We establish new lower bounds on the distance between two points of a minimum energy configuration of $N$ points in $\mathbb{R}^3$ interacting according to a pairwise potential function. For the Lennard-Jones case, this bound is 0.67985 (and 0.7633 in the planar case). A similar argument yields an estimate for the minimal distance in Morse clusters, which improves previously known lower bounds. Moreover, we prove that the optimal configuration cannot be two-dimensional, and establish an upper bound for the distance to the nearest neighbour of every particle, which depends on the position of this particle. On the boundary of the optimal configuration polytope, this is unity while in the interior, this bound depends on the potential function. In the Lennard-Jones case, we get the value $\sqrt[6]{\frac{11}5}\approx 1.1404$. Also, denoting by $V_N$ the global minimum in an $N$ point minimum energy configuration, we prove in Lennard-Jones clusters $\frac{V_N}N\ge-41.66$ for all $N\ge2$, while asymptotically $\lim_{N\to\infty}\frac{V_N}N\le-8.611$ holds (as opposed to $\frac{V_N}N\ge-8.22$ in the planar case, confirming non-planarity for large $N$).