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arxiv: 2604.07046 · v1 · submitted 2026-04-08 · ⚛️ physics.chem-ph · cond-mat.mtrl-sci

Self-consistent Hessian-level meta-generalized gradient approximation

Pooria Dabbaghi , Juan Maria Garc\'ia Lastra , Piotr de Silva This is my paper

Pith reviewed 2026-05-10 17:13 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.mtrl-sci
keywords density functional theorymeta-GGAdensity Hessianexchange-correlation functionalprojector augmented-waveorbital-independentchemisorption
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0 comments X

The pith

The full density Hessian distinguishes atomic and bonding one-electron limits that standard indicators mix together.

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

The paper reformulates the ϑ-MGGA class of functionals as Hessian-level meta-GGAs that draw on the complete set of second-order spatial density derivatives instead of kinetic-energy density or the Laplacian. This descriptor separates single-center atomic densities from two-center bonds more cleanly than conventional iso-orbital indicators. The authors introduce a simplified non-empirical functional called ϑ-PBE and sketch its self-consistent implementation inside the projector augmented-wave method. Benchmarks on molecular and solid datasets show useful accuracy for chemisorption energies and molecular properties, although bulk lattice constants remain difficult. The work establishes that orbital-independent, Hessian-based exchange-correlation functionals are both physically motivated and practically realizable.

Core claim

By employing the full density Hessian, the functional's descriptor successfully separates distinct one-electron density limits such as single-center atomic densities and two-center bonds that standard iso-orbital indicators often conflate. A simplified non-empirical functional ϑ-PBE is presented together with a roadmap for its self-consistent projector augmented-wave implementation, and benchmarks confirm accurate chemisorption energies and molecular properties across the tested sets while lattice constants in solids continue to pose challenges.

What carries the argument

Hessian-level meta-generalized gradient approximation whose exchange-correlation energy is built directly from the complete set of spatial second-order density derivatives.

If this is right

  • Accurate chemisorption energies and molecular properties become available without orbital dependence.
  • Self-consistent projector augmented-wave calculations are feasible for both molecules and periodic systems.
  • The orbital-independent construction removes a major computational bottleneck for large-scale simulations.
  • Further refinement of the functional form is needed to improve bulk lattice constants.

Where Pith is reading between the lines

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

  • The same Hessian information could be reused to build response properties or time-dependent extensions.
  • Replacing the base functional with other non-empirical forms might systematically improve solid-state performance.
  • The descriptor's ability to separate density limits suggests it could be combined with range-separated hybrids for broader accuracy.

Load-bearing premise

The Hessian-based indicator must remain numerically stable and computationally tractable when used self-consistently in the projector augmented-wave method for the full range of molecular and periodic systems.

What would settle it

A self-consistent calculation on a standard diatomic molecule or simple solid that fails to converge or produces unstable density derivatives would show the approach is not yet practical.

Figures

Figures reproduced from arXiv: 2604.07046 by Juan Maria Garc\'ia Lastra, Piotr de Silva, Pooria Dabbaghi.

Figure 1
Figure 1. Figure 1: FIG. 1. Exchange energy per particle along the internuclear axis of the Na [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Fitting of the scaling parameter [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Errors in calculated lattice-constants (in Å) relative to experimental values for the LC20 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

The $\vartheta$-MGGA class of density functionals is formally reformulated as Hessian-level meta-generalized gradient approximations (HL-MGGAs). In contrast to standard meta-GGAs that rely on the orbital-dependent kinetic-energy density or the density Laplacian, HL-MGGAs utilize the full density Hessian. We introduce a simplified, non-empirical functional, $\vartheta$-PBE, and present a roadmap for its self-consistent implementation within the projector augmented-wave (PAW) method. By utilizing the complete set of spatial second-order density derivatives, the functional's underlying descriptor successfully distinguishes between distinct one-electron density limits, such as single-center atomic densities and two-center bonds, that standard iso-orbital indicators often conflate. Benchmarks across molecular and solid-state datasets reveal that while $\vartheta$-PBE delivers accurate chemisorption energies and molecular properties, challenges remain in predicting bulk lattice constants. Ultimately, this work demonstrates the physical utility and feasibility of designing orbital-independent, Hessian-based exchange-correlation functionals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript reformulates the ϑ-MGGA class as Hessian-level meta-GGAs (HL-MGGAs) that use the full density Hessian in place of the kinetic-energy density or Laplacian. It introduces the simplified non-empirical functional ϑ-PBE and supplies a roadmap for self-consistent PAW implementation. The central claim is that the complete set of second-order density derivatives allows the underlying descriptor to distinguish single-center atomic densities from two-center bond densities more effectively than standard iso-orbital indicators. Benchmarks on molecular and solid-state data sets are reported to yield accurate chemisorption energies and molecular properties, while bulk lattice constants remain challenging.

Significance. If the self-consistent PAW calculations ultimately confirm numerical stability of the Hessian components and reproduce the quoted accuracies without ad-hoc smoothing, the work would provide a concrete demonstration that orbital-independent functionals can exploit higher-order density information to resolve physically distinct one-electron limits. The parameter-free construction and explicit roadmap for PAW are constructive contributions. At present, however, the absence of completed self-consistent results across the benchmark sets leaves the practical advantage of the Hessian descriptor unverified.

major comments (1)
  1. [Implementation roadmap and benchmarks] The manuscript presents benchmark results for chemisorption energies and molecular properties yet describes the PAW self-consistent implementation only as a roadmap. Because the central claim rests on the Hessian descriptor distinguishing density limits in the functional's own converged density, the lack of explicit verification that the six independent Hessian components remain stable (without smoothing that would erase the claimed distinction) is load-bearing for the performance assertions.
minor comments (1)
  1. The abstract states that challenges remain with bulk lattice constants but does not quantify the errors or identify which data sets were used; adding a brief table or sentence with the relevant mean absolute errors would improve clarity.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the constructive comments and for recognizing the potential of the Hessian-level meta-GGA approach. We concur that self-consistent calculations are necessary to fully verify the practical advantages. The present work establishes the formal framework, introduces the non-empirical ϑ-PBE functional, and supplies a detailed implementation roadmap, while reporting non-self-consistent benchmarks to demonstrate initial performance. We provide a point-by-point response to the major comment and indicate the revisions made.

read point-by-point responses

  1. Referee: [Implementation roadmap and benchmarks] The manuscript presents benchmark results for chemisorption energies and molecular properties yet describes the PAW self-consistent implementation only as a roadmap. Because the central claim rests on the Hessian descriptor distinguishing density limits in the functional's own converged density, the lack of explicit verification that the six independent Hessian components remain stable (without smoothing that would erase the claimed distinction) is load-bearing for the performance assertions.

    Authors: We appreciate the referee pointing out the distinction between the roadmap and the completed benchmarks. The performance data for chemisorption energies and molecular properties were indeed generated by applying the ϑ-PBE functional to self-consistent densities obtained from a reference GGA functional. This procedure allows assessment of the functional's form without the full self-consistent loop. The core claim concerning superior distinction of one-electron density limits is supported by the analytic properties of the Hessian-based iso-orbital indicator, which we demonstrate using prototypical density models (atomic vs. bonding). These properties are intrinsic to the descriptor and do not depend on the density being self-consistent with ϑ-PBE, although consistency would be ideal. In the revised manuscript, we have clarified that the benchmarks are non-self-consistent and have expanded the discussion of the roadmap to include explicit statements on the numerical stability of the six Hessian components, confirming that no ad-hoc smoothing is required that would diminish the descriptor's resolution. We note that full self-consistent results will be pursued in subsequent studies. revision: partial

standing simulated objections not resolved
  • The provision of fully self-consistent PAW calculations and associated stability verification for the Hessian components on the complete benchmark sets, as this constitutes future implementation work outlined in the roadmap.

Circularity Check

0 steps flagged

Reformulation of ϑ-MGGA class into HL-MGGA with non-empirical ϑ-PBE exhibits only minor self-citation

full rationale

The manuscript reformulates the existing ϑ-MGGA class as Hessian-level MGGAs and introduces the simplified non-empirical ϑ-PBE functional. The central descriptor distinction between single-center and two-center one-electron limits is presented as a direct consequence of using the full density Hessian rather than iso-orbital indicators. Benchmarks on molecular and solid-state data are reported independently of any fitted parameters in the present work. The self-citation to prior ϑ-MGGA work is present but not load-bearing for the new claims or the PAW implementation roadmap; no equation reduces a reported performance metric or physical distinction to a parameter fit or self-referential definition by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Hessian reformulation and the superiority of the new descriptor; no free parameters are introduced and no new physical entities are postulated beyond standard DFT assumptions.

axioms (1)
  • domain assumption Standard Kohn-Sham DFT framework and the existence of an exchange-correlation functional depending on density and its derivatives
    Invoked throughout the reformulation of ϑ-MGGA into HL-MGGA.

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages

  1. [1]

    Ionic relaxations were performed using the BFGS algorithm until forces dropped below 0.05 eV/Å (0.08 eV/Å for Ni)

    with a width of 0.2 eV. Ionic relaxations were performed using the BFGS algorithm until forces dropped below 0.05 eV/Å (0.08 eV/Å for Ni). For adsorption calculations, the lower two layers were fixed at bulk positions, whereas for the clean reference slabs, all atoms were allowed to relax. Adsorbates were initialized at atop, bridge, hollow, and fcc sites...

    work page 2022

  2. [2]

    Inhomogeneous electron gas.Physical review, 136(3B):B864, 1964

    Pierre Hohenberg and Walter Kohn. Inhomogeneous electron gas.Physical review, 136(3B):B864, 1964

    work page 1964

  3. [3]

    Self-consistent equations including exchange and correlation effects.Physical review, 140(4A):A1133, 1965

    Walter Kohn and Lu Jeu Sham. Self-consistent equations including exchange and correlation effects.Physical review, 140(4A):A1133, 1965

    work page 1965

  4. [4]

    Perdew and Wang Yue

    John P. Perdew and Wang Yue. Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation.Phys. Rev. B, 33:8800–8802, Jun 1986

    work page 1986

  5. [5]

    A. D. Becke. Density-functional exchange-energy approximation with correct asymptotic be- havior.Phys. Rev. A, 38:3098–3100, Sep 1988

    work page 1988

  6. [6]

    Generalized gradient approximation made simple.Physical review letters, 77(18):3865, 1996

    John P Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized gradient approximation made simple.Physical review letters, 77(18):3865, 1996

    work page 1996

  7. [7]

    Improved adsorption energetics within density-functional theory using revised perdew-burke-ernzerhof functionals.Physical review B, 59(11):7413, 1999

    BHLB Hammer, Lars Bruno Hansen, and Jens Kehlet Nørskov. Improved adsorption energetics within density-functional theory using revised perdew-burke-ernzerhof functionals.Physical review B, 59(11):7413, 1999

    work page 1999

  8. [8]

    Perdew, Adrienn Ruzsinszky, Gábor I

    John P. Perdew, Adrienn Ruzsinszky, Gábor I. Csonka, Oleg A. Vydrov, Gustavo E. Scuse- ria, Lucian A. Constantin, Xiaolan Zhou, and Kieron Burke. Restoring the density-gradient expansion for exchange in solids and surfaces.Phys. Rev. Lett., 100:136406, Apr 2008

    work page 2008

  9. [9]

    Non-empirical improve- ment of pbe and its hybrid pbe0 for general description of molecular properties.The Journal of chemical physics, 136(10), 2012

    Jorge M del Campo, José L Gázquez, SB Trickey, and Alberto Vela. Non-empirical improve- ment of pbe and its hybrid pbe0 for general description of molecular properties.The Journal of chemical physics, 136(10), 2012

    work page 2012

  10. [10]

    EduardoFabiano, LucianAConstantin, andFabioDellaSala. Generalizedgradientapproxima- tion bridging the rapidly and slowly varying density regimes: A pbe-like functional for hybrid interfaces.Physical Review B—Condensed Matter and Materials Physics, 82(11):113104, 2010

    work page 2010

  11. [11]

    A local exchange-correlation potential for the spin polarized case

    U von Barth and L Hedin. A local exchange-correlation potential for the spin polarized case. i.Journal of Physics C: Solid State Physics, 5(13):1629, Jul 1972. 19

    work page 1972

  12. [12]

    Perdew and Yue Wang

    John P. Perdew and Yue Wang. Accurate and simple analytic representation of the electron-gas correlation energy.Phys. Rev. B, 45:13244–13249, Jun 1992

    work page 1992

  13. [13]

    P. S. Svendsen and U. von Barth. Gradient expansion of the exchange energy from second-order density response theory.Phys. Rev. B, 54:17402–17413, Dec 1996

    work page 1996

  14. [14]

    Correlation energy of an electron gas with a slowly varying high density.Physical Review, 165(1):18, 1968

    KEITH A BRUECKNER et al. Correlation energy of an electron gas with a slowly varying high density.Physical Review, 165(1):18, 1968

    work page 1968

  15. [15]

    Pedroza, Antonio J

    Philipp Haas, Fabien Tran, Peter Blaha, Luana S. Pedroza, Antonio J. R. da Silva, Mariana M. Odashima, and Klaus Capelle. Systematic investigation of a family of gradient-dependent functionals for solids.Phys. Rev. B, 81:125136, Mar 2010

    work page 2010

  16. [16]

    John P. Perdew. Accurate density functional for the energy: Real-space cutoff of the gradient expansion for the exchange hole.Phys. Rev. Lett., 55:1665–1668, Oct 1985

    work page 1985

  17. [17]

    A. D. Becke and M. R. Roussel. Exchange holes in inhomogeneous systems: A coordinate-space model.Phys. Rev. A, 39:3761–3767, Apr 1989

    work page 1989

  18. [18]

    Phase-spaceapproachtotheexchange-energyfunctional of density-functional theory.Phys

    SwapanK.GhoshandRobertG.Parr. Phase-spaceapproachtotheexchange-energyfunctional of density-functional theory.Phys. Rev. A, 34:785–791, Aug 1986

    work page 1986

  19. [19]

    Scuseria

    Troy Van Voorhis and Gustavo E. Scuseria. A novel form for the exchange-correlation energy functional.The Journal of Chemical Physics, 109(2):400–410, 07 1998

    work page 1998

  20. [20]

    Climbing the density functional ladder: Nonempirical meta–generalized gradient approximation designed for molecules and solids.Physical review letters, 91(14):146401, 2003

    Jianmin Tao, John P Perdew, Viktor N Staroverov, and Gustavo E Scuseria. Climbing the density functional ladder: Nonempirical meta–generalized gradient approximation designed for molecules and solids.Physical review letters, 91(14):146401, 2003

    work page 2003

  21. [21]

    Strongly constrained and appropriately normed semilocal density functional.Physical review letters, 115(3):036402, 2015

    Jianwei Sun, Adrienn Ruzsinszky, and John P Perdew. Strongly constrained and appropriately normed semilocal density functional.Physical review letters, 115(3):036402, 2015

    work page 2015

  22. [22]

    Garza, Alexis T

    Alejandro J. Garza, Alexis T. Bell, and Martin Head-Gordon. Nonempirical meta-generalized gradient approximations for modeling chemisorption at metal surfaces.Journal of Chemical Theory and Computation, 14(6):3083–3090, 2018. PMID: 29746113

    work page 2018

  23. [23]

    Accurate and numerically efficient r2scan meta-generalized gradient approximation.The journal of physical chemistry letters, 11(19):8208–8215, 2020

    James W Furness, Aaron D Kaplan, Jinliang Ning, John P Perdew, and Jianwei Sun. Accurate and numerically efficient r2scan meta-generalized gradient approximation.The journal of physical chemistry letters, 11(19):8208–8215, 2020

    work page 2020

  24. [24]

    Balancing the contributions to the gradient expansion: Accurate binding and band gaps with a nonempirical meta-gga.Phys

    Timo Lebeda, Thilo Aschebrock, and Stephan Kümmel. Balancing the contributions to the gradient expansion: Accurate binding and band gaps with a nonempirical meta-gga.Phys. Rev. Lett., 133:136402, Sep 2024. 20

    work page 2024

  25. [25]

    Daniel Mejia-Rodriguez and S. B. Trickey. Deorbitalization strategies for meta-generalized- gradient-approximation exchange-correlation functionals.Phys. Rev. A, 96:052512, Nov 2017

    work page 2017

  26. [26]

    Daniel Mejía-Rodríguez and S. B. Trickey. Meta-gga performance in solids at almost gga cost. Phys. Rev. B, 102:121109, Sep 2020

    work page 2020

  27. [27]

    Kaplan and John P

    Aaron D. Kaplan and John P. Perdew. Laplacian-level meta-generalized gradient approxima- tion for solid and liquid metals.Phys. Rev. Mater., 6:083803, Aug 2022

    work page 2022

  28. [28]

    CFZ Von Weizäcker. Phys. 1935 96 431 von weizäcker, cf.Z. Phys, 96:431, 1935

    work page 1935

  29. [29]

    Application of statistical gas methods to electronic systems.Atti accad

    E Fermi. Application of statistical gas methods to electronic systems.Atti accad. naz. Lincei, 6:602–607, 1927

    work page 1927

  30. [30]

    Seidl, A

    A. Seidl, A. Görling, P. Vogl, J. A. Majewski, and M. Levy. Generalized kohn-sham schemes and the band-gap problem.Phys. Rev. B, 53:3764–3774, Feb 1996

    work page 1996

  31. [31]

    Piotr de Silva, Jacek Korchowiec, and Tomasz A Wesolowski. Revealing the bonding pattern from the molecular electron density using single exponential decay detector: an orbital-free alternative to the electron localization function.ChemPhysChem, 13(15):3462–3465, 2012

    work page 2012

  32. [32]

    Extracting information about chemical bonding from molecular electron densities via single exponential decay detector (sedd).Chimia, 67(4):253–253, 2013

    Piotr de Silva, Jacek Korchowiec, Nirmal Ram JS, and Tomasz A Wesolowski. Extracting information about chemical bonding from molecular electron densities via single exponential decay detector (sedd).Chimia, 67(4):253–253, 2013

    work page 2013

  33. [33]

    Wesolowski

    Piotr de Silva, Jacek Korchowiec, and Tomasz A. Wesolowski. Atomic shell structure from the single-exponential decay detector.The Journal of Chemical Physics, 140(16):164301, 04 2014

    work page 2014

  34. [34]

    Simultaneous visualization of covalent and nonco- valent interactions using regions of density overlap.Journal of Chemical Theory and Compu- tation, 10(9):3745–3756, Sep 2014

    Piotr de Silva and Clémence Corminboeuf. Simultaneous visualization of covalent and nonco- valent interactions using regions of density overlap.Journal of Chemical Theory and Compu- tation, 10(9):3745–3756, Sep 2014

    work page 2014

  35. [35]

    Communication: A new class of non-empirical explicit density functionals on the third rung of jacob’s ladder.The Journal of Chemical Physics, 143(11):111105, 2015

    Piotr de Silva and Clémence Corminboeuf. Communication: A new class of non-empirical explicit density functionals on the third rung of jacob’s ladder.The Journal of Chemical Physics, 143(11):111105, 2015

    work page 2015

  36. [36]

    Maier, Matthias Haasler, Alexei V

    Toni M. Maier, Matthias Haasler, Alexei V. Arbuznikov, and Martin Kaupp. New approaches for the calibration of exchange-energy densities in local hybrid functionals.Phys. Chem. Chem. Phys., 18:21133–21144, 2016

    work page 2016

  37. [37]

    Local hybrid functionals with orbital-free mixing functions and balanced elimination of self-interaction error.The Journal of chemical physics, 142(7), 2015

    Piotr De Silva and Clémence Corminboeuf. Local hybrid functionals with orbital-free mixing functions and balanced elimination of self-interaction error.The Journal of chemical physics, 142(7), 2015. 21

    work page 2015

  38. [38]

    P. E. Blöchl. Projector augmented-wave method.Phys. Rev. B, 50:17953–17979, Dec 1994

    work page 1994

  39. [39]

    Higher-order gradient corrections for exchange- correlation functionals.Chemical physics letters, 266(1-2):16–22, 1997

    Ralf Neumann and Nicholas C Handy. Higher-order gradient corrections for exchange- correlation functionals.Chemical physics letters, 266(1-2):16–22, 1997

    work page 1997

  40. [40]

    Improved lower bound on the indirect coulomb energy

    Elliott H Lieb and Stephen Oxford. Improved lower bound on the indirect coulomb energy. International Journal of Quantum Chemistry, 19(3):427–439, 1981

    work page 1981

  41. [41]

    Electronic structure of solids’ 91, 1991

    John P Perdew, P Ziesche, and H Eschrig. Electronic structure of solids’ 91, 1991

    work page 1991

  42. [42]

    Workhorse semilocal density functional for condensed matter physics and quantum chemistry

    JohnP.Perdew, AdriennRuzsinszky, GáborI.Csonka, LucianA.Constantin, andJianweiSun. Workhorse semilocal density functional for condensed matter physics and quantum chemistry. Phys. Rev. Lett., 103:026403, Jul 2009

    work page 2009

  43. [43]

    G. L. Oliver and J. P. Perdew. Spin-density gradient expansion for the kinetic energy.Phys. Rev. A, 20:397–403, Aug 1979

    work page 1979

  44. [44]

    Glenn Moynihan, Gilberto Teobaldi, and David D. O’Regan. Inapplicability of exact con- straints and a minimal two-parameter generalization to the dft+u based correction of self- interaction error.Phys. Rev. B, 94:220104, Dec 2016

    work page 2016

  45. [45]

    Challenges for density functional theory

    Aron J Cohen, Paula Mori-Sánchez, and Weitao Yang. Challenges for density functional theory. Chemical reviews, 112(1):289–320, 2012

    work page 2012

  46. [46]

    Ivanov, Alireza Taghizadeh, Andrew Peterson, Anubhab Haldar, Asmus Ougaard Dohn, Christian Schäfer, Elvar Örn Jónsson, Eric D

    Jens Jørgen Mortensen, Ask Hjorth Larsen, Mikael Kuisma, Aleksei V. Ivanov, Alireza Taghizadeh, Andrew Peterson, Anubhab Haldar, Asmus Ougaard Dohn, Christian Schäfer, Elvar Örn Jónsson, Eric D. Hermes, Fredrik Andreas Nilsson, Georg Kastlunger, Gianluca Levi, Hannes Jónsson, Hannu Häkkinen, Jakub Fojt, Jiban Kangsabanik, Joachim Sødequist, Jouko Lehtomäk...

    work page 2024

  47. [47]

    Self-consistent implementation of hessian-level meta-ggas.https://gitlab.com/gpaw/gpaw/ -/merge_requests/3232. 22

    work page

  48. [48]

    The matrix cookbook.Technical University of Denmark, 7(15):510, 2008

    Kaare Brandt Petersen, Michael Syskind Pedersen, et al. The matrix cookbook.Technical University of Denmark, 7(15):510, 2008

    work page 2008

  49. [49]

    Projector augmented wave method: ab initio molecular dynamics with full wave functions.Bulletin of Materials Science, 26(1):33– 41, 2003

    Peter E Blöchl, Clemens J Först, and Johannes Schimpl. Projector augmented wave method: ab initio molecular dynamics with full wave functions.Bulletin of Materials Science, 26(1):33– 41, 2003

    work page 2003

  50. [50]

    Kresse and D

    G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method.Phys. Rev. B, 59:1758–1775, Jan 1999

    work page 1999

  51. [51]

    The projector augmented-wave method, 2009

    Carsten Rostgaard. The projector augmented-wave method, 2009

    work page 2009

  52. [52]

    Chelikowsky, N

    James R. Chelikowsky, N. Troullier, K. Wu, and Y. Saad. Higher-order finite-difference pseu- dopotential method: An application to diatomic molecules.Phys. Rev. B, 50:11355–11364, Oct 1994

    work page 1994

  53. [53]

    Smallrepresentativebenchmarksforthermochemical calculations.The Journal of Physical Chemistry A, 107(42):8996–8999, 2003

    BenjaminJLynchandDonaldGTruhlar. Smallrepresentativebenchmarksforthermochemical calculations.The Journal of Physical Chemistry A, 107(42):8996–8999, 2003

    work page 2003

  54. [54]

    Yan Zhao, Núria González-García, and Donald G Truhlar. Benchmark database of bar- rier heights for heavy atom transfer, nucleophilic substitution, association, and unimolecu- lar reactions and its use to test theoretical methods.The Journal of Physical Chemistry A, 109(9):2012–2018, 2005

    work page 2012

  55. [55]

    Theoretical reference values for the ae6 and bh6 test sets from explicitly correlated coupled-cluster theory.Theoretical Chemistry Accounts, 131(2):1112, 2012

    Robin Haunschild and Wim Klopper. Theoretical reference values for the ae6 and bh6 test sets from explicitly correlated coupled-cluster theory.Theoretical Chemistry Accounts, 131(2):1112, 2012

    work page 2012

  56. [56]

    Lars Goerigk, Andreas Hansen, Christoph Bauer, Stephan Ehrlich, Asim Najibi, and Stefan Grimme. A look at the density functional theory zoo with the advanced gmtkn55 database for generalmaingroupthermochemistry, kineticsandnoncovalentinteractions.Physical Chemistry Chemical Physics, 19(48):32184–32215, 2017

    work page 2017

  57. [57]

    Csonka, Adrienn Ruzsinszky, Pan Hao, Yoon-Suk Kim, Georg Kresse, and John P

    Jianwei Sun, Martijn Marsman, Gábor I. Csonka, Adrienn Ruzsinszky, Pan Hao, Yoon-Suk Kim, Georg Kresse, and John P. Perdew. Self-consistent meta-generalized gradient approxi- mation within the projector-augmented-wave method.Phys. Rev. B, 84:035117, Jul 2011

    work page 2011

  58. [58]

    Monkhorst and James D

    Hendrik J. Monkhorst and James D. Pack. Special points for brillouin-zone integrations.Phys. Rev. B, 13:5188–5192, Jun 1976

    work page 1976

  59. [59]

    Daniel Mejia-Rodriguez and S. B. Trickey. Deorbitalized meta-gga exchange-correlation func- tionals in solids.Phys. Rev. B, 98:115161, Sep 2018. 23

    work page 2018

  60. [60]

    Alchagirov, John P

    Alim B. Alchagirov, John P. Perdew, Jonathan C. Boettger, R. C. Albers, and Carlos Fiolhais. Energy and pressure versus volume: Equations of state motivated by the stabilized jellium model.Phys. Rev. B, 63:224115, May 2001

    work page 2001

  61. [61]

    The atomic simulation envi- ronment—a python library for working with atoms.Journal of Physics: Condensed Matter, 29(27):273002, 2017

    Ask Hjorth Larsen, Jens Jørgen Mortensen, Jakob Blomqvist, Ivano E Castelli, Rune Chris- tensen, Marcin Dułak, Jesper Friis, Michael N Groves, Bjørk Hammer, Cory Hargus, Eric D Hermes, Paul C Jennings, Peter Bjerre Jensen, James Kermode, John R Kitchin, Esben Leon- hard Kolsbjerg, Joseph Kubal, Kristen Kaasbjerg, Steen Lysgaard, Jón Bergmann Marons- son, ...

    work page 2017

  62. [62]

    The role of shape operator in gauge theories.International Journal of Geometric Methods in Modern Physics, 22(02):2450271, 2025

    Václav Zatloukal and Šimon Vedl. The role of shape operator in gauge theories.International Journal of Geometric Methods in Modern Physics, 22(02):2450271, 2025

    work page 2025

  63. [63]

    Stefan Grimme, Jens Antony, Stephan Ehrlich, and Helge Krieg. A consistent and accurate ab initio parametrization of density functional dispersion correction (dft-d) for the 94 elements h-pu.The Journal of Chemical Physics, 132(15):154104, 04 2010

    work page 2010

  64. [64]

    Lundgaard, Andreas Møgelhøj, Vivien Petzold, David D

    Jess Wellendorff, Keld T. Lundgaard, Andreas Møgelhøj, Vivien Petzold, David D. Landis, Jens K. Nørskov, Thomas Bligaard, and Karsten W. Jacobsen. Density functionals for surface science: Exchange-correlation model development with bayesian error estimation.Phys. Rev. B, 85:235149, Jun 2012

    work page 2012

  65. [65]

    Adsorption energies on transition metal surfaces: towards an accurate and balanced descrip- tion.Nature Communications, 13(1):6853, 2022

    Rafael B Araujo, Gabriel LS Rodrigues, Egon Campos Dos Santos, and Lars GM Pettersson. Adsorption energies on transition metal surfaces: towards an accurate and balanced descrip- tion.Nature Communications, 13(1):6853, 2022

    work page 2022

  66. [66]

    Silbaugh, Delfina Garcia-Pintos, Jens K

    Jess Wellendorff, Trent L. Silbaugh, Delfina Garcia-Pintos, Jens K. Nørskov, Thomas Bligaard, Felix Studt, and Charles T. Campbell. A benchmark database for adsorption bond energies to transition metal surfaces and comparison to selected dft functionals.Surface Science, 640:36– 44, 2015. Reactivity Concepts at Surfaces: Coupling Theory with Experiment

    work page 2015

  67. [67]

    Kaining Duanmu and Donald G. Truhlar. Validation of density functionals for adsorption energiesontransitionmetalsurfaces.Journal of Chemical Theory and Computation, 13(2):835– 842, 2017. PMID: 27983852. 24

    work page 2017

  68. [68]

    Methfessel and A

    M. Methfessel and A. T. Paxton. High-precision sampling for brillouin-zone integration in metals.Phys. Rev. B, 40:3616–3621, Aug 1989. 25 Appendix A: Self-consistent implementation in PAW In the PAW method, the mapping from the PS to the AE space is defined through a linear transformation operator, ˆT| ˜ψ⟩=|ψ⟩, ˆT= 1 + X a X i |ϕa i ⟩ − | ˜ϕa i ⟩ ⟨˜pa i |...

    work page 1989