  Equal-mass two-component Fermi gases under spherically symmetric external
harmonic confinement with large s-wave scattering length are considered. Using
the stochastic variational approach, we determine the lowest 286 and 164
relative eigenenergies of the (2,2) and (3,1) systems at unitarity as a
function of the range $r_0$ of the underlying two-body potential and
extrapolate to the $r_0 \rightarrow 0$ limit. Our calculations include all
states with vanishing and finite angular momentum $L$ (and natural and
unnatural parity $\Pi$) with relative energy up to $10.5 \hbar \Omega$, where
$\Omega$ denotes the angular trapping frequency of the external confinement.
Our extrapolated zero-range energies are estimated to have uncertainties of
0.1% or smaller. The (2,2) and (3,1) energies are used to determine the
fourth-order virial coefficient of the trapped unitary two-component Fermi gas
in the low-temperature regime. Our results are compared with recent predictions
for the fourth-order virial coefficient of the homogeneous system. We also
calculate small portions of the energy spectra of the (3,2) and (4,1) systems
at unitarity.
