Nanocrystal Programmable Assembly Beyond Hard Spheres (or Shapes) and Other (Simple) Potentials

Ligands are the key to almost any strategy in the assembly of programmable nanocrystals (or nanoparticles) and must be accurately considered in any predictive model. Hard Spheres (or Shapes) provide the simplest and yet quite successful approach to assembly, with remarkable sophisticated predictions verified in experiments. There are, however, many situations where hard spheres/shapes predictions fail. This prompts three important questions: {\em In what situations should hard spheres/shapes models be expected to work?} and when they do not work, {\em Is there a general model that successfully corrects hard sphere/shape predictions?} and given other successful models where ligands are included explicitly, and of course, numerical simulations, {\em can we unify hard sphere/shape models, explicit ligand models and all atom simulations?}. The Orbifold Topological Model (OTM) provides a positive answer to these three questions. In this paper, I give a detailed review of OTM, describing the concept of ligand vortices and how it leads to spontaneous valence and nanoparticle "eigenshapes" while providing a prediction of the lattice structure, without fitting parameters, which accounts for many body effects not captured in (two-body) potentials. I present a thorough survey of experiments and simulations and show that, to this date, they are in full agreement with the OTM predictions. I will conclude with a discussion on whether NC superlattices are equilibrium structures and some significant challenges in structure prediction.