Enter atom coordinates (Å) in the molecule-fixed frame. The inertia tensor is diagonalised to yield principal moments and rotational constants A, B, C. These seed the spectroscopic constant table. Place the C₂ axis along z, molecule in the xz-plane.
| SYM | X (Å) | Y (Å) | Z (Å) | MASS |
|---|
Select the molecular PI group G. The character table induces matrix representations M(g) for vibrational ladder operators {a†, a} and rotational components {Jx, Jy, Jz}. Each normal mode is assigned its symmetry species Γᵢ. Time-reversal symmetry (T J T⁻¹ = −J) forbids odd-rank J operators.
Derived from group vibrational representation.
Φ_vib: (1/|G|)Σ_g 1/det(I−t·D_vib(g)) counts A₁ vibrational monomials. Φ_rot: 1/(1−t²)³ — only even degrees (time-reversal). Hadamard Φ_rovib = Φ_vib ⊙ Φ_rot gives cₙ slots per order n. λ-Scaling: Cost(a,a†)=1, Cost(J)=2 defines expansion depth O(λ⁶).
⊘ Odd-degree terms are zero (time-reversal forbidden)
A raw pool of all products of {a, a†, Jx, Jy, Jz} up to O(λ⁶) is generated. The PI group projection operator P̂ = (1/|G|)Σ_{g∈G} χ_{A₁}(g) R̂_g is applied — monomials projecting to zero are discarded. Surviving operators are normal-ordered (a† left of a) using [a,a†]=1 and J commutators [Ji,Jj]=−iℏεijk Jk to identify unique candidates.
Run Phase 2 to populate.
Run Phase 2 to populate.
For each projected operator Ô_i, the numerical signature Vᵢ = ⟨v,J,k|Ô_i|v,J,k⟩ is computed over a test basis. Candidates are grouped by λ-order n. For each order, an SVD on the signature matrix selects the cₙ operators with the largest singular values — the most physically important and least correlated set.
Waiting for Phase 3 to run…
The final symmetry-allowed, normal-ordered, SVD-selected operator candidates for the effective rovibrational Hamiltonian. These are the terms to include in H_eff = Σᵢ cᵢ Ôᵢ, ordered by λ-scaling importance. Slot budget is enforced by the Molien Hadamard product cₙ at each order.
Complete Phases 2 and 3 to generate the final report.
Upload a HITRAN .par file (or use the built-in H₂O lines) to fit the SVD-selected operator set against experimental line positions. The BIC stepwise selector adds one operator at a time, keeping only those that reduce the Bayesian Information Criterion — penalising high-rank terms unless they significantly improve the fit. The final model is then refined by extended coordinate-descent fitting.
Run fitter to populate.
Final fitted spectroscopic constants for the BIC-selected operator set. Observed−Calculated residuals for each matched HITRAN line. The RMSE (unbiased) is the true fit quality metric. Lines with |O−C| > 3σ are flagged in red. The chart below overlays experimental .par line positions (stick heights ∝ log₁₀ line strength) with the model stick spectrum from the fitted effective Hamiltonian.
Complete Phase 4 to generate the fit report.