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arxiv: 1906.08402 · v1 · pith:QEJM3LIEnew · submitted 2019-06-20 · ❄️ cond-mat.mtrl-sci

Polarons from first principles, without supercells

Weng Hong Sio , Carla Verdi , Samuel Ponce , Feliciano Giustino This is my paper

Pith reviewed 2026-05-25 20:06 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords polaronsfirst principlesdensity functional perturbation theoryelectron-phonon couplingLiFLi2O2secular equation
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0 comments X

The pith

Polarons are computed from first principles by solving a secular equation using DFPT phonons and electron-phonon matrix elements, without supercells.

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

The paper develops a method to calculate polaron properties in insulators and semiconductors by solving a secular equation that incorporates phonons and electron-phonon couplings obtained from density-functional perturbation theory. This replaces the need for large supercells that are typically required to accommodate the lattice distortion around the polaron. The resulting approach yields polaron wavefunctions, formation energies, and spectral decompositions. It is shown to treat both large and small polarons within the same framework, as demonstrated through explicit calculations on LiF and Li2O2. The method therefore makes first-principles polaron studies feasible for a wider range of materials where supercell sizes would otherwise be prohibitive.

Core claim

We develop a formalism and a computational method to study polarons in insulators and semiconductors from first principles. Unlike in standard calculations requiring large supercells, we solve a secular equation involving phonons and electron-phonon matrix elements from density-functional perturbation theory, in a spirit similar to the Bethe-Salpeter equation for excitons. We show that our approach describes seamlessly large and small polarons, and we illustrate its capability by calculating wavefunctions, formation energies, and spectral decomposition of polarons in LiF and Li2O2.

What carries the argument

The secular equation constructed from DFPT phonons and electron-phonon matrix elements, which is solved to obtain the polaron states and energies.

If this is right

  • Polaron wavefunctions and formation energies become accessible without constructing supercells whose size scales with the polaron radius.
  • Large (delocalized) and small (localized) polarons are obtained from the same eigenvalue problem.
  • Spectral decomposition of the polaron states can be extracted directly from the eigenvectors of the secular equation.
  • The computational cost is set by the DFPT calculation on the primitive cell rather than by supercell size.

Where Pith is reading between the lines

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

  • The approach could be applied to materials with small primitive cells where supercell methods become impractical due to computational scaling.
  • Direct comparison of the computed polaron binding energies against measured values in LiF would provide an external test of the secular equation's accuracy.
  • Because the formalism is built on standard DFPT outputs, it can be implemented in existing first-principles codes that already generate phonon and electron-phonon data.

Load-bearing premise

The secular equation built from DFPT phonons and electron-phonon matrix elements is sufficient to capture the essential polaron physics for the materials considered.

What would settle it

A side-by-side comparison of the formation energies and wavefunction spreads obtained from this secular-equation method versus converged large-supercell calculations for the same materials LiF and Li2O2.

Figures

Figures reproduced from arXiv: 1906.08402 by Carla Verdi, Feliciano Giustino, Samuel Ponce, Weng Hong Sio.

Figure 1
Figure 1. Figure 1: (f) shows the polaron eigenvalue ε and for￾mation energy ∆Ef as a function of supercell size. For supercells smaller than 12×12×12 unit cells the nonlinear eigenproblem in Eqs. (4)-(5) does not admit localized so￾lutions. This can be understood as a manifestation of the Mott transition [63] at a critical density of 4 · 1019 cm−3 . Below this critical density we find localized solutions of the type shown in… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Small electron polaron in Li [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We develop a formalism and a computational method to study polarons in insulators and semi-conductors from first principles. Unlike in standard calculations requiring large supercells, we solve a secular equation involving phonons and electron-phonon matrix elements from density-functional perturbation theory, in a spirit similar to the Bethe-Salpeter equation for excitons. We show that our approach describes seamlessly large and small polarons, and we illustrate its capability by calculating wavefunctions, formation energies, and spectral decomposition of polarons in LiF and Li2O2.

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 / 1 minor

Summary. The paper develops a formalism for calculating polaron properties in insulators and semiconductors by solving a secular equation that incorporates phonons and electron-phonon matrix elements obtained from density-functional perturbation theory (DFPT). This approach is presented as an alternative to supercell-based methods and is claimed to handle both large and small polarons seamlessly. The method is illustrated through calculations of wavefunctions, formation energies, and spectral decompositions for polarons in LiF and Li2O2.

Significance. If the central construction holds, the work offers a computationally efficient first-principles route to polaron studies that could enable investigations in a broader range of materials. The seamless description across polaron sizes is a potential strength, and the use of standard DFPT quantities is advantageous for reproducibility with existing codes.

major comments (2)
  1. [Secular equation construction] The kernel of the secular equation is built from harmonic phonons and linear electron-phonon matrix elements. For the small polaron in LiF, where lattice distortions are large and localized, this linear approximation may not suffice; the manuscript should demonstrate that higher-order terms do not affect the reported formation energies by more than the claimed precision.
  2. [Results for LiF] No explicit validation against a fully relaxed supercell calculation for the same material is provided to test the assumption that the linear-response basis captures the essential physics.
minor comments (1)
  1. [Abstract] The abstract mentions 'spectral decomposition of polarons' without defining what this quantity represents in the context of the method.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting potential limitations of the linear-response framework. We address each major comment below.

read point-by-point responses

  1. Referee: [Secular equation construction] The kernel of the secular equation is built from harmonic phonons and linear electron-phonon matrix elements. For the small polaron in LiF, where lattice distortions are large and localized, this linear approximation may not suffice; the manuscript should demonstrate that higher-order terms do not affect the reported formation energies by more than the claimed precision.

    Authors: The secular equation is constructed from the harmonic phonon frequencies and the first-order electron-phonon matrix elements obtained via DFPT, as is standard for this class of methods. The self-consistent solution of the secular equation permits substantial lattice relaxation through the polaron coefficients even within the linear coupling. For the small polaron in LiF the reported formation energy is given to the numerical precision of the underlying DFPT data. We agree that anharmonic and higher-order electron-phonon contributions could become relevant for strongly localized distortions; the present work does not include such terms. In the revised manuscript we will add an explicit paragraph discussing the linear-response approximation and its expected range of validity for the systems considered. revision: partial

  2. Referee: [Results for LiF] No explicit validation against a fully relaxed supercell calculation for the same material is provided to test the assumption that the linear-response basis captures the essential physics.

    Authors: A central motivation of the formalism is to obtain polaron properties without recourse to supercells, by working directly with DFPT quantities in reciprocal space. A fully relaxed supercell calculation for the small polaron in LiF would require cells large enough to isolate the localized distortion, rendering such a benchmark both computationally prohibitive and subject to its own finite-size errors. The internal consistency of the results—namely the seamless description of both large and small polarons in LiF and Li2O2—provides the primary validation within the scope of the paper. We will insert a short clarifying paragraph explaining why a direct supercell comparison is not performed. revision: partial

standing simulated objections not resolved
  • Quantitative demonstration that higher-order (anharmonic or nonlinear) terms do not shift the reported formation energies beyond the stated precision for the small polaron in LiF, as this would require extending the formalism beyond the linear DFPT framework used throughout the manuscript.

Circularity Check

0 steps flagged

Derivation self-contained from standard DFPT inputs

full rationale

The paper constructs a secular equation from DFPT phonons and first-order electron-phonon matrix elements to obtain polaron wavefunctions, formation energies, and spectral weights. This is presented as a direct first-principles method analogous to the Bethe-Salpeter equation, applied to LiF and Li2O2. No quoted equations or steps reduce the output to a fitted parameter, self-citation chain, or input by construction. The central claim remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are specified.

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Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [1]

    L. D. Landau, Phys. Z. Sowjet. 3, 664 (1933)

    work page 1933

  2. [2]

    J. G. Bednorz and K. A. M¨ uller, Z. Phys. B 64, 189 (1986)

    work page 1986

  3. [3]

    Verdi, F

    C. Verdi, F. Caruso, and F. Giustino, Nat. Commun. 8, 15769 (2017)

    work page 2017

  4. [4]

    Nat. Mater. 15, 835 (2016)

    work page 2016

  5. [5]

    Cancellieri, A

    C. Cancellieri, A. S. Mishchenko, U. Aschauer, A. Fil- ippetti, C. Faber, O. Bariˇ si´ c, V. Rogalev, T. Schmitt, N. Nagaosa, and V. N. Strocov, Nat. Commun. 7, 10386 (2016)

    work page 2016

  6. [6]

    C. Chen, J. Avila, E. Frantzeskakis, A. Levy, and M. C. Asensio, Nat. Commun. 6, 8585 (2015)

    work page 2015

  7. [7]

    Y. F. Nie, D. Di Sante, S. Chatterjee, P. D. C. King, M. Uchida, S. Ciuchi, D. G. Schlom, and K. M. Shen, Phys. Rev. Lett. 115, 096405 (2015)

    work page 2015

  8. [8]

    H. Zhu, K. Miyata, Y. Fu, J. Wang, P. P. Joshi, D. Nies- ner, K. W. Williams, S. Jin, and X.-Y. Zhu, Science 353, 1409 (2016)

    work page 2016

  9. [9]

    R. P. Feynman, Phys. Rev. 97, 660 (1955)

    work page 1955

  10. [10]

    Pekar, J

    S. Pekar, J. Exp. Theor. Phys 16, 341 (1946)

    work page 1946

  11. [11]

    Bahrami, A

    M. Bahrami, A. Großardt, S. Donadi, and A. Bassi, New J. Phys. 16, 115007 (2014)

    work page 2014

  12. [12]

    Penrose, Found

    R. Penrose, Found. Phys. 44, 557 (2014)

    work page 2014

  13. [13]

    Fr¨ ohlich, H

    H. Fr¨ ohlich, H. Pelzer, and S. Zienau, Lond. Edinb. Dubl. Phil. Mag. J. Sci. 41, 221 (1950)

    work page 1950

  14. [14]

    T. D. Lee, F. E. Low, and D. Pines, Phys. Rev. 90, 297 (1953)

    work page 1953

  15. [15]

    Holstein, Ann

    T. Holstein, Ann. Phys. 8, 325 (1959)

    work page 1959

  16. [16]

    L´ epine and Y

    Y. L´ epine and Y. Frongillo, Phys. Rev. B 46, 14510 (1992)

    work page 1992

  17. [17]

    L´ epine, J

    Y. L´ epine, J. Phys. Condens. Matter6, 6611 (1994)

    work page 1994

  18. [18]

    A. S. Alexandrov and P. E. Kornilovitch, Phys. Rev. Lett. 82, 807 (1999)

    work page 1999

  19. [19]

    A. S. Alexandrov, Phys. Rev. B 61, 12315 (2000)

    work page 2000

  20. [20]

    A. S. Alexandrov and B. Y. Yavidov, Phys. Rev. B 69, 073101 (2004)

    work page 2004

  21. [21]

    Perroni, V

    C. Perroni, V. Cataudella, and G. De Filippis, J. Phys. Condens. Matter 16, 1593 (2004)

    work page 2004

  22. [22]

    R¨ osch and O

    O. R¨ osch and O. Gunnarsson, Phys. Rev. Lett. 92, 146403 (2004)

    work page 2004

  23. [23]

    Cruzeiro-Hansson, J

    L. Cruzeiro-Hansson, J. Eilbeck, J. Marın, and F. Rus- sell, Phys. Lett. A 266, 160 (2000)

    work page 2000

  24. [24]

    A. S. Alexandrov, Theory of superconductivity: from weak to strong coupling (IOP, Bristol, 2003)

    work page 2003

  25. [25]

    Ortmann, F

    F. Ortmann, F. Bechstedt, and K. Hannewald, Phys. Rev. B 79, 235206 (2009)

    work page 2009

  26. [26]

    Fehske and S

    H. Fehske and S. Trugman, in Polarons in Advanced Ma- terials, Springer Series in Material Sciences 103, edited by A. S. Alexandrov (Springer Verlag, Dordrecht, 2007) pp. 393–461

    work page 2007

  27. [27]

    Jeckelmann and S

    E. Jeckelmann and S. R. White, Phys. Rev. B 57, 6376 (1998)

    work page 1998

  28. [28]

    Grusdt, Phys

    F. Grusdt, Phys. Rev. B 93, 144302 (2016)

    work page 2016

  29. [29]

    Kornilovitch, in Polarons in Advanced Materials , Springer Series in Material Sciences 103, edited by A

    P. Kornilovitch, in Polarons in Advanced Materials , Springer Series in Material Sciences 103, edited by A. S. Alexandrov (Springer Verlag, Dordrecht, 2007) pp. 192– 230

    work page 2007

  30. [30]

    A. S. Mishchenko, N. V. Prokof’ev, A. Sakamoto, and B. V. Svistunov, Phys. Rev. B 62, 6317 (2000)

    work page 2000

  31. [31]

    Prokof’ev and B

    N. Prokof’ev and B. Svistunov, Phys. Rev. B 77, 020408 (2008)

    work page 2008

  32. [32]

    T. Hahn, S. Klimin, J. Tempere, J. T. Devreese, and C. Franchini, Phys. Rev. B 97, 134305 (2018)

    work page 2018

  33. [33]

    A. S. Alexandrov, Polarons in advanced materials , Springer Series in Material Sciences, Vol. 103 (Springer, Dordrecht, 2007)

    work page 2007

  34. [34]

    J. T. Devreese and A. S. Alexandrov, Rep. Prog. Phys. 72, 066501 (2009)

    work page 2009

  35. [35]

    A. S. Alexandrov and J. T. Devreese, Advances in polaron physics (Springer, Berlin, 2010)

    work page 2010

  36. [36]

    Emin, Polarons (Cambridge University Press, Cam- bridge, 2013)

    D. Emin, Polarons (Cambridge University Press, Cam- bridge, 2013)

    work page 2013

  37. [37]

    J. T. Devreese, arXiv:1611.06122

    work page arXiv

  38. [38]

    Franchini, G

    C. Franchini, G. Kresse, and R. Podloucky, Phys. Rev. Lett. 102, 256402 (2009)

    work page 2009

  39. [39]

    Setvin, C

    M. Setvin, C. Franchini, X. Hao, M. Schmid, A. Jan- 6 otti, M. Kaltak, C. G. Van de Walle, G. Kresse, and U. Diebold, Phys. Rev. Lett. 113, 086402 (2014)

    work page 2014

  40. [40]

    Himmetoglu, A

    B. Himmetoglu, A. Janotti, L. Bjaalie, and C. G. Van de Walle, Phys. Rev. B 90, 161102 (2014)

    work page 2014

  41. [41]

    Bondarenko, O

    N. Bondarenko, O. Eriksson, and N. V. Skorodumova, Phys. Rev. B 92, 165119 (2015)

    work page 2015

  42. [42]

    Reticcioli, M

    M. Reticcioli, M. Setvin, X. Hao, P. Flauger, G. Kresse, M. Schmid, U. Diebold, and C. Franchini, Phys. Rev. X 7, 031053 (2017)

    work page 2017

  43. [43]

    Reticcioli, M

    M. Reticcioli, M. Setvin, M. Schmid, U. Diebold, and C. Franchini, Phys. Rev. B 98, 045306 (2018)

    work page 2018

  44. [44]

    J. M. Morbec and G. Galli, Phys. Rev. B 93, 035201 (2016)

    work page 2016

  45. [45]

    Kokott, S

    S. Kokott, S. V. Levchenko, P. Rinke, and M. Scheffler, New J. Phys. 20, 033023 (2018)

    work page 2018

  46. [46]

    Spreafico and J

    C. Spreafico and J. VandeVondele, Phys. Chem. Chem. Phys. 16, 26144 (2014)

    work page 2014

  47. [47]

    Erhart, A

    P. Erhart, A. Klein, D. ˚Aberg, and B. Sadigh, Phys. Rev. B 90, 035204 (2014)

    work page 2014

  48. [48]

    W. H. Sio, C. Verdi, S. Ponc´ e, and F. Giustino, (un- published)

    work page

  49. [49]

    Onida, L

    G. Onida, L. Reining, R. W. Godby, R. Del Sole, and W. Andreoni, Phys. Rev. Lett. 75, 818 (1995)

    work page 1995

  50. [50]

    Rohlfing and S

    M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 81, 2312 (1998)

    work page 1998

  51. [51]

    Giustino, Rev

    F. Giustino, Rev. Mod. Phys. 89, 015003 (2017)

    work page 2017

  52. [52]

    Iadonisi, Riv

    G. Iadonisi, Riv. Nuovo Cimento 7, 1 (1984)

    work page 1984

  53. [53]

    Z. Feng, V. Timoshevskii, A. Mauger, C. M. Julien, K. H. Bevan, and K. Zaghib, Phys. Rev. B 88, 184302 (2013)

    work page 2013

  54. [54]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996)

    work page 1996

  55. [55]

    Giannozzi et al., J

    P. Giannozzi et al., J. Phys. Condens. Matter 29, 465901 (2017)

    work page 2017

  56. [56]

    D. R. Hamann, Phys. Rev. B 88, 085117 (2013)

    work page 2013

  57. [57]

    Baroni, S

    S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Gian- nozzi, Rev. Mod. Phys. 73, 515 (2001)

    work page 2001

  58. [58]

    Marzari, A

    N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012)

    work page 2012

  59. [59]

    A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 185, 2309 (2014)

    work page 2014

  60. [60]

    Giustino, M

    F. Giustino, M. L. Cohen, and S. G. Louie, Phys. Rev. B 76, 165108 (2007)

    work page 2007

  61. [61]

    Ponc´ e, E

    S. Ponc´ e, E. R. Margine, C. Verdi, and F. Giustino, Comput. Phys. Commun. 209, 116 (2016)

    work page 2016

  62. [62]

    Bokdam, T

    M. Bokdam, T. Sander, A. Stroppa, S. Picozzi, D. Sarma, C. Franchini, and G. Kresse, Sci. Rep. 6, 28618 (2016)

    work page 2016

  63. [63]

    N. F. Mott, Rev. Mod. Phys. 40, 677 (1968)

    work page 1968

  64. [64]

    Madelung, Phys

    E. Madelung, Phys. Z. 19, 524 (1918)

    work page 1918

  65. [65]

    Makov and M

    G. Makov and M. C. Payne, Phys. Rev. B 51, 4014 (1995)

    work page 1995

  66. [66]

    Verdi and F

    C. Verdi and F. Giustino, Phys. Rev. Lett. 115, 176401 (2015)

    work page 2015

  67. [67]

    J. Kang, Y. S. Jung, S.-H. Wei, and A. C. Dillon, Phys. Rev. B 85, 035210 (2012)

    work page 2012