Charge Symmetry Beyond Wyckoff Equivalence

Qiu-Shi Huang; Su-Huai Wei; Xin-Gao Gong

arxiv: 2605.19383 · v1 · pith:CJCR422Enew · submitted 2026-05-19 · ❄️ cond-mat.mtrl-sci

Charge Symmetry Beyond Wyckoff Equivalence

Qiu-Shi Huang , Xin-Gao Gong , Su-Huai Wei This is my paper

Pith reviewed 2026-05-20 04:55 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords charge transferWyckoff positionspressure effectssodiumhidden symmetryelectronic equivalencemetal-insulator transition

0 comments

The pith

Crystallographically equivalent sites can become charge-inequivalent under compression while inequivalent sites can stay equivalent due to hidden symmetry.

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

The paper argues that usual expectations from crystal symmetry about electronic equivalence do not always hold. Atoms on the same Wyckoff position can develop different charges when compressed, while atoms on different positions can share the same charge at low pressure because of an emergent hidden symmetry. The authors illustrate this with two cases in sodium, where pressure drives charge transfer that either breaks or maintains equivalence in ways not predicted by positions alone. If this holds, predictions of electronic properties under high pressure must account for these additional mechanisms rather than relying solely on crystallographic data.

Core claim

The central claim is that pressure-induced charge transfer destabilizes charge-equivalent states when intersite Coulomb gains exceed onsite costs, as described by a minimal Landau theory. In BCC Na this produces an electronically symmetry-broken CsCl-type state on an unchanged BCC framework, while in hP4 Na an emergent gauge equivalence keeps distinct Wyckoff sites charge-equivalent at low pressure until compression splits near-Fermi doublets and drives a metal-insulator transition. These cases show lattice symmetry constrains but does not fix the electronic equivalence structure.

What carries the argument

Minimal Landau theory of pressure-induced charge transfer, in which compression increases the intersite Coulomb energy gained by redistribution until it overcomes the onsite charging cost.

If this is right

In BCC Na, compression drives charge transfer between neighboring sites to produce an electronically symmetry-broken CsCl-type state on the BCC ionic framework.
In hP4 Na, compression breaks the emergent hidden equivalence, splits near-Fermi doublets, and induces a metal-insulator transition.
Electronic equivalence can fall below or rise above what Wyckoff positions suggest.
Lattice symmetry constrains but does not uniquely determine the equivalence structure of the electronic state.

Where Pith is reading between the lines

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

Similar pressure-driven charge redistributions could appear in other compressed alkali metals or compounds.
Charge-density maps from high-pressure experiments on sodium would directly test for the predicted site differences.
The mechanism offers a route to pressure-tunable electronic states in materials where crystallographic analysis alone appears insufficient.

Load-bearing premise

Compression enhances the intersite Coulomb energy from charge redistribution until it overcomes the onsite charging cost and destabilizes the charge-equivalent state.

What would settle it

Observation or calculation of no charge difference developing between neighboring sites in compressed BCC sodium, or no splitting of near-Fermi doublets and no metal-insulator transition in compressed hP4 sodium.

Figures

Figures reproduced from arXiv: 2605.19383 by Qiu-Shi Huang, Su-Huai Wei, Xin-Gao Gong.

**Figure 2.** Figure 2: FIG. 2. Electronic spontaneous symmetry breaking in a BCC prototype. (a) Conventional BCC structure [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Structural and charge evolution of hP4 Na. (a) ABAC (hP4) unit cell with two Wyckoff sites: Na1 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The band structure evolution of the ABAC structure of pressure is shown. To clearly illustrate the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Band structures of the (a) ABAC (hP4), (b) ABAB (AB supercell), and (c) AB stackings at ambient [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

read the original abstract

Crystallographic symmetry is usually taken as a guide to electronic equivalence in crystals: atoms on the same Wyckoff position are expected to have the same charge, whereas atoms on different Wyckoff positions are expected to be electronically distinct. Here we show that both expectations can fail in oppo-site ways: crystallographically equivalent sites can become charge-inequivalent under compression, whereas crystallographically inequivalent sites can remain charge-equivalent at low pressure because of an emergent hidden symmetry. We develop a minimal Landau theory of pressure-induced charge transfer, in which compression enhances the intersite Coulomb energy gained by charge redistribution until it overcomes the onsite charging cost and destabilizes the charge-equivalent state. In BCC Na, all sites are charge-equivalent at low pressure, but compression drives charge transfer between neighboring sites, pro-ducing an electronically symmetry-broken CsCl-type state on an unchanged BCC ionic framework. In hP4 Na, the opposite anomaly occurs: two Na sites occupy distinct Wyckoff positions, yet remain charge-equivalent at low pressure because of an emergent gauge equivalence in the low-energy manifold, giving rise to near-Fermi doublets that appear accidental in conventional space-group analysis. Upon compres-sion, pressure-induced charge transfer breaks this hidden equivalence, splits the near-Fermi doublets, and drives a metal-insulator transition. These two complementary cases establish pressure-induced charge transfer as a mechanism by which electronic equivalence can either fall below or rise above what Wyckoff positions alone would suggest, showing that lattice symmetry constrains but does not uniquely determine the equivalence structure of the electronic state.

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 identifies pressure-driven charge transfer as a way to break or create electronic symmetries beyond Wyckoff positions in sodium phases using a minimal Landau model, though the volume scaling of the terms needs more microscopic support.

read the letter

The key takeaway is that compression in sodium can drive charge redistribution that either breaks crystallographic equivalence in BCC or preserves it through hidden symmetry in hP4, via a minimal Landau theory of competing Coulomb terms. The paper does a good job laying out these two opposite cases as examples where electronic equivalence deviates from Wyckoff expectations. It frames the BCC case as pressure-induced charge transfer leading to an electronically broken state on the same lattice, and the hP4 case as an emergent gauge equivalence at low pressure that gets broken under compression, splitting doublets and triggering a metal-insulator transition. This complementary view is a nice way to highlight the mechanism. What stands out is the focus on how lattice symmetry sets bounds but does not fix the charge pattern completely. The minimal theory keeps things simple and points to intersite Coulomb gain winning over onsite cost as volume decreases. The main limitation is that the theory remains phenomenological. It assumes the intersite coefficient in the free energy increases faster with decreasing volume than the onsite term, but the paper does not provide a microscopic derivation tying this to hopping, Madelung sums, or explicit DFT results for sodium. Without that, the volume dependence is postulated rather than derived, which weakens the predictive power for these specific structures. No error analysis or validation data is mentioned at the abstract level, though the full text might include some supporting calculations. This paper is aimed at researchers in high-pressure materials science and those working on electronic symmetry breaking in metals. Readers who follow Landau theories or charge ordering under pressure would find it relevant. It shows honest engagement with the literature on sodium phases and clear reasoning on the symmetry implications. I would bring this to a reading group for discussion on the assumptions. It deserves peer review because the claims are specific and the anomalies are worth checking against calculations. My recommendation is to send it to referees, with attention to strengthening the justification for the model's volume dependence.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that crystallographic Wyckoff positions do not uniquely determine electronic charge equivalence in crystals. Using a minimal Landau theory of pressure-induced charge transfer, it shows two complementary anomalies in sodium: in BCC Na, compression destabilizes the charge-equivalent state and drives charge transfer between neighboring sites, producing an electronically symmetry-broken CsCl-type state on an unchanged BCC framework; in hP4 Na, distinct Wyckoff sites remain charge-equivalent at low pressure due to an emergent hidden gauge symmetry that produces near-Fermi doublets, but compression breaks this equivalence, splits the doublets, and induces a metal-insulator transition. The theory posits that decreasing volume enhances the intersite Coulomb energy gain relative to the onsite charging cost until the uniform-charge state is destabilized.

Significance. If the central claims hold, the work is significant for understanding high-pressure phases of simple metals. It demonstrates that electronic equivalence can fall below or rise above Wyckoff expectations via pressure-induced charge transfer and hidden symmetries, offering a mechanism for anomalies in alkali metals and a phenomenological framework that could inform DFT studies or experiments on metal-insulator transitions. The complementary BCC and hP4 cases provide falsifiable predictions about charge ordering and band splitting under compression.

major comments (1)

[§3] §3 (minimal Landau theory): The central assumption that the intersite Coulomb coefficient grows faster with decreasing volume than the onsite charging penalty is introduced phenomenologically without an explicit microscopic derivation or volume-dependent mapping from hopping integrals, Madelung sums, or DFT parameters for Na. This volume dependence is load-bearing for both the BCC charge-symmetry breaking and the hP4 hidden-symmetry breaking claims, yet remains unvalidated.

minor comments (2)

[Abstract] Abstract: hyphenation artifacts such as 'oppo-site' and 'pro-ducing' should be corrected for readability.
The manuscript would benefit from a brief comparison to existing literature on charge ordering or hidden symmetries in compressed alkali metals to clarify novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and positive evaluation of the significance of our manuscript. We address the major comment on the minimal Landau theory below.

read point-by-point responses

Referee: [§3] §3 (minimal Landau theory): The central assumption that the intersite Coulomb coefficient grows faster with decreasing volume than the onsite charging penalty is introduced phenomenologically without an explicit microscopic derivation or volume-dependent mapping from hopping integrals, Madelung sums, or DFT parameters for Na. This volume dependence is load-bearing for both the BCC charge-symmetry breaking and the hP4 hidden-symmetry breaking claims, yet remains unvalidated.

Authors: We appreciate the referee's emphasis on this foundational aspect of our approach. Our minimal Landau theory is constructed to be phenomenological by design, capturing the essential competition between onsite charging costs and intersite Coulomb gains under compression without relying on a full ab initio parameterization. The volume dependence—specifically that the intersite term strengthens more rapidly—is motivated by general physical considerations: as volume decreases, the lattice contracts, enhancing the relative importance of interatomic Coulomb interactions (which scale with inverse interatomic distances in Madelung sums) compared to more localized onsite terms. This is consistent with trends observed in high-pressure studies of simple metals. Nevertheless, we agree that a more explicit connection to microscopic parameters would be valuable. In the revised version, we have expanded §3 to include a brief discussion of this motivation, referencing relevant literature on volume-dependent effective interactions in alkali metals, and we have clarified that the functional form is illustrative of the generic instability rather than a quantitative fit to Na. We believe this addresses the concern while preserving the minimal character of the theory. Full microscopic validation would require extensive DFT-based mapping, which lies beyond the scope of the present work but could be pursued in follow-up studies. revision: partial

Circularity Check

1 steps flagged

Minimal Landau theory postulates volume-dependent intersite Coulomb dominance without microscopic derivation, rendering charge-transfer predictions equivalent to the input ansatz.

specific steps

self definitional [Abstract]
"We develop a minimal Landau theory of pressure-induced charge transfer, in which compression enhances the intersite Coulomb energy gained by charge redistribution until it overcomes the onsite charging cost and destabilizes the charge-equivalent state."

The model is constructed by fiat with the premise that compression strengthens the intersite gain relative to the onsite penalty; the subsequent claims of pressure-driven charge transfer and symmetry breaking are therefore true by the definition of the free-energy functional rather than derived from more fundamental electronic-structure inputs.

full rationale

The paper introduces a phenomenological model whose central volume dependence (intersite term growing faster than onsite under compression) is stated as part of the theory definition rather than obtained from hopping, Madelung sums, or DFT parameters. Consequently the predicted destabilization of the charge-equivalent state in BCC Na and the breaking of hidden gauge equivalence in hP4 Na follow directly from that postulate. No self-citations or data-fitting steps appear in the supplied text, but the absence of an independent mapping from microscopic quantities to the Landau coefficients produces moderate circularity in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on a minimal Landau theory whose energy competition terms are described conceptually in the abstract; no explicit free parameters or additional axioms are stated.

free parameters (1)

onsite charging cost
Described as the energy barrier that must be overcome by intersite Coulomb gain under compression.

axioms (1)

domain assumption Compression enhances intersite Coulomb energy until it exceeds onsite charging cost
This is the key premise of the minimal Landau theory of pressure-induced charge transfer stated in the abstract.

pith-pipeline@v0.9.0 · 5823 in / 1132 out tokens · 44331 ms · 2026-05-20T04:55:22.717069+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.

minimal Landau theory of pressure-induced charge transfer, in which compression enhances the intersite Coulomb energy gained by charge redistribution until it overcomes the onsite charging cost
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Madelung lattice sum … Meff(P)/r(P)

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

32 extracted references · 32 canonical work pages

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Siringo, R

F. Siringo, R. Pucci, and G. G. N. Angilella, Are light alkali metals still metals under high pressure?, in Correlations in Condensed Matter under Extreme Conditions: A Tribute to Renato Pucci on the Occasion of His 70th Birthday (Springer International Publishing, Cham, 2017), pp. 257-265

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A. F. Goncharov, V . V . Struzhkin, H.-K. Mao, and R. J. Hemley, Spectroscopic evidence for broken- symmetry transitions in dense lithium up to megabar pressures, Phys. Rev. B 71, 184114 (2005)

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E. He, C. M. Wilson, and R. Ganesh, Metallic bonding in close-packed structures: Structural frus- tration from a hidden gauge symmetry, Phys. Rev. Lett. 133, 256401 (2024)

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Liu and A

Q. Liu and A. Zunger, Predicted realization of cubic Dirac fermion in quasi-one-dimensional tran- sition-metal monochalcogenides, Phys. Rev. X 7, 021019 (2017)

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Yang, H.-Y

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Lazicki, A

A. Lazicki, A. F. Goncharov, V . V . Struzhkin, R. E. Cohen, Z. Liu, E. Gregoryanz, C. Guillaume, H.-K. Mao, and R. J. Hemley, Anomalous optical and electronic properties of dense sodium, Proc. Natl. Acad. Sci. U.S.A. 106, 6525 (2009)

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Cordero, V

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Blaha, K

P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k: An augmented plane wave + local orbitals program for calculating crystal properties (Technische Universitat Wien, Austria, 2001)

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Birch, Finite elastic strain of cubic crystals, Phys

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Katsura and Y

T. Katsura and Y . Tange, A simple derivation of the Birch-Murnaghan equations of state (EOSs) and comparison with EOSs derived from other definitions of finite strain, Minerals 9, 745 (2019)

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[21]

J. B. Neaton and N. W. Ashcroft, On the constitution of sodium at higher densities, Phys. Rev. Lett. 86, 2830 (2001)

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Gregoryanz, L

E. Gregoryanz, L. F. Lundegaard, M. I. McMahon, C. Guillaume, R. J. Nelmes, and M. Mezouar, Structural diversity of sodium, Science 320, 1054 (2008)

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Gatti, I

M. Gatti, I. V . Tokatly, and A. Rubio, Sodium: A charge-transfer insulator at high pressures, Phys. Rev. Lett. 104, 216404 (2010)

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Racioppi, C

S. Racioppi, C. V . Storm, M. I. McMahon, and E. Zurek, On the electride nature of Na-hP4, Angew. Chem. Int. Ed. 62, e202310802 (2023)

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C. V . Storm, S. Racioppi, M. J. Duff, J. D. McHardy, E. Zurek, and M. I. McMahon, Experimental signatures of interstitial electron density in transparent dense sodium, Commun. Mater. 6, 201 (2025)

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[26]

P. P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Ann. Phys. 369, 253 (1921)

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W. A. Harrison, Simple calculation of Madelung constants, Phys. Rev. B 73, 212103 (2006)

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G. Hagg, Some notes on MX2 layer lattices with close-packed X atoms, Ark. Kemi Mineral. Geol. 16B, 1 (1943)

work page 1943
[29]

M. I. McMahon and R. J. Nelmes, High-pressure structures and phase transformations in elemental metals, Chem. Soc. Rev. 35, 943 (2006)

work page 2006
[30]

A. R. Oganov, J. Chen, C. Gatti, Y . Ma, Y . Ma, C. W. Glass, Z. Liu, T. Yu, O. O. Kurakevych, and V . L. Solozhenko, Ionic high-pressure form of elemental boron, Nature 457, 863 (2009)

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P. W. Anderson, More is different: Broken symmetry and the nature of the hierarchical structure of science, Science 177, 393 (1972)

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A. W. Overhauser, Spin density waves in an electron gas, Phys. Rev. 128, 1437 (1962)

work page 1962

[1] [1]

Siringo, R

F. Siringo, R. Pucci, and G. G. N. Angilella, Are light alkali metals still metals under high pressure?, in Correlations in Condensed Matter under Extreme Conditions: A Tribute to Renato Pucci on the Occasion of His 70th Birthday (Springer International Publishing, Cham, 2017), pp. 257-265

work page 2017

[2] [2]

H. T. Hall, L. Merrill, and J. D. Barnett, High pressure polymorphism in cesium, Science 146, 1297 (1964)

work page 1964

[3] [3]

Olijnyk and W

H. Olijnyk and W. Holzapfel, Phase transitions in K and Rb under pressure, Phys. Lett. A 99, 381 (1983)

work page 1983

[4] [4]

Olinger and J

B. Olinger and J. W. Shaner, Lithium, compression and high-pressure structure, Science 219, 1071 (1983)

work page 1983

[5] [5]

Hanfland, I

M. Hanfland, I. Loa, and K. Syassen, Sodium under pressure: bcc to fcc structural transition and pressure-volume relation to 100 GPa, Phys. Rev. B 65, 184109 (2002)

work page 2002

[6] [6]

A. F. Goncharov, V . V . Struzhkin, H.-K. Mao, and R. J. Hemley, Spectroscopic evidence for broken- symmetry transitions in dense lithium up to megabar pressures, Phys. Rev. B 71, 184114 (2005)

work page 2005

[7] [7]

Zhang, Y

L. Zhang, Y . Wang, J. Lv, and Y . Ma, Materials discovery at high pressures, Nat. Rev. Mater. 2, 17005 (2017)

work page 2017

[8] [8]

Marques, M

M. Marques, M. I. McMahon, E. Gregoryanz, M. Hanfland, C. L. Guillaume, C. J. Pickard, G. J. Ackland, and R. J. Nelmes, Crystal structures of dense lithium: A metal -semiconductor-metal transition, Phys. Rev. Lett. 106, 095502 (2011)

work page 2011

[9] [9]

Y . Ma, M. I. Eremets, A. R. Oganov, Y . Xie, I. A. Trojan, S. A. Medvedev, A. O. Lyakhov, M. Valle, and V . B. Prakapenka, Transparent dense sodium, Nature 458, 182 (2009)

work page 2009

[10] [10]

E. He, C. M. Wilson, and R. Ganesh, Metallic bonding in close-packed structures: Structural frus- tration from a hidden gauge symmetry, Phys. Rev. Lett. 133, 256401 (2024)

work page 2024

[11] [11]

Liu and A

Q. Liu and A. Zunger, Predicted realization of cubic Dirac fermion in quasi-one-dimensional tran- sition-metal monochalcogenides, Phys. Rev. X 7, 021019 (2017)

work page 2017

[12] [12]

Yang, H.-Y

H.-A. Yang, H.-Y . Wei, and B.-Y . Cao, Symmetry-enforced planar nodal-chain phonons in non- symmorphic materials, J. Appl. Phys. 132, 224401 (2022)

work page 2022

[13] [13]

Lazicki, A

A. Lazicki, A. F. Goncharov, V . V . Struzhkin, R. E. Cohen, Z. Liu, E. Gregoryanz, C. Guillaume, H.-K. Mao, and R. J. Hemley, Anomalous optical and electronic properties of dense sodium, Proc. Natl. Acad. Sci. U.S.A. 106, 6525 (2009)

work page 2009

[14] [14]

Pyykko and M

P. Pyykko and M. Atsumi, Molecular single-bond covalent radii for elements 1-118, Chem. Eur. J. 15, 186 (2009). 18

work page 2009

[15] [15]

Cordero, V

B. Cordero, V . Gomez, A. E. Platero-Prats, M. Reves, J. Echeverria, E. Cremades, F. Barragan, and S. Alvarez, Covalent radii revisited, Dalton Trans., 2832 (2008)

work page 2008

[16] [16]

Kittel, Introduction to Solid State Physics, 8th ed

C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, New York, 2005)

work page 2005

[17] [17]

Blaha, K

P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H. Madsen, and L. D. Marks, WIEN2k: An APW+lo program for calculating the properties of solids, J. Chem. Phys. 152, 074101 (2020)

work page 2020

[18] [18]

Blaha, K

P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k: An augmented plane wave + local orbitals program for calculating crystal properties (Technische Universitat Wien, Austria, 2001)

work page 2001

[19] [19]

Birch, Finite elastic strain of cubic crystals, Phys

F. Birch, Finite elastic strain of cubic crystals, Phys. Rev. 71, 809 (1947)

work page 1947

[20] [20]

Katsura and Y

T. Katsura and Y . Tange, A simple derivation of the Birch-Murnaghan equations of state (EOSs) and comparison with EOSs derived from other definitions of finite strain, Minerals 9, 745 (2019)

work page 2019

[21] [21]

J. B. Neaton and N. W. Ashcroft, On the constitution of sodium at higher densities, Phys. Rev. Lett. 86, 2830 (2001)

work page 2001

[22] [22]

Gregoryanz, L

E. Gregoryanz, L. F. Lundegaard, M. I. McMahon, C. Guillaume, R. J. Nelmes, and M. Mezouar, Structural diversity of sodium, Science 320, 1054 (2008)

work page 2008

[23] [23]

Gatti, I

M. Gatti, I. V . Tokatly, and A. Rubio, Sodium: A charge-transfer insulator at high pressures, Phys. Rev. Lett. 104, 216404 (2010)

work page 2010

[24] [24]

Racioppi, C

S. Racioppi, C. V . Storm, M. I. McMahon, and E. Zurek, On the electride nature of Na-hP4, Angew. Chem. Int. Ed. 62, e202310802 (2023)

work page 2023

[25] [25]

C. V . Storm, S. Racioppi, M. J. Duff, J. D. McHardy, E. Zurek, and M. I. McMahon, Experimental signatures of interstitial electron density in transparent dense sodium, Commun. Mater. 6, 201 (2025)

work page 2025

[26] [26]

P. P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Ann. Phys. 369, 253 (1921)

work page 1921

[27] [27]

W. A. Harrison, Simple calculation of Madelung constants, Phys. Rev. B 73, 212103 (2006)

work page 2006

[28] [28]

Hagg, Some notes on MX2 layer lattices with close-packed X atoms, Ark

G. Hagg, Some notes on MX2 layer lattices with close-packed X atoms, Ark. Kemi Mineral. Geol. 16B, 1 (1943)

work page 1943

[29] [29]

M. I. McMahon and R. J. Nelmes, High-pressure structures and phase transformations in elemental metals, Chem. Soc. Rev. 35, 943 (2006)

work page 2006

[30] [30]

A. R. Oganov, J. Chen, C. Gatti, Y . Ma, Y . Ma, C. W. Glass, Z. Liu, T. Yu, O. O. Kurakevych, and V . L. Solozhenko, Ionic high-pressure form of elemental boron, Nature 457, 863 (2009)

work page 2009

[31] [31]

P. W. Anderson, More is different: Broken symmetry and the nature of the hierarchical structure of science, Science 177, 393 (1972)

work page 1972

[32] [32]

A. W. Overhauser, Spin density waves in an electron gas, Phys. Rev. 128, 1437 (1962)

work page 1962