  Based on an exposition of the underlying physics and applied mathematics in
arXiv:1603.00899, this paper in five separate parts presents a description of
the properties of the amplitude functions of the hydrogen atom according to
wave mechanics in the coordinate representation in all four systems of
coordinates in which Schroedinger's equations are separable, explicitly I
spherical polar coordinates, II paraboloidal coordinates, III ellipsoidal
coordinates and IV spheroconical coordinates; part V discusses the implications
of these multiple systems in which these atomic orbitals can be directly
calculated. Most published plots of orbitals are inaccurate. In particular,
parts I - IV contain many accurate plots of the surfaces of amplitude functions
in the various systems with a common criterion for the amplitude at which the
surface is constructed. Although the plots are generated from amplitude
functions in the specified systems of coordinates, the plots are translated to
cartesian coordinates that the eye can recognise. The surfaces of the amplitude
functions in ellipsoidal and spheroconical functions are previously unreported,
with the accurate surfaces in spherical polar and paraboloidal coordinates for
a direct comparison.
