Atomic Orbital Ontology

FLOW ALGEBRA
ATOMIC ORBITAL ONTOLOGY
Version: 1.1
Source: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/atomic-entities/atomic-orbital-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library | https://kirethresearch.org
Status: Scientific synthesis; source page not independently verified

01. PURPOSE
Represent atomic orbitals as quantum-mechanical one-electron states or basis functions, their quantum numbers, probability distributions, energies, and relationships to atomic structure. Distinguish orbitals from classical electron trajectories and from occupied electron states in many-electron atoms.

02. ONTOLOGY IDENTITY
ontology_id: kireth:atomic-orbital
entity_type: AtomicOrbital
parent: AtomicEntities
related: Atom, ElectronCloud, Electron, QuantumState, ElectronConfiguration, AtomicSpectrum
scope: Nonrelativistic orbital theory with explicit extensions and limitations

03. TYPED ENTITIES
Atom(id, Z:int, N:int?, charge:int)
Nucleus(atom_id, nuclear_charge:int)
Electron(id, spin:half_integer, charge:real)
Orbital(id, atom_id, n:int, l:int, m:int, basis_model:str)
SpinOrbital(id, orbital_id, ms:half_integer)
Wavefunction(id, domain:SpatialCoordinates, codomain:Complex)
ProbabilityDensity(id, orbital_id, density:NonnegativeReal)
Hamiltonian(id, operator:LinearOperator)
EnergyLevel(id, energy:Energy, approximation:str)
ElectronConfiguration(id, atom_id, occupations:Map)
QuantumNumber(id, name:str, value:number)
Measurement(id, observable:str, value:number, uncertainty:number?, unit:str)

04. RELATIONSHIP SIGNATURES
HAS_ORBITAL: Atom x Orbital -> Boolean
HAS_WAVEFUNCTION: Orbital x Wavefunction -> Boolean
HAS_QUANTUM_NUMBER: Orbital x QuantumNumber -> Boolean
HAS_ENERGY: Orbital x EnergyLevel -> Boolean
HAS_DENSITY: Orbital x ProbabilityDensity -> Boolean
OCCUPIES: Electron x SpinOrbital -> Boolean [model-dependent]
BELONGS_TO_SHELL: Orbital x Shell -> Boolean
BELONGS_TO_SUBSHELL: Orbital x Subshell -> Boolean
ORTHOGONAL_TO: Orbital x Orbital -> Boolean [given orthonormal basis]
TRANSITIONS_TO: QuantumState x QuantumState -> Boolean [subject to selection rules]

05. CORE AXIOMS
A1: n is a positive integer for hydrogenic bound orbitals.
A2: l is an integer with 0 <= l < n.
A3: m is an integer with -l <= m <= l.
A4: electron spin projection ms is +1/2 or -1/2 in the nonrelativistic model.
A5: normalized orbital psi satisfies integral |psi(r)|^2 d^3r = 1.
A6: orbital density rho(r) = |psi(r)|^2 is nonnegative.
A7: distinct spin-orbitals in a Slater determinant obey fermionic antisymmetry.
A8: occupation of any given spin-orbital is at most one electron (Pauli exclusion).
A9: orbitals are not fixed classical paths.
A10: many-electron orbitals are model-dependent approximations; exact many-electron states are not generally products of independent orbitals.

06. QUANTUM NUMBER SEMANTICS
n: principal quantum number; shell label for hydrogenic bound states.
l: orbital angular momentum quantum number.
m: magnetic quantum number (projection relative to chosen axis).
ms: spin projection; applies to spin-orbitals rather than spatial orbitals alone.
Orbital angular momentum squared eigenvalue: l(l+1) hbar^2.
Orbital angular momentum z-component eigenvalue: m hbar.

07. SUBSHELL LABELS
l=0 -> s
l=1 -> p
l=2 -> d
l=3 -> f
l=4 -> g
Higher labels continue alphabetically with conventions.
For fixed l, there are 2l+1 spatial orbitals with distinct m.
With two spin projections, the subshell has capacity 2(2l+1) electrons.

08. HYDROGENIC WAVEFUNCTION
psi_nlm(r,theta,phi) = R_nl(r) Y_lm(theta,phi)
H psi_nlm = E_n psi_nlm, for the ideal nonrelativistic Coulomb Hamiltonian.
E_n = -(mu Z^2 e^4)/(2 (4 pi epsilon_0)^2 hbar^2 n^2)
where mu is reduced mass and Z is nuclear charge.
This n-only degeneracy does not generally persist for many-electron atoms.

09. SCHRODINGER MODEL
H = -(hbar^2/(2 mu)) nabla^2 + V(r)
H psi = E psi
For a Coulomb nucleus, V(r) = -Z e^2/(4 pi epsilon_0 r).
Boundary conditions: normalizable bound states, regularity, appropriate asymptotic decay.

10. PROBABILITY INTERPRETATION
P(r in region Omega) = integral_Omega |psi(r)|^2 d^3r.
Radial probability density for a separated hydrogenic orbital:
P_radial(r) = r^2 |R_nl(r)|^2, with integral_0^infinity P_radial(r) dr = 1.
The wavefunction amplitude is not itself a probability.

11. NODES
Radial nodes for hydrogenic bound states: n-l-1.
Angular nodes (conventional count): l.
Total nodes (conventional count): n-1.
Node geometry depends on orbital basis and representation; complex spherical harmonics differ visually from real orbital combinations.

12. ORBITAL SHAPES
s: spherically symmetric probability density for hydrogenic orbitals.
p: directional patterns in commonly used real combinations.
d and f: higher angular structures.
These visualisations represent amplitude or density surfaces, not electron tracks.

13. OCCUPATION RULES
Pauli: at most two electrons per spatial orbital when opposite spin projections are available.
Hund: for approximately degenerate orbitals, maximum multiplicity often favoured in ground-state configurations; context-dependent energetic principle.
Aufbau: useful approximate filling heuristic, with known exceptions.
Occupation assignments are not universal exact observables independent of the chosen orbital model.

14. ATOMIC ELECTRON CONFIGURATION
Configuration maps subshell labels to occupancy counts.
Example hydrogen ground state: 1s^1.
Example helium ground state: 1s^2.
Example carbon approximate ground configuration: 1s^2 2s^2 2p^2.
Configuration does not fully specify a correlated many-electron wavefunction.

15. MANY-ELECTRON MODELS
Hartree-Fock: self-consistent single-particle spin-orbitals and Slater determinant.
Density functional theory: Kohn-Sham orbitals are auxiliary constructs reproducing density in an exact formulation; their eigenvalues generally are not all direct excitation energies.
Configuration interaction: linear combination of determinants.
Natural orbitals: eigenfunctions of one-particle reduced density matrix.
Model provenance is mandatory for computed orbital energies and shapes.

16. RELATIVISTIC EXTENSIONS
Dirac spinors replace simple scalar spatial orbitals when relativistic effects matter.
Spin-orbit coupling can require total angular momentum j and projection mj.
The labels n,l,m,ms are not always simultaneously good quantum numbers under spin-orbit coupling.

17. EXTERNAL FIELDS
Electric fields can cause Stark shifts and mixing.
Magnetic fields can cause Zeeman splitting.
Selection rules depend on symmetry, coupling scheme, and perturbation.

18. SPECTROSCOPIC TRANSITIONS
Transition energy DeltaE = E_final - E_initial.
Absorption photon energy h nu = DeltaE > 0.
Emission photon energy h nu = E_initial - E_final > 0.
For electric-dipole transitions in the hydrogenic orbital approximation: Delta l = +/-1; Delta m = 0,+/-1, with polarisation dependence.
These are approximate selection rules under stated assumptions, not universal rules for all transitions.

19. SPATIAL ORBITAL OVERLAP
S_ij = integral psi_i*(r) psi_j(r) d^3r.
Orthonormal basis: S_ij = delta_ij.
Atomic orbitals on different centres need not be orthogonal.
Overlap supports molecular orbital construction and bonding analysis.

20. MATHEMATICAL OPERATIONS
NORMALIZE(psi) -> psi / sqrt(integral |psi|^2 dV) [if norm finite and nonzero]
EXPECTATION(A,psi) -> integral psi* A psi dV [if defined]
OVERLAP(psi_i,psi_j) -> integral psi_i* psi_j dV
PROJECT(psi,basis) -> expansion coefficients
TRANSFORM_BASIS(orbitals,U) -> equivalent span if U unitary
CALCULATE_DENSITY(psi) -> |psi|^2

21. INFERENCE RULES
R1: Orbital(n,l,m) and valid quantum numbers -> HydrogenicOrbitalCandidate.
R2: l=0 -> subshell=s.
R3: l=1 -> subshell=p.
R4: l=2 -> subshell=d.
R5: fixed l -> spatial_degeneracy=2l+1 in central-potential angular basis.
R6: normalized psi -> total_probability=1.
R7: spin-orbital occupied -> cannot be occupied by a second identical fermion in same state.
R8: density(r)=0 -> probability of detection in infinitesimal volume there is zero.
R9: exact hydrogenic bound state with principal n -> E_n follows Coulomb formula under model assumptions.
R10: transition dipole matrix element zero -> electric-dipole transition forbidden within that approximation.

22. WORKED EXAMPLE: 2p
n=2; l=1; m in {-1,0,1}.
Number of spatial 2p orbitals = 3.
Spin-orbital capacity = 6 electrons.
Radial nodes = 2-1-1 = 0.
Angular nodes = 1.

23. WORKED EXAMPLE: 3d
n=3; l=2; m in {-2,-1,0,1,2}.
Number of spatial 3d orbitals = 5.
Spin-orbital capacity = 10 electrons.
Radial nodes = 3-2-1 = 0.
Angular nodes = 2.

24. WORKED EXAMPLE: 3s
n=3; l=0; m=0.
Number of spatial 3s orbitals = 1.
Radial nodes = 3-0-1 = 2.
Angular nodes = 0.

25. DIMENSIONAL CHECKS
Wavefunction psi in 3D position representation has units length^(-3/2).
Density |psi|^2 has units length^(-3).
Integral |psi|^2 d^3r is dimensionless.
Orbital energies have units joules or electronvolts.
Angular momentum has units joule-second.

26. VALIDATION
V01: n >= 1.
V02: 0 <= l < n.
V03: -l <= m <= l and all n,l,m are integers.
V04: ms in {-1/2,+1/2} for spin-1/2 electrons.
V05: normalization integral approximately 1 within declared numerical tolerance.
V06: probability density nonnegative.
V07: subshell occupation between 0 and 2(2l+1).
V08: computed energy includes model, basis, units and provenance.
V09: hydrogenic formulas only applied under their assumptions.
V10: orbital shape claims specify basis and plotting convention.
V11: distinguish exact observables from model-dependent orbital assignments.
V12: conservation and selection-rule statements include approximation context.

27. SEMANTIC EQUIVALENCE
Atomic orbital ~ one-electron orbital function [context: quantum chemistry / atomic physics].
Spatial orbital != spin-orbital.
Orbital != classical orbit.
Electron cloud != single atomic orbital.
Atomic orbital != molecular orbital, though atomic orbitals can be used as molecular basis functions.
Degenerate orbitals may be mixed by unitary basis transformations without changing their spanned subspace.

28. CONFIDENCE MODEL
Established quantum number constraints: high confidence within nonrelativistic hydrogenic framework.
Calculated orbital energies: confidence conditional on model and numerical method.
Approximate occupation rules: conditional confidence, with documented exceptions.
Experimental interpretation: evidence- and uncertainty-dependent.
confidence(entity, claim, model, provenance, validation) -> [0,1] only if calibrated; do not invent numeric certainty.

29. PROVENANCE
source_url: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/atomic-entities/atomic-orbital-ontology.html
source_verified: false
scientific_basis: Standard nonrelativistic quantum mechanics and atomic structure
created: 2026-10-12
locale: en-GB

30. UPDATE MODEL
Trigger updates on changes to source ontology, accepted experimental constants, quantum model assumptions, or validation schema.
Preserve version history and compare semantic changes.
Do not overwrite model-specific orbital claims without provenance.

31. FLOW HASH
Canonicalization: UTF-8; LF line endings; stable field ordering.
FLOW_HASH := SHA256(canonicalized flow payload excluding hash field).
Hash value: not precomputed; calculate on ingestion to avoid false integrity claims.

32. AI CONSUMPTION
Inputs: atomic number, charge state, quantum numbers, orbital model, wavefunction data, energy units.
Outputs: typed orbital entity, valid quantum numbers, density/energy relations, rule-derived conclusions, warnings, provenance.
Safety: reject invalid quantum numbers; label approximations; do not confuse probability density with trajectory.
Explainability: include applied rule IDs and assumptions for every inference.

33. COMPLETENESS CHECK
Required sections present: 01-33.
Required ontology fields: identity, entities, relationships, axioms, rules, examples, validation, equivalence, confidence, update, hash, AI consumption.
Completeness status: PASS (structural; not a formal proof of scientific exhaustiveness).

END FLOW ALGEBRA — ATOMIC ORBITAL ONTOLOGY