We provide a characterization of the spectral minimum for a random Schrödinger operator of the form $H=-\Delta + \sum_i {\in Z^d} q(x-i-\omega_i)$ in $L^2(R^d)$, where the single site potential $q$ is reflection symmetric, compactly supported in the unit cube centered at $0$, and the displacement parameters $\omega_i$ are restricted so that adjacent single site potentials do not overlap. In particular, we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible $2^d$-clusters of single site potentials. This is joint work with Günter Stolz and Jeff Baker.