Limitation of Bohr's atom model

However successful it was at explaining the hydrogen spectrum, Bohr's model has real limits:

• It only works for one-electron species. The clean derivation behind equations (2.3)-(2.5) applies exclusively to hydrogen and hydrogen-like ions such as $\text{Li}^{2+}$ - anything with exactly one electron orbiting the nucleus. The moment a second electron is added, electron-electron repulsion enters the picture and the simple circular-orbit picture no longer works; Bohr's model has nothing to say about multi-electron atoms.

• It cannot explain spectral line splitting in external fields. When a hydrogen sample is placed in a magnetic field, each spectral line splits into several closely spaced lines - the Zeeman effect. The same kind of splitting happens in a strong electric field, called the Stark effect. Bohr's model, which allows only one single energy for a given $n$, has no mechanism to produce this splitting, since it doesn't account for the different possible orientations an orbit could have in space. …