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arxiv: 2503.18811 · v2 · pith:Z3YE6IWPnew · submitted 2025-03-24 · ❄️ cond-mat.mtrl-sci

Rototranslational sum rules for nuclear dynamics via traveling pseudopotentials

Massimiliano Stengel , Miquel Royo , Emilio Artacho This is my paper

Pith reviewed 2026-05-22 22:57 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords rototranslational sum rulespseudopotentialsGalilean covariancenonadiabatic regimeforce constantselectromagnetic susceptibilityLarmor theoremDrude weight
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0 comments X

The pith

Adapting pseudopotentials with velocity-dependent nonlocal operators restores Galilean covariance and rototranslational sum rules in nonadiabatic nuclear dynamics.

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

The paper derives exact sum rules that connect interatomic force constants directly to the frequency-dependent electromagnetic susceptibility of solids or molecules. These rules extend the familiar rototranslational symmetries into the nonadiabatic regime where nuclei move dynamically. Standard pseudopotential implementations break the rules because they do not account for nuclear motion. The solution is to let the nonlocal part of each pseudopotential travel with its nucleus by adding explicit velocity dependence. This single change makes the Schrödinger equation Galilean covariant again and equates mechanical rotations or translations with electromagnetic perturbations, removing inconsistencies such as violations of the Larmor theorem and mismatched definitions of the Drude weight.

Core claim

We establish a set of exact sum rules that relate the interatomic force constants to the frequency-dependent electromagnetic susceptibility of a solid or molecule, thereby generalizing the long-established principles of rototranslational symmetry to the nonadiabatic regime. Crucially, we show that in practical numerical implementations these sum rules are violated, unless special precautions are taken in the treatment of the atomic pseudopotentials. We solve these issues once and for all by correctly adapting the pseudopotential to the motion of the corresponding nucleus, with a velocity dependence of the nonlocal operator. This prescription restores the correct Galilean covariance of theSch

What carries the argument

The traveling pseudopotential: a velocity-dependent nonlocal operator that moves with its nucleus to enforce Galilean covariance of the Schrödinger equation.

If this is right

  • The sum rules between force constants and electromagnetic susceptibility hold exactly once the pseudopotential travels with the nucleus.
  • The Larmor theorem is satisfied in linear-response calculations of solids and molecules.
  • Inertial and electrical definitions of the Drude weight become equivalent in metals.
  • Mechanical rototranslations are identical to electromagnetic perturbations in the nonadiabatic regime.

Where Pith is reading between the lines

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

  • The same velocity adaptation could be applied to other nonlocal operators used in electronic-structure codes to remove similar hidden symmetry violations.
  • Numerical tests on isolated atoms or diatomic molecules would provide a minimal check that the restored sum rules are now satisfied to machine precision.
  • The approach supplies a concrete route to consistent calculations of coupled phonon and dielectric responses without ad-hoc corrections.

Load-bearing premise

Introducing velocity dependence into the nonlocal pseudopotential operator preserves the accuracy and transferability of the pseudopotential approximation across regimes without introducing new artifacts or requiring reparameterization.

What would settle it

A linear-response calculation of the Drude weight in a simple metal or small molecule performed both with and without the velocity-dependent pseudopotential; persistent mismatch between the inertial and electrical values after the change would falsify the restoration claim.

read the original abstract

We establish a set of exact sum rules that relate the interatomic force constants to the frequency-dependent electromagnetic susceptibility of a solid or molecule, thereby generalizing the long-established principles of rototranslational symmetry to the nonadiabatic regime. Crucially, we show that in practical numerical implementations these sum rules are violated, unless special precautions are taken in the treatment of the atomic pseudopotentials. We solve these issues once and for all by correctly adapting the pseudopotential to the motion of the corresponding nucleus, with a velocity dependence of the nonlocal operator. This prescription restores the correct Galilean covariance of the Schr\"odinger equation, and the expected identity between mechanical rototranslations and electromagnetic perturbations. These results conclusively fix a number of worrisome inconsistencies that were pointed out over the years in the context of linear-response theory restoring, e.g., the validity of the Larmor theorem, and the equivalence between the inertial and electrical definitions of the Drude weight in metals.

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 establishes exact sum rules relating interatomic force constants to the frequency-dependent electromagnetic susceptibility of solids or molecules, generalizing rototranslational symmetry to the nonadiabatic regime. It identifies violations of these sum rules in standard numerical implementations due to the treatment of atomic pseudopotentials and proposes a fix by adapting the pseudopotential to nuclear motion via a velocity-dependent nonlocal operator. This restores Galilean covariance of the Schrödinger equation and the identity between mechanical rototranslations and electromagnetic perturbations, resolving inconsistencies such as the Larmor theorem and the equivalence of inertial and electrical Drude weights in metals.

Significance. If the derivations hold, the work is significant for computational materials science and linear-response theory. It provides a symmetry-based, parameter-free correction (relying on Galilean covariance rather than fitted quantities) that addresses long-standing inconsistencies in pseudopotential-based calculations of nuclear dynamics and electromagnetic responses. This could improve reliability of phonon, dielectric, and transport predictions without introducing new free parameters.

major comments (2)
  1. [Abstract] The central claim (abstract) that velocity dependence of the nonlocal operator in the pseudopotential restores Galilean covariance and sum-rule validity is load-bearing; the manuscript must supply the explicit derivation or proof of this restoration, as the abstract states the result without detailing the steps or showing how the velocity term enters the Schrödinger equation.
  2. The approach rests on the assumption that the velocity-dependent nonlocal operator preserves pseudopotential accuracy and transferability without new artifacts or reparameterization (reader's weakest assumption); this needs explicit demonstration via a test case or argument in the main text, as it directly affects whether the fix can be applied in practice across regimes.
minor comments (2)
  1. Clarify notation for the velocity-dependent operator and ensure all equations are numbered with cross-references.
  2. Add a brief discussion of how the proposed operator is implemented numerically in existing codes.

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.

read point-by-point responses

  1. Referee: [Abstract] The central claim (abstract) that velocity dependence of the nonlocal operator in the pseudopotential restores Galilean covariance and sum-rule validity is load-bearing; the manuscript must supply the explicit derivation or proof of this restoration, as the abstract states the result without detailing the steps or showing how the velocity term enters the Schrödinger equation.

    Authors: The explicit derivation showing how the velocity-dependent term enters the Schrödinger equation and restores Galilean covariance is provided in Sections III and IV of the manuscript. The abstract is a concise summary, as is conventional. To make the logical steps more immediately accessible, we have added a short outline paragraph in the introduction that references the relevant equations and sections. revision: yes

  2. Referee: The approach rests on the assumption that the velocity-dependent nonlocal operator preserves pseudopotential accuracy and transferability without new artifacts or reparameterization (reader's weakest assumption); this needs explicit demonstration via a test case or argument in the main text, as it directly affects whether the fix can be applied in practice across regimes.

    Authors: We have added an argument in the revised discussion section explaining that the operator is constructed to be identical to the original pseudopotential in the instantaneous rest frame of each nucleus, thereby preserving matrix elements, accuracy, and transferability by design without requiring reparameterization. A dedicated numerical test case lies outside the scope of the present theoretical work but could be pursued in follow-up studies. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives rototranslational sum rules from Galilean covariance and symmetry requirements applied to the Schrödinger equation with adapted pseudopotentials. These are external physical constraints, not self-definitions or fitted inputs. No step reduces a claimed prediction to a parameter fit or self-citation chain; the velocity-dependent nonlocal operator is introduced to enforce covariance rather than being defined circularly from the target sum rules. The approach is self-contained against external benchmarks like Larmor theorem and Drude weight equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard quantum-mechanical symmetries and domain assumptions from electronic structure theory; the velocity-dependent operator is introduced as a methodological adjustment rather than a new physical entity.

axioms (2)
  • standard math Galilean covariance must hold for the Schrödinger equation in the presence of moving nuclei
    Invoked as the physical requirement that the pseudopotential adaptation must satisfy.
  • domain assumption Rototranslational symmetry holds exactly in the adiabatic limit and extends to the nonadiabatic regime via sum rules
    Basis for generalizing the sum rules to frequency-dependent susceptibility.
invented entities (1)
  • velocity-dependent nonlocal operator in pseudopotential no independent evidence
    purpose: To enforce sum rules and restore Galilean invariance in numerical implementations
    Introduced as the technical solution to observed violations; no independent evidence provided in abstract.

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    G. B. Bachelet, D. R. Hamann, and M. Schl¨ uter, Pseudopotentials that work: From h to pu, Phys. Rev. B 26, 4199 (1982)

    work page 1982

  2. [2]

    D. R. Hamann, M. Schl¨ uter, and C. Chi- ang, Norm-conserving pseudopotentials, Phys. Rev. Lett. 43, 1494 (1979)

    work page 1979

  3. [3]

    Troullier and J

    N. Troullier and J. L. Martins, Efficient pseudopoten- tials for plane-wave calculations, Phys. Rev. B 43, 1993 (1991)

    work page 1993

  4. [4]

    Kleinman and D

    L. Kleinman and D. M. Bylander, Efficacious form for model pseudopotentials, Phys. Rev. Lett. 48, 1425 (1982)

    work page 1982

  5. [5]

    Vanderbilt, Soft self-consistent pseudopotentials in a generalized eigenvalue formalism, Phys

    D. Vanderbilt, Soft self-consistent pseudopotentials in a generalized eigenvalue formalism, Phys. Rev. B , 7892 (1990)

    work page 1990

  6. [6]

    P. E. Bl¨ ochl, Projector augented-wave method, Phys. Rev. B 50, 17953 (1994)

    work page 1994

  7. [7]

    D. R. Hamann, Optimized norm-conserving vanderbilt pseudopotentials, Phys. Rev. B 88, 085117 (2013)

    work page 2013

  8. [11]

    C. J. Pickard and F. Mauri, All-electron magnetic re- sponse with pseudopotentials: Nmr chemical shifts, Phys. Rev. B 63, 245101 (2001)

    work page 2001

  9. [14]

    D. R. Bates, H. S. W. Massey, and A. Stewart, Inelastic collisions between atoms i. general theoretical considera - tions, Proc. Royal Soc. London A 216, 437 (1953)

    work page 1953

  10. [15]

    Crothers and J

    D. Crothers and J. G. Hughes, H (2s) formation and the lyman- α polarization in 1-7-kev h+-h collisions, Phys. Rev. Lett. 43, 1584 (1979)

    work page 1979

  11. [16]

    Kimura and C

    M. Kimura and C. Lin, Unified treatment of slow atom-atom and ion-atom collisions, Phys. Rev. A 31, 590 (1985)

    work page 1985

  12. [17]

    Kwong, Coupling between quantal and classical degrees of freedom in atomic collisions, J

    N.-H. Kwong, Coupling between quantal and classical degrees of freedom in atomic collisions, J. Phys. B: Atom. Molec. Phys. 20, L647 (1987)

    work page 1987

  13. [18]

    Todorov, Time-dependent tight binding, J

    T. Todorov, Time-dependent tight binding, J. Phys. Condens. Matter 13, 10125 (2001)

    work page 2001

  14. [19]

    For a periodic crystal, it boils down to operating a shif t in momentum space as k → k − me ˙Rlκ/ℏ

    work page

  15. [20]

    See Supplemental Material at http://link for more in- formation about theoretical derivations, computational details and results,

    work page

  16. [22]

    C. A. Mead and D. G. Truhlar, On the de- termination of born–oppenheimer nuclear mo- tion wave functions including complications due to conical intersections and identical nuclei, The Journal of Chemical Physics 70, 2284 (1979), https://pubs.aip.org/aip/jcp/article-pdf/70/5/2284/18916340/2284 1

    work page 1979

  17. [24]

    X. Bian, Z. Tao, Y. Wu, J. Rawlinson, R. G. Littlejohn, and J. E. Subotnik, Total angular momentum conserva- tion in ab initio born-oppenheimer molecular dynamics, Phys. Rev. B 108, L220304 (2023)

    work page 2023

  18. [26]

    (15b) can be equivalently writ- ten as −iωσαβ(ω)

    Since the susceptibility is related to the conductivit y via χ = iσ/ω, the rhs of Eq. (15b) can be equivalently writ- ten as −iωσαβ(ω). This indicates that a minus sign is missing in Eq. (5) of Ref. 9

    work page

  19. [27]

    Santerv´ as-Arranz, M

    N. Santerv´ as-Arranz, M. Stengel, and E. Artacho, Excess energy and countercurrents after a quantum kick (2025), arXiv:2502.09469 [cond-mat.mtrl-sci]. 6

    work page arXiv 2025

  20. [28]

    Trifunovic, S

    L. Trifunovic, S. Ono, and H. Watanabe, Geo- metric orbital magnetization in adiabatic processes, Phys. Rev. B 100, 054408 (2019)

    work page 2019

  21. [30]

    Ceresoli and E

    D. Ceresoli and E. Tosatti, Berry-phase calculation of magnetic screening and rotational g factor in molecules and solids, Phys. Rev. Lett. 89, 116402 (2002)

    work page 2002

  22. [31]

    Gonze, B

    X. Gonze, B. Amadon, P.-M. Anglade, J.-M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Cara- cas, M. Cˆ ot´ e, T. Deutsch, L. Genovese, P. Ghosez, M. Gi- antomassi, S. Goedecker, D. Hamann, P. Hermet, F. Jol- let, G. Jomard, S. Leroux, M. Mancini, S. Mazevet, M. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G.-M. Rignanese, D. Sangalli, R. S...

    work page 2009

  23. [32]

    Keith and R

    T. Keith and R. Bader, Calculation of magnetic response properties using atoms in molecules, Chemical Physics Letters 194, 1 (1992)

    work page 1992

  24. [33]

    Royo and M

    M. Royo and M. Stengel, First-principles theory of spat ial dispersion: Dynamical quadrupoles and flexoelectricity, Phys. Rev. X 9, 021050 (2019)

    work page 2019

  25. [34]

    Rototranslational sum rules for nuclear dynamics via traveling pseudopotentials

    S. Ren, J. Bonini, M. Stengel, C. E. Dreyer, and D. Vanderbilt, Adiabatic dynamics of cou- pled spins and phonons in magnetic insulators, Phys. Rev. X 14, 011041 (2024). arXiv:2503.18811v1 [cond-mat.mtrl-sci] 24 Mar 2025 Supplemental material for “Pseudopotentials that move: From inertia to electromagne tism” Massimiliano Stengel, 1, 2 Miquel Royo, 1 and ...

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  26. [35]

    C. E. Dreyer, M. Stengel, and D. Vanderbilt, Current- density implementation for calculating flexoelectric coef - ficients, Phys. Rev. B 98, 075153 (2018)

    work page 2018

  27. [36]

    Gonze and C

    X. Gonze and C. Lee, Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomi c force constants from density-functional perturbation the - ory, Phys. Rev. B 55, 10355 (1997)

    work page 1997

  28. [37]

    Baroni, S

    S. Baroni, S. de Gironcoli, and A. D. Corso, Phonons and related crystal properties from density-functional pertu r- bation theory, Rev. Mod. Phys. 73, 515 (2001)

    work page 2001

  29. [38]

    C. E. Dreyer, S. Coh, and M. Stengel, Nonadiabatic born effective charges in metals and the drude weight, Phys. Rev. Lett. 128, 10.1103/physrevlett.128.095901 (2022)

    work page doi:10.1103/physrevlett.128.095901 2022

  30. [39]

    Schiaffino, C

    A. Schiaffino, C. E. Dreyer, D. Vanderbilt, and M. Stengel, Metric wave approach to flexoelectric- ity within density functional perturbation theory, Phys. Rev. B 99, 085107 (2019)

    work page 2019

  31. [40]

    Stengel and D

    M. Stengel and D. Vanderbilt, Quantum theory of me- chanical deformations, Phys. Rev. B 98, 125133 (2018)

    work page 2018

  32. [41]

    C. J. Pickard and F. Mauri, Nonlocal pseudopotentials and magnetic fields, Phys. Rev. Lett. 91, 196401 (2003)

    work page 2003

  33. [42]

    Vanderbilt, Berry phases in electronic structure theory (Cambridge University Press, Cambridge, UK, 2018)

    D. Vanderbilt, Berry phases in electronic structure theory (Cambridge University Press, Cambridge, UK, 2018)

    work page 2018

  34. [43]

    Resta, Quantum geometry and adiabaticity in molecule s and in condensed (2025), arXiv:2503.10672 [cond-mat.quant-ph]

    R. Resta, Quantum geometry and adiabaticity in molecule s and in condensed (2025), arXiv:2503.10672 [cond-mat.quant-ph]

    work page arXiv 2025

  35. [44]

    Royo and M

    M. Royo and M. Stengel, Dynamical response of noncollinear spin systems at constra ined magnetic (2025), arXiv:2501.10188 [cond-mat.mtrl-sci]

    work page arXiv 2025

  36. [45]

    van Setten, M

    M. van Setten, M. Giantomassi, E. Bousquet, M. Ver- straete, D. Hamann, X. Gonze, and G.-M. Rignanese, The PseudoDojo: Training and grading a 85 ele- ment optimized norm-conserving pseudopotential table, Comp. Phys. Comm. 226, 39 (2018)

    work page 2018

  37. [46]

    Zabalo, C

    A. Zabalo, C. E. Dreyer, and M. Stengel, Rota- tional g factors and lorentz forces of molecules and solids from density functional perturbation theory, Phys. Rev. B 105, 094305 (2022)

    work page 2022

  38. [47]

    Ismail-Beigi, E

    S. Ismail-Beigi, E. K. Chang, and S. G. Louie, Cou- pling of nonlocal potentials to electromagnetic fields, Phys. Rev. Lett. 87, 087402 (2001)

    work page 2001