Angular (θ) extension: coupled partial-wave channels — research direction¶

Status: not implemented. This is a captured research direction, recorded 2026-07-23 for later work. No code, spec, or plan exists yet; the brainstorming around the physics and the model construction is still to be done. Relates to: docs/physics/n2-resonance.md, docs/physics/n2-cross-section.md, docs/physics/nd-tensor-hamiltonian.md. Units: atomic units throughout.

The question to answer¶

Every N₂ model in this repo so far fixes a single electronic partial wave: the ²Π_g shape resonance is represented by one l (the d-wave), and the electron-molecule interaction -lambda(R) * exp(-alpha_c * r^2) is isotropic in the electron’s angular coordinate. The direction recorded here adds theta (the electron’s polar angle relative to the molecular axis) by expanding in spherical harmonics and solving the resulting coupled-channel problem.

The purpose is not a better N₂ number. It is the same purpose as the rest of the program: validate an approximation by computing the thing it approximates. Concretely —

Treat the single-partial-wave model as an approximation of the multi-wave model, and measure how the fixed-l decomposition into a resonant model affects the observable.

That is a well-posed validation question of exactly the same shape as “LCP vs. the exact 2-D solution”: one model is the oracle, the other is under test, and the difference between them is the result. It is currently unanswerable, because with one channel there is nothing to compare against.

Why the decomposition is sound¶

For a linear molecule in the body-fixed frame, the projection Lambda of the electronic angular momentum on the molecular axis is exactly conserved (D_inf_h symmetry). Restricting to a single Lambda block therefore costs nothing — it is a symmetry, not an approximation. Within that block, expanding the interaction in Legendre components and projecting onto spherical harmonics gives

\[\begin{split} \begin{aligned} V(r, \theta_e, R) &= \sum_\lambda v_\lambda(r, R)\, P_\lambda(\cos\theta_e) \\ V_{l l'}(r, R) &= \sum_\lambda \langle Y_{l\Lambda} \vert P_\lambda \vert Y_{l'\Lambda} \rangle\, v_\lambda(r, R) \end{aligned} \end{split}\]

which is exact until the sum over \(l\) is truncated. For the ²Π_g resonance, \(\Lambda = 1\) and \(l = 2, 4, 6, \dots\). The truncation order \(N_l\) then becomes an ordinary convergence parameter, studied exactly like box size \(R_\mathrm{max}\), the ECS radius \(R_0\), and the ECS angle \(\theta\) already are — which is the standard this repo already holds itself to.

The truncation criterion — the one thing to get right¶

The tempting rule is “keep the partial waves that host a resonance, discard the rest.” Do not use that rule. The non-resonant waves carry the background scattering, and it is the interference between the resonance and that background which shapes the profile (the Fano-like asymmetry). Discarding them would reintroduce precisely the deficiency the exact-2-D work exists to expose — a missing elastic background is one of the two documented NOTEs against LCP.

The criterion must be observable convergence: keep adding \(l\) until \(\sigma_{0\to v'}(E)\) stops moving by more than tolerance. The two criteria give different answers, and the resonance-hosting one quietly bakes in an approximation of the same character as the one under test, which would make the result circular.

What it would buy¶

A single-l model cannot, by construction, test whether the single-discrete-state Feshbach assumption underlying both LCP and the nonlocal resonance model survives when the resonance can leak across several partial waves. Multi-channel leakage is a genuinely new failure mode — not a more accurate number — and it is invisible to every model currently in the repo.

Consequences to plan for¶

It breaks a library assumption. qscat.dvr.potential_nd / hamiltonian_nd assume a diagonal potential; a coupled V_{l l'} is exactly what is not diagonal. qscat.linalg.kron_sum is unaffected (the channel index enters as a dimension carrying a zero kinetic operator), so the shape is

\[H = \mathbb{1}_l \otimes (T_r \oplus T_R) + \sum_{l,l'} E_{l l'} \otimes \mathrm{diag}\,V_{l l'}(r, R)\]

i.e. a block/channel-coupled assembler alongside the diagonal one — still sparse, still Kronecker-structured. An addition, not a rewrite, but one worth designing deliberately rather than bolting on.
Cost scales as N_l^2 in nonzeros. Dimension grows by N_l, nnz by roughly N_l^2. At N ~ 1e5 with four partial waves that is 4e5 unknowns and ~16x the nonzeros. Sparse-LU fill-in is where this bites, which makes the MMD_AT_PLUS_A ordering result load-bearing and may force iterative/preconditioned solvers from optional to necessary.
It adds channels, not dimensions. This remains a 2-D continuum problem (r, R) with N_l channels. It is complementary to, not the same as, the higher-dimensional direction (a second nuclear coordinate — a bend, or a triatomic stretch — giving a true (r, R_1, R_2) problem). Only the latter exercises the D-general library layer.
It costs the external oracle. The current interaction is isotropic and generates no coupling at all, so the anisotropy would have to be designed (at minimum a lambda = 0 and a lambda = 2 component). That is legitimate — the model is chosen, not derived — but Houfek’s independent CSVE.V00.J00 data would no longer certify the solver. The cheap replacement check: switching the anisotropy off must reproduce the current single-channel model exactly.

Lead worth scouting first¶

reference/eMoScat contains source/Model2d/CoupledModel2d.cpp, include/Model2d/CoupledModel2d.h, and input/coupled/ decks — coupled-model machinery that exists but was apparently never exercised for N₂. Run port-scout over it before designing the coupling structure; it may already encode the intended v_lambda form and channel bookkeeping.

Open questions for the brainstorming session¶

What functional form for v_lambda(r, R)? How is the anisotropy strength chosen, and against what is it calibrated?
How many lambda terms, and does the lambda truncation converge independently of the l truncation?
Does the resonance pole stay a single pole under coupling, or split? (The two-spectrum matcher qscat.ecs.find_resonance_pole assumes one.)
Is the comparison run at fixed model parameters, or is the single-wave model re-fitted to best match the multi-wave one? (The second is the fairer test of “how good can the approximation be made,” the first of “how good is it as used.”)
Does the TD route (docs/physics/n2-td-cross-section.md) extend unchanged? Crank-Nicolson does not care that H is block-structured, so it probably does — which would give a second independent numerical route for free.