By examining all the lines in the spectrum of the hydrogen atoms, an empirical model was derived that explained the pattern of the emission. The specific wavelengths (or frequencies or energies) could be predicted based upon a constant and two integers. The interpretation was that one integer represented the initial state and one integer the final state. The wavelength (or frequency or energy) was related to the change that occurred moving between these two states.
The original formula related inverse wavelength (known as "wavenumber") to the integers that were related to the initial and final states.
\[{ 1 \over \lambda} = {\cal R} \left({1 \over n_f^2} - {1 \over n_i^2}\right)\]
Where \({ \cal R}\) is equal to \(1.097\times 10^7\) m-1 and \(n_f\) and \(n_i\) are integers (1,2,3,....) that describe the initial and final states of the electron.
While the original formula derived by Rydberg did not look directly at energies, we can rewrite the formula to have these units. Under these conditions, the change in energy of the electron is given by
\[\Delta E = {\cal R} \left({1 \over n_f^2} - {1 \over n_i^2}\right)\]
Now the constant \({\cal R}\) (the Rydberg constant) is equal to \(2.178\times 10^{-18}\) J. Note now the units are energy. \(n_f\) and \(n_i\) are integers (1,2,3,....) still describe the initial and final states of the electron. Many of you may have seen this in the past so you just take it for granted. However, this is quite remarkable. There are fixed energy states that can be related to integers (1,2,3....) for the energies of the electrons in a hydrogen atom. One energy level is exactly (1/22) different in energy than another. Another is different by (1/52). Two others have a ratio of (32/172). Everything is related to the square of an integer!
Note the formula here is given in terms of energy changes. The original formula from Rydberg simply related the inverse of the wavelength to two integers. However, we can convert wavelength to frequency to energy. So the above equation combines the Empirical ideas of Rydberg with the relation of energy to frequency derived from the photoelectric effect.
Also note that the above formula is for energy differences between two energy levels (\(\Delta E\)). If you want (or need) to calculate the potential energy value (\(E_n\)) of the individual quantum levels (\(n=1,2,3,...\)), then you use the formula:
\[ E_n = -{\cal R}\left({1 \over n}\right)^2\]
One more important thing... On the absolute energy scale, we assign a free electron with nothing near it (floating in space) a value of ZERO joules. This would correspond to an infinite level in the atom, or \(E_\infty\). This also means that all energy levels in the atom are below this value and are therefore negative.