Introduction

The oxidation state (OS) of an element is a fundamental concept in chemistry and materials science, underpinning our understanding of charge distribution, electronic structure, and resulting material properties in both molecular and extended systems1,2,3,4,5. The emergence of nontraditional OSs is often associated with atypical structural motifs and unconventional charge-transfer mechanisms, which can lead to emergent physical phenomena and expand the accessible chemical space6,7,8,9,10,11,12,13. As such, the exploration of elemental OSs remains a central and enduring theme in chemical and materials research14,15,16, driven by the promise of discovering exotic bonding configurations and extraordinary functionalities. Traditionally, access to extreme OSs has relied on the use of strong oxidants to stabilize highly positive states8,11,17,18 or strong reductants to access deeply negative ones10,19. While these approaches have enabled landmark discoveries, such as the +IX state in [IrO4]+ cation1 and the −II state in Cs2Pt20, the stabilization of more negative oxidation states (NOSs), particularly for transition metals, remains highly restricted due to intrinsic electronic and structural constraints.

At ambient pressure, the most negative OS ever observed in isolated anionic species does not exceed –IV14,15, due to conflicting requirements. Achieving deeper NOSs necessitates central atoms that simultaneously exhibit strong electronegativity, large atomic radii, and high electron-accommodating capacity—an inherently contradictory set of criteria. For instance, electronegative p-block elements typically have small atomic radii and limited capacity to accept additional electrons, whereas transition metals offer extensive orbital availability (e.g., unfilled nd and unoccupied (n + 1)p orbitals) but lack sufficient electronegativity. Ligand atoms also present trade-offs: strongly electropositive s-block metals tend to be large, limiting coordination due to steric hindrance, while smaller-radius ligands exhibit reduced electropositivity. Consequently, excess negative charge in known NOS-containing compounds is usually stabilized within the anionic framework itself14,21, leaving little room for more exotic bonding configurations such as electride states.

Electrides, where excess electrons are localized in interstitial voids and act as nuclear-free anions (IAEs)22, provide an alternative platform for hosting unconventional charge states23,24. However, these IAEs are typically weakly confined and spatially delocalized—characteristics unfavorable for stabilizing the highly localized electron density required for deep metal reduction. In such systems, excess electrons preferentially occupy lattice voids rather than being transferred to metal centers, impeding the formation of deeper reduced OSs. Nonetheless, electrides containing negatively charged metal species have exhibited remarkable physical properties under pressure,25,26,27,28, such as superionic conductivity and superconductivity, as exemplified by high-pressure Li6Al19. This apparent incompatibility—between the delocalized nature of IAEs and the localized requirements for deep transition-metal reduction—raises a fundamental question: can a condensed-phase system be designed in which reduced transition metals and IAEs coexist stably?

Here, we address this challenge by employing a contrasting design principle: rather than relying on a single reductive or oxidative agent, we introduce a chemically polarized combination of a strong reductant (Li) and a strong oxidant (F) under high pressure. This dual-driven framework enables significant charge redistribution, allowing simultaneous stabilization of a −VII OS of gold (Au−7) and the formation of paired IAEs—realizing a ternary electride state. Gold, a late transition metal with strong relativistic effects29,30, possesses the highest electronegativity among metals and a small energy gap between its 5 d and 6 s/6p orbitals. These features enable Au to access various OSs21,31,32,33,34 and unique hybridizations35,36,37,38 in condensed phases, making it a promising candidate for realizing unusual deep NOSs under suitable conditions.

Using this design principle and first-principles calculations, we identify a thermodynamically stable compound, Li10AuF, above 150 GPa, in which Au adopts a near-closed-shell configuration analogous to radon (Rn), while two excess electrons occupy well-defined interstitial voids. Comparative studies on related compounds, Li10AuI and Li10AuP, reveal tunable charge-transfer behavior, orbital delocalization, and even superconductivity in Li10AuP. These findings demonstrate the efficacy of combining strong redox elements to push the boundaries of oxidation-state chemistry and offer a promising platform for designing quantum materials with exotic charge states and functionalities.

Results

Stability and crystal structure of electride Li10AuF

Structure prediction based on first-principles swarm-intelligence algorithms was performed over 31 Li-/F-rich stoichiometries in Li-Au-F system, yielding two previously unreported Li-rich compounds—Li10AuF and Li6AuF—as well as an energetically more favorable phase of LiAuF6 at 200 GPa (Fig. 1a and Supplementary Figs. S1a and b and S2). Among these, Li10AuF was identified as the most Li-enriched composition and is thus the focus of this work. Li10AuF crystallizes into a monoclinic structure with P21 symmetry and contains two formula units (f.u.) per unit cell (Fig. 1b). Remarkably, the Au atom is encapsulated by 21 Li atoms, forming a highly coordinated AuLi21 polyhedron that is spatially separated from neighboring Au atoms by more than 4.1 Å, effectively suppressing Au–Au interactions. The AuLi21 polyhedra form a quasi-hexagonal arrangement within the ab plane and stack in an AA-like fashion along the c axis (Supplementary Fig. S3). The interstitial regions are filled by two distinct types of zigzag polyhedral chains (Fig. 1b): distorted FLi9 tri-capped trigonal prisms (violet) and hollow Li10 hexadecahedra (yellow), which face-share with each other and with the AuLi21 units to form a dense, close-packed framework. Notably, the Li10 hexadecahedron hosts two more vertices than the Na8 cube in Na2He electride24 and the Li8 cube in Li8Au electride39, reflecting its enhanced capacity for IAEs accommodation.

Fig. 1: Stability, structure, and electride characteristics of monoclinic P21-Li10AuF.
Fig. 1: Stability, structure, and electride characteristics of monoclinic P21-Li10AuF.
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a Convex hull represented by the formation enthalpy (ΔH, unit: eV atom-1) of Li-Au-F system at 150 GPa. b Crystal structure of Li10AuF. c Partial charge density distribution corresponding to interstitial anionic electrons (IAEs) in the energy range of −2.76 to −2.10 eV (with the Fermi level set to zero), and d electron localization function (ELF) map on the (001) plane at 150 GPa. e Phonon dispersion curves of Li10AuF at 50 GPa. f The averaged mean-squared displacements (MSD) of Li, Au, and F atoms as a function of temperature at 150 GPa. g Atomic trajectories from the ab initio molecular dynamics simulations at 500 K.

Given the structural similarity to known electrides, the presence of IAEs was rigorously examined. The calculated partial charge density clearly reveals ellipsoidal electron centered within the Li10 polyhedra (Fig. 1c), strongly suggesting the formation of an electride state. This is further corroborated by the highly localized features in the corresponding electron localization functions (ELF) map (Fig. 1d), and the presence of non-nuclear attractor (NNA) characterized by an electron density of ρNNA = 0.047 e/bohr3 and a negative Laplacian of ▽2ρNNA = − 0.041 e/bohr5 (Supplementary Fig. S4), comparable to those of hP4 Na40. The ellipsoidal topology of the IAEs indicates s-like orbital character, stabilized via coulombic attraction to the surrounding Li+ framework. In addition, quasi-spherical electron localization features around the Au and F atoms (Supplementary Fig. S5) point to a dominantly ionic bonding character, consistent with the charge density distribution (Supplementary Fig. S6).

Phonon dispersion calculations confirm that Li10AuF is dynamically stable down to 50 GPa (Fig. 1e), and thermodynamically stable to at least 150 GPa (Fig. 1a). Finite-temperature ab initio molecular dynamics (AIMD) simulations in a 1 × 2 × 2 supercell indicate thermal stability up to 500 K, as evidenced by near-zero mean square displacements (MSD) of all atomic species (Fig. 1f and g). Upon heating beyond 1500 K, Li atoms diffuse significantly, while Au and F remain fixed near their equilibrium sites (Fig. 1f and Supplementary Fig. S7), indicating an incipient transition to a superionic phase. The optimized structure information and initial/final structures of AIMD are provided in the Supplementary Data 1.

Oxidation state of Li10AuF electride

For non-elemental electrides, their formation can be generally interpreted as a signature that the electronegative atoms in the compound have reached their lowest possible NOS. Conseqently, the excess electrons provided by the electropositive atoms become confined within lattice interstices, forming IAEs, as exemplified by N3− in Ca2N41, P3− in Li6P42, and C4− in Li5C43. On the other hand, electronic band structure analysis offers an effective approach to assessing formal OSs in ionic compounds21,44,45. The amount of fully occupied bands below the Fermi level (EF) directetly correspond to the number of electron-filled atomic and intersitial orbitals. This principle, well illustrated in NaCl (Supplementary Fig. S8), provides a rigorous framework for quantifying the charge distribution in ionic solids. The meaning and scientific rationale for the theoretical investigation of high-pressure OSs are presented in Supplementary Note 1.

We therefore analyze the orbital-resolved electronic structure of Li10AuF to elucidate the electron occupation across atomic valence orbitals and IAE sites, enabling the assignment of formal OSs and a mechanistic understanding of IAEs formation. As shown in Fig. 2a, Li10AuF is a semiconductor with an indirect bandgap of 0.56 eV at the PBE level (larger bandgap under reduced pressure or within the the revised Heyd-Scuseria-Ernzerhof screened hybrid functional, Supplementary Fig. S9). Below the EF, the projected band structure reveals 6 bands primarily derived from Au 6p orbitals (−2.15 to 0.0 eV), 2 bands from IAEs (−2.80 to −2.15 eV), 10 bands from Au 5 d orbitals, and 2 bands from Au 6 s orbitals (−7.20 to −6.00 eV), while other bands appear at deeper energies (Supplementary Fig. S10). Notably, the number of these bands precisely matches the total number of corresponding orbitals per unit cell: 5 ×  2 = 10 for Au 5 d, 1 × 2 = 2 for Au 6 s, 3 × 2 = 6 for Au 6p, and 1 × 2 = 2 for IAE-s orbitals in two formula units.

Fig. 2: Electronic structures and oxidation state analysis of electride Li10AuF.
Fig. 2: Electronic structures and oxidation state analysis of electride Li10AuF.
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a, d Electronic band structures projected by partial atomic orbitals in Li10AuF and the hypothetical model of Li10AuFO, respectively. The number of electronic bands is marked with corresponding colors on the right side. b Band-decomposed charge density corresponding to the three highest-energy Au 6p bands and the higher-energy IAE band, as labeled i-iv in (a). c Projected density of density (PDOS) of Li10AuF at 150 GPa (F 2p states are shown in the inset). e Schematic valence electron configurations of Au, IAE, F, and O atoms in Li10AuF and Li10AuFO. Intrinsic valence electrons are denoted in black, while electrons donated by Li atoms are shown in red. f PDOS comparison between X 6p in Li0XF0 (X = Po, At, and Rn) and Au 6p in Li10AuF at 200 GPa.

To further verify the orbital attribution, we analyze the band-decomposed charge density of the four highest-energy Au and IAE bands (labeled i-iv), which correspond the three quasi-orthogonal dumbbell-shaped distributions around Au (6p orbitals) and one ellipsoidal distribution at the IAE site (s-orbital character) (Fig. 2b), also illustrated by the band-resolved maximally-localized Wannier functions (supplementary Fig. S11). These features confirm the electronic occupation of Au 6p and IAE-s orbitals. The projected density of states (PDOS, Fig. 2c) also supports this conclusion: Li 2 s/2p and F 2p states show negligible contributions near EF, indicating near-complete electron donation by Li and full 2p occupancy for F (with F 2p states mainly below −11 eV, Fig. 2c). In contrast, the Au 6 s, 5 d, and 6p orbitals, as well as IAE-s states, exhibit strong localization below EF. Importantly, hybridization between Au 5 d and its 6 s/6p orbitals—enhanced by relativistic effects46, s-d orbital penetration36, and pressure-induced orbital reshaping—lowers the energy of the 6 s/6p states, thereby promoting deep electron accommodation. From this analysis, we assign a formal OS of +I for Li, −I for F, and an unknown −VII for Au, reflecting essentially complete filling of its 6 s and 6p orbitals. The IAEs appear as paired electrons confined in Li10 cages (Fig. 2e).

To corroborate this electron-counting scheme, we construct two hypothetical models. In Li10AuFO, one oxygen atom is introduced into each IAE site of Li10AuF. This leads to the disappearance of the 2 IAE bands and the appearance of 6 O 2p bands at deeper energy levels, while retaining semiconducting character (Fig. 2d). This confirms that each O atom receives two electrons at the IAE site to fill its 2p orbitals (Fig. 2e), further validating that IAEs exist as electron pairs in Li10AuF. Similar behavior is observed in hP4 Na and Na2He electrides and their oxygen-implanted counterparts24 (Supplementary Figs. S12 and S13). The second model, Li0XF0 (X = Po, At, and Rn), is created by replacing all Au with X and removing all Li and F atoms. Among them, Li0RnF0 display a PDOS profile for Rn 6p orbitals closely resembling that of Au 6p orbitals in Li10AuF (Fig. 2f), confirming the near-closed-shell configuration of Au 6p orbitals in the original compound.

As a result, a quasi-quantitative electron-counting scenario emerges: in Li10AuF per f.u., 10 electrons are donated by Li atoms, 1 is accepted by F to fill its 2p orbital, 7 are accommodated by Au atom (filling 6 s and 6p orbitals), and 2 are localized as a pair within the Li10 cavity. This balanced electron distribution satisfies the octet rule for all species, supporting the semiconducting nature of Li10AuF. This insight aligns with known insulating electrides like Na2He24 and hP4-Na23. Crucially, unlike conventional 18-electron complexes47 and half-Heusler compounds14 with the only partial (n + 1)s and (n + 1)p filling, Li10AuF exhibits nearly complete occupation of Au 6p orbitals, establishing a −VII OS. This finding not only redefines the NOS limit of Au atom33,39 and other transition metals15, but also exceeds the previously accepted lower bound (−V) of IIIA-group elements based on atomic shell theory48. While Geng et al. recently proposed highly reduced states of Be8−, Mg8−, Fe8−, and Al7− in body-centered IAE sites of face-centered cubic Li framework49, their assignments are inconsistent with the actual stoichiometry (Li4X), undermining their validity. Lastly, the considerable electron gain by Au in Li10AuF leads to a significant increase in its atomic radius, reduces steric hindrance, and enables the observed hypercoordination, as further visualized in the expanded electronic shell structure of the Au 6p partial charge density (Supplementary Fig. S14).

Heteromorphic Li10AuX (X = F, I, P) electrides

To elucidate the role of fluorine in modulating charge transfer and stabilizing the crystal structure in Li10AuF, we substitute the F atoms with I and P atoms. These substitutions were motivated by the larger atomic radius and lower electronegativity of iodine, and the known larger electron-accepting of phosphorus. Although replacement of F in Li10AuF by I and P leads to dynamically unstable structures, extensive structure searches in the Li-Au-I and Li-Au-P systems uncover two heteromorphic electrides at 200 GPa: a stable Li10AuI with Pmc21 symmetry and a metastable Li10AuP with P-1 symmetry (Supplementary Figs. S1c-f, S15-S17), which are structurally heteromorphic to Li10AuF.

Electronic structure analysis via PDOS demonstrates prominent Au 6p and IAEs states near the EF in both Li10AuI and Li10AuP (Fig. 3a). From Li10AuF to Li10AuI to Li10AuP, these states display progressive broadening and increasing overlap with the EF, indicative of enhanced delocalization, which is corroborated by the ELF maps (Supplementary Fig. S18). This trend drives a gradual evolution in electronic character—from semiconductor (Li10AuF), to semimetal (Li10AuI), and finally to metal/superconductor (Li10AuP). Simultaneously, the PDOS intensity of the Au 6p and IAE states decreases in Li10AuI and Li10AuP, suggesting a reduced electron occupation in these orbitals. In particular, the Au atom in Li10AuP exhibits a 6p electron occupation profile remarkably similar to that of a Bi atom (Fig. 3e), implying a significantly delocalized yet partially filled 6p shell. Moreover, the emergence of finite I 5 d and P 3 d states below EF points to fractional OSs for I and P atoms that are lower than the nominal −I and −III, respectively. These conclusions are well supported by Bader charge analysis (Fig. 3b and Table S1).

Fig. 3: Charge modulation mechanism in heteromorphic Li10AuX electrides.
Fig. 3: Charge modulation mechanism in heteromorphic Li10AuX electrides.
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a PDOS of partial atomic orbitals. b Bader charge and the coordination number of Au, X, and IAE in Li10AuX at 200 GPa. c Schematic diagram of X modulating the gained electrons by Au and IAE sites in Li10AuX electrides, assuming that each Li atom donates one electron per formula unit. The area of circles denotes the quantity of gained electron by X, and Au atoms as well as IAE sites (Scircle ~ Ne-). The gained electrons by F, Au, and IAE site in Li10AuF are denoted with red, and the ultimate electron configurations are shown in brackets with red. d ELF mapping on the (3.194 1 −9.286) plane in Li10AuP. e Comparison of PDOS for Au 6p orbital in Li10AuP and Bi 6p orbital in the derived Li0BiP0 model. f Phonon dispersion curves weighted by partial electron-phonon coupling λq,v (the size of red circle is proportional to λq,v), phonon density of states (PHDOS), and Eliashberg spectral function α2F(ω), and frequency-dependent electron-phonon coupling parameter λ(ω) of Li10AuP at 200 GPa. g Illustration of the relative orbital levels and electronic occupation of Li, Au, P, I, and F atoms, and the interstitial site (Inter) at 200 GPa. The area of blue-filled and unfilled regions represents the number of occupied and admissible electrons in the respective atomic orbitals.

In addition, Li atoms in Li10AuF present a slightly higher degree of charge loss than those in Li10AuP and Li10AuI (Supplementary Fig. S19 and Table S1), consistent with the strong ionization tendency induced by the high electronegativity of F. As the X element changes from P to I to F, the electron-accommodating ability of X decreases, resulting in a larger fraction of electrons being transferred to the Au atom and IAE sites. This trend enables the electronic configuration of Au and IAEs to approach those of the corresponding noble gas elements (Fig. 3c), in accordance with the principle of charge conservation. Besides, the weakened ionization of Li atoms, decreased electron gaining by Au atoms and metallization of Au 6p bands are also illustrated in Li-Au binaries and other constructed ternary phases (Supplementary Notes 2–4 and Fig. S20–S22).

Notably, in Li10AuP, the IAEs exhibit a dumbbell-like topology and form spatially extended bonds with extranuclear electrons from Au and P atoms (Fig. 3d), resembling the IAE characteristics observed in superconducting Li6P50 and Li9S51. The combination of electron-phonon coupling (EPC) calculations and the Allen-Dynes-McMillan equation52 gives a EPC parameter λ = 0.84, and a logarithmic average frequency ωlog = 141.37 K, resulting in a predicted superconducting critical temperature of Tc = 7.1 K at 200 GPa (Coulomb pseudopotential parameter µ* = 0.1). This Tc is comparable to measured 7 K of Bi-III and 8 K of Bi-V phase53. Based on the phonon density of states (PHDOS) (Fig. 3f), the low-frequency phonons dominated by Au atoms (below 6.3 THz) make a primary contribution of 62.5% to total λ, whereas the medium/high-frequency phonons associated to the coupled vibrations of Li and P atoms (above 6.3 THz) contribute 37.5%. Accordingly, metallized Au with 6p attribute dominates the superconductivity of Li10AuP.

Discussion

The underlying mechanism governing the charge modulation in the heteromorphic Li10AuX (X = F, I, P) electrides can be elucidated through the lens of atomic shell theory (Fig. 3g) in combination with the local coordination environment (Fig. 3b). At ambient pressure, the outmost valence orbitals—2p for F, 5p for I, and 3p for P—serve as primary electron acceptors, with intrinsic electron acquisition limits of approximately 1 for F and I, and 3 for P, repsectively. Under high pressure, however, orbital broadening and energy reshaping result in the I 5 d and P 3 d orbitals attaining energy levels comparable to the Au 6p and IAE-1s orbitals (Fig. 3g), thus enabling competitive electron acquisition. In contrast, such energetic proximity is forbidden for F, whose 3 s orbital significantly higher in energy, thus restricting its ability to compete for electrons. These electronic trends are further supported by the simulations of atomic orbital levels under pressure (Supplementary Fig. S23). Similtaneously, variations in anionic radius across the X series (P > I > F) induce a monotonic decrease in the coordination number of the X atoms. This structural evolution leads to a corresponding increase in Li coordination for both Au atom and IAE (Fig. 3b), which in turn facilitates enhanced charge transfer toward the central sites48.

With respect to binary Li-Au and other tenary Li-Au-X compounds, the unique ability of Li10AuF to stabilize a −VII OS in gold arises from a synergistic interplay of factors. (i) Structural modulation, the incorporation of F facilitates the formation of Li–F polyhedra, which spatially separate the Au-centered motifs and effectively increase the Li coordination number around Au. Also, the presence of F enlarges the IAE-centered Li cavities, facilitating the formation and identification of IAE pairs. (ii) Electronic polarization, F’s participation in Li–F bonding strengthens Li ionization and promotes internal charge redistribution owing to its high electronegativity, meanwhile, F atom only gains one electron, minimizing its participation in electron competition. On the structral side, high pressure is essential to achieving the requisite coordination geometry and orbital alignment, as evidenced by the reduced coordination numbers of Au and F, dispersive distribution of IAEs, and the band mixing of Au 6p and IAE-s bands with conduction bands in Li10AuF at 0 GPa (Supplementary Fig. S24). Besides, the small energy gap and mixing between the Au 6p and 5 d orbitals contribute favorably to the population of the Au 6p orbitals, completing the quasi-Rn electronic configuration.

Collectively, the Li–F polarity, in concert with pressure-induced electronic restructuring, plays a critical role in stabilizing this distinct NOS. In contrast, the substitution of F with I or P progressively weakens this polarity, reduces Au-IAE coordination, and reconfigures orbital interactions, leading to reduced Au electron gain and increased delocalization of IAEs. These transitions underpin the electronic evolution from an electride semiconductor (Li10AuF) to a semimetal (Li10AuI) and ultimately to a superconductor (Li10AuP), thereby illustrating the profound implications of chemically polarized environments in dictating charge distribution and emergent functionalities.

In conclusion, we have demonstrated, through a combination of orbital-resolved electronic structure calculations and electron counting analysis, that a record −VII OS of Au—nominally isoelectronic with Rn—is stabilized in a pressure-induced electride semiconductor, Li10AuF. This unique phase, identified via extensive structure searches in the Li-Au-F system, also hosts two paired IAEs, forming a highly localized and insulating electronic environment. Comparative analysis with the analogs Li10AuI and Li10AuP reveals a continuous reduction in the charge states of both Au and IAEs, accompanied by enhanced electron delocalization, which drives a progressive transition from semiconductor to semimetal and ultimately to a superconducting phase. This evolution stems from the intrinsic differences in orbital alignment and chemical polarity among the X-site elements. Our findings highlight the decisive role of Li-F chemical polarity and external pressure in stabilizing unusual oxidation states of transition metals, thus broadening the chemical understanding of gold and offering a viable design strategy for realizing highly charged quantum states in solid-state materials. This work paves the way toward the rational discovery of materials with emergent and tunable electronic functionalities through targeted manipulation of element polarity under extreme conditions.

Methods

Structure search and optimization

Using the CALYPSO structure search method in conjunction with first-principles calculations54,55, we systematically explored the crystal structures of LixAuyXz (X = F, P, and I, x = 1–6, y = 1-2, z = 1–3; x = 6-10, y = z = 1) at 100 and 200 GPa. The structural relaxations and electronic properties calculations were performed within the framework of density functional theory as implemented in the VASP (version 5.4.4) code. The generalized gradient approximation (GGA)56 within the Perdew-Burke-Ernzerhof (PBE)57 framework was adopted for the exchange-correlation functional. The projector augmented wave (PAW) potentials58 were used to describe the electron-ion interaction, where the valence configurations were treated with 1s22s1, 5d106s1, 3s23p3, 5s25p5, 2s22p5 for Li, Au, P, I, and F atoms, respectively. The cutoff energy of 700 eV and the Monkhorst-Pack k meshes with spacing of 2π × 0.03 Å−1 were used to ensure that all the enthalpy calculations with well convergence. The localized Wannier functions were through applying the WANNIER90 package59. The characterization of non-nuclear attractor (NNA) and its electron density (ρNNA) and Laplacian (▽2ρNNA) was performed in the Critic2 code60,61. Crystal structures were visualized with VESTA62.

Stability estimation

Phonon dispersion calculations were performed to determine the dynamical stability as done in the Phonopy code63. First-principles molecular dynamics simulations in NVT ensemble using the Nosé heat bath method64 lasted for 10 ps with a time step of 1.0 fs.

IAE-related electronic structures

The IAE-projected density of states was calculated by projecting the charge density and wavefunction of the compounds onto empty spheres located at the IAE sites, as implemented in the VASP (version 5.4.4) code. To derive the IAEs-projected band structures, the electronic wavefunction of the pristine structure was firstly calculated, and then the volumes of IAEs were filled with random pseudo atoms with the same Wigner-Seitz radius of IAE blocks. The wavefunction is subsequently projected to the pseudo atoms. Notably, as the wavefunction is derived from the pristine structure, the choice of pseudo-atoms does not affect the electronic bands. They serve solely as atomic spacers to enable the projection of the wavefunction for IAEs.

Superconductivity estimation

The electron-phonon coupling (EPC) was calculated based on density functional perturbation theory, as implemented in the Quantum ESPRESSO (version 6.5) code65. The ultrasoft pseudopotentials were employed with 1s22s1, 5d106s1, and 3s23p3 as valence electrons for Li, Au, and P atoms, respectively. The kinetic energy cutoff for the wave-function expansion was chosen as 70 Ry. The dense k-meshes (16 × 9 × 8) for electronic Brillouin zone integration and enough q-meshes (4 × 3 × 2) for evaluating average contributions from the phonon modes were sampled for P−1 Li10AuP. The superconducting Tc is estimated from the McMillan-Allen-Dynes formula with Coulomb pseudopotential (μ* = 0.1)66,67.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.