Rydberg formula ($1/\lambda =R(1/n_f^2-1/n_i^2)$)

The Rydberg formula relates wavelengths of hydrogenic spectral lines to pairs of principal quantum numbers. For hydrogen emission from $n_i$ to $n_f<n_i$, $1/\lambda =R_H(1/n_f^2-1/n_i^2)$, where $R_H$ includes the electron–proton reduced mass correction to the infinite-nuclear-mass Rydberg constant.

Holding the lower level fixed generates a series: $n_f=1$ gives the Lyman series and $n_f=2$ the Balmer series. Higher upper levels accumulate toward a series limit because their binding energies approach zero. For example, $3\to 2$ gives $1/\lambda =5R_H/36$, whereas the Balmer limit has $1/\lambda =R_H/4$.

The Bohr model reproduces this gross energy pattern, while a more complete treatment includes fine structure and other corrections. A precise measured line position therefore requires more specification than two principal quantum numbers alone.

The inverse-length constant in the infinite-nuclear-mass hydrogenic spectrum, $R_{\infty }=m_e{e}^4/[8{ϵ}_0^2{h}^{3}c]$. A nucleus of finite mass replaces it by $R_M=R_{\infty }\mu /m_e$, where $\mu$ is the electron-nucleus reduced mass. Some spectroscopic fits call $Z^2{R}_M$ an effective Rydberg constant; the charge-inclusive and reduced-mass-only conventions must be distinguished.