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Relativistic Slater–Dirac–Lamb Modelling of Atomic Ionization Energies, Electron Affinities, Covalent Radii and Debye Temperatures: Validation on 56 NIST Reference Indicators

This paper introduces and rigorously validates a closed-form, first-principles physics model for predicting key atomic properties, achieving a mean absolute percentage error of 0.17% across 56 NIST reference indicators while providing comprehensive documentation, comparative analysis, and links to sustainable development goals.

Original authors: Lucas Lima Freitag

Published 2026-09-18

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CC BY 4.0

Original authors: Lucas Lima Freitag

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Science often relies on a delicate balance between two opposing needs: the desire for perfect accuracy and the need for speed. When engineers design a new battery, a solar cell, or a heat-resistant alloy, they must know the fundamental properties of the atoms involved, such as how tightly an atom holds its electrons or how large it is. Traditionally, getting these numbers with high precision required running massive, time-consuming computer simulations that could take hours or days for a single element. These simulations are powerful, but they are too slow for the rapid screening needed to discover new materials quickly. On the other side of the spectrum are simple, fast formulas that give quick answers but often miss the subtle physical effects that make real-world materials behave the way they do. For decades, scientists have struggled to find a middle ground: a method that is both fast enough to run millions of times in a second and accurate enough to be trusted for critical design work.

A new study by Lucas Lima Freitag at the Federal University of Rio Grande do Norte in Brazil offers a solution to this long-standing problem. The researcher has developed a complete, closed-form mathematical model that predicts key atomic properties with extraordinary speed and precision. Unlike previous attempts that either hid their formulas behind complex computer code or relied on fitting data to specific cases, this work lays out every single equation used, ensuring that the physics behind each prediction is transparent and verifiable. The model focuses on four specific types of atomic behavior: the energy required to strip an electron from an atom, the energy released when an atom gains one, the physical size of the atom when it bonds with others, and the temperature at which the atom's vibrations change character. These properties are fundamental to understanding how materials conduct electricity, resist corrosion, or store energy.

To test the reliability of this new approach, the researcher compared its predictions against a massive set of 56 trusted reference values. These reference numbers come from authoritative sources like the National Institute of Standards and Technology and other major scientific handbooks, representing the best measurements currently available. The results were striking. The model predicted all 56 indicators with an average error of just 0.17 percent. To put this in perspective, the most accurate predictions were off by less than one-tenth of a percent, and even the least accurate prediction was still within a quarter of a percent of the true value. Every single one of the 56 tests passed the strict threshold for what scientists consider an excellent result. This level of accuracy was achieved without using any arbitrary "tuning knobs" or hidden factors that would only work for the specific data points tested; instead, the model relies on first principles of physics, incorporating corrections for the effects of relativity and quantum mechanics that become important as atoms get heavier.

The study also carefully examined where the model works best and where it struggles slightly. The predictions were most accurate for atoms with simple, stable electron structures, such as the noble gases and alkali metals, where the physics is straightforward. The model showed slightly larger, though still very small, errors for atoms with complex, partially filled electron shells, where the interactions between electrons are more chaotic and difficult to pin down. This honest mapping of the model's strengths and weaknesses is a crucial part of the work, as it tells future users exactly where they can trust the numbers and where they might need to be cautious. By comparing this new method against five other established models, the study demonstrated that it outperforms them all, achieving a level of precision that previously required much slower, more computationally expensive techniques.

The implications of this work extend beyond just a better way to calculate numbers. Because the model is so fast and accurate, it can be used to screen vast numbers of potential materials for clean energy applications, industrial manufacturing, and climate solutions without the need for expensive trial-and-error experiments. The researcher explicitly links these capabilities to global goals for sustainable development, suggesting that this tool could accelerate the discovery of better batteries, more efficient solar panels, and materials that use fewer rare or critical elements. The study concludes by providing a fully open and reproducible framework, including the specific equations and the data sources used, inviting other scientists to verify the results and build upon them. In a field often dominated by "black box" computer models, this work offers a clear, transparent, and highly effective path forward for understanding the building blocks of matter.

Technical Summary: Relativistic Slater–Dirac–Lamb Modelling of Atomic Properties

Problem Statement
Current materials design often relies on expensive trial-and-error or "black-box" surrogate models that lack transparency. While high-accuracy methods like Hartree–Fock, DFT, and CCSD(T) exist, they are computationally prohibitive for rapid screening and often require empirical corrections. Conversely, classical analytic models (e.g., Slater rules, Dulong–Petit) are fast and transparent but fail to capture critical physics such as relativity, electron correlation, and anharmonicity, typically resulting in errors of 1–6%. Machine learning surrogates interpolate well within training data but frequently lack closed-form equations, dimensional checks, or out-of-distribution guarantees, limiting their utility for safety-critical certification. There is a gap for a single-domain analytic model that combines full equation disclosure, rigorous validation against authoritative references, and explicit error attribution.

Methodology
The author proposes a fully specified, closed-form physics model for quantum atomic properties, validated against 56 independent reference indicators from NIST, CODATA, CRC, ASM, ASTM, and CCCBDB/JANAF. The model avoids arbitrary gain factors, relying instead on first-principles-based analytic expressions with second- and third-order corrections.

The theoretical framework includes:

Ionization Energies ( $E_{ion}$ ): Modeled by combining Slater screening with Dirac relativity and the Lamb shift (QED). The formula incorporates the Rydberg energy, effective nuclear charge ( $Z_{eff}$ ), a Dirac relativistic factor, and a QED shift term.
Electron Affinities ( $E_A$ ): Calculated using a Hartree–Fock gap augmented by a coupled-cluster (CCSD(T)) correlation increment, spin–orbit terms, and zero-point energy differences.
Covalent Radii ( $r_{cov}$ ): Derived from the Cordero scheme with coordination and hybridization corrections.
Debye Temperatures ( $\Theta_D$ ): Constructed from elastic velocities using the Debye construction, accounting for longitudinal and transverse modes.

The validation protocol (Algorithm 1) evaluates predictions against reference values ( $y_{obs}$ ) using Mean Absolute Percentage Error (MAPE), RMSE, MAE, $R^2$ , and bias. Measurement uncertainties from handbooks are propagated as GUM floors to prevent over-claiming accuracy where reference data itself is uncertain.

Key Contributions
The paper presents six specific contributions:

Closed-Form Multi-Term Model: All constants, ranges, and equations are explicitly stated with dimensional and limit checks (e.g., recovering hydrogen values at $Z \to 1$ ).
Rigorous Validation: Performance is measured on 56 independent indicators using multiple metrics (MAPE, RMSE, etc.).
Head-to-Head Comparison: The model is benchmarked against five established literature models (Slater rules, Hartree–Fock, DFT-PBE, CCSD(T)) on the identical set of indicators.
Ablation Study: The marginal gain of second- and third-order terms is quantified, showing the reduction of error from base forms to the final model.
Error Attribution: A detailed map identifies where the model performs best (closed-shell/alkali atoms) and worst (open d-shell atoms), attributing residuals to specific physical limitations (e.g., multi-reference character).
SDG Linkage: Quantitative connections are drawn between the model's utility and UN Sustainable Development Goals (SDGs) 7, 9, 12, and 13.

Results
On the frozen reference set of 56 indicators, the model achieves:

Mean MAPE: 0.17%
Range: 0.08% to 0.25%
Consistency: 100% of indicators (56/56) fall below 0.30%, and all fall below 0.50%.

Performance Breakdown:

Most Accurate: Closed-shell and alkali atoms (e.g., He, Ne, Na, K) and halogen affinities, where screening groups are clean and spin–orbit corrections are well-tabulated. Covalent radii for sp-elements (C, Si) also show excellent agreement.
Least Accurate: Open d-shell atoms (Sc, Ti, Cr) and Hydrogen affinity. The author attributes these larger residuals (up to 0.25%) to multi-reference character, core–valence correlation beyond single-reference CCSD(T), and finite-nuclear-size effects, which are coarsely approximated by the perturbative Lamb term.
Bias: Residuals are symmetric about zero with no systematic over- or under-prediction (homoscedastic errors).

Comparison with Literature
The proposed model significantly outperforms established baselines on the same dataset:

Slater rules only: 2.5–3.0% MAPE
Hartree–Fock (Koopmans): 3.0–4.0% MAPE
DFT-PBE: 1.5–2.0% MAPE
CCSD(T)/aug-cc-pVQZ alone: 0.5–0.8% MAPE
This Work: 0.17% MAPE

Ablation analysis reveals that the base closed form yields ~1.10% MAPE. Adding second-order terms (relativity, anharmonicity, dispersion, etc.) reduces this to 0.25%, and third-order refinements bring it to the final 0.17%.

Significance and Claims
The paper claims to fill a critical gap in materials modeling by providing a transparent, reproducible, and highly accurate analytic alternative to both expensive numerics and opaque machine learning. By ensuring every equation has units, limits, and source citations (DOIs), the model enables millisecond-speed screening for safety-critical applications (e.g., pressure vessels, nuclear data) where "black-box" models are insufficient. The author links this capability to accelerating the discovery of materials for clean energy, low-carbon industry, responsible resource use, and climate action. The work is presented not as a universal predictor for all materials science, but as a fully validated solution for a specific domain of quantum atomic properties.

Technical Summary: Relativistic Slater–Dirac–Lamb Modelling of Atomic Properties

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