  The accuracy of calculations of atomic Rydberg excitations cannot be judged
by the usual measures, such as mean unsigned errors of many transitions. We
show how to use quantum defect theory to (a) separate errors due to approximate
ionization potentials, (b) extract smooth quantum defects to compare with
experiment, and (c) quantify those defects with a few characteristic
parameters. The particle-particle random phase approximation (pp-RPA) produces
excellent Rydberg transitions that are an order of magnitude more accurate than
those of time-dependent density functional theory with standard approximations.
We even extract reasonably accurate defects from the lithium Rydberg series,
despite the reference being open-shell. Our methodology can be applied to any
Rydberg series of excitations with 4 transitions or more to extract the
underlying threshold energy and characteristic quantum defect parameters. Our
pp-RPA results set a demanding challenge for other excitation methods to match.
