Hamiltonian and sources
H = T + ½ħω0ρt2[1 − ⅔ε2P2(cos θt) + 2ε4P4(cos θt)] − κħω̊0[2ℓt·s + μ(ℓt2 − ⟨ℓt2⟩N)], ⟨ℓt2⟩N = N(N+3)/2, in stretched coordinates ξ = x(Mω⊥/ħ)1/2, ζ = z(Mωz/ħ)1/2, ω⊥ = ω0(1 + ε2/3), ωz = ω0(1 − ⅔ε2). Ragnarsson & Nilsson, Shapes and Shells in Nuclear Structure (Cambridge, 2005), Secs. 8.3 to 8.5; the same form is Eq. (vosc) of Möller et al. (2016) at γ = 0.
κ, μ by oscillator shell N: Bengtsson & Ragnarsson, Nucl. Phys. A 436 (1985) 14, as tabulated in Table 6.2 of Shapes and Shells (protons N = 4: 0.065, 0.57; N = 5: 0.060, 0.65; N = 6: 0.054, 0.69; neutrons N = 4: 0.070, 0.39; N = 5: 0.062, 0.43; N = 6: 0.062, 0.34; the N ≥ 8 row for all higher shells). ω0(ε2, ε4) keeps the volume inside every equipotential fixed; ħω̊0 = 41A−1/3 MeV.
Basis: stretched spherical oscillator |N ℓ Λ Σ⟩, one block per Ω and parity. With ε4 = 0 each (N, Ω) block is exact. ρ2P4 couples all shells of one parity, so ε4 ≠ 0 is diagonalized in N ≤ Nmax. At the chosen deformation Nmax grows two shells at a time until the 13 levels around the Fermi level move by less than 10−4 ħω0; the change reached is printed under the diagram. The curves use a smaller basis, checked along the path the same way, with unconverged ranges shaded. An independent solver in a Cartesian basis agrees to 3×10−14 ħω0.
Labels Ωπ[N nz Λ]: in each (N, Ω) block of the pure ε2 problem levels never cross, so the k-th one connects to the k-th asymptotic state (largest nz lowest for prolate, smallest for oblate); labels are assigned one-to-one, greedily: all overlaps² between the levels of an Ωπ block and the asymptotic block states are sorted, largest first, and a pair is taken when neither member is taken yet. A level can therefore get a state that is not its own largest overlap; such a label, or one with overlap² below 0.5, is marked with ?.
Ground-state deformations: FRDM(2012), P. Möller et al., At. Data Nucl. Data Tables 109–110 (2016) 1, arXiv:1508.06294 (ε2, ε4 of the axial ground state; ε3, ε6 not used here). β2 is computed from the surface drawn, by their Eq. (betaconv); to first order ε2 ≈ 0.95β2. Measured Jπ: NUBASE2020, Kondev et al., Chin. Phys. C 45, 030001 (2021).