  We determine the energetically lowest lying states in the BEC-BCS crossover
regime of s-wave interacting two-component Fermi gases under harmonic
confinement by solving the many-body Schrodinger equation using two distinct
approaches. Essentially exact basis set expansion techniques are applied to
determine the energy spectrum of systems with N=4 fermions. Fixed-node
diffusion Monte Carlo methods are applied to systems with up to N=20 fermions,
and a discussion of different guiding functions used in the Monte Carlo
approach to impose the proper symmetry of the fermionic system is presented.
The energies are calculated as a function of the s-wave scattering length a_s
for N=2-20 fermions and different mass ratios \kappa of the two species. On the
BEC and BCS sides, our energies agree with analytically-determined first-order
correction terms. We extract the scattering length and the effective range of
the dimer-dimer system up to \kappa = 20. Our energies for the
strongly-interacting trapped system in the unitarity regime show no shell
structure, and are well described by a simple expression, whose functional form
can be derived using the local density approximation, with one or two
parameters. The universal parameter \xi for the trapped system for various
\kappa is determined, and comparisons with results for the homogeneous system
are presented.
