  The $\nu=5/2$ fractional quantum Hall effect is a system of intense
experimental and theoretical interest as its ground state may host non-abelian
excitations, but the exact nature of the ground state is still undetermined. We
present the results of an exact diagonalization study of an electron system in
the disk configuration including the effects of Landau level (LL) mixing and
the finite thickness of the quantum well confining the electrons. The
degeneracy between the two leading candidates for the ground state, the
Pfaffian and anti-Pfaffian, is broken by interactions with a neutralizing
background, in addition to the inclusion of two- and three-body interactions
via LL mixing. As a result of the neutralizing background in the disc
configuration, there is a phase transition from the anti-Pfaffian to the
Pfaffian as LL mixing is turned on, in stark contrast to what is observed in a
spherical geometry. The LL mixing leads to an increased charge $e/4$ quasihole
size. LL mixing interactions are also shown to overcome the effects of edge
reconstruction. Due to finite thickness effects, these properties are enhanced
dramatically. We also find that only the Pfaffian and anti-Pfaffian states
continue to possess energy gaps at finite width, while gaps for compressible
stripe states close.
