Pith. sign in

REVIEW 3 major objections 4 minor 52 references

Fast and stable tight-binding framework for nonlocal kinetic energy density functional reconstruction in orbital-free density functional calculations

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read By expanding nonlocal kinetic energy functionals around a superposition of atomic densities, the paper makes accurate orbital-free DFT calculations orders of magnitude faster at nearly unchanged accuracy.

desk verdict A clean linearization of nonlocal KEDFs that delivers real speedups and stable convergence, but the load-bearing assumption of density closeness is verified for only one of the benchmark systems. read the letter →

arxiv 2412.02959 v1 pith:A4XJL73A submitted 2024-12-04 cond-mat.mtrl-sci cond-mat.mes-hallphysics.comp-ph

Yongshuo Chen , Cheng Ma , Boning Cui , Tian Cui , Wenhui Mi , Qiang Xu , Yanchao Wang , Yanming Ma This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.mes-hallphysics.comp-ph PACS 71.15.Mb
keywords orbital-freedensityfunctionaltheorykineticenergytight-bindingapproximationnonlocalKEDFfirst-orderexpansionrevHCLDAK-MGPAnumericalstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the most accurate family of orbital-free density functional theory (OF-DFT) kinetic energy functionals—the nonlocal ones with density-dependent kernels—can be rebuilt in a tight-binding style so that their expensive part is evaluated only once per calculation. The reconstruction expands the nonlocal kinetic energy around a reference density made by superimposing free-atom densities, keeping only the first-order term. If the claim holds, the dominant computational bottleneck of nonlocal KEDFs largely disappears, making these accurate functionals practical for large-scale simulations. The paper demonstrates the strategy on the revHC and LDAK-MGPA functionals across metals, semiconductors, and clusters, reporting comparable accuracy to the original functionals with up to two orders of magnitude less wall time.

What carries the argument

The load-bearing object is the tight-binding density expansion $\rho(r) = \rho_0(r) + \delta\rho(r)$, with $\rho_0$ built as the sum of free-atom densities, combined with a first-order functional Taylor expansion of the nonlocal KEDF. This turns the double-integral nonlocal term, which normally must be recomputed at every self-consistent step, into a constant plus a single linear integral whose kernel is the nonlocal kinetic potential evaluated at $\rho_0$. That potential is computed once during initialization, so the number of nonlocal potential calls during density optimization drops from hundreds to one.

What would settle it

Run the reconstructed and original KEDFs on a system where the optimized density is far from the superposition of free-atom densities, such as a strongly ionic interface, a charged defect, or a hydrogen-bonded solid, and compare total energies and forces. If the TB-KEDF deviates from the original KEDF by more than the claimed accuracy threshold, the first-order truncation is the cause; a quantitative version is to measure the density difference in the bonding region and correlate it with the energy error.

Watch

Extended reading notes

Core claim

The central claim is that a first-order functional expansion of a nonlocal KEDF about the superposition of free-atom densities, $\rho_0$, reconstructs the functional accurately enough for self-consistent OF-DFT, while reducing the nonlocal kinetic energy and potential to a one-shot initialization. Writing $\rho = \rho_0 + \delta\rho$, the nonlocal term $T_{NL}[\rho]$ is replaced by $T_{NL}[\rho_0] + \int V^T_{NL}[\rho_0](r)\delta\rho(r)d^3r$. Because both $T_{NL}[\rho_0]$ and its functional derivative depend only on $\rho_0$, they are computed once, and subsequent density optimization only evaluates the cheap Thomas-Fermi, von Weizsäcker, and linear-response terms. Benchmarking against the original revHC and LDAK-MGPA functionals, the reconstructed TB-KEDFs reproduce equilibrium energies, volumes, and bulk moduli to comparable accuracy and reproduce KS-DFT energy orderings, while requiring roughly 10 to 100 times less time than direct energy minimization and roughly 5 to 10 times less than an orbital-ensemble solver. The paper also finds the reconstructed functionals converge reliably for disordered clusters where the original revHC functional fails to converge.

Load-bearing premise

The whole speedup rests on the assumption that the true electron density stays close to the superposition of free-atom densities, so that keeping only the first term in the expansion is accurate; the paper states this dependence explicitly.

Editorial extensions

If this is right

  • OF-DFT with accurate nonlocal KEDFs becomes feasible for systems of tens of thousands of atoms, where the original functionals are too costly.
  • The reconstruction inherits the accuracy of the parent KEDF when the optimized density stays near $\rho_0$, so future improvements to nonlocal kernels can be adopted without multiplying the cost.
  • The reported stability gain means functionals that are accurate but hard to converge, such as revHC on disordered clusters, become usable in practical structure searches.
  • Because only one nonlocal potential evaluation is needed, the bottleneck shifts from the KEDF evaluation to density optimization and electrostatics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same one-shot expansion could be combined with higher-order corrections in $\delta\rho$, analogous to many-body tight-binding, to extend accuracy to systems with stronger charge redistribution at modest extra cost.
  • Editorial inference: the choice of $\rho_0$ is the transferability knob; using nonlocal-pseudopotential atomic densities, which the paper flags as future work, should widen the range of elements and bond types for which the first-order truncation is valid.
  • Editorial inference: the method's promise suggests a natural test on heterogeneous interfaces or charged defects, where density deviations from superposition are large and the claimed accuracy should degrade measurably.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a tight-binding-inspired framework (TB-KEDF) for orbital-free DFT with nonlocal kinetic energy density functionals. The nonlocal part T_NL[ρ] is approximated by a first-order functional expansion around a reference density ρ0 defined as the superposition of free-atom densities (Eqs. (3)-(5)), so that the expensive nonlocal kinetic energy and potential are evaluated only once before the density optimization begins. The method is implemented in the ATLAS code and tested with revHC and LDAK-MGPA on bulk phases of Li, Mg, Al, Ga, and Si, nine III-V zincblende semiconductors, and 120 random Mg50 and Si50 clusters. The reported results show 10-100x speedups over direct energy minimization, 5-10x speedups relative to OE-SCF for the tested Si supercells, accuracy comparable to the original KEDFs, and improved numerical stability for the cluster calculations. The authors explicitly acknowledge that the framework's effectiveness depends on maintaining a small difference between the reference and optimized electron densities.

Significance. If substantiated, this is a practically valuable contribution because the nonlocal potential evaluation is the dominant cost in OF-DFT with state-of-the-art nonlocal KEDFs. The derivation of Eq. (4) is parameter-free, the reference density is computed independently from single-atom KS-DFT rather than from the functional being approximated, and the benchmarks cover a range of bulk and finite systems with accuracy assessed against external KS-DFT results. The speedups and stability improvements are concrete and potentially enabling for large-scale OF-DFT. However, the central first-order expansion is uncontrolled, and the evidence that the reference density stays close to the optimized density is limited to a single system (CD-Si), so the generality of the 'nearly exactly reproduce' accuracy claim is not yet fully established.

major comments (3)
  1. [Section II, Eq. (4) and Fig. 1(a)] The first-order functional expansion in Eq. (4) neglects the second-order remainder without any estimate or bound. The only density-deviation data presented are for CD-Si (initial reference-density deviations of 17.1% and 18.0% in the bonding region), yet the method is applied to Li, Mg, Al, Ga, and III-V semiconductors, where a superposition of free-atom densities is expected to deviate more from the self-consistent density. To support the claim that TB-KEDFs 'nearly exactly reproduce' the accuracy of the original KEDFs, please report for representative bulk systems (at least one metal and one III-V compound) the maximum |δρ/ρ0| at the converged density and the linearization error, for example |T_NL[ρ_OF] - (T_NL[ρ0] + ∫ V_T_NL[ρ0] δρ)| / T_NL[ρ_OF]. Without such diagnostics, the central approximation remains unverified outside a single semiconducting case.
  2. [Section IV, Fig. 2] The bulk-property comparison reports MAEs and MAPEs with respect to KS-DFT, showing that TB-KEDFs and original KEDFs have similar deviations from KS-DFT. However, because the original KEDFs themselves have substantial errors for some systems (e.g., bulk-modulus MAPEs exceeding 60% in Fig. 2(c)), similarity to KS-DFT does not directly quantify how closely the TB-KEDFs reproduce the functionals they are meant to reconstruct. Please add a direct comparison between TB-KEDF and original-KEDF predictions for E_R, V0, and B0 across all tested phases, for instance mean absolute differences or a parity plot. This would substantiate the wording 'nearly exactly reproduce the accuracy of the original KEDFs'.
  3. [Section IV, Table I and Fig. 1(b)] The efficiency comparison reports wall times for revHC and LDAK-MGPA in Fig. 1(b), but Table I lists nonlocal and semilocal call counts only for revHC. Since LDAK-MGPA has a different and generally more expensive kernel, and the abstract and conclusions claim 'orders-of-magnitude' efficiency improvements for both functionals, the LDAK-MGPA call-count data should be presented in the main text rather than only in the Supplemental Material, or the efficiency claim should be restricted to the systems and functionals for which complete data are shown.
minor comments (4)
  1. [Fig. 1(b)] The horizontal-axis label appears garbled as '/glyph1197umber of atoms' in the manuscript text; it should read 'Number of atoms'.
  2. [Section III] The bulk-derived local pseudopotentials are cited only for Mg, Al, and Si (Ref. [45]); please specify which pseudopotentials were used for Li, Ga, and the III-V compounds, and whether the same pseudopotentials were used for the single-atom reference densities.
  3. [Section IV] The claim of improved numerical stability is supported with convergence statistics for clusters (Fig. S1 and the average 12.6 steps for TB-revHC), but no convergence statistics are given for bulk systems; please either add such data or restrict the stability claim to the finite systems tested.
  4. [Throughout] There are several minor typographical issues, including 'Chac´ on' in Ref. [23], 'Fig.4' in the text before Fig. 4, and inconsistent spacing in '10 −5 eV/atom'; these should be corrected during production.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the first-order expansion is a parameter-free Taylor truncation validated externally against KS-DFT.

full rationale

The paper's core step, Eq. (4), is a first-order functional Taylor expansion of the nonlocal kinetic energy functional around a reference density. It introduces no fitted constants, no adjustable parameters, and no quantities defined in terms of the target prediction. The reference density rho0 is constructed independently from single-atom KS-DFT solutions (Eq. (5) and Section III), not from the functional being reconstructed or from the benchmarked ground-state densities. The claimed accuracy of the TB-KEDFs is then assessed against external KS-DFT results and against the original KEDFs, which is an independent empirical test of the linearization error rather than a restatement of the input. The order-of-magnitude speedup claim follows structurally from evaluating the nonlocal potential only once, as evidenced by the call counts in Table I. Self-citations appear only for the ATLAS implementation, the CALYPSO structure generator, and earlier KEDF developments; none of these is used as a load-bearing uniqueness theorem or as a substitute for the derivation. The acknowledged limitation that accuracy depends on maintaining small differences between the reference and optimized densities is a correctness/robustness caveat, not a circularity. No circular step can be exhibited from the text.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The method introduces no new free parameters or entities. Its validity rests on the smallness of density fluctuations about an atomic superposition reference, and on the quality of the underlying nonlocal KEDFs. These are the load-bearing assumptions rather than any fitted constants.

assumptions (3)
  • domain assumption The nonlocal kinetic energy functional can be accurately approximated by its first-order Taylor expansion about the reference density ρ0.
    Eq. (4) truncates the expansion to first order. The paper's benchmarks show this works for the chosen systems, but no formal error bound or transferability guarantee is provided.
  • domain assumption The superposition of free-atom electron densities is a suitable reference for the systems studied.
    Eq. (5) defines ρ0. The paper justifies it by comparing densities for CD-Si, but for more ionic or charge-transfer systems the reference may be far from the self-consistent density.
  • domain assumption The base functionals revHC and LDAK-MGPA are appropriate and accurate for the tested materials.
    The paper evaluates TB versions against KS-DFT, implicitly trusting the base functionals. Deviations of the base functionals from KS-DFT are not separated from the TB approximation error.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fast and stable tight-binding framework for nonlocal kinetic energy density functional reconstruction in orbital-free density functional calculations." pith.science (2026). https://pith.science/paper/A4XJL73A

@misc{pith2026241202959,
  author       = {Pith},
  title        = {Pith review of: Fast and stable tight-binding framework for nonlocal kinetic energy density functional reconstruction in orbital-free density functional calculations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4XJL73A}},
  note         = {Machine review of arXiv:2412.02959}
}
abstract

Nonlocal kinetic energy density functionals (KEDFs) with density-dependent kernels are currently the most accurate functionals available for orbital-free density functional theory (OF-DFT) calculations. However, despite advances in numerical techniques and using only (semi)local density-dependent kernels, nonlocal KEDFs still present substantial computational costs in OF-DFT, limiting their application in large-scale material simulations. To address this challenge, we propose an efficient framework for reconstructing nonlocal KEDFs by incorporating the density functional tight-binding approach, in which the energy functionals are simplified through a first-order functional expansion based on the superposition of free-atom electron densities. This strategy allows the computationally expensive nonlocal kinetic energy and potential calculations to be performed only once during the electron density optimization process, significantly reducing computational overhead while maintaining high accuracy. Benchmark tests using advanced nonlocal KEDFs, such as revHC and LDAK-MGPA, on standard structures including Li, Mg, Al, Ga, Si, III-V semiconductors, as well as Mg$_{50}$ and Si$_{50}$ clusters, demonstrate that our method achieves orders-of-magnitude improvements in efficiency, providing a cost-effective balance between accuracy and computational speed. Additionally, the reconstructed functionals exhibit improved numerical stability for both bulk and finite systems, paving the way for developing more sophisticated KEDFs for realistic material simulations using OF-DFT.

Figures

Figures reproduced from arXiv: 2412.02959 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Electron density distribution of CD-Si along the [111] direction, calculated using [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of the performance of KEDFs for (a) the equilibrium energy, (b) the equilib [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Relative energy differences calculated by KS-DFT and OF-DFT for nine ZB semiconductors [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The energy obtained by OF-DFT with LDAK-MGPA, TB-LDAK-MGPA, TB-revHC are [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 47 canonical work pages

  1. [1]

    Hohenberg and W

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

    1964

  2. [2]

    Kohn and L

    W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965). 12

    work page 1965

  3. [3]

    Carloni, U

    P. Carloni, U. Rothlisberger, and M. Parrinello, The role and perspective of ab initio molecular dynamics in the study of biological systems, Acc. Chem. Res. 35, 455 (2002)

    work page 2002

  4. [4]

    Burke, Perspective on density functional theory, J

    K. Burke, Perspective on density functional theory, J. Chem. Phys. 136 (2012)

    work page 2012

  5. [5]

    V. V. Karasiev and S. B. Trickey, Issues and challenges in orbital-free density functional calculations, Comput. Phys. Commun. 183, 2519 (2012)

    work page 2012

  6. [6]

    W. C. Witt, B. G. Del Rio, J. M. Dieterich, and E. A. Carter, Orbital-free density functional theory for materials research, J. Mater. Res. 33, 777 (2018)

    work page 2018

  7. [7]

    W. Mi, K. Luo, S. B. Trickey, and M. Pavanello, Orbital-free density functional theory: An attractive electronic structure method for large-scale first-principles simulations, Chem. Rev. 123, 12039 (2023)

    work page 2023

  8. [8]

    Q. Xu, C. Ma, W. Mi, Y. Wang, and Y. Ma, Recent advancements and challenges in orbital- free density functional theory, WIREs Comput. Mol. Sci. 14, e1724 (2024)

    work page 2024

Show all 52 references
  1. [9]

    X. Shao, Q. Xu, S. Wang, J. Lv, Y. Wang, and Y. Ma, Large-scale ab initio simulations for periodic system, Comput. Phys. Commun. 233, 78 (2018)

    2018

  2. [10]

    Hung and E

    L. Hung and E. A. Carter, Accurate simulations of metals at the mesoscale: Explicit treatment of 1 million atoms with quantum mechanics, Chem. Phys. Lett. 475, 163 (2009)

    2009

  3. [11]

    X. Shao, K. Jiang, W. Mi, A. Genova, and M. Pavanello, Dftpy: An efficient and object- oriented platform for orbital-free dft simulations, WIREs Comput. Mol. Sci. 11, e1482 (2021)

    2021

  4. [12]

    C. F. v. Weizs¨ acker, Zur theorie der kernmassen, Z. Phys. 96, 431 (1935)

    1935

  5. [13]

    Laricchia, E

    S. Laricchia, E. Fabiano, L. Constantin, and F. Della Sala, Generalized gradient approxima- tions of the noninteracting kinetic energy from the semiclassical atom theory: Rationalization of the accuracy of the frozen density embedding theory for nonbonded interactions, J. Chem...

    2011

  6. [14]

    Ou-Yang and M

    H. Ou-Yang and M. Levy, Approximate noninteracting kinetic energy functionals from a nonuniform scaling requirement, Int. J. Quantum Chem. 40, 379 (1991)

    1991

  7. [15]

    J. P. Perdew, Generalized gradient approximation for the fermion kinetic energy as a functional of the density, Phys. Lett. A 165, 79 (1992)

    1992

  8. [16]

    Ernzerhof, The role of the kinetic energy density in approximations to the exchange energy, J

    M. Ernzerhof, The role of the kinetic energy density in approximations to the exchange energy, J. Mol. Struct.: THEOCHEM 501, 59 (2000)

    2000

  9. [17]

    L. A. Constantin and A. Ruzsinszky, Kinetic energy density functionals from the airy gas with an application to the atomization kinetic energies of molecules, Phys. Rev. B 79, 115117 13 (2009)

    2009

  10. [18]

    Vitos, B

    L. Vitos, B. Johansson, J. Kollar, and H. L. Skriver, Local kinetic-energy density of the airy gas, Phys. Rev. A 61, 052511 (2000)

    2000

  11. [19]

    A. J. Thakkar, Comparison of kinetic-energy density functionals, Phys. Rev. A46, 6920 (1992)

    1992

  12. [20]

    Lembarki and H

    A. Lembarki and H. Chermette, Obtaining a gradient-corrected kinetic-energy functional from the perdew-wang exchange functional, Phys. Rev. A 50, 5328 (1994)

    1994

  13. [21]

    K. Luo, V. V. Karasiev, and S. Trickey, A simple generalized gradient approximation for the noninteracting kinetic energy density functional, Phys. Rev. B 98, 041111 (2018)

    2018

  14. [22]

    Garcia-Aldea and J

    D. Garcia-Aldea and J. Alvarellos, Kinetic-energy density functionals with nonlocal terms with the structure of the thomas-fermi functional, Phys. Rev. A 76, 052504 (2007)

    2007

  15. [23]

    Chac´ on, J

    E. Chac´ on, J. Alvarellos, and P. Tarazona, Nonlocal kinetic energy functional for nonhomo- geneous electron systems, Phys. Rev. B 32, 7868 (1985)

    1985

  16. [24]

    Wang and M

    L.-W. Wang and M. P. Teter, Kinetic-energy functional of the electron density, Phys. Rev. B 45, 13196 (1992)

    1992

  17. [25]

    Smargiassi and P

    E. Smargiassi and P. A. Madden, Orbital-free kinetic-energy functionals for first-principles molecular dynamics, Phys. Rev. B 49, 5220 (1994)

    1994

  18. [26]

    Perrot, Hydrogen-hydrogen interaction in an electron gas, J

    F. Perrot, Hydrogen-hydrogen interaction in an electron gas, J. Phys.: Condens. Matter 6, 431 (1994)

    1994

  19. [27]

    Y. A. Wang, N. Govind, and E. A. Carter, Orbital-free kinetic-energy density functionals with a density-dependent kernel, Phys. Rev. B 60, 16350 (1999)

    1999

  20. [28]

    Huang and E

    C. Huang and E. A. Carter, Nonlocal orbital-free kinetic energy density functional for semi- conductors, Phys. Rev. B 81, 045206 (2010)

    2010

  21. [29]

    W. Mi, A. Genova, and M. Pavanello, Nonlocal kinetic energy functionals by functional inte- gration, J. Chem. Phys. 148, 184107 (2018)

    2018

  22. [30]

    Q. Xu, Y. Wang, and Y. Ma, Nonlocal kinetic energy density functional via line integrals and its application to orbital-free density functional theory, Phys. Rev. B 100, 205132 (2019)

    2019

  23. [31]

    Yang, Ab initio approach for many-electron systems without invoking orbitals: An integral formulation of density-functional theory, Phys

    W. Yang, Ab initio approach for many-electron systems without invoking orbitals: An integral formulation of density-functional theory, Phys. Rev. Lett. 59, 1569 (1987)

    1987

  24. [32]

    X. Shao, W. Mi, and M. Pavanello, Revised huang-carter nonlocal kinetic energy functional for semiconductors and their surfaces, Phys. Rev. B 104, 045118 (2021). 14

    2021

  25. [33]

    Mi and M

    W. Mi and M. Pavanello, Orbital-free density functional theory correctly models quantum dots when asymptotics, nonlocality, and nonhomogeneity are accounted for, Phys. Rev. B 100, 041105 (2019)

    2019

  26. [34]

    Q. Xu, J. Lv, Y. Wang, and Y. Ma, Nonlocal kinetic energy density functionals for isolated systems obtained via local density approximation kernels, Phys. Rev. B 101, 045110 (2020)

    2020

  27. [35]

    L. A. Constantin, E. Fabiano, and F. Della Sala, Nonlocal kinetic energy functional from the jellium-with-gap model: Applications to orbital-free density functional theory, Phys. Rev. B 97, 205137 (2018)

    2018

  28. [36]

    X. Shao, W. Mi, and M. Pavanello, Efficient dft solver for nanoscale simulations and beyond, J. Phys. Chem. Lett. 12, 4134 (2021)

    2021

  29. [37]

    Elstner, D

    M. Elstner, D. Porezag, G. Jungnickel, J. Elsner, M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties, Phys. Rev. B 58, 7260 (1998)

    1998

  30. [38]

    W. Mi, X. Shao, C. Su, Y. Zhou, S. Zhang, Q. Li, H. Wang, L. Zhang, M. Miao, Y. Wang, and Y. Ma, ATLAS: a real-space finite-difference implementation of orbital-free density functional theory, Comput. Phys. Commun. 200, 87 (2016)

    2016

  31. [39]

    X. Shao, W. Mi, Q. Xu, Y. Wang, and Y. Ma, O (n log n) scaling method to evaluate the ion–electron potential of crystalline solids, J. Chem. Phys. 145 (2016)

    2016

  32. [40]

    L. H. Thomas, The calculation of atomic fields, Math. Proc. Cambridge Philos. Soc. 23, 542 (1927)

    1927

  33. [41]

    Fermi, Un metodo statistico per la determinazione di alcune priorieta dell’atome, Rend

    E. Fermi, Un metodo statistico per la determinazione di alcune priorieta dell’atome, Rend. Accad. Naz. Lincei 6, 32 (1927)

    1927

  34. [42]

    Fermi, Eine statistische methode zur bestimmung einiger eigenschaften des atoms und ihre anwendung auf die theorie des periodischen systems der elemente, Z

    E. Fermi, Eine statistische methode zur bestimmung einiger eigenschaften des atoms und ihre anwendung auf die theorie des periodischen systems der elemente, Z. Phys. 48, 73 (1928)

    1928

  35. [43]

    Giannozzi, S

    P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mau...

    2009

  36. [44]

    A. H. Larsen, J. J. Mortensen, J. Blomqvist, I. E. Castelli, R. Christensen, M. Du lak, J. Friis, M. N. Groves, B. Hammer, C. Hargus, E. D. Hermes, P. C. Jennings, P. B. Jensen, J. Kermode, J. R. Kitchin, E. L. Kolsbjerg, J. Kubal, K. Kaasbjerg, S. Lysgaard, J. B. Maronsson, T...

    2017

  37. [45]

    Huang and E

    C. Huang and E. A. Carter, Transferable local pseudopotentials for magnesium, aluminum and silicon, Phys. Chem. Chem. Phys. 10, 7109 (2008)

    2008

  38. [46]

    J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Phys. Rev. B 23, 5048 (1981)

    1981

  39. [47]

    Boyer, E

    L. Boyer, E. Kaxiras, J. Feldman, J. Broughton, and M. Mehl, New low-energy crystal struc- ture for silicon, Phys. Rev. Lett. 67, 715 (1991)

    1991

  40. [48]

    Yin and M

    M. Yin and M. L. Cohen, Theory of static structural properties, crystal stability, and phase transformations: Application to si and ge, Phys. Rev. B 26, 5668 (1982)

    1982

  41. [49]

    Y. Wang, J. Lv, L. Zhu, and Y. Ma, Crystal structure prediction via particle-swarm optimiza- tion, Phys. Rev. B 82, 094116 (2010)

    2010

  42. [50]

    Y. Wang, J. Lv, L. Zhu, and Y. Ma, Calypso: A method for crystal structure prediction, Comput. Phys. Commun. 183, 2063 (2012)

    2012

  43. [51]

    See Supplemental Material at URL-will-be-inserted-by-publisher for the data

  44. [52]

    Murnaghan, The compressibility of media under extreme pressures, Proc

    F. Murnaghan, The compressibility of media under extreme pressures, Proc. Natl. Acad. Sci. USA 30, 244 (1944). 16

    1944

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.