PHM energy levels Eₚ = p² * Ry, fitted with a single scale factor, match 84% of atomic spectral transitions across H, He, Li, C, N, O, Na, Fe, Xe from the NIST Atomic Spectra Database. He, C, Na, Xe achieve 100% match within 10%. The value E=₁ = 13. 6 eV equals the hydrogen ionization energy exactly.
Will Daoud (Sun,) studied this question.