Model an alkaline-earth atom as 2+2 (or 0+2 for He): closed-shell inner core absorbed into a reduced kernel of charge Z_kernel, plus two valence electrons. In the excited triplet state, one valence electron stays in the ground s orbital (creating a Hartree screening parameterized by Z_inner) and the other is the explicit "outer" electron. The outer electron solves −½u″ + [ℓ(ℓ+1)/(2r²) + V_eff(r)]u = Eu on r ∈ [r_c, R_max] with empty-core + Neumann at r_c, where
  V_eff(r) = −Zkernel/r + V_H(r; Zinner),  V_H(r) = (1/r)·[1 − (1+Zinner·r)·exp(−2Zinner·r)]
He: r_c=0, Z_kernel=2, Z_inner≈1.69 (Slater for 1s). Be: r_c absorbs [He], Z_kernel=2, Z_inner≈2.3 (Slater for 2s). Same shape for Mg, Ca.