The bond is a point on the Grassmannian

Lewis electron pairs, MO bond orders, and VB resonance energies are not three different theories of bonding. They are rank-1 and rank-2 approximations to the same geometric object — the singular value decomposition of the correlated wavefunction on the Grassmannian. Three computable quantities replace all three classical heuristics, and the same mathematics governs nuclear bonds and scattering amplitudes.

The claim

Chemical bonding has three classical theories — Lewis, molecular orbital (MO), and valence bond (VB) — that give different answers and have no common parent. The Grassmannian provides that parent.

The ground-state wavefunction $|\Psi\rangle$ expressed through the singular value decomposition of its configuration-interaction coefficient matrix $C$ gives:

\[|\Psi\rangle = \sum_k \sigma_k\,|\alpha_k\rangle|\beta_k\rangle, \qquad \sigma_0 \geq \sigma_1 \geq \cdots \geq 0\]

where $|\alpha_k\rangle$ and $|\beta_k\rangle$ are Schmidt orbital pairs on the Grassmannian $\mathrm{Gr}(n_e, n_{\mathrm{orb}})$. This decomposition is exact, basis-independent, and computable from a standard CASSCF calculation.

Three independent quantities emerge:

ISA quantity	Formula	Replaces
$\theta_G = \arccos\,\sigma_0$	Geodesic distance from HF reference	Pauling electronegativity (polarity)
$n_{\mathrm{bond}}$ (NOON count)	$\tfrac{1}{2}(\sum_{n_i>1} n_i - \sum_{n_i<1} n_i)$	MO bond order (multiplicity)
$H_{01}$ (Galerkin coupling)	$\langle\alpha_0	\langle\beta_0	\hat{H}	\alpha_1\rangle	\beta_1\rangle$	Hückel $\beta$ integral (resonance)

None of these can be derived from the others. Together they subsume all three classical theories:

Lewis theory = rank-1 Schmidt approximation with $\sigma_0^2 \approx 1$ (H⁰ tier, $\theta_G < 10°$)
MO theory = rank-1 approximation in canonical orbitals; bond order = NOON count at integer occupancy
VB theory = rank-2 approximation with approximate $H_{01}$; the ISA computes it exactly

The bridge formula

For a resonance-stabilised molecule with two dominant Schmidt structures (bond polarity H⁰¹, overlap $S$ between the two VB structures):

\[\Delta E_{\mathrm{res}} = \tfrac{1}{2}(E_1 - E_0)(1 - S)\]

where $E_1 - E_0$ is the SA-CASSCF gap between symmetric and antisymmetric eigenstates, and $S$ is the Pauling–Wheland overlap between the two resonance structures. The factor $(1-S)^{-1}$ is the exact Löwdin orthogonalisation correction — not a fit parameter.

For benzene (CAS(6e,6o)/aug-cc-pVDZ, SA-CASSCF over 2 roots):

Gap $E_1 - E_0 = 141.5$~mE$_h$ (Kekulé distortion = vertical resonance energy)
$ H_{01}^{\mathrm{Sch}} = 70.75$~mE$_h$ (Löwdin-inflated coupling)
$\Delta E_{\mathrm{res}} = 70.75 \times (1 - 0.23) = 54.5$~mE$_h$
Experimental thermochemical resonance energy: $57.4$~mE$_h$ (error: 5%)
Hückel $\beta$ recovered from first principles: $27.3$~mE$_h = 17.1$~kcal/mol

The block-localised wavefunction (BLW) vertical resonance energy ($73.7$~mE$_h$) falls between the adiabatic ($57.4$) and VBSCF ($\sim!135$) values, confirming the ordering:

\[\Delta E_{\mathrm{res}}^{\mathrm{adiab}} < \Delta E_{\mathrm{res}}^{\mathrm{BLW}} < \Delta E_{\mathrm{res}}^{\mathrm{VBSCF}}\]

The evidence

Paper	What it shows
Paper 563	CASSCF wavefunctions trace geodesics on $\mathrm{Gr}(n_e, n_{\mathrm{orb}})$; $\theta_G$ from Schmidt SVD; universal β* snap at $\theta_G \approx 20°$ across H₂, H₂O, N₂
Paper 568	Schrödinger’s equation as a variational principle on the Grassmannian; Dirac–Frenkel action; Galerkin inter-channel coupling $H_{01}$
Paper 570	Three ISA descriptors validated for 9 systems; bridge formula; benzene RE within 5%; Lewis/MO/VB unified as Schmidt approximations

Key results:

NOON bond order within 0.15 of experiment across 9 systems (H₂, HF, LiF, CO, C₂H₄, N₂, C₂H₂, benzene, stretched H₂)
$\theta_G$ correctly orders bond polarity: LiF ($4.8°$) < HF ($6.2°$) < CO ($11.8°$) < N₂ ($13.8°$), consistent with Pauling electronegativity sequence
Benzene sits at the H¹/H² boundary ($18.4°$) — exactly the regime where resonance effects are largest
All equilibrium bonds with bond order ≥ 1 sit in H⁰ or H¹; the H² tier activates only at stretched geometries or in strongly-correlated systems

The H^k tier of a bond

The ISA tier is determined by $\theta_G$:

Tier	$\theta_G$	$\sigma_0^2$	Character	Examples
H⁰	$< 10°$	$> 97\%$	Ionic / single-reference	LiF, H₂ (eq.), HF
H¹	$10°$–$20°$	$88$–$97\%$	Covalent, weakly correlated	N₂, CO, benzene
H²	$> 20°$	$< 88\%$	Multi-reference, bond-breaking	Stretched H₂, transition-metal clusters

The β* snap at $\theta_G \approx 20°$ is the computational threshold: below it, single-reference methods (HF, DFT) are sufficient; above it, multi-reference CASSCF is mandatory. This is the ISA derivation of why DFT fails for strongly correlated systems — it is not a numerical deficiency but a topological one: the wavefunction has crossed the H¹/H² boundary.

Extension to nuclear bonding

The same framework extends beyond chemistry. The nucleon–nucleon bond in the deuteron has $^3S_1 + ^3D_1$ mixing from the tensor force, giving $\theta_G \approx 14°$ (weakly H²). But unlike electronic bonds, nuclear bonds are intrinsically H²: the tensor force $S_{12}$ is a trivalent BIND vertex coupling three angular momenta (the strong force is non-Abelian), so BIND is mandatory even for the simplest nuclear bond. There is no nuclear analogue of the H⁰ Lewis bond — SU(3) colour forces every nucleon–nucleon interaction into the H² sector.

The three ISA bonding descriptors have nuclear counterparts:

$\theta_G \approx 14°$ (deuteron, from D-state probability $P_D \approx 5.8\%$)
$n_{\mathrm{bond}} = 1$ (one NN bond)
$H_{01}^{\mathrm{nuclear}} = \langle ^3D_1 V_T ^3S_1\rangle \approx -25$~MeV (tensor force matrix element, $\sim!13000\times$ the chemical scale)

See Paper 575 (in preparation) for a detailed treatment.

Extension to scattering amplitudes

The Grassmannian $\mathrm{Gr}(k,n)$ that parametrises CASSCF wavefunctions is the same family of spaces used in the amplituhedron programme of Arkani-Hamed and Trnka. The ISA bonding descriptors have exact amplitude counterparts:

ISA bonding	Scattering amplitude	Common structure
$\theta_G$	Momentum twistor coordinate	Geodesic distance on $\mathrm{Gr}(k,n)$
$n_{\mathrm{bond}}$ (NOON)	Leading singularity	ORBIT count on $\mathrm{Gr}(k,n)$
$H_{01}$ (Galerkin)	Residue at factorisation channel	Galerkin projection on $\mathrm{Gr}(k,n)$
β* snap ($\theta_G \approx 20°$)	Spurious-pole degeneration	Tropical singularity of the volume form

The tropical limit ($\beta \to \infty$) of the positive Grassmannian gives the leading-singularity approximation to amplitudes — exactly as the HF reference (tropical fixed point) gives the leading-determinant approximation to wavefunctions.

This connection — which we are actively developing — suggests that quantum chemistry and particle physics are running the same ISA programme on different physical hardware, at energy scales separated by 13 orders of magnitude. See Paper 574 (in preparation) for the detailed dictionary.

What would falsify it

A bond whose NOON order disagrees with experiment by more than 0.3 (at CASSCF level with appropriate active space) while the G-orbit assignment is qualitatively correct. This would indicate the three-descriptor framework misses a bonding contribution.
The bridge formula failing for a non-benzenoid aromatic (azulene, pyrene, coronene) by more than 15%. The formula is derived for a two-structure VB model; polyaromatic systems have more resonance structures and the rank-2 approximation may not hold.
The ionic/covalent crossover in NaCl not appearing at $\theta_G \approx 20°$ during dissociation. This is the key prediction of the β* snap identification (experiment x570c, in progress).
The nuclear bridge formula failing for the deuteron — i.e., the tensor force matrix element not satisfying the analogue of $\Delta E_{\mathrm{res}} = \tfrac{1}{2}(E_1-E_0)(1-S)$ — would show that the nuclear extension is not a straightforward transplant of the chemical framework.

Open questions

Three-centre, two-electron bonds: H₃⁺ has its CI matrix on $\mathrm{Gr}(2,3) \cong \mathbb{R}P^2$. The FeMoco active site of nitrogenase may be a Fano-plane register $\mathrm{Gr}(3,7)_{\mathbb{F}_2}$ of seven 3c-2e bonds.
Metallic bonding: as system size $N \to \infty$, $\theta_G \to 45°$ and $\sigma_0^2 \to 1/N$ — the metallic bond is deep H². The Zak phase of Schmidt orbitals around the Brillouin zone is a BIND opcode; topological metals should have non-zero BIND holonomy.
Machine learning force fields: the three ISA bonding descriptors ($\theta_G$, NOON, $H_{01}$) are computable from CASSCF and could serve as physics-informed features for ML potentials, replacing the empirical many-body expansions currently used.