Non-Overlapping Arrangement of Identical Objects: An insight for molecular close packing

In this study, we investigate the arrangement of identical objects to achieve non-overlapping configurations, a fundamental challenge across various scientific and engineering disciplines, particularly in the context of molecular close-packing. We introduce the concept of "Maximal Boundary Distance", defined as the largest distance between extreme points of an object in a given direction. This length ensures that aligned objects touch without overlapping. Our approach is validated through mathematical proofs and illustrated with both, two-dimensional and three-dimensional examples, emphasizing applications in molecular systems. We further explore the practical implications of this for the packing of molecular crystals by identifying optimal lattice constants. The stability of these configurations is further analyzed using the Density Functional Tight Binding method. This work not only advances our understanding of object arrangement but also has significant implications for the strategic positioning of molecules in close-packed structures.