Interatomic Electron Accumulation

• Interatomic electron accumulation is the buildup of electron density in regions between atomic centers, driven by quantum-mechanical effects rather than conventional orbital overlap.
• Mechanisms such as potential-barrier affinity (PBA) and interatomic Coulombic electron capture (ICEC) amplify barrier-region wavefunctions and induce electron capture cascades with distinct scaling laws.
• This phenomenon influences material properties, enabling the design of tailored superconductivity, magnetism, and conductivity through engineered barrier modulations and interface controls.

Interatomic electron accumulation is the phenomenon by which electron density builds up in regions between atomic centers—typically outside the conventional atomic spheres—due to quantum-mechanical effects. This effect fundamentally controls chemical bonding, electronic microstructure, and materials properties across systems including metals, covalent solids, electrides, polar interfaces, and weakly bound clusters. Recent research has revealed that a range of mechanisms—including potential-barrier affinity (PBA) and interatomic Coulombic electron capture (ICEC) processes—can drive interatomic electron accumulation, overturning the traditional narrative of bound-state orbital overlap and potential-well constraints as the necessary ingredients for interstitial electron density. Below is a comprehensive treatment of the physical mechanisms, mathematical framework, key regimes, and materials implications of interatomic electron accumulation.

1. Physical Mechanisms of Interatomic Electron Accumulation

1.1 Potential-Barrier Affinity (PBA) in Periodic Systems

The PBA effect occurs in periodic solids when single-particle energies $E_k$ cross above the maximum of the periodic potential barrier $V_\mathrm{max}$. Contrary to the classical intuition—which dictates electron localization in potential wells for $E < V_\mathrm{max}$—PBA predicts that, due to the required continuity of the wavefunction and its derivative at well-barrier junctions, long-wavelength components of $\psi_k(r)$ are strongly amplified in the barrier (interatomic) region for $E_k > V_\mathrm{max}$ (Xu et al., 14 Nov 2025). This leads to substantial density in "non-nuclear" regions, even for unbound or near-free states—without invoking multicenter hybridization.

One-dimensional illustration (Kronig–Penney model):

For a square barrier of height $V_0$, as $E \to V_0^+$, the barrier-region wave amplitude diverges as $(1 - V_0/E)^{-1/2}$, leading to $|\psi^{B}_{nk}| \gg |\psi^{A}_{nk}|$ and a spike in $\rho(x)$ between atoms.

1.2 Interatomic Coulombic Electron Capture (ICEC) and Cascades

ICEC is a two-center quantum process where an incident electron is captured by center $A$ and the excess energy is transferred radiationlessly (by Coulomb interaction) to neighbor $B$, causing ionization of $B$ and emission of a secondary electron. Double ICEC (DICEC) involves a cascade: the initial ICEC event is followed by a second ICEC, producing net electron accumulation on the capturing center (often denoted $A$), with cross-sections that display strong $R^{-14}$ scaling with the internuclear distance $R$ (Ellerbrock et al., 9 Sep 2024). Resonant enhancements occur in three-center systems when intermediate states match dipole-allowed transitions, resulting in amplification of electron accumulation probabilities by 6–8 orders of magnitude (Remme et al., 2022).

1.3 Electrostatic Polar Discontinuities and Interface Effects

At interfaces such as LaMnO$_3$/SrTiO$_3$, the discontinuity between charged and neutral layer stacking creates a built-in polar field that, above a critical thickness, drives electronic reconstruction: electrons accumulate in interfacial layers until the field is neutralized, resulting in measurable changes in valence and potentially leading to emergent phenomena such as interface superconductivity or magnetism (Chen et al., 2017).

1.4 Surface Band Bending and Two-Dimensional Accumulation

On semiconductor surfaces such as InAs(111), downward band-bending induced by Fermi-level pinning below the bulk conduction band minimum engenders strong surface-localized accumulation of electrons in quantized subbands. The standing-wave patterns of these states directly modulate the local density between atomic layers (Vrubel et al., 2020).

2. Quantum and Mathematical Formalism

2.1 Schrödinger Equation and Boundary Effects

The periodic one-electron Schrödinger equation,

$$H ψ_k(r) = [-\frac{\hbar^2}{2m}\nabla^2 + V(r)] ψ_k(r) = E_k ψ_k(r)$$

with $V(r+R)=V(r)$ (lattice periodicity), leads by Bloch's theorem to

$ψ_k(r) = e^{i k·r}u_k(r),$

where $u_k(r)$ is periodic. For $E_k > V_\mathrm{max}$, the wavefunction does not exhibit exponential decay in barrier regions, but instead remains oscillatory and amplifies at the well-barrier interfaces due to continuity constraints. The resulting charge density,

$ρ(r) = \sum_{k}^{\rm occ} |\psi_k(r)|^2,$

can be partitioned into contributions from bound and unbound (PBA) states (Xu et al., 14 Nov 2025).

2.2 ICEC, DICEC, and Cross-Section Scaling

The ICEC differential cross section between centers $A$ and $B$ at large $R$ is

$$dσ_{ICEC}^{(AB)} = \frac{1}{4π^2} \frac{p'}{p} |\langle φ_g^A χ_{p'}^B | V_{AB} | φ_p^A χ_g^B \rangle|^2 dΩ_{p'},$$

with $V_{AB}$ the dipole-dipole Coulomb interaction. The total cross section factorizes (neglecting exchange and higher multipoles),

$$σ_{ICEC}^{(AB)}(E) = \frac{α c^4}{ω_A^4 R_{AB}^6} σ_{RR}^{(A)}(E) σ_{PI}^{(B)}(ω_A),$$

where $σ_{RR}$ and $σ_{PI}$ are single-center recombination and photoionization cross sections, respectively (Ellerbrock et al., 9 Sep 2024).

For DICEC, the cross section scales as $\sim 1/R^{14}$, sharply confining the effect to short to moderate-range neighbors unless resonant enhancements (e.g., via autoionizing intermediate states) are available.

2.3 Time-Resolved Probes: Ultrafast X-ray Scattering

Ultrafast resonant x-ray scattering (RXS) at sub-femtosecond pulse durations allows direct imaging of interatomic current and electron accumulation. The antisymmetric (imaginary) part of the Fourier-transformed scattering pattern encodes the instantaneous, pairwise interatomic current, $\mathbf{j}_{qr}(t)$, providing a real-space and time-resolved picture of bonding and charge flow (Popova-Gorelova et al., 2015).

3. Comparison of Mechanisms and Regimes

Mechanism or Regime	Key Condition	Physical Signal/Consequence
Potential-Barrier Affinity (PBA)	$E_k > V_\mathrm{max}$	Interstitial density spikes; unified view of bond electrons
ICEC/DICEC (Two-/Three-center)	Occupied centers; energy resonance	Cross sections $\sim R^{-6}$/$R^{-14}$; electron flux/accum.
Polar-catastrophe interface	Charge discontinuity; critical $d_c$	Accum. in $\sim$2 UC; controls interface magnetism/supercond.
Surface band bending (2DEG)	Downward $E_C$ bend; Fermi-level pin	Sub-band quantization; STM-detected charge near surface

Traditional models (potential-well, hybrid orbitals) attribute interatomic accumulation to bound-state overlap and exponential decay in interstitial regions. PBA and ICEC-based frameworks reveal substantial interatomic density in the absence of bound or hybridized states, fundamentally due to quantum continuity or intercenter Coulomb correlations.

4. Prototypical Materials and Analytical Results

4.1 Electrides and Conventional Crystals

In high-pressure Na-hP4, Kohn–Sham DFT yields an interstitial barrier maximum $V_M \approx 4.4$ eV and Fermi level $E_F \approx 16$ eV, so all valence electrons contribute to interstitial peaks in the ELF and $\rho(r)$ consistent with PBA (Xu et al., 14 Nov 2025). Metallic fcc Al shows nearly uniform PBA-driven electron clouds between atoms; in diamond, PBA enhances electron density along directional C–C bonds.

4.2 Interface and Surface Systems

In LaMnO$_3$/SrTiO$_3$ heterostructures, DFT and XAS/XMCD experiments confirm electron accumulation onset at 2 unit cells—linked to polar-catastrophe physics—and its impact on emergent magnetism (Chen et al., 2017). At InAs(111) surfaces, coupled Schrödinger–Poisson and DFT+$U$+SOC treatments reproduce STM-observed charge enhancement at atomic protrusions, with 2DEG densities $Q_s \sim 10^{13}$ cm$^{-2}$ (Vrubel et al., 2020).

4.3 Weakly Bound Clusters and Dimers

ICEC and DICEC dictate electron localization and redistribution in rare gas dimers, quantum-dot assemblies, and even biological environments (e.g., influencing low-energy electron accumulation and damage in solvated clusters) (Ellerbrock et al., 9 Sep 2024, Jahr et al., 8 May 2025). In HeNe dimers, ICEC cross sections reach $10^4$ Mb near threshold energies, dramatically exceeding photorecombination by $10^3$–$10^6$ (Jahr et al., 8 May 2025).

5. Consequences for Material Science and Design

PBA provides a unifying quantum-mechanical principle for understanding electron accumulation in metals, covalents, ionic compounds, and electrides. The microstructure of $\rho(r)$ determined by PBA and related effects is inextricably linked to physical properties such as mechanical hardness, superconductivity, and band topology. Interfacial electron accumulation governs emergent properties at oxide and semiconductor boundaries, modulating phenomena from magnetism (via Mn valence reconstruction) to interface superconductivity.

Design strategies emerge: by engineering barrier heights ($V_\mathrm{max}$) through atomic substitution, strain, or external fields, one can tune interatomic electron densities and control functional properties—enabling the rational realization of custom conductive, catalytic, or quantum materials (Xu et al., 14 Nov 2025).

6. Dynamical Probes and Imaging of Interatomic Currents

Ultrafast RXS provides time-resolved, real-space access to dynamical interatomic electron currents that underlie accumulation and transport phenomena. In model systems such as KBr$_{108}$ (rocksalt) and Ge$_{83}$ (sp$^3$ networks), the antisymmetric component of the Fourier-transformed x-ray scattering pattern yields direct signatures of electron flow and accumulation along specific bonds on sub-femtosecond scales (Popova-Gorelova et al., 2015). Thus, both the static (ground-state $\rho(r)$) and dynamical aspects of interatomic electron accumulation become experimentally accessible for detailed mapping and functional exploitation.

7. Outlook and Open Directions

Recognition of PBA and correlated interatomic capture mechanisms as the core drivers of electron accumulation dissolves the distinction between metallic, covalent, ionic, and interstitial electron behaviors. The capacity to quantify, image, and manipulate these regimes through quantum design and advanced probing technologies opens targeted pathways for the atomic-scale control of material microstructure and emergent functional phenomena, ranging from high-temperature superconductivity to energy conversion and nanoscale electronics. Any advances in experimental precision (ultrafast spectroscopy, high-resolution STM, electron energy-loss mapping) and multiscale simulation spanning quantum, electrostatic, and many-body effects will further consolidate this unifying framework for interatomic electron accumulation.