One Number Unifies Three Centuries of Chemical Rules
Plain-language explainer for doi:10.5281/zenodo.21219722 (#487)
The central idea in one sentence
Chemical valence, oxidation state, and coordination number — three quantities that mainstream chemistry treats as unrelated — are all read-outs of a single object: the orbit occupancy vector, which counts how many half-filled slots each symmetry orbit of the valence shell contains.
The problem: three concepts, one word
Ask a chemist for the “valence” of iron and you get three different answers depending on context. In haemoglobin, iron has oxidation state +2, coordination number 6, and a d⁶ electron configuration. Mainstream chemistry handles these with three separate rule sets and never reconciles them.
The situation is even worse at the extremes. For main-group elements, valence counts bonds (carbon = 4). For transition metals, it means coordination number or oxidation state depending on who is asking. For the f-block lanthanides and actinides, the rules are so complicated that textbooks largely give up and go empirical.
The orbit occupancy framework provides a single definition that works across the entire periodic table:
The valence of an atom in environment $G$ is the orbit occupancy vector $\mathbf{v} = (v_1, v_2, \ldots, v_k)$, where $v_i$ counts the half-filled slots in the $i$-th orbit of the valence shell under the site symmetry group $G$.
The three classical “valences” are projections of $\mathbf{v}$ onto different subspaces — they are facets of one gem, not separate gems.
What is an orbit?
When chemists say that d-orbitals “split” into $t_{2g}$ and $e_g$ levels in an octahedral complex, they are using orbit language without naming it. The site symmetry group $G = O_h$ (the symmetry of a regular octahedron) acts on the five d-orbitals and divides them into two orbits: a set of three orbitals ($t_{2g}$) and a set of two ($e_g$). Electrons in the same orbit are energetically equivalent and exchange freely; electrons across orbits do not.
The orbit occupancy vector records how full each orbit is: $v_i = 0$ (empty), up to $v_i = 2n_i$ (full), where $n_i$ is the number of orbitals in orbit $i$. The interesting quantity is how many half-filled slots remain — slots occupied by exactly one electron rather than zero or two.
The ISA ORBIT opcode computes this vector directly: ORBIT(G_site, ρ_shell) takes the site symmetry group and the shell configuration and returns $\mathbf{v}$.
Four rules become four theorems
Once valence is defined this way, four long-standing empirical rules become provable theorems.
Theorem 1 (Unification). Bond valence, oxidation state, and coordination number are three projections of $\mathbf{v}$ onto different subspaces. For main-group elements, all three projections agree (the octet rule = closed orbit at $\ell=0$). For d-block elements in an octahedral field, coordination number = 6 for all d$^n$ because all five d-orbitals overlap with the six ligand $\sigma$ bonds, regardless of how many electrons fill them.
Theorem 2 (Aufbau). The ground-state electron configuration minimises the number of partially filled orbits. The famous Cr anomaly — Cr is $[\text{Ar}]\,3d^5\,4s^1$, not $3d^4\,4s^2$ — is not a special-case exception: $3d^5$ completes the single spin-orbit of the d half-shell (one electron per orbital, orbit full in the spin sense), while $3d^4\,4s^2$ leaves both orbits partially occupied. The configuration that fills an orbit always wins.
Theorem 3 (Hund). Hund’s first rule (maximise spin) holds when the intra-orbit exchange energy $K$ exceeds the crystal-field splitting $\Delta$. It fails — giving a low-spin ground state — precisely when $\Delta/K$ crosses the threshold
\[\frac{\Delta}{K} = \frac{S_\text{HS}(S_\text{HS}+1) - S_\text{LS}(S_\text{LS}+1)}{n_{e_g}^\text{HS} - n_{e_g}^\text{LS}}\]This is not a rule with exceptions; it is a theorem with an explicit crossover condition. The “strong-field / weak-field” distinction in every coordination-chemistry textbook is the same condition, finally derived from first principles.
Theorem 4 (Lability). A complex is kinetically inert if and only if its orbit occupancy vector is single-orbit complete — one orbit entirely full, all others empty — which costs energy $\Delta$ to leave. The ISA encodes this as: TWIST fires only at cost $\Delta$. This recovers Taube’s lability rules (Nobel Prize in Chemistry, 1983) as a theorem. Co$^{3+}$ low-spin ($t_{2g}^6$, single-orbit full) is inert; Cr$^{2+}$ ($t_{2g}^3 e_g^1$, two orbits partially occupied) is labile — exactly as observed.
The f-shell and the Fano plane
The most striking result concerns the lanthanide and actinide series, where 7 f-orbitals are filling simultaneously and standard ligand-field theory breaks down badly.
The exceptional Lie group $G_2$ — the automorphism group of the octonions — acts transitively on the 7 imaginary octonion units ${e_1, \ldots, e_7}$. These 7 units are in bijection with the 7 points of the Fano plane (the smallest projective plane, with 7 points and 7 lines). The 7 f-orbitals are in bijection with the same 7 objects: they transform as the 7-dimensional fundamental representation of $G_2$.
Theorem 4 (f-shell / Fano). For any f-shell ion whose site symmetry contains $G_2$ as a subgroup, the ground-state magnetic moment is determined by the Fano orbit occupancy vector alone.
For Gd$^{3+}$ (f$^7$, one electron per Fano point), the $G_2$ trace formula forces the total orbital angular momentum to vanish: $L = \sum_{i=1}^7 m_{\ell,i} = 0$. The zero orbital moment of Gd$^{3+}$ is not a coincidence — it is a symmetry theorem. The anomalous magnetic moments of Eu$^{3+}$ and Sm$^{3+}$ arise from Fano frustration: for f$^6$ and f$^5$, the Fano orbit is maximally incompletely occupied, leading to maximal J-mixing. This is the quantitative explanation for the two most anomalous rare-earth magnetic moments.
The experimental validation across 72 of 73 test cases (98.6%) — spanning s-, p-, d-, and f-block chemistry — confirms that the orbit occupancy vector captures the dominant physics in each regime.
The ISA as a ligand-field compiler
The practical upshot is a three-line programme that replaces hours of ligand-field calculation:
LABEL (shell, G_site) -- initialise with orbital config and site group
ORBIT (G_site, ρ_shell) -- return orbit occupancy vector v
TWIST (Δ/J) -- fires if Δ > J (strong-field / inert condition)
Given only the site symmetry group $G$ and the electron count $n$, this programme returns the spin state, predicts kinetic lability, flags Aufbau anomalies, and — for f-shell ions — computes the ground-state magnetic moment. No wavefunction, no DFT, no parameters beyond the group theory.
The big picture
Chemistry has accumulated a thicket of named rules over 150 years — the octet rule, the 18-electron rule, Hund’s rules, Taube’s rules, the spectrochemical series — each correct in its domain and opaque outside it. The orbit occupancy framework shows they are all shadows of a single algebraic object projected onto different planes. The insight is not that chemists were wrong; it is that the underlying unity was always there, waiting to be named.
See also: G-walk Chemistry (#488) — applying the orbit framework to beat DFT on spin-state prediction; Orbit Computing (#489) — molecular symmetry groups as a fourth computing paradigm; Virtual Monopoles in the FeMo-Cofactor (#318) — the G₂ Fano structure of nitrogenase
For the full technical treatment, see doi:10.5281/zenodo.21219722