Operable Atomic Properties (OAPs)

• Operable Atomic Properties (OAPs) are atomic-level quantities that are rigorously defined and measurable, directly linking computational and experimental methods.
• They are derived using frameworks like Atomic Cluster Expansion, coupled-cluster theory, and statistical learning, ensuring accurate modeling and reliable uncertainty quantification.
• OAPs underpin practical applications in spectroscopy, optical clocks, and intermolecular force fields, bridging high-precision quantum methods with real-world atomic and materials research.

Operable Atomic Properties (OAPs) denote atomic-level quantities that are rigorously defined, computationally accessible, and immediately utilizable in theory, experiment, data curation, and modeling workflows. Distinct from more abstract atomic descriptors, OAPs comprise quantities such as energy levels, matrix elements, transition rates, polarizabilities, and tensorial properties, together with their estimated uncertainties. Their theoretical, computational, and data-platform infrastructures span high-precision quantum many-body methods, data-driven compressions, and modern machine-learning pipelines, positioning OAPs as the backbone for both fundamental quantum mechanics and practical applications in atomic physics, chemistry, and materials science.

1. Definitions and Formal Categories of OAPs

At the core of the OAP concept is unambiguous operationalization: OAPs are quantities that can be measured, calculated ab initio, or extracted from authority databases, and that translate directly into the equations of motion, response functions, and interaction Hamiltonians of atomic, molecular, and condensed-matter theory. Key formal instances include:

• Spectroscopic and Structural: Excitation energies $E(n,J)$, ionization potentials, transition wavelengths $\lambda = hc/[E(f)-E(i)]$, level degeneracies.
• Electromagnetic and Hyperfine: Electric/magnetic dipole ($\langle n'J'||D||nJ \rangle$), quadrupole moments $\Theta(J)$, hyperfine constants ($A_{hfs}, B_{hfs}$), oscillator strengths.
• Dynamical and Response: Static and frequency-dependent scalar/tensor polarizabilities $\alpha(0), \alpha(\omega)$, dispersion coefficients ($C_6, C_8, C_{10}$), radiative lifetimes $\tau$, branching ratios $b_{i\to f}$.
• Many-Body and Model-Ready: Elemental-mode low-dimensional embeddings (Herr et al., 2018), compressed OAP codes for machine learning; cluster expansion coefficients for multivariate atomic observables (Drautz, 2020).

This operability extends to vectorial and tensorial OAPs, such as the local (site-resolved) electric dipole, force vectors, or stress tensors, and includes quantum numbers, partial charges, and local magnetic moments—all central in both classical and quantum atomistic simulations (Drautz, 2020, Barakhshan et al., 2022).

2. Theoretical Frameworks for OAP Construction

Multiple rigorous frameworks systematically derive, compute, and interpret OAPs:

• Atomic Cluster Expansion (ACE): ACE provides a complete, orthonormal, and systematically improvable basis for scalar, vectorial, and tensorial OAPs, treating atomic positions, species, partial charges, and local magnetic moments on equal footing. The per-site property is expanded as

$$G_i = \sum_{N=0}^{N_{\max}} \sum_{\substack{v_1 \ge \cdots \ge v_N \ L_1,\ldots,L_N}} c^{(N)}_{p_i; v_1 \cdots v_N ; L_1 \cdots L_N} \; B^{(N)}_{i, \{v_k\}, \{L_k\}},$$

where the $B^{(N)}$ are body-order, symmetry-adapted basis invariants, and expansion includes all relevant degrees of freedom (Drautz, 2020).

• Coupled-Cluster Many-Body Theory: State-of-the-art atomic OAPs—transition matrix elements, polarizabilities, lifetimes—are obtained from linearized coupled-cluster theory (SD/SDpT) with ab initio or correlation-corrected amplitudes. Uncertainty is estimated by internal spread among variants, total size of correlation corrections, and benchmarking (Barakhshan et al., 2022, Paez et al., 2016).
• Quantum Drude Oscillator Models: QDO/OQDO maps key OAPs (e.g., $\alpha_1, C_6, R_{vdW}$) analytically to a three-parameter effective Hamiltonian, yielding closed-form expressions for multipolar response (e.g., $C_8, C_{10}$) and producing transferable "force-field-ready" OAP parameterizations (Góger et al., 2022).
• Statistical Learning and Compression: Machine learning approaches generate compressed OAP embeddings (elemental modes (Herr et al., 2018)), or select minimal yet maximally predictive OAP sets for chemical and spectroscopic tasks via Gaussian process regression (Ibrahim et al., 2023).

3. Computation, Curation, and Uncertainty Quantification

Precision, verifiability, and reproducibility are central to OAPs' utility:

• Data Sources and Curation Protocols: OAP databases (e.g., Atom portal (Barakhshan et al., 2022)) incorporate both theoretical all-order calculations and experimental values, prioritizing the latter when uncertainties $\Delta$ are stricter. Experimental values are tagged with bibliographic metadata and, where appropriate, direct links to major authorities (e.g., NIST-ASD).
• Uncertainty Propagation: For computed OAPs, rigorous error estimates are propagated through subsequent derived quantities (e.g., $$\Delta \langle n'J'||D||nJ \rangle \to \Delta A_{i \to f} \to \Delta \tau$$), ensuring reliability for downstream modeling. When OAPs are used as regression targets or features, standard or cross-validation errors summarize predictive confidence (Barakhshan et al., 2022, Ibrahim et al., 2023, Paez et al., 2016).
• Software Architecture: OAP dissemination platforms employ simple ASCII/CSV ingest for new values, static-site or future RESTful API delivery, and scripting hooks for rapid extraction and integration into research workflows. State labels are serialized in dot notation to support automated processing.

4. OAPs in Data-Driven and Machine Learning Frameworks

Contemporary molecular and materials modeling leverages OAPs as fundamental features or as compressed representations:

• Elemental Modes: Vectors of 10 fundamental atomic properties per species (atomic number, mass, shell occupation, radius, electronegativity, IE, EA, polarizability) are compressed via autoencoders to low-dimensional "elemental modes," which serve as differentiable, transferable OAP embeddings for input to neural networks and kernel models. These span the chemical space with smooth, periodic-trend-respecting coordinates (Herr et al., 2018).
• Feature Selection and Regression: For diatomic spectroscopic prediction, OAP selection from group/period numbers, atomic numbers, and reduced mass underpins GPR models for bond lengths, frequencies, and dissociation energies, achieving test errors competitive with high-level ab initio results without explicit quantum-chemical inputs (Ibrahim et al., 2023).
• Machine-Learning Potentials: OAPs inform symmetry function channels and bias terms, enabling scale-efficient, species-transferable models (e.g., Behler-Parrinello NNPs with elemental-mode augmentation), and facilitate "alchemical" transformations via latent-space interpolation (Herr et al., 2018).

5. Advanced Tensorial, Many-Body, and Unified OAP Descriptions

State-of-the-art theory and modeling demand OAPs that encompass high-rank and coupled-body observables with explicit symmetry properties and multivariate dependency:

• ACE for Tensorial OAPs: The generalized ACE formalism provides systematic construction of invariant expansions for scalar ($L=0$), vectorial ($L=1$), and higher-tensor ($L\ge2$) OAPs, such as forces, torques, polarizabilities, and local stress, with atomic species, charges, and magnetic moments incorporated identically to geometry (Drautz, 2020).
• Symmetry and Permutation Invariance: OAP representations automatically build in permutation symmetry (e.g., through sum-form atomic bases), translation invariance (via use of relative coordinates), and rotation invariance (via Clebsch–Gordan–coupled bases). These properties ensure physical consistency under all allowable atom relabelings and coordinate transformations.
• Connections to Other Descriptors: Moment Tensor Potentials (MTP), SOAP/SNAP descriptors, and similar approaches are shown to be subsets of or re-expandable in the ACE/OAP framework, enabling unified convergence analysis and hierarchical body-order control (Drautz, 2020).

6. Application Domains and Case Studies

OAPs enable predictive modeling and data curation across a diverse suite of platforms:

• Atomic Data Portals: The Atom portal distributes OAPs for neutrals, ions, and highly charged ions, including reduced matrix elements, polarizabilities, transition rates, and hyperfine constants with uncertainties, facilitating state-of-the-art atomic, plasma, astrophysical, and quantum-optics computations (Barakhshan et al., 2022).
• Diatomic Spectroscopy: OAP-based GPR models deliver sub-picometer and sub-30 cm⁻¹ accuracy in bond-length and frequency predictions, rationalize isotopic shifts, and permit classification of chemical bonding regimes (Ibrahim et al., 2023).
• Optical Atomic Clocks: OAPs (e.g., level energies, transition probabilities, differential dynamic polarizabilities) underlie the systematic evaluation of clock transitions, blackbody radiation shifts, and probe-induced AC Stark effects in systems such as Lu⁺ (Paez et al., 2016).
• Intermolecular Force Fields and Polarization: OAP-parametrized Drude oscillators (OQDO) connect measured α₁, C₆, and R_vdW to analytical model Hamiltonians, yielding accurate dispersion, induction, and polarizability descriptors for simulation across the periodic table (Góger et al., 2022).

7. Outlook and Future Directions

OAPs represent a paradigm of rigor, accessibility, and interoperability in atomistic and molecular science:

• Increasing Granularity and Tensoriality: Next-generation OAP infrastructures will expand to higher-order tensor properties (e.g., hyperpolarizabilities, multipole coupling), explicitly nonadiabatic corrections, and charge/magnetic/mass-coupled expansions.
• Automated, Large-Scale Curation and Delivery: Dynamic API endpoints, SQL/NoSQL-backed OAP databases, and bulk-access protocols are under active development, targeting the integration of millions of computed and experimental OAP entries (Barakhshan et al., 2022).
• Data-Driven Discovery and Inverse Design: With smooth, compressed, and differentiable OAP representations, ML models can interpolate, extrapolate, and optimize species and structure properties "on demand," accelerating inverse design of molecules and materials (Herr et al., 2018, Ibrahim et al., 2023).
• Unified Theoretical-Experimental Feedback: OAP ecosystems link experimental measurement, high-accuracy theory, and data-driven surrogates, with uncertainties and data provenance tracked throughout, ensuring traceability and reliability of all OAP deployments.

The OAP formalism thus constitutes a foundational infrastructure for atomic, molecular, and condensed-matter science, unifying theoretical completeness, computational accessibility, and data-driven application (Drautz, 2020, Barakhshan et al., 2022, Góger et al., 2022, Herr et al., 2018, Ibrahim et al., 2023, Paez et al., 2016).