Determining Molecular Structure, Coordination Geometry, and Molecular Symmetry

Chemists have a strong language describing and defining idealized polyhedra P and symmetry point groups G, but no efficient measure to correlate these to real molecular structures Q. Symmetry and structure have been ascribed through experience, but this approach is error-prone and provides no measure that can correlate molecular structure to molecular properties. Quantification of molecular structure and symmetry is crucial, and this requires two tools: a tool that orients a molecular structure to a symmetry axis, and a tool that measures the deviation of the molecular structure from symmetry. The first approach developed was the Continuous Symmetry Measure (CSM), which does both tasks and is exact across all symmetries as well as mathematically elegant. Unfortunately as a tool the CSM is close to impossible to implement. This led to the development of the Continuous Shape Measure (CShM) that measures how close a molecular structure Q is to a set of selected structures P’. The CShM tool have severe limitations, and we suggest that the Continuous Symmetry operation Measure (CSoM) should be the preferred tool. The CSoM tool can quantify the symmetry of any structure that can be described as a list of points in space. Here, we describe the CSM, CShM, and CSoM approaches and demonstrate how these can be used as tools to determine molecular structure, coordination geometry, and symmetry of water, organic molecules, transition metal complexes, and lanthanide compounds.