N₂ electron-impact ²Π_g shape resonance: fixed-R pole search¶

Location: projects/n2_resonance/ (toy model: potential.py, grid_n2.py, pole.py), qscat.ecs.find_resonance_pole (the promoted general matcher), validation/n2/ (model.py, resonance.py, experiment.py — the benchmark harness, Group B). Origin: LCP (local complex potential) model extracted from reference/eMoScat/input/experimental/N2-model.json; see the eMoScat LCP model. Builds on the FEM-DVR-ECS grid (docs/physics/femdvr-ecs.md). Units: atomic units throughout (energy in Hartree, length in Bohr).

Physical picture¶

Electron scattering off N₂ near ~2 eV is dominated by the $^2\Pi_g$ shape resonance — a metastable N₂⁻ state (the well-known “boomerang” resonance) that decays back to N₂ + e⁻. In the local complex potential model, the electron sees a fixed-nuclei (“clamped-R”) effective potential $$V_\mathrm{eff}^\mathrm{el}(r, R) = v_\mathrm{int}(r, R) + \frac{l(l+1)}{2r^2}, \qquad l = 2$$

and the resonance shows up as a complex pole $E(R) = E_\mathrm{res}(R) - i\Gamma(R)/2$ of the associated scattering problem: $E_\mathrm{res}(R)$ is the resonance position (vertical attachment energy) and $\Gamma(R)$ the autodetachment width, both functions of the N₂ bond length $R$.

v0(R), lam(R), v_int(r, R), v_eff_el(r, R) are closed-form (Morse neutral curve, Gaussian well in $r$ scaled by an $R$-dependent strength — see validation/n2/model.py, the validated “single source” these formulas are kept in lockstep with). $E_\mathrm{res}(R)$ and $\Gamma(R)$ are not closed-form: they require diagonalizing the fixed-$R$ electronic Hamiltonian $H_\mathrm{el}(R) = T + \mathrm{diag}\,V_\mathrm{eff}^\mathrm{el}(r, R)$ on an ECS-rotated radial grid and locating the pole in its spectrum.

Method: two-angle ECS pole matching¶

Exterior complex scaling (docs/physics/femdvr-ecs.md) turns the divergent continuum wavefunctions of the unscaled problem into decaying ones by rotating the radial coordinate beyond a pivot $R_0$ into the complex plane at angle $\theta$. A resonance pole becomes (nearly) $\theta$-independent once $\theta$ is large enough to fully expose it, while the discretized continuum eigenvalues keep rotating with $\theta$.

This gives a direct way to find the pole without a theta-stabilization scan: diagonalize the same $H_\mathrm{el}(R)$ on two grids built at two different (but both “large enough”) ECS angles, restrict both spectra to a search window, and find the pair of eigenvalues — one from each spectrum — that agree most closely. That pair is the angle-stable pole; nearby discretized continuum eigenvalues, having rotated by different amounts on the two grids, do not match this closely. Formalized in qscat.ecs.find_resonance_pole(eigs_a, eigs_b, window):

\[\begin{split} \begin{aligned} E_\mathrm{pole} &= 0.5\,(e_a + e_b) &&\text{$e_a$, $e_b$: the closest-matching pair}\\ \mathrm{residual} &= |e_a - e_b| &&\text{small } (\ll \Gamma) \Rightarrow \text{genuine angle-stable pole}\\ E_\mathrm{res} &= \operatorname{Re} E_\mathrm{pole} \\ \Gamma &= \max(0,\, -2\operatorname{Im} E_\mathrm{pole}) \end{aligned} \end{split}\]

projects/n2_resonance/pole.find_pole(R, grid_a, grid_b, window) assembles $H_\mathrm{el}(R)$ on each grid (electronic_hamiltonian, qscat.dvr.hamiltonian + eigen) and calls find_resonance_pole on the resulting spectra; it is the thin, physics-specific caller of the promoted general matcher. validation/n2/resonance.py mirrors the same grid construction and calls find_pole equivalent logic against model.v_eff_el independently, so the harness (Group B) does not import from projects/.

R0 result (validated)¶

At the N₂ equilibrium bond length R0 = 2.01943 Bohr, with grids built at $\theta = 35°$ and $44°$ (grid_n2.n2_electronic_grid, r_pivot=10, n_real=n_complex=8, quadrature=8) and search window $\operatorname{Re} \in [0.04, 0.16]$ Ha, $\operatorname{Im} \in [-0.05, 0]$ Ha:

\[E_\mathrm{res}(R_0) = 2.445\ \text{eV}, \qquad \Gamma(R_0) = 0.455\ \text{eV}, \qquad \text{residual} \sim 3 \times 10^{-6}\ \text{Ha}\]

Both lie inside the literature plausibility bands ($E_\mathrm{res} \in [2.3, 2.5]$ eV, $\Gamma \in [0.35, 0.55]$ eV — Schulz; Berman/Domcke; see validation/n2/reference.py), and the match residual (~1e-6 to 1e-5 Ha) is far smaller than $\Gamma$ (~0.017 Ha), the signature of a genuine angle-stable pole rather than an accidental near-miss between rotating continuum states. The pole is also resolution-stable: a coarser grid (n_real=n_complex=6) reproduces $E_\mathrm{pole}$ to within a few percent (projects/n2_resonance/test_pole.py, V1/V2).

R-scan: `E_res(R)`, `Gamma(R)`, `V_d(R)`¶

projects/n2_resonance/pole.resonance_curve(R_grid, grid_a, grid_b) traces the pole across a range of bond lengths (e.g. R ∈ [1.6, 3.0] Bohr, np.linspace(1.6, 3.0, 15)), returning (E_res, Gamma, V_d) with V_d(R) = v0(R) + E_res(R) — the anion/resonance curve measured from the neutral Morse potential.

A single fixed search window cannot span this whole range: at short R the pole is a genuine complex shape resonance (as at R0), but as R stretches past roughly 2.3–2.4 Bohr, Gamma(R) narrows to zero and the state crosses into a real, angle-independent bound state — the standard dissociative-electron-attachment picture, where the anion curve dips below the neutral one at large R. resonance_curve handles this with a continuation walk: starting from the R_grid index nearest the R0 region (where the fixed seed window Re ∈ [0.04, 0.16] Ha is valid), it walks outward in both directions, each step recentering the search window on a ±0.05 Ha box around the previous step’s matched pole. This tracks the pole smoothly through the resonance-to-bound crossing without mode-hopping onto a neighboring (continuum or other-pole) eigenvalue.

Over R ∈ [1.6, 3.0] Bohr (validated in projects/n2_resonance/test_curve.py, V3):

E_res(R) decreases smoothly and monotonically from ~5.7 eV (R=1.6) to ~-2.9 eV (R=3.0) — a well-behaved, mode-hop-free curve (curvature $|\Delta\Delta E_\mathrm{res}|$ stays a small multiple of the local step throughout).
Gamma(R) decreases monotonically from ~3.2 eV (R=1.6) to exactly 0 for R ≳ 2.4 Bohr, matching the expected resonance-to-bound-state closure.
Gamma(R) >= 0 everywhere by construction (Gamma = max(0, -2*Im(E_pole))).

Model caveat: neutral N₂ Morse is a model potential, not a spectroscopic fit¶

v0(R) (validation/n2/model.py, projects/n2_resonance/potential.py) is eMoScat’s neutral N₂ Morse curve, extracted as-is from reference/eMoScat/input/experimental/N2-model.json. Its dissociation energy D_0 = 0.75102 Ha (=~20.4 eV) is =~2x real N₂’s actual dissociation energy (=~9.8 eV). Consequently the model’s neutral vibrational spacing (omega_e =~ 0.0125 Ha analytic; FEM-DVR eps1-eps0 =~ 0.0124 Ha, see projects/n2_ti_cross_section/test_vibrational.py) is =~16% larger than real N₂’s (0.01074 Ha / 2358 cm⁻¹).

This is a deliberate, accepted property of the model (maintainer decision), not a bug: the FEM-DVR-ECS vibrational solver is verified correct against the analytic Morse spectrum of this same potential (residuals ~1e-14 Ha), so the gap lives entirely in the model’s D_0/alpha_0 choice, not in the numerics. Meanwhile the model’s resonance parameters E_res(R0) and $\Gamma$ at R0 (=~2.44 eV / 0.46 eV, see the “R0 result” section above) DO match real N₂ electron-scattering data — only the neutral vibrational ladder departs from real N₂ spectroscopy. This model-vs-reality gap is folded into the LCP-vs-Houfek-2D differences seen in the time-independent (TI) cross-section benchmark (projects/n2_ti_cross_section/; method, resonance-peak agreement, and the two structural LCP limitations are documented in docs/physics/n2-cross-section.md).

Simplifications / out of scope¶

Diagonal-potential DVR approximation (inherited from qscat.dvr): V is represented by its values on the grid points, not exact quadrature of ⟨i|V|j⟩. Adequate here because v_eff_el is smooth away from r=0 (the centrifugal term diverges there, but the grid does not place a point at the origin).
Fixed, single ECS angle per grid, hand-picked window/angle pair: the two-angle match at (35°, 44°) with the Re ∈ [0.04, 0.16], Im ∈ [-0.05, 0] Ha window is validated at R0 and used as the continuation seed for the R-scan; it is not an automated theta-stabilization scan (which would locate the pole without any prior window guess). Automating that is future work, as noted in docs/physics/femdvr-ecs.md’s limitations.
Fixed reduced-mass / no nuclear dynamics here: this is the electronic (fixed-R) problem only. E_res(R)/$\Gamma(R)$/V_d(R) are inputs to a nuclear-motion (vibrational excitation, dissociative attachment) solve; the time-independent version of that solve is now implemented and validated (docs/physics/n2-cross-section.md, validation/n2/README.md Group C5 — 4/4 gated anchors PASS); the time-dependent (wavepacket propagation) version remains out of scope / PENDING (Group D).
Continuation window widths are hand-tuned (re_half_width = im_half_width = 0.05 Ha in resonance_curve), chosen empirically to track the pole across the observed drift rate without catching a neighboring branch (branches sit ~0.02–0.05 Ha apart on this grid); a different R_grid spacing or a much wider R range could require re-tuning.

Validation¶

projects/n2_resonance/test_pole.py:
- V1 — E_res(R0) ∈ [2.3, 2.5] eV, Gamma(R0) ∈ [0.35, 0.55] eV.
- V2 — match residual < 1e-3 Ha (angle-stability); resolution stability (coarser grid agrees to <5%).
projects/n2_resonance/test_curve.py:
- V3 — Gamma(R) >= 0 everywhere; E_res(R) monotonically non-increasing (no mode-hop) across the scan; curvature of E_res(R) bounded by a small multiple of the local step; Gamma(R) monotonically non-increasing to 0.
libs/qscat/tests/test_find_resonance_pole.py: the promoted matcher on synthetic spectra (a shared “pole” eigenvalue plus differing “continuum” eigenvalues; empty-window ValueError).
validation/n2/experiment.py Group B (B1): E_res(R0) computed independently via validation/n2/resonance.py against the literature window — PASS.
The nuclear-motion time-independent cross-section solve built on top of E_res(R)/$\Gamma(R)$ (Group C5) is documented separately in docs/physics/n2-cross-section.md.