  The calculation of higher-order binding corrections to bound systems is a
fundamental problem of theoretical physics. For any nonrelativistic expansion,
one needs the Foldy-Wouthuysen Transformation which disentangles the particle
and the antiparticle degrees of freedom. This transformation is carried out
here to eighth order in the momenta, or, to eighth order in the momentum
operators, which is equivalent to the eighth order of the fine-structure
constant. Matrix elements of the eighth-order terms are evaluated for F_5/2 and
F_7/2 states in hydrogenlike ions and compared with the Dirac-Coulomb energy
levels.
