  We analytically derived the effective two-body interaction for a finite
thickness quantum Hall system with a harmonic perpendicular confinement and an
in-plane magnetic field. The anisotropic effective interaction in the lowest
Landau level (LLL) and first Landau level (1LL) are expanded in the basis of
the generalized pseudopotentials (PPs), and we analyze how the coefficients of
some prominent isotropic and anisotropic PPs depend on the thickness of the
sample and the strength of the in-plane magnetic field. We also investigate the
stability of the topological quantum Hall states, especially the Laughlin state
and its emergent guiding center metric, which we can now compute analytically.
An interesting reorientation of the anisotropy direction of the Laughlin state
in the 1LL is revealed, and we also discuss various possible experimental
ramifications for this quantum Hall system with broken rotational symmetry.
