory through a limiting procedure after developing the inversion in a periodic homogeneous Sobolev space, with the proximal mapping playing a central role.

What carries the argument

The proximal mapping of the Moreau-Yosida regularization applied to the non-interacting kinetic-energy functional inside the periodic homogeneous Sobolev space.

If this is right

  • The inversion recovers the exchange-correlation potential for periodic Kohn-Sham systems.
  • The same scheme applies to the Gross-Pitaevskii equation.
  • Algorithmic evaluation of the proximal mapping yields a practical numerical method.
  • The lower-semicontinuity result supplies the analytical justification for the limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The periodic Sobolev setting could be relaxed to other function spaces if analogous semicontinuity can be shown.
  • The proximal-mapping iteration might be combined with existing plane-wave DFT codes to reconstruct potentials from output densities.
  • The same regularization pattern may extend to time-dependent or finite-temperature periodic problems.

Load-bearing premise

The non-interacting kinetic-energy functional is lower semicontinuous in the chosen periodic homogeneous Sobolev topology.

What would settle it

A sequence of densities that converges in the periodic homogeneous Sobolev norm but violates the lower-semicontinuity inequality for the kinetic-energy functional would invalidate the limiting recovery step.

Figures

Figures reproduced from arXiv: 2606.19471 by Andre Laestadius, Markus Penz, Michael F. Herbst, Vebj{\o}rn H. Bakkestuen, Vegard Falm{\aa}r.

Figure 1
Figure 1. Figure 1: FIG. 1. The convergence of the inversion scheme for different methods of computing the proximal density. (top) The convergence [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Real space plot of the potentials [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The mean computational time for the proximal point algorithm given in Eq. (31) in order to reach the desired tolerance [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (top) Real-space plot of the reference xc potential for sodium chloride (NaCl) at the level of PBE [86] with pseu [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Convergence of KS inversion for Si ( [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Convergence of KS inversion for Si using different guiding functionals, [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The convergence of the KS inversion with references density [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
read the original abstract

Density-potential inversion for periodic systems within Moreau-Yosida-regularised density-functional theory is investigated, both theoretically and numerically. We develop the framework in a periodic homogeneous Sobolev space and use it to recover the exchange-correlation potential of Kohn-Sham theory through a limiting procedure. A key analytical ingredient is the proof of lower semicontinuity of the non-interacting kinetic-energy functional in the chosen topology. The proximal mapping, together with its algorithmic evaluation, plays a central role in the resulting inversion scheme. Numerical experiments illustrate the performance and properties of the method for both the Kohn-Sham and Gross-Pitaevskii equations.

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

1 major / 2 minor

Summary. The paper develops a density-potential inversion framework for periodic systems using Moreau-Yosida regularization of density-functional theory, formulated in a periodic homogeneous Sobolev space. It proves lower semicontinuity of the non-interacting kinetic-energy functional in this topology as the key analytical step, employs the proximal mapping (with algorithmic evaluation) to construct the inversion scheme, and recovers the exchange-correlation potential of Kohn-Sham theory via a limiting procedure. Numerical experiments demonstrate the method on both Kohn-Sham and Gross-Pitaevskii equations.

Significance. If the lower semicontinuity result and limiting recovery hold, the work supplies a mathematically rigorous route to stable density inversion for periodic systems, with the proximal-mapping construction offering a concrete algorithmic tool. The combination of functional-analytic proof and numerical illustration strengthens the case for applicability in computational materials modeling.

major comments (1)
  1. [Section on limiting procedure] The manuscript states that the limiting procedure recovers the XC potential, but the precise conditions under which the limit coincides with the exact XC potential (without additional regularity assumptions on the density) are not fully spelled out; a short remark clarifying this would strengthen the central claim.
minor comments (2)
  1. [Introduction] Notation for the periodic homogeneous Sobolev space could be introduced earlier with an explicit definition of the norm to aid readability.
  2. [Numerical experiments] The numerical section would benefit from a brief statement of the discretization parameters and convergence tolerance used in the proximal-mapping iterations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Section on limiting procedure] The manuscript states that the limiting procedure recovers the XC potential, but the precise conditions under which the limit coincides with the exact XC potential (without additional regularity assumptions on the density) are not fully spelled out; a short remark clarifying this would strengthen the central claim.

    Authors: We agree that a clarifying remark would strengthen the presentation. In the revised manuscript we will insert a short paragraph immediately following the statement of the limiting procedure. The remark will explicitly state that, within the periodic homogeneous Sobolev-space setting already adopted, the limit of the regularized potentials recovers the exact Kohn-Sham exchange-correlation potential for any density in the admissible set, without invoking further regularity assumptions on the density beyond those required for the lower-semicontinuity result and the proximal-mapping construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper develops an inversion framework in periodic homogeneous Sobolev space, proves lower semicontinuity of the non-interacting kinetic-energy functional as an original analytical step, and recovers the exchange-correlation potential via a limiting procedure using the proximal mapping as an algorithmic construct. No load-bearing step reduces by construction to fitted inputs, self-definitions, or self-citation chains; the central proof and limiting argument are self-contained against the stated topology and functional properties, with numerical illustrations following directly from the derived scheme.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and properties of the proximal mapping in the chosen function space and on the validity of the limiting procedure that recovers the XC potential; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Lower semicontinuity of the non-interacting kinetic-energy functional holds in the periodic homogeneous Sobolev topology.
    Explicitly identified as the key analytical ingredient required for the limiting procedure.

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