 Solution

For two decays from a common upper energy level, their photon wavenumber difference equals the separation of the two lower energy levels. Treating both quoted wavelengths as vacuum values initially giveswhose difference $11443.96\mathrm{c}{\mathrm{m}}^{-1}$ is close to $21648-10192=11456\mathrm{c}{\mathrm{m}}^{-1}$. The only matching lower pair among the supplied energy levels is ${}^1{D}_2,{}^1{S}_0$, with the shorter-wavelength photon ending on ${}^1{D}_2$. The two independent upper energy estimates areThusThe electric-dipole selection rules require odd parity, $S=0$, $L=1$ and $J=1$ to connect to both singlet lower energy levels. The apparent discrepancy has a concrete convention explanation. The NIST carbon line table lists $193.0906\mathrm{nm}$ under vacuum wavelengths and $247.8561\mathrm{nm}$ under air wavelengths, reproducing the source's rounded numbers.^[ref] The second number must therefore be converted to a vacuum wavelength before using $E=hc/\lambda$. For the tabulated upper $61981.82\mathrm{c}{\mathrm{m}}^{-1}$ and lower $21648.02\mathrm{c}{\mathrm{m}}^{-1}$ energy levels, its vacuum wavelength is $247.931\mathrm{nm}$, corresponding to an air refractive index about $1.00030$. This removes the spurious $12\mathrm{c}{\mathrm{m}}^{-1}$ discrepancy. The first calculation alone, which correctly uses a vacuum wavelength, already gives the intended upper energy level to the accuracy of the rounded data. The independent upper value is $61981.82\mathrm{c}{\mathrm{m}}^{-1}$.^[ref]