Semi-classical analysis of real atomic spectra beyond Gutzwiller's approximation

Real atomic systems, like the hydrogen atom in a magnetic field or the helium atom, whose classical dynamics are chaotic, generally present both discrete and continuous symmetries. In this letter, we explain how these properties must be taken into account in order to obtain the proper (i.e. symmetry projected) $\mathrm{\hslash }$ expansion of semiclassical expressions like the Gutzwiller trace formula. In the case of the hydrogen atom in a magnetic field, we shed light on the excellent agreement between present theory and exact quantum results.