H₂⁺ — the first ionic target

H₂⁺ is the only ion in this repository. Every molecule before it — N₂, NO, F₂ — is a neutral electron–diatomic scattering problem; H₂⁺ is e⁻ + H₂⁺(v) → H + H, dissociative recombination (DR), and the scattering electron sees the ionic core’s +1 charge. qscat.model.H2P is verified to be an IonicResonanceModel with charge = -1 (the neutral models — N2, NO, F2 — are all DiatomicResonanceModel with charge = 0). That single sign changes three things at once: the incident wave becomes a long-range Coulomb wave rather than a free one, the electronic grid has to reach ~1300 bohr to hold it, and instead of one dissociation channel there is a whole Rydberg series of exit channels — the neutral H₂’s bound electronic states, accumulating at the continuum edge, cut off at a measurable number (N_CHANNELS = 3 here). It is also, for exactly that grid reason, the first model in this repository that does not run on a laptop.

The model — form and parameters

qscat.model.H2P (IonicResonanceModel), values printed directly from the model object:

parameter	value
mu (nuclear reduced mass)	918.076
ell (fixed partial wave)	1
charge	−1
V0 (ion Morse well depth)	0.1027
R0 (ion Morse equilibrium bond length, bohr)	2.0
alpha (ion Morse range)	0.69
a1 (σ-capture amplitude)	1.6435
a2	6.2
a3	0.0125
a4	1.15
max_nuclear_ecs_angle_deg	22.5

Reproduce with:

from qscat.model import H2P
print({k: v for k, v in vars(H2P).items() if not k.startswith("_")})

The ion Morse v0(R) folds in the 1/R proton–proton repulsion rather than carrying it as a separate term; the electronic surface is v0(R) + v_int(r,R) + ℓ(ℓ+1)/(2r²) + charge/r, where charge/r = −1/r is the ionic electron–core Coulomb attraction. mu = 918.076 is m_p/2 for the modern proton mass, given by three independent published sources (Hvizdoš 2016 Table 1.1; Váňa 2017 Table 1.2; Hvizdoš et al., Phys. Rev. A 97, 022704 (2018) §II A) — eMoScat’s own JSON deck instead carries 918.25, which those three sources contradict; the port originally inherited eMoScat’s value and it was corrected on 2026-08-15.

The first non-laptop model

The Coulomb tail’s long range forces a large electronic grid — the full deck reaches 1300 bohr electronically and 14 bohr nuclearly, giving ~1.15 million unknowns (1,150,108 measured, 19,507,356 nonzeros). That is Docker/MUMPS territory, not laptop-runnable: a measured full-deck energy point costs 17.4 s on a 32-core, 123 GB x86-64 host under MUMPS (peak RSS 4.34 GB), so a σ(E) sweep costs roughly an hour per 200 energies — an overnight job, not a multi-day one. A smaller config.proxy_grid() (electronic → 60 bohr) gives a laptop-feasible well-posedness/threshold gate, not a converged cross section. No independent golden σ_DR data ships with this repository (eMoScat’s own output file is absent from the snapshot), so — as with NO/F₂’s dissociative attachment — the exact solver is the oracle here too.

The DR1 (n=0) channel peaks at E ≈ 6.31×10⁻³ Ha, σ ≈ 1.54×10⁻³ bohr² above a ~10⁻¹⁰ background; DR2 (n=1) is ~10⁻⁶; DR3 (n=2) is closed in this window (threshold ≈ 0.0426 Ha).

Dissociative recombination, exactly

The Coulomb-generalized driven Lippmann–Schwinger solve, looped over the Rydberg exit series. Checked against a prior external reference sweep: the published ω_i^j level table (53 levels) agrees to ≤4e-6 Ha, but σ_DR itself carries two open discrepancies — a ~30% systematic deficit in the DR1 channel, and a runaway near a vibrational threshold (7.6×/3.7× too high, DR0/DR1) that grows to 100–700× too large right at a Rydberg channel’s opening.

H₂⁺ dissociative recombination (the first ionic model)

Resonance states, verified

Poles of the full 2-D S-matrix, checked against a Born–Oppenheimer basis by overlap rather than trusted on angle stability alone. Detail below.

Exact resonance states of H₂⁺, against the Born–Oppenheimer picture

Angle stability is necessary — and not sufficient

qscat.core.exact_resonance_states finds poles by two-angle ECS stability: an eigenvalue that does not move when either the electronic or the nuclear ECS rotation angle does. On H₂⁺’s three published DR energy windows that test found 57 angle-stable states. Overlapping each one against a Born–Oppenheimer basis (qscat.core.bo/assignment, the c-product overlap) turned up two failure modes angle stability alone cannot see:

Four of the 57 are not resonances at all. They pass angle stability — the eigenvalue genuinely does not move as the contour rotates — but score overlaps of 6×10⁻⁴ to 7×10⁻³ against every state in the Born–Oppenheimer basis, where the genuine resonances score 0.87–0.99. They are rotated-continuum eigenvalues that happen to sit in a stable corner of the spectrum, not physical states. (At E = 0.004479 Ha the best overlap is 0.0006 — three orders of magnitude below even the low end of the genuine-state range.)

A further 18 of the 57 are box-limited, which the overlap is structurally blind to. The c-product cancels the rotated ECS tail by construction — that is what makes it the correct pairing rule — so a state with 97% of its probability outside the unscaled region still overlaps at 0.99 with the Born–Oppenheimer product it genuinely is. real_weight (the fraction of |Ψ|² inside the unscaled region) is the separate check the overlap cannot make: across one window it collapses from 0.682 at Ry₁₁ to 0.010 at Ry₁₆, because those high-n Rydberg orbitals (⟨r⟩ ~ n²) are simply larger than the 300-bohr box. 18 of 57 fall below a 0.5 real_weight threshold — reported, not deleted, since the identification still stands; nothing about their shift or width is quotable until the box grows.

Fig. 1 The exact 2-D pole at E_tot = −0.093680 Ha (E ≈ 0.0039 Ha), Γ = 5.6×10⁻⁷ Ha.

Fig. 2 The Born–Oppenheimer product φ_Ry4(r;R)·χ_v=2(R) at E_BO = −0.093564 Ha — the state the approximation asserts the one above is. They share a core (five radial lobes, two nodes in R, the same turning surface); the exact state carries extra amplitude near r ≈ 130–160 bohr the product has nowhere to put.

Fig. 3 ω₁⁹ (Ry₉, v=1): diffuse, ~9 radial lobes reaching past 250 bohr. Overlap 0.970 — nearly pure, and it barely moves against the Born-Oppenheimer level.

Fig. 4 ω₃³ (Ry₃, v=3): compact, confined inside ~70 bohr. Overlap 0.783 — strongly mixed, and it carries almost the whole splitting of a near-degeneracy with the pair above.

Once the four non-resonances and the 18 box-limited states are set aside, the BO error itself sorts by regime, not by level index — an eightfold split between diffuse high-n Rydberg states (median 0.457 meV) and compact low-n ones (median 3.702 meV), both larger than N₂’s 0.22 meV (Exact (non-Born-Oppenheimer) resonance states of the 2-D model). The Born-Oppenheimer levels and the exact poles, marked on the DR sweep that measures both:

What is established, and what is not

Quoted directly from the resonance-states note:

claim	status
exact poles land on the published peaks, 0.2–0.3 widths, all three windows	measured against data the poles were not fitted to
4 of 57 angle-stable poles are not resonances	overlaps 6e-4…7e-3 against a basis proven complete at those energies
the BO shift splits by regime, 0.457 vs 3.702 meV median	24 quotable pairs
18 of 57 poles are box-limited (real_weight < 0.5)	the high-n Rydberg orbitals are larger than the 300-bohr box; nothing about them is quotable
pole POSITIONS are box-converged	r_max 300 → 600 moved them 3e-9 or less
the pole COUNT is not converged	18→22, 13→10, 14→13 across the three windows — not even monotone
per-level shifts for the 6 basis-limited poles	not quoted — their partner is outside the basis
σ_DR near a threshold	wrong, 100–700× too large

Both level/pole figures above are computed at μ = 918.25 (the value the σ_DR reference sweep itself used), not the model’s shipped 918.076 — a deliberate choice so the marks and the curve are on the same footing; the ~2×10⁻⁶ Ha the mass correction is worth would otherwise read as ~10% of a resonance width.

Where to read more

The Coulomb channel, the discretisation, and the full DR cross section including its two open discrepancies against an external reference sweep: H₂⁺ dissociative recombination (the first ionic model). The exact resonance-state verification — angle stability, the overlap test, and the box-limited states: Exact resonance states of H₂⁺, against the Born–Oppenheimer picture. The general two-angle method this is built on, and its first (N₂, neutral) application: Exact (non-Born-Oppenheimer) resonance states of the 2-D model, N₂ — the benchmark target. The Born–Oppenheimer resonance levels these exact poles are compared against: BO/LCP resonance levels — quasi-bound vibrational states of the anion.