  We study the local optimality of periodic point sets in $\mathbb{R}^n$ for
energy minimization in the Gaussian core model, that is, for radial pair
potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable
parameter spaces for $m$-periodic sets, we can locally rigorously analyze the
energy of point sets, within the family of periodic sets having the same point
density. We derive a characterization of periodic point sets being
$f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained
in the set. Especially for $2$-periodic sets like the family $\mathsf{D}^+_n$
we obtain expressions for the hessian of the energy function, allowing to
certify $f_c$-optimality in certain cases. For odd integers $n\geq 9$ we can
hereby in particular show that $\mathsf{D}^+_n$ is locally $f_c$-optimal among
periodic sets for all sufficiently large~$c$.
