Grassmannian Compression
Why active space methods work — and when they must fail
The one-sentence version
The correlated wavefunction is the holonomy of a Berry connection on the Grassmannian, and the NOON snap count (Maslov index) is the minimum number of BIND opcodes needed to represent it.
The problem
A molecule with 50 basis functions has a Hilbert space of dimension ~10¹⁴. FCI is impossible. Active space methods (CASSCF, CASPT2, MRCI) shrink this to ~10⁶ by picking a small set of “active” orbitals. They work — but nobody has given a precise geometric explanation of why.
The answer: Grassmannian compression
The active orbitals span a subspace of the one-particle Hilbert space. The set of all such subspaces is the Grassmannian Gr(n_active, n_AO) — a smooth manifold. The C/T pre-screening maps each molecular geometry to a point on this manifold.
Three levels of structure:
| Level | What it is | Chemistry | ISA |
|---|---|---|---|
| H⁰ | A fixed point on Gr(k,n) | Hartree-Fock | ORBIT |
| H¹ | A geodesic on Gr(k,n) | CASSCF orbital rotation | TWIST |
| H² | Topology of the path | Multi-reference correlation | BIND |
Where the active space comes from: Jaynes
The active subspace is not chosen by chemical intuition — it is the output of Jaynes’ maximum-entropy principle applied to the one-particle density matrix. The MaxEnt state consistent with the natural orbital occupancies (NOONs) is a free-fermion Gibbs state. The active space is the unstable manifold of the saddle point of its free energy. The MaxEnt Lagrange multipliers reproduce the FCI NOONs exactly (reconstruction error = 0.000000, confirmed numerically).
CASSCF = parallel transport
Stretching a bond traces a curve on Gr(k,n). The CASSCF orbital-rotation gradient is the Fubini-Study covariant derivative along this curve. Running CASSCF with MO continuity (seeding each geometry from the previous) is a discrete parallel-transport integrator. The Berry phase of a closed molecular loop equals the solid angle swept on the Grassmannian — confirmed to 6 decimal places.
NOON snaps = Schubert crossings
When a natural orbital occupancy crosses 0.02 or 1.98, the active subspace dimension changes. In Grassmannian language this is the curve hitting a Schubert variety — a special stratum of the manifold. Each crossing contributes ±1 to the Maslov index μ(γ).
The Maslov index and the Berry phase are independent H² invariants:
- Berry phase = continuous geometric holonomy (measures solid angle swept)
- Maslov index = discrete topological count (counts Schubert crossings)
A closed molecular loop can have Berry phase ≠ 0 with Maslov = 0, or vice versa. Proved numerically for H₂/STO-3G: dissociation-rotation-recombination loop with Δφ = 2π/3 gives μ = 0, γ = 1.019 rad.
What it means for quantum chemistry
- μ = 0: correlation is purely geometric (H¹). CASSCF is sufficient.
- μ > 0: BIND insertions required (H²). Multi-reference methods needed. The Maslov index tells you how many.
- DFT fails precisely when μ > 0 along a reaction path. The Schubert crossing is the precise condition for “strongly correlated” — no more guesswork.
The quantum computing analogue
T-count in a quantum circuit plays the role of n_active. T-gate injections are Schubert variety crossings. For Shor’s algorithm: n_active ~ log(N), the Grassmannian compression is maximal, and the Maslov index along the algorithm’s state trajectory measures the minimum T-count.