A time and memory efficient recipe for fast normal mode computations of complexes with icosahedral symmetry

With the recent breakthroughs in experimental technologies, structure determination of extremely large assemblies, many with icosahedral symmetry, has been rapidly accelerating. Computational studies of their dynamics are important to deciphering their functions as well as to structural refinement but are challenged by their extremely large size, which ranges from hundreds of thousands to even millions of atoms. Group theory can be used to significantly speed up the normal mode computations of these symmetric complexes, but the derivation is often obscured by the complexity of group theory and consequently is not widely accessible. To address this problem, this work presents an easy recipe for normal mode computations of complexes with icosahedral symmetry. The recipe details how the Hessian matrix in symmetry coordinates can be constructed in a few easy steps of matrix multiplications, without going through the complexity of group theory. All the "ingredient" matrices required in the recipe are fully provided in the Supplemental Information for easy reproduction. The work is timely considering the expected large in-flux of many more icosahedral assemblies in the near future. The recipe uses a minimum amount of memory and solves the normal modes in a significantly reduced amount of time, making it feasible to perform normal mode computations of these assemblies on most computer systems.