Stability of Z²configurations in 3D

Inspired by the issue of stability of molecular structures, we investigate the strict minimality of point sets with respect to configurational energies featuring two- and three-body contributions. Our main focus is on characterizing those configurations which cannot be deformed without changing distances between first neighbours or angles formed by pairs of first neighbours. Such configurations are called angle-rigid. We tackle this question in the class of finite configurations in Z2, seen as planar three-dimensional point sets. A sufficient condition preventing angle-rigidity is presented. This condition is also proved to be necessary when restricted to specific subclasses of configurations.