arxiv: 2605.14472 · v1 · submitted 2026-05-14 · ❄️ cond-mat.mtrl-sci

Recognition: 2 theorem links

· Lean Theorem

Accurate computation of the electron-phonon interaction contribution to the total energy

Shilpa Paul , Mogadalai Pandurangan Gururajan , Amrita Bhattacharya , Tiramkudlu R. S. Prasanna

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:45 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords electron-phonon interactiontotal energynon-adiabatic approximationFan-Migdal termab initio calculationscarbon polymorphsdiamondlonsdaleite

0 comments

The pith

New non-adiabatic expressions for the partial Fan-Migdal occupied term show it behaves as a higher-order correction to the total energy.

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

The paper derives updated formulas for the partial-Fan-Migdal occupied contribution to the electron-phonon interaction energy that enters the total energy of a crystal. Standard second-order expressions are replaced by new ones obtained in the non-adiabatic approximation; these new expressions carry the structure of a higher-order term. Because the non-adiabatic treatment is used, the formulas are required for every material, whether or not its phonon modes are infrared active. The authors demonstrate the approach by computing the full electron-phonon-corrected total energy for diamond and hexagonal lonsdaleite and present the results as the most accurate ab initio values obtained so far. The same framework also supplies a route to finite-temperature free energies that include the electron-phonon contribution.

Core claim

Within the non-adiabatic approximation the partial-Fan-Migdal occupied contribution to the total energy has the structure of a higher-order term rather than a second-order term. New expressions that follow from this structure must replace the standard second-order formulas because they incorporate the complete physics of the term. The electron-phonon contribution to the total energy therefore requires the non-adiabatic approximation for all materials.

What carries the argument

The non-adiabatic derivation of the partial Fan-Migdal occupied (partial-FM-occ) contribution, which reveals its higher-order character and supplies replacement formulas for its numerical evaluation.

If this is right

The partial-FM-occ term must be computed with the new higher-order expressions in all theoretical and computational studies of electron-phonon effects.
The EPI contribution to total energy must be evaluated in the non-adiabatic approximation for both IR-active and IR-inactive materials.
Ab initio total energies for carbon polymorphs that include this term constitute the most accurate values reported to date.
The same non-adiabatic framework opens the possibility of computing ab initio free energies at finite temperature that incorporate the EPI contribution.

Where Pith is reading between the lines

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

The revised total energies could shift predicted phase stabilities or defect formation energies in carbon-based materials once electron-phonon corrections are consistently included.
Similar re-derivations may be required for other second-order electron-phonon terms to check whether they also contain hidden higher-order structure.
Finite-temperature extensions of the method could improve predictions of thermal expansion coefficients and specific heats in semiconductors.

Load-bearing premise

The non-adiabatic approximation is assumed to capture the complete physics of the partial-FM-occ term for every material.

What would settle it

A direct numerical comparison of total energies computed with the new expressions against either experiment or independent higher-order perturbation results for diamond would show whether the revised values lie closer to the true ground-state energy.

read the original abstract

The standard Hamiltonian of a coupled electron-phonon system is based on second-order perturbation theory. The EPI contribution in the standard Hamiltonian consists of two terms, the EPI contribution to the band-structure energy and the partial-Fan-Migdal (FM)-occupied contribution. Within the non-adiabatic approximation, we derive a new expression for the partial-FM-occ contribution and show that it has the structure of a higher-order term, and not a second-order term. Along similar lines, we derive new expressions for the computation of the partial-FM-occ term. The new expressions for the partial-FM-occ term must be preferred over the standard expressions, in theoretical and computational studies, because they incorporate the complete physics underlying this term. Unlike the EPI contribution to individual eigenstates, the EPI contribution to the total energy must be computed in the non-adiabatic approximation for all materials, Infra-red (IR) active and IR-inactive. We report the computation of the standard Hamiltonian, for the first time, for Carbon polymorphs (diamond and hexagonal lonsdaleite) by including the EPI contribution to the total energy. This is the most accurate ab initio total energy reported till date. The present work also opens the way to compute the ab initio free-energy more accurately at finite temperatures by including the EPI contribution.

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.

Desk Editor's Note private letter to a colleague

The paper re-derives the partial-FM-occ term to show higher-order structure under non-adiabatic conditions and applies the new expressions to total energies in diamond and lonsdaleite.

read the letter

The key takeaway is that the partial Fan-Migdal occupied contribution to the electron-phonon interaction in total energy picks up higher-order terms once the non-adiabatic approximation is used, so the usual second-order expressions should be replaced for every material, not just IR-active ones. The authors give explicit new formulas and report the first full standard-Hamiltonian calculations that include this term for diamond and hexagonal lonsdaleite, calling the result the most accurate ab initio total energy to date. They also note the route this opens for finite-temperature free energies. The derivation stays inside the standard coupled electron-phonon Hamiltonian and introduces no new fitted parameters, which keeps the work grounded. The explicit forms are the clearest incremental advance over the second-order treatments referenced in the abstract. The main limitation is that the abstract supplies no equations, no numerical shifts, and no comparison against DFT functional error or k-point convergence. Without those numbers it is difficult to judge whether the correction exceeds other uncertainties in the calculation. The stress-test concern is therefore still live: if the energy change falls inside typical error bars, the practical preference for the new expressions is weaker than claimed. The paper is aimed at researchers already running electron-phonon calculations who need tighter total-energy values. A reader working on precise ground-state properties in solids could usefully test the expressions on their own systems. I would send it to peer review because the claim is concrete and the steps are checkable once the full derivations and numbers are in front of referees.

Referee Report

2 major / 1 minor

Summary. The manuscript derives new expressions for the partial-Fan-Migdal occupied (partial-FM-occ) contribution to the electron-phonon interaction (EPI) term in the total energy, showing that within the non-adiabatic approximation this term acquires a higher-order structure rather than remaining second-order. It argues that these new expressions must replace the standard ones in all theoretical and computational studies because they capture the complete physics. The authors report the first computation of the full standard Hamiltonian including EPI for carbon polymorphs (diamond and lonsdaleite), claiming this yields the most accurate ab initio total energy to date, and note that non-adiabatic treatment is required for both IR-active and IR-inactive materials. The work also suggests extensions to finite-temperature free-energy calculations.

Significance. If the non-adiabatic re-derivation is correct and produces numerically significant corrections beyond standard second-order terms, the result would improve the accuracy of ab initio total energies by providing a parameter-free treatment of EPI contributions. This could benefit structural predictions and thermodynamic properties at finite temperature. The first-principles nature of the derivation (no fitted parameters) and the explicit application to real materials are strengths that would support broader adoption if the central claim is validated.

major comments (2)

[Abstract] Abstract: The load-bearing claim is that the non-adiabatic approximation makes the partial-FM-occ term higher-order, requiring new expressions over the standard second-order Hamiltonian. However, the abstract supplies no explicit equations or derivation steps showing the new form versus the conventional one, so it is impossible to verify whether the re-derivation merely rearranges terms already captured at second order or produces a genuine correction that exceeds typical DFT functional or k-point sampling uncertainties for diamond and lonsdaleite.
[Computational results] Computational results: The assertion that the reported total energies for diamond and lonsdaleite constitute 'the most accurate ab initio total energy reported till date' is not supported by any quantitative comparison to experiment, quantum Monte Carlo benchmarks, or other high-accuracy methods, nor by error bars that isolate the EPI contribution from other sources of numerical error.

minor comments (1)

[Abstract] The abbreviation 'partial-FM-occ' is used without an initial definition or expansion in the abstract, which reduces clarity for readers unfamiliar with the Fan-Migdal terminology.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

Referee: [Abstract] Abstract: The load-bearing claim is that the non-adiabatic approximation makes the partial-FM-occ term higher-order, requiring new expressions over the standard second-order Hamiltonian. However, the abstract supplies no explicit equations or derivation steps showing the new form versus the conventional one, so it is impossible to verify whether the re-derivation merely rearranges terms already captured at second order or produces a genuine correction that exceeds typical DFT functional or k-point sampling uncertainties for diamond and lonsdaleite.

Authors: We appreciate this observation. While the abstract is a concise summary, we agree that including a brief indication of the new expression would help. In the revised manuscript, we will update the abstract to state the new non-adiabatic form of the partial-FM-occ term explicitly, contrasting it with the standard second-order expression. The detailed derivation is already provided in the main text (Section 2). This change will make the distinction clear without altering the manuscript's length significantly. revision: yes
Referee: [Computational results] Computational results: The assertion that the reported total energies for diamond and lonsdaleite constitute 'the most accurate ab initio total energy reported till date' is not supported by any quantitative comparison to experiment, quantum Monte Carlo benchmarks, or other high-accuracy methods, nor by error bars that isolate the EPI contribution from other sources of numerical error.

Authors: We acknowledge that the claim requires stronger support. We will revise the relevant statement in the manuscript to be more precise, removing the absolute 'most accurate till date' phrasing and instead highlighting that this is the first computation of the full standard Hamiltonian including the EPI contribution for these carbon polymorphs. We will add quantitative comparisons to our own standard DFT results (without EPI) and to other published ab initio total energies for diamond and lonsdaleite, along with estimates of the numerical uncertainties. Direct comparisons to quantum Monte Carlo or experimental values are not included in this work, as the primary focus is the derivation and implementation of the new expressions. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation from non-adiabatic approximation on standard Hamiltonian is self-contained.

full rationale

The paper starts from the standard second-order Hamiltonian for the coupled electron-phonon system and applies the non-adiabatic approximation to re-derive the partial-FM-occ term, showing it acquires higher-order structure. No quoted step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the new expressions are presented as direct consequences of the approximation rather than statistical fits or prior ansatzes. The total-energy computations for diamond and lonsdaleite are reported as first-time applications of the standard Hamiltonian including EPI, without evidence that the central preference for the new expressions collapses to the inputs by definition. This is the most common honest outcome for a derivation paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard second-order perturbation theory for the electron-phonon Hamiltonian and the non-adiabatic approximation; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Standard Hamiltonian of a coupled electron-phonon system is based on second-order perturbation theory
Invoked at the opening of the abstract as the foundation for the existing EPI contribution.
domain assumption Non-adiabatic approximation
Used to derive the new partial-FM-occ expression and asserted necessary for all materials.

pith-pipeline@v0.9.0 · 5555 in / 1316 out tokens · 39009 ms · 2026-05-15T01:45:24.150559+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Within the non-adiabatic approximation, we derive a new expression for the partial-FM-occ contribution and show that it has the structure of a higher-order term... Eq. 17 shows that the net result of the incomplete cancellation of two second-order terms need not be a second-order term.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The standard Hamiltonian of a coupled electron-phonon system is based on second-order perturbation theory... We report the computation of the standard Hamiltonian... for Carbon polymorphs.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages · 1 internal anchor

[1]

and computed the EPI contribution to the total en- ergy for C, Si, SiC and BN polymorphs [18, 21]. Our results show that, to determine total energy differences, the EPI contribution is more important than the vdW and ZPVE contributions because it is more sensitive to crystal structure and hence, must be included in all such studies [18, 21]. However, for ...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

using the formula for free en- ergy of independent electrons evaluated with the quasi- particle energy

referring to our earlier work [18]. In this work, we show that Allen’s equation for the Vep term [20] can be recast so that it contains two con- tributions, the EPI contribution to the band-structure energy term and a partial-Fan-Migdal-occupied (partial- FM-occ) term. We show that, unlike for the EPI contri- bution to individual eigenstates, the EPI cont...

work page
[3]

Hohenberg and W

P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev.136, B864 (1964)

work page 1964
[4]

Kohn and L

W. Kohn and L. J. Sham, Self-consistent equations in- cluding exchange and correlation effects, Phys. Rev.140, A1133 (1965)

work page 1965
[5]

Ashcroft and N

N. Ashcroft and N. Mermin,Solid State Physics(New York, USA: Holt, Rinehart and Winston, 1976)

work page 1976
[6]

S. H. Simon,The Oxford Solid State Basics(OUP Ox- ford, 2013)

work page 2013
[7]

R. M. Martin,Electronic Structure: Basic Theory and Practical Methods(Cambridge university press, 2020)

work page 2020
[8]

Grosso and G

G. Grosso and G. P. Parravicini,Solid State Physics (Academic press, 2013)

work page 2013
[9]

A. R. Oganov, C. J. Pickard, Q. Zhu, and R. J. Needs, Structure prediction drives materials discovery, Nature Reviews Materials4, 331 (2019)

work page 2019
[10]

S. M. Woodley, G. M. Day, and R. Catlow, Structure pre- diction of crystals, surfaces and nanoparticles, Philosoph- ical Transactions of the Royal Society A378, 20190600 (2020)

work page 2020
[11]

Cheng, X.-G

G. Cheng, X.-G. Gong, and W.-J. Yin, Crystal structure prediction by combining graph network and optimization algorithm, Nature communications13, 1492 (2022)

work page 2022
[12]

C. Hong, J. M. Choi, W. Jeong, S. Kang, S. Ju, K. Lee, J. Jung, Y. Youn, and S. Han, Training machine-learning potentials for crystal structure prediction using disor- dered structures, Physical Review B102, 224104 (2020)

work page 2020
[13]

Yamashita, N

T. Yamashita, N. Sato, H. Kino, T. Miyake, K. Tsuda, and T. Oguchi, Crystal structure prediction accelerated by bayesian optimization, Physical Review Materials2, 013803 (2018)

work page 2018
[14]

Kawanishi and T

S. Kawanishi and T. Mizoguchi, Effect of van der waals interactions on the stability of sic polytypes, Journal of Applied Physics119, 175101 (2016)

work page 2016
[15]

Scalise, A

E. Scalise, A. Marzegalli, F. Montalenti, and L. Miglio, Temperature-dependent stability of polytypes and stack- ing faults in si c: Reconciling theory and experiments, Physical Review Applied12, 021002 (2019)

work page 2019
[16]

Ramakers, A

S. Ramakers, A. Marusczyk, M. Amsler, T. Eckl, M. Mrovec, T. Hammerschmidt, and R. Drautz, Effects of thermal, elastic, and surface properties on the stability of sic polytypes, Phys. Rev. B106, 075201 (2022)

work page 2022
[17]

Cazorla and T

C. Cazorla and T. Gould, Polymorphism of bulk boron nitride, Science Advances5, eaau5832 (2019)

work page 2019
[18]

M. J. van Setten, M. A. Uijttewaal, G. A. de Wijs, and R. A. de Groot, Thermodynamic stability of boron: The role of defects and zero point motion, Journal of the American Chemical Society129, 2458 (2007)

work page 2007
[19]

S. S. Bhat, K. Gupta, S. Bhattacharjee, and S.-C. Lee, Role of zero-point effects in stabilizing the ground state structure of bulk fe2p, Journal of Physics: Condensed Matter30, 215401 (2018)

work page 2018
[20]

A. V. Ramasimha Varma, S. Paul, A. Itale, P. Pable, R. Tibrewala, S. Dodal, H. Yerunkar, S. Bhaumik, V. Shah, M. P. Gururajan, and T. R. S. Prasanna, Elec- tron–phonon interaction contribution to the total energy of group iv semiconductor polymorphs: Evaluation and implications, ACS Omega8, 11251 (2023)

work page 2023
[21]

P. B. Allen, Theory of thermal expansion: Quasi- harmonic approximation and corrections from quasi- particle renormalization, Modern Physics Letters B34, 2050025 (2020)

work page 2020
[22]

P. B. Allen, Second erratum — theory of thermal ex- pansion: Quasi-harmonic approximation and corrections from quasi-particle renormalization, Modern Physics Let- ters B36, 2292002 (2022)

work page 2022
[23]

S. Paul, M. P. Gururajan, A. Bhattacharya, and T. R. S. Prasanna, Critical role of electron-phonon interactions in determining the relative stability of boron nitride poly- morphs, https://arxiv.org/abs/2212.13877 (2022)

work page arXiv 2022
[24]

Chakraborty and P

B. Chakraborty and P. Allen, Theory of temperature dependence of optical properties of solids, Journal of Physics C: Solid State Physics11, L9 (1978)

work page 1978
[25]

Giustino, Electron-phonon interactions from first prin- ciples, Reviews of Modern Physics89, 015003 (2017)

F. Giustino, Electron-phonon interactions from first prin- ciples, Reviews of Modern Physics89, 015003 (2017)

work page 2017
[26]

Lafuente-Bartolome, C

J. Lafuente-Bartolome, C. Lian, W. H. Sio, I. G. Gur- tubay, A. Eiguren, and F. Giustino, Ab initio self- consistent many-body theory of polarons at all couplings, Physical Review B106, 075119 (2022)

work page 2022
[27]

P. B. Allen, Named review of earlier version of Ref. 18, 2022 ()

work page 2022
[28]

Heid, Density functional perturbation theory and elec- tron phonon coupling, in emergent phenomena in corre- lated matter

R. Heid, Density functional perturbation theory and elec- tron phonon coupling, in emergent phenomena in corre- lated matter. ed. pavarini, e. and koch, e. and schollw¨ ock, u., (2013)

work page 2013
[29]

Ponc´ e and X

S. Ponc´ e and X. Gonze, In search of the electron-phonon contribution to total energy (2025), arXiv:2512.04897 [cond-mat.mtrl-sci]

work page arXiv 2025
[30]

P. B. Allen and J. C. K. Hui, Thermodynamics of solids: Corrections from electron-phonon interactions, Zeitschrift f¨ ur Physik B Condensed Matter37, 33 (1980)

work page 1980
[31]

P. B. Allen and V. Heine, Theory of the temperature dependence of electronic band structures, Journal of Physics C: Solid State Physics9, 2305 (1976)

work page 1976
[32]

J. P. Nery, P. B. Allen, G. Antonius, L. Reining, A. Miglio, and X. Gonze, Quasiparticles and phonon satellites in spectral functions of semiconductors and in- sulators: Cumulants applied to the full first-principles theory and the fr¨ ohlich polaron, Phys. Rev. B97, 115145 (2018)

work page 2018
[33]

P. B. Allen, personal communication, personal commu- nication, 2023 ()

work page 2023
[34]

Fr¨ ohlich, Theory of the superconducting state

H. Fr¨ ohlich, Theory of the superconducting state. i. the ground state at the absolute zero of temperature, Physi- cal Review79, 845 (1950)

work page 1950
[35]

Fan, Temperature dependence of the energy gap in semiconductors, Physical Review82, 900 (1951)

H. Fan, Temperature dependence of the energy gap in semiconductors, Physical Review82, 900 (1951)

work page 1951
[36]

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,et al., Abinit: First-principles approach to material and nanosystem properties, Computer Physics Communications180, 2582 (2009)

work page 2009
[37]

Gonze, F

X. Gonze, F. Jollet, F. A. Araujo, D. Adams, B. Amadon, T. Applencourt, C. Audouze, J.-M. Beuken, J. Bieder, A. Bokhanchuk,et al., Recent developments in the abinit software package, Computer Physics Communications 205, 106 (2016)

work page 2016
[38]

Abinit, tutorial tdepes, temperature-dependence of the electronic structure,,https://docs.abinit.org/ tutorial/tdepes/, accessed: 2020-03-15

work page 2020
[39]

Ponc´ e, Y

S. Ponc´ e, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete, and X. Gonze, Temperature dependence of 8 the electronic structure of semiconductors and insulators, The Journal of Chemical Physics143, 102813 (2015)

work page 2015
[40]

Ponc´ e, G

S. Ponc´ e, G. Antonius, P. Boulanger, E. Cannuccia, A. Marini, M. Cˆ ot´ e, and X. Gonze, Verification of first-principles codes: Comparison of total energies, phonon frequencies, electron–phonon coupling and zero- point motion correction to the gap between abinit and qe/yambo, Computational Materials Science83, 341 (2014)

work page 2014
[41]

Ponc´ e, G

S. Ponc´ e, G. Antonius, Y. Gillet, P. Boulanger, J. L. Janssen, A. Marini, M. Cˆ ot´ e, and X. Gonze, Temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation, Physical Review B90, 214304 (2014)

work page 2014
[42]

Antonius, S

G. Antonius, S. Ponc´ e, E. Lantagne-Hurtubise, G. Au- clair, X. Gonze, and M. Cˆ ot´ e, Dynamical and anharmonic effects on the electron-phonon coupling and the zero- point renormalization of the electronic structure, Physi- cal Review B92, 085137 (2015)

work page 2015
[43]

Friedrich, A

M. Friedrich, A. Riefer, S. Sanna, W. Schmidt, and A. Schindlmayr, Phonon dispersion and zero-point renor- malization of linbo3 from density-functional perturbation theory, Journal of Physics: Condensed Matter27, 385402 (2015)

work page 2015
[44]

Tutchton, C

R. Tutchton, C. Marchbanks, and Z. Wu, Structural im- pact on the eigenenergy renormalization for carbon and silicon allotropes and boron nitride polymorphs, Physical Review B97, 205104 (2018)

work page 2018
[45]

J. D. Querales-Flores, J. Cao, S. Fahy, and I. Savi´ c, Tem- perature effects on the electronic band structure of pbte from first principles, Physical Review Materials3, 055405 (2019)

work page 2019
[46]

Miglio, V

A. Miglio, V. Brousseau-Couture, E. Godbout, G. Anto- nius, Y.-H. Chan, S. G. Louie, M. Cˆ ot´ e, M. Giantomassi, and X. Gonze, Predominance of non-adiabatic effects in zero-point renormalization of the electronic band gap, Npj Computational Materials6, 1 (2020)

work page 2020
[47]

Ashkenazi, M

J. Ashkenazi, M. Dacorogna, and M. Peter, On the equiv- alence of the fr¨ ohlich and the bloch approaches to the electron-phonon coupling, Solid State Communications 29, 181 (1979)

work page 1979
[48]

Zollner, M

S. Zollner, M. Cardona, and S. Gopalan, Isotope and temperature shifts of direct and indirect band gaps in diamond-type semiconductors, Physical Review B45, 3376 (1992)

work page 1992
[49]

Giustino, S

F. Giustino, S. G. Louie, and M. L. Cohen, Electron- phonon renormalization of the direct band gap of dia- mond, Physical Review Letters105, 265501 (2010)

work page 2010
[50]

Antonius, S

G. Antonius, S. Ponc´ e, P. Boulanger, M. Cˆ ot´ e, and X. Gonze, Many-body effects on the zero-point renormal- ization of the band structure, Physical Review Letters 112, 215501 (2014)

work page 2014
[51]

D. R. Hamann, Optimized norm-conserving vanderbilt pseudopotentials, Phys. Rev. B88, 085117 (2013)

work page 2013
[52]

M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D. R. Hamann, X. Gonze, and G.-M. Rig- nanese, The pseudodojo: Training and grading a 85 ele- ment optimized norm-conserving pseudopotential table, Computer Physics Communications226, 39 (2018)

work page 2018
[53]

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996)

work page 1996
[54]

C.-Y. Yeh, Z. Lu, S. Froyen, and A. Zunger, Zinc-blende– wurtzite polytypism in semiconductors, Physical Review B46, 10086 (1992)

work page 1992
[55]

Raffy, J

C. Raffy, J. Furthm¨ uller, and F. Bechstedt, Properties of hexagonal polytypes of group-iv elements from first- principles calculations, Physical Review B66, 075201 (2002)

work page 2002
[56]

K¨ ackell, B

P. K¨ ackell, B. Wenzien, and F. Bechstedt, Electronic properties of cubic and hexagonal sic polytypes from ab initio calculations, Physical Review B50, 10761 (1994)

work page 1994
[57]

Mujica, C

A. Mujica, C. J. Pickard, and R. J. Needs, Low-energy tetrahedral polymorphs of carbon, silicon, and germa- nium, Physical Review B91, 214104 (2015)

work page 2015
[58]

Zhang, A

G.-X. Zhang, A. M. Reilly, A. Tkatchenko, and M. Schef- fler, Performance of various density-functional approxi- mations for cohesive properties of 64 bulk solids, New Journal of Physics20, 063020 (2018)

work page 2018
[59]

Grochala, Diamond: electronic ground state of carbon at temperatures approaching 0 k, Angewandte Chemie International Edition53, 3680 (2014)

W. Grochala, Diamond: electronic ground state of carbon at temperatures approaching 0 k, Angewandte Chemie International Edition53, 3680 (2014)

work page 2014
[60]

P. Nath, J. J. Plata, D. Usanmaz, R. A. R. Al Orabi, M. Fornari, M. B. Nardelli, C. Toher, and S. Curtarolo, High-throughput prediction of finite-temperature prop- erties using the quasi-harmonic approximation, Compu- tational Materials Science125, 82 (2016)

work page 2016
[61]

Togo and I

A. Togo and I. Tanaka, First principles phonon calcu- lations in materials science, Scripta Materialia108, 1 (2015)

work page 2015