The energy levels of the hydrogen atom , taking account only of the Coulomb interaction between the electron and proton, are shown on the left. Perturbed energy levels , also including spin-orbit interaction and relativistic corrections, produce the so-called fine structure, as shown on the right.
Atomic structure, specifically for the hydrogen atom, is determined principally by Coulomb interactions among electrons and the nucleus. This leads to the unperturbed energy . There also exist smaller contributions to the energy, most notably from spin-orbit interactions. These are interactions between orbital and spin magnetic moments of the electron, represented by the Hamiltonian
where and are the orbital and spin angular momenta, respectively. A secondary perturbation comes from relativistic corrections to electron kinetic energy, represented by a term in the Hamiltonian of the form
The total fine structure is then represented by the perturbation
with a first-order energy correction
This gives the total energy of the state , , to first order in perturbation theory,
where is the principal quantum number, is the total electronic angular momentum quantum number and is the fine-structure constant .