  We consider a system of $N$ spinless fermions, interacting with each other
via a power-law interaction $\epsilon/r^n$, and trapped in an external harmonic
potential $V(r) = r^2/2$, in $d=1,2,3$ dimensions. For any $0 < n < d+2$, we
obtain the ground-state energy $E_N$ of the system perturbatively in
$\epsilon$, $E_{N}=E_{N}^{\left(0\right)}+\epsilon
E_{N}^{\left(1\right)}+O\left(\epsilon^{2}\right)$. We calculate
$E_{N}^{\left(1\right)}$ exactly, assuming that $N$ is such that the "outer
shell" is filled. For the case of $n=1$ (corresponding to a Coulomb interaction
for $d=3$), we extract the $N \gg 1$ behavior of $E_{N}^{\left(1\right)}$,
focusing on the corrections to the exchange term with respect to the
leading-order term that is predicted from the local density approximation
applied to the Thomas-Fermi approximate density distribution. The leading
correction contains a logarithmic divergence, and is of particular importance
in the context of density functional theory. We also study the effect of the
interactions on the fermions' spatial density. Finally, we find that our result
for $E_{N}^{\left(1\right)}$ significantly simplifies in the case where $n$ is
even.
