One reasonable and memorable approximation excluding any fine structure is:

E_{n} = - \frac{1 3 . 6 e V}{n ^{2}}

Hydrogen spectral series mnemonic

Hydrogen 1-2 spectral line (121.6 nm, 10.2e V)

E_{n} = - 13.6 e V [\frac{1}{2 ^{2}} - \frac{1}{1 ^{2}}] = 10.2 e V

which with Planck-Einstein relation gives about 121.6 nm (

2.47 \times 1 0^{1} 5

Hz), which is a reasonable match with the value of 121.567... from the NIST Atomic Spectra Database.

Rydberg formula (1888)

Kind of a synonym for hydrogen emission spectrum not very clear if fine structure is considered by this term or not.

Formula discovered in 1885, was it the first set to have an empirical formula?

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