On the αZ expansion for the Dirac energy of a one-electron atom

A procedure for directly prescribing a term of arbitrary order in an alpha Z expansion of the Dirac energy of a one-electron atom is presented, and utilised to obtain higher-order corrections to the Dirac fine-structure formula. These can then be combined with terms not included in the Dirac formalism and applied, for example, to semi-empirical charge-screening parametrisations of multi-electron atoms.