With,
\(C=\cfrac { \sqrt [ 3 ]{ n_{ 2 } } -\sqrt [ 3 ]{ n_{ 1 } } }{ \sqrt [ 3 ]{ n_{ 1 }n_{ 2 } } } -\cfrac { n_{ 2 }^{ 2 }-n_{ 1 }^{ 2 } }{ (n_{ 1 }n_{ 2 })^{ 2 } }\)
Could it be that because the spectrum observations are conducted at high temperature,
\(\cfrac { \sqrt [ 3 ]{ n_{ 2 } } -\sqrt [ 3 ]{ n_{ 1 } } }{ \sqrt [ 3 ]{ n_{ 1 }n_{ 2 } } }\)
is due to temperature particles. And,
\(\cfrac { n_{ 2 }^{ 2 }-n_{ 1 }^{ 2 } }{ (n_{ 1 }n_{ 2 })^{ 2 } }\)
orbiting particles not necessarily electrons.
\(C=C_{\small{T}}+C_{\small{B}}\)
where \(C_{\small{T}}\) is due to the energy density \(\psi\) of interacting temperature particles and \(C_{\small{B}}\) due to Bohr model not necessarily of electrons, with quantized momenta.
Temperature particles orbiting around electron orbits that in resonance produce the infrared spectrum and the ultraviolet spectrum was proposed previously "Lemmings Over The Cliff..." dated 18 May 2016. Spectra lines in the visible spectrum will come from transitions between energy levels in the ultraviolet spectrum.
Electrons may not be involved! Unless the observations are also conducted in the presence of a high electric field, electron are not affected. If this is the case, temperature should be raised without the use of an electric field. Within the confines of the experiment, there should be only one strong field affecting one type of particles.
Let electrons not be involved. Maybe then spectra lines can be indicative of the existence of temperature particles in orbit around electron orbits as proposed in the post "Capturing \(T^{+}\) Particles" dated 15 May 2016.
In which case, electron ionization energies has no direct relation with spectra lines (due to temperature particles), not even in the simplest model of Hydrogen atoms.
Have a nice day...
