arxiv: 2605.11724 · v1 · submitted 2026-05-12 · ❄️ cond-mat.mtrl-sci

Recognition: 2 theorem links

· Lean Theorem

Theory and Discovery of Electrides

Chengcheng Xiao , Nicholas Bristowe , Arash A. Mostofi

Authors on Pith no claims yet

Pith reviewed 2026-05-13 05:53 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords electridesinterstitial electronstheoretical frameworkdescriptorshigh-throughput discoveryfirst-principles calculationsinorganic materials

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The pith

A theoretical framework explains the origin of interstitial electrons in electrides and supplies descriptors to discover new candidates via first-principles calculations.

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

The paper introduces a theoretical framework that accounts for why electrons localize in the empty spaces of a crystal lattice rather than attaching to atoms. This framework successfully reproduces the properties of known electrides and derives simple rules, or descriptors, that can be computed from first principles to screen large numbers of materials for new electrides. Such discovery would matter because electrides are candidates for applications in catalysis, electron emission, and superconductivity. The approach also covers high-pressure and organic electrides and reframes F-center defects and solvated electrons in similar terms.

Core claim

We present a theoretical framework for the origin of interstitial electrons in electrides. We demonstrate that this theory can explain electride-like behavior in prototypical electrides, and we use it to develop descriptors for the high-throughput discovery of new inorganic electride candidates from first principles. We also show that the same concepts can explain electride-like behavior in other classes of material, including high-pressure electrides and organic electrides and, more broadly, provide an alternative understanding of F-center defects and solvated electrons.

What carries the argument

The theoretical framework for the origin of interstitial electrons, which is used to derive predictive descriptors for electride behavior.

If this is right

Descriptors enable efficient high-throughput computational searches for new inorganic electrides.
Electride-like behavior in high-pressure and organic materials can be explained by the same framework.
The framework offers an alternative perspective on F-center defects and solvated electrons.
New electrides identified this way could be tested for applications in catalysis and superconductivity.

Where Pith is reading between the lines

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

Materials predicted by the descriptors could be synthesized and tested to validate the theory experimentally.
The descriptors might be adapted to guide the design of electrides with specific electronic properties.
Similar interstitial electron phenomena in other condensed matter systems could be reinterpreted using this framework.

Load-bearing premise

The framework accurately captures the physical origin of interstitial electrons and the descriptors derived from it are general enough to apply across different classes of materials.

What would settle it

Finding a material that satisfies the descriptors but shows no interstitial electron localization in calculations or experiments, or identifying a known electride that the framework cannot explain.

Figures

Figures reproduced from arXiv: 2605.11724 by Arash A. Mostofi, Chengcheng Xiao, Nicholas Bristowe.

**Figure 2.** Figure 2: FIG. 2. (a) Upper panel: Electronic charge density associated with the band that crosses the Fermi level in Ca [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Schematic workflow for the electride figure of merit [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Scatter plot of the ELF basin charge [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a) Left panel: ELF of Hf [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

read the original abstract

Electrides are materials with electrons localized at interstitial regions of the crystal lattice and have been identified as promising candidates for a variety of applications, including catalysis, electron emission, and superconductivity. We present a theoretical framework for the origin of interstitial electrons in electrides. We demonstrate that this theory can explain electride-like behavior in prototypical electrides, and we use it to develop descriptors for the high-throughput discovery of new inorganic electride candidates from first principles. We also show that the same concepts can explain electride-like behavior in other classes of material, including high-pressure electrides and organic electrides and, more broadly, provide an alternative understanding of F-center defects and solvated electrons.

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

This paper supplies a theoretical framework for interstitial electrons that explains known electrides and supplies descriptors for first-principles discovery of new ones.

read the letter

The main takeaway is that the authors lay out a framework for why electrons localize in interstitial sites, show it accounts for the standard inorganic cases, and convert the same ideas into descriptors that can be used for high-throughput screening from DFT calculations. They also apply the picture to high-pressure electrides, organic electrides, F-centers, and solvated electrons, which gives the work some reach beyond a single material class. That unification and the move toward systematic search are the concrete advances here. The high-throughput angle is the part that could actually change how people look for these materials in practice, rather than relying on serendipity or narrow intuition. The validation on prototypical examples appears to hold together without obvious internal contradictions, and the stress-test found no load-bearing gaps in the argument chain. Soft spots are limited. The abstract gives little quantitative detail on how sharply the descriptors separate electrides from non-electrides or on error rates in the screening, so the full paper needs to show those numbers clearly. Generality across very different bonding types will also need concrete test cases rather than just conceptual extension. Minor issues like that are normal and fixable. This is the sort of paper that materials theorists and people running large-scale materials searches would read. It is coherent enough and grounded enough to go to peer review rather than a desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript presents a theoretical framework for the origin of interstitial electrons in electrides. It demonstrates that this theory explains electride-like behavior in prototypical electrides, develops descriptors for high-throughput discovery of new inorganic electride candidates from first principles, and extends the same concepts to high-pressure electrides, organic electrides, F-center defects, and solvated electrons.

Significance. If the framework and descriptors hold, the work could advance understanding of interstitial electrons and enable systematic discovery of new electrides with applications in catalysis, electron emission, and superconductivity. The unification across material classes and defect types broadens potential impact in condensed-matter materials science. No machine-checked proofs or open reproducible code are mentioned, but the first-principles basis aligns with standard practice in the field.

minor comments (2)

Abstract: the summary of validation on prototypical electrides and the performance of the derived descriptors would benefit from one or two quantitative metrics (e.g., agreement with known structures or screening success rate) to allow readers to gauge support for the central claims without reading the full text.
The manuscript would be strengthened by explicit discussion of the range of validity of the descriptors across different bonding types or by a limitations paragraph addressing cases where the framework may not apply.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript on the theoretical framework for interstitial electrons in electrides, the development of descriptors for high-throughput discovery, and the unification across material classes. We appreciate the recommendation for minor revision. No specific major comments were provided in the report, so we have no point-by-point responses to address. We will incorporate any minor improvements in the revised version to enhance clarity and presentation.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives a theoretical framework for interstitial electrons from first-principles considerations of electron localization, validates its explanatory power on known prototypical electrides, and extracts general descriptors that are then applied predictively to screen new candidates. No equation or descriptor reduces to a self-referential definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain; the central claims remain independent of the target results and are externally falsifiable via computation on unseen materials.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities can be identified. The work references first-principles methods but supplies no further detail on assumptions or new postulates.

pith-pipeline@v0.9.0 · 5409 in / 1059 out tokens · 30906 ms · 2026-05-13T05:53:52.545775+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We consider interstitial regions defined by N atomic sites ... regular polygon (2D) or polyhedron (3D) ... evaluate the gradient and Hessian of the lowest energy bonding linear combination ... requiring ... a maximum at the interstitial center.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

The value of b_c depends on the elemental species ... but not on the number of atomic sites N ... analytical form of hydrogenic s-orbitals

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

63 extracted references · 63 canonical work pages

[1]

Fur- ther details of the calculation method and relaxed lattice parameters and atomic coordinates are given in Sec

with iterative Hirshfeld partitioning [36] to account for the van der Waals interactions between layers. Fur- ther details of the calculation method and relaxed lattice parameters and atomic coordinates are given in Sec. II of the SM [34]. The octahedral interstitial sites with Ca atoms at the corners and located between Ca2N layers (Fig. 2(a), bot- tom p...

work page
[2]

super- atom molecular orbitals

direction with respect to the Au layer. Fig. 5(a) [left panels] shows the atomic structure as well as an ELF isosurface. The ELF is localized in tetrahedral interstitial sites that are formed by Hf atoms from adjacent Hf 2Au layers and which have a cage size of 1.92 ˚A [see right panel of Fig. 5(a)]. This is less than the critical cage size b[3D] c = 2.37...

work page doi:10.14469/hpc/2232
[3]

J. L. Dye,Electrons as Anions, Science301, 607 (2003)

work page 2003
[4]

S. W. Kim, Y. Toda, K. Hayashi, M. Hirano, and H. Hosono,Synthesis of a Room Temperature Stable 12CaO·7Al2O3 Electride from the Melt and Its Applica- tion as an Electron Field Emitter, Chem. Mater.18, 1938 (2006). 7

work page 1938
[5]

R. H. Huang and J. L. Dye,Low temperature (−80°C) Thermionic Electron Emission from Alkalides and Elec- trides, Chem. Phys. Lett.166, 133 (1990)

work page 1990
[6]

J. Hu, B. Xu, S. A. Yang, S. Guan, C. Ouyang, and Y. Yao,2D Electrides as Promising Anode Materials for Na-Ion Batteries from First-Principles Study, ACS Appl. Mater. Interfaces7, 24016 (2015)

work page 2015
[7]

J. Hou, K. Tu, and Z. Chen,Two-Dimensional Y 2C Elec- tride: A Promising Anode Material for Na-Ion Batteries, J. Phys. Chem. C120, 18473 (2016)

work page 2016
[8]

H. M. Yamamoto, M. Nakano, M. Suda, Y. Iwasa, M. Kawasaki, and R. Kato,A Strained Organic Field- Effect Transistor with a Gate-Tunable Superconducting Channel, Nat. Commun.4, 2379 (2013)

work page 2013
[9]

Kitano, Y

M. Kitano, Y. Inoue, Y. Yamazaki, F. Hayashi, S. Kan- bara, S. Matsuishi, T. Yokoyama, S.-W. Kim, M. Hara, and H. Hosono,Ammonia Synthesis Using a Stable Elec- tride as an Electron Donor and Reversible Hydrogen Store, Nat. Chem.4, 934 (2012)

work page 2012
[10]

T.-N. Ye, J. Li, M. Kitano, and H. Hosono,Unique Nanocages of 12CaO·7Al2O3 Boost Heterolytic Hydrogen Activation and Selective Hydrogenation of Heteroarenes Over Ruthenium Catalyst, Green Chem.19, 749 (2017)

work page 2017
[11]

Zhang, B

Y. Zhang, B. Wang, Z. Xiao, Y. Lu, T. Kamiya, Y. Uwa- toko, H. Kageyama, and H. Hosono,Electride and Su- perconductivity Behaviors in Mn5Si3-type Intermetallics, npj Quantum Mater.2, 45 (2017)

work page 2017
[12]

Z. S. Pereira, G. M. Faccin, and E. Z. da Silva,Predicted Superconductivity in the Electride Li 5C, J. Phys. Chem. C125, 8899 (2021)

work page 2021
[13]

Miyakawa, S

M. Miyakawa, S. W. Kim, M. Hirano, Y. Kohama, H. Kawaji, T. Atake, H. Ikegami, K. Kono, and H. Hosono,Superconductivity in an Inorganic Electride 12CaO·7Al2O3:e−, J. Am. Chem. Soc.129, 7270 (2007)

work page 2007
[14]

Hirayama, S

M. Hirayama, S. Matsuishi, H. Hosono, and S. Mu- rakami,Electrides as a New Platform of Topological Ma- terials, Phys. Rev. X8, 31067 (2018)

work page 2018
[15]

Huang, K.-H

H. Huang, K.-H. Jin, S. Zhang, and F. Liu,Topological Electride Y2C, Nano Lett.18, 1972 (2018)

work page 1972
[16]

Zhang, R

X. Zhang, R. Guo, L. Jin, X. Dai, and G. Liu,Intermetal- lic Ca3Pb: A Topological Zero-dimensional Electride Ma- terial, J. Mater. Chem. C6, 575 (2018)

work page 2018
[17]

Matsuishi, Y

S. Matsuishi, Y. Toda, M. Miyakawa, K. Hayashi, T. Kamiya, M. Hirano, I. Tanaka, and H. Hosono,High- Density Electron Anions in a Nanoporous Single Crystal: [Ca24Al28O64]4+(4e−), Science301, 626 (2003)

work page 2003
[18]

Y. Ma, M. Eremets, A. R. Oganov, Y. Xie, I. Tro- jan, S. Medvedev, A. O. Lyakhov, M. Valle, and V. Prakapenka,Transparent Dense Sodium, Nature458, 182 (2009)

work page 2009
[19]

K. Lee, S. W. Kim, Y. Toda, S. Matsuishi, and H. Hosono,Dicalcium Nitride as a Two-dimensional Electride with an Anionic Electron Layer, Nature494, 336 (2013)

work page 2013
[20]

Zhang, Z

X. Zhang, Z. Xiao, H. Lei, Y. Toda, S. Matsuishi, T. Kamiya, S. Ueda, and H. Hosono,Two-Dimensional Transition-Metal Electride Y2C, Chem. Mater.26, 6638 (2014)

work page 2014
[21]

Park, J.-Y

J. Park, J.-Y. Hwang, K. H. Lee, S.-G. Kim, K. Lee, and S. W. Kim,Tuning the Spin-Alignment of Interstitial Electrons in Two-Dimensional Y 2C Electride via Chem- ical Pressure, J. Am. Chem. Soc.139, 17277 (2017)

work page 2017
[22]

Zheng, T

Q. Zheng, T. Feng, J. A. Hachtel, R. Ishikawa, Y. Cheng, L. Daemen, J. Xing, J. C. Idrobo, J. Yan, N. Shibata, Y. Ikuhara, B. C. Sales, S. T. Pantelides, and M. Chi, Direct Visualization of Anionic Electrons in an Electride Reveals Inhomogeneities, Sci. Adv.7, eabe6819 (2021)

work page 2021
[23]

M. Y. Redko, J. E. Jackson, R. H. Huang, and J. L. Dye,Design and Synthesis of a Thermally Stable Organic Electride, J. Am. Chem. Soc.127, 12416 (2005)

work page 2005
[24]

Ellaboudy, J

A. Ellaboudy, J. L. Dye, and P. B. Smith,Cesium 18- crown-6 Compounds. A Crystalline Ceside and a Crys- talline Electride, J. Am. Chem. Soc.105, 6490 (1983)

work page 1983
[25]

C. Liu, S. A. Nikolaev, W. Ren, and L. A. Burton,Elec- trides: A review, J. Mater. Chem. C8, 10551 (2020)

work page 2020
[26]

S. G. Dale and E. R. Johnson,Theoretical Descriptors of Electrides, J. Phys. Chem. A122, 9371 (2018)

work page 2018
[27]

T. Tada, S. Takemoto, S. Matsuishi, and H. Hosono, High-Throughput ab Initio Screening for Two- Dimensional Electride Materials, Inorg. Chem.53, 10347 (2014)

work page 2014
[28]

J. Zhou, L. Shen, M. Yang, H. Cheng, W. Kong, and Y. P. Feng,Discovery of Hidden Classes of Layered Elec- trides by Extensive High-Throughput Material Screening, Chem. Mater.31, 1860 (2019)

work page 2019
[29]

L. A. Burton, F. Ricci, W. Chen, G.-M. Rignanese, and G. Hautier,High-Throughput Identification of Electrides from All Known Inorganic Materials, Chem. Mater.30, 7521 (2018)

work page 2018
[30]

Miao and R

M.-S. Miao and R. Hoffmann,High Pressure Electrides: A Predictive Chemical and Physical Theory, Acc. Chem. Res.47, 1311 (2014)

work page 2014
[31]

Miao and R

M.-s. Miao and R. Hoffmann,High-Pressure Electrides: The Chemical Nature of Interstitial Quasiatoms, J. Am. Chem. Soc.137, 3631 (2015)

work page 2015
[32]

Dong and A

X. Dong and A. R. Oganov, Electrides and Their High- Pressure Chemistry, inCorrelations in Condensed Matter under Extreme Conditions, edited by G. G. N. Angilella and A. L. Magna (Springer International Publishing,

work page
[33]

1st ed., Chap. 6, pp. 69–84

work page
[34]

Modak and A

P. Modak and A. K. Verma,Pressure Induced Multi- centre Bonding and Metal–insulator Transition in PtAl 2, Phys. Chem. Chem. Phys.21, 13337 (2019)

work page 2019
[35]

A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson,Commentary: The Materials Project: A materials Genome Approach to Accelerating Materials Innovation, APL Mater.1, 011002 (2013)

work page 2013
[36]

super-atom molecular orbitals

In principle, our model can be further extended to other atomic orbitals (e.g., p and d) or molecular orbitals such as “super-atom molecular orbitals” (SAMOs) [55–57], as well as irregular atomic cages

work page
[37]

See Supplemental Material atURL_will_be_inserted_ by_publisherfor detailed derivation of the critical cage size, DFT details, details of band structure projection schemes, the workflow of the electride descriptor, details of the electron localization function and bonding analysis of more prototypical electride systems

work page
[38]

Tkatchenko and M

A. Tkatchenko and M. Scheffler,Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data, Phys. Rev. Lett. 102, 073005 (2009)

work page 2009
[39]

Bultinck, C

P. Bultinck, C. Van Alsenoy, P. W. Ayers, and R. Carb´ o- Dorca,Critical Analysis and Extension of the Hirshfeld Atoms in Molecules, J. Chem. Phys.126, 144111 (2007)

work page 2007
[40]

J. K. Burdett and T. A. McCormick,Electron Localiza- tion in Molecules and Solids: The Meaning of ELF, J. Phys. Chem. A102, 6366 (1998). 8

work page 1998
[41]

A. D. Becke and K. E. Edgecombe,A Simple Measure of Electron Localization in Atomic and Molecular Systems, J. Chem. Phys.92, 5397 (1990)

work page 1990
[42]

Y. Grin, A. Savin, and B. Silvi, The ELF Perspective of chemical bonding, inThe Chemical Bond, edited by G. Frenking and S. Shaik (Wiley-VCH Verlag GmbH & Co. KGaA, 2014) Chap. 10, pp. 345–382

work page 2014
[43]

S. Zhao, E. Kan, and Z. Li,Electride: From Computa- tional Characterization to Theoretical Design, Wiley In- terdiscip. Rev.: Comput. Mol. Sci.6, 430 (2016)

work page 2016
[44]

B. Wan, Y. Lu, Z. Xiao, Y. Muraba, J. Kim, D. Huang, L. Wu, H. Gou, J. Zhang, F. Gao, H.-k. Mao, and H. Hosono,Identifying quasi-2D and 1D Electrides in Yttrium and Scandium Chlorides via Geometrical Iden- tification, npj Comput. Mater.4, 77 (2018)

work page 2018
[45]

Zhang, H

Y. Zhang, H. Wang, Y. Wang, L. Zhang, and Y. Ma, Computer-Assisted Inverse Design of Inorganic Elec- trides, Phys. Rev. X7, 011017 (2017)

work page 2017
[46]

X. Dong, A. R. Oganov, A. F. Goncharov, E. Stavrou, S. Lobanov, G. Saleh, G.-R. Qian, Q. Zhu, C. Gatti, V. L. Deringer, R. Dronskowski, X.-F. Zhou, V. B. Prakapenka, Z. Konˆ opkov´ a, I. A. Popov, A. I. Boldyrev, and H.-T. Wang,A Stable Compound of Helium and Sodium at High Pressure, Nat. Chem.9, 440 (2017)

work page 2017
[47]

R. F. W. Bader,A Quantum Theory of Molecular Struc- ture and Its Applications, Chem. Rev.91, 893 (1991)

work page 1991
[48]

Nesper,Bonding Patterns in Intermetallic Com- pounds, Angew

R. Nesper,Bonding Patterns in Intermetallic Com- pounds, Angew. Chem., Int. Ed. Engl.30, 789 (1991)

work page 1991
[49]

Y. Wang, L. D. Calvert, E. J. Gabe, and J. B. Taylor, Structure of Two Forms of Europium Arsenide Eu5As3, Acta Crystallogr. B: Struct. Sci. Cryst. Eng. Mater.34, 2281 (1978)

work page 1978
[50]

E. A. Leon-Escamilla and J. D. Corbett,Hydrogen in Polar Intermetallics. Binary Pnictides of Divalent Metals with Mn 5Si−3-type Structures and Their Iso- typic Ternary Hydride Solutions, Chem. Mater.18, 4782 (2006)

work page 2006
[51]

Eisenmann and H

B. Eisenmann and H. Sch¨ afer,Die Kristallstrukturen der Verbindungen Ca 2Sb und Ca 2Bi, Zeitschrift f¨ ur Natur- forschung B29, 13 (1973)

work page 1973
[52]

Schubert, H

K. Schubert, H. G. Meissner, M. P¨ otzschke, W. Ross- teutscher, and E. Stolz,Einige Strukturdaten metallis- cher Phasen (7), Naturwissenschaften49, 57 (1962)

work page 1962
[53]

Nuss and M

J. Nuss and M. Jansen,New Ternary La 2Sb-Type Com- pounds, ScRESb (RE=La, Ce, Pr, Nd, Sm, Tb), and the Oxygen Stuffed Variant Sc 4Yb4Sb4O, Z. Anorg. Allg. Chem.640, 713 (2014)

work page 2014
[54]

Le Roy, J.-M

J. Le Roy, J.-M. Moreau, D. Paccard, and E. Parth´ e, Rare-earth (and Yttrium)–Iridium and –Platinum Com- pounds with the Fe 3C Structure Type, Acta Crystallo- graphica Section B Structural Crystallography and Crys- tal Chemistry35, 1437 (1979)

work page 1979
[55]

Martin and A

A. Martin and A. Moore,The Structure of Beryl- lium, with Particular Reference to Temperatures Above 1200°C, J. Less-Common Met.1, 85 (1959)

work page 1959
[56]

Y. Lu, J. Li, T. Tada, Y. Toda, S. Ueda, T. Yokoyama, M. Kitano, and H. Hosono,Water Durable Electride Y5Si3: Electronic Structure and Catalytic Activity for Ammonia Synthesis, J. Am. Chem. Soc.138, 3970 (2016)

work page 2016
[57]

Chanhom, K

P. Chanhom, K. E. Fritz, L. A. Burton, J. Kloppenburg, Y. Filinchuk, A. Senyshyn, M. Wang, Z. Feng, N. Insin, J. Suntivich, and G. Hautier,Sr 3CrN3: A New Electride with a Partially Filledd-Shell Transition Metal, J. Am. Chem. Soc.141, 10595 (2019)

work page 2019
[58]

M. Feng, J. Zhao, and H. Petek,Atomlike, Hollow- Core-Bound Molecular Orbitals of C 60, Science320, 359 (2008)

work page 2008
[59]

M. Feng, J. Zhao, T. Huang, X. Zhu, and H. Petek, The Electronic Properties of Superatom States of Hollow Molecules, Acc. Chem. Res.44, 360 (2011)

work page 2011
[60]

J. O. Johansson, E. Bohl, and E. E. B. Campbell,Super- atom Molecular Orbital Excited Atates of Fullerenes, Phi- los. Trans. R. Soc. Math. Phys. Eng. Sci.374, 20150322 (2016)

work page 2016
[61]

Kulichenko, N

M. Kulichenko, N. Fedik, D. Steglenko, R. M. Minyaev, V. I. Minkin, and A. I. Boldyrev,Periodic F-defects on the MgO Surface as Potential Single-defect Catalysts with Non-linear Optical Properties, Chem. Phys.532, 110680 (2020)

work page 2020
[62]

non-atomic

J. Lan, V. Kapil, P. Gasparotto, M. Ceriotti, M. Ian- nuzzi, and V. V. Rybkin,Simulating the Ghost: Quan- tum Dynamics of the Solvated Electron, Nat. Commun. 12, 766 (2021) 1 –Supplemental Material– Theory and Discovery of Electrides Chengcheng Xiao,1 Nicholas Bristowe,2 and Arash A. Mostofi 1 1Departments of Materials and Physics, and the Thomas Young Ce...

work page 2021
[63]

localization

is located: P σσ cond (⃗ r1, ⃗ r2) ⃗ r2→⃗ r1 = P σσ (⃗ r1, ⃗ r2) ρσ(⃗ r1) ⃗ r2→⃗ r1 ,(S12) here,P σσ (⃗ r1, ⃗ r2) is the reduced density matrix which represents the pair probability of simultaneously finding electron 1 with spinσat⃗ r 1 and electron 2 with spinσat⃗ r 2,ρ σ(⃗ r1) is the one-body charge density of electron 1 with spinσ. When⃗ r2 is close to...

work page 2021