  We present the first results from the Rhapsody cluster re-simulation project:
a sample of 96 "zoom-in" simulations of dark matter halos of 10^14.8 +- 0.05
Msun/h, selected from a 1 (Gpc/h)^3 volume. This simulation suite is the first
to resolve this many halos with ~5x10^6 particles per halo in the cluster-mass
regime, allowing us to statistically characterize the distribution of and
correlation between halo properties at fixed mass. We focus on the properties
of the main halos and how they are affected by formation history, which we
track back to z=12, over five decades in mass. We give particular attention to
the impact of the formation history on the density profiles of the halos. We
find that the deviations from the Navarro-Frenk-White (NFW) model and the
Einasto model depend on formation time. Late-forming halos tend to have
considerable deviations from both models, partly due to the presence of massive
subhalos, while early-forming halos deviate less but still significantly from
the NFW model and are better described by the Einasto model. We find that the
halo shapes depend only moderately on formation time. Departure from spherical
symmetry impacts the density profiles through the anisotropic distribution of
massive subhalos. Further evidence of the impact of subhalos is provided by
analyzing the phase-space structure. A detailed analysis of the properties of
the subhalo population in Rhapsody is presented in a companion paper.
