Simple, reliable, and universal metrics of molecular planarity

Planarity is a very important structural character of molecules, which is closely related to many molecular properties. Unfortunately, there is currently no simple, universal, and robust way to measure molecular planarity. In order to fill this evident gap, we propose two metrics of molecular planarity, namely molecular planarity parameter (MPP) and span of deviation from plane (SDP), to quantitatively characterize planarity of molecules. MPP reflects the overall degree of deviation of the structure from a plane, while SDP represents the span of the structural deviation relative to the fitting plane; respectively, they are complementary to each other. The examples in this article demonstrate that these metrics have strong rationality and practicality. They can not only be used to investigate the planarity of the entire molecule, but also measure the planarity of local structures, and they can even be employed to study variation of molecular planarity during a dynamic process. In addition, we also propose a new representation, namely coloring atoms according to their signed deviation distance to the fitting plane. This kind of map allows researchers to intuitively and quickly recognize position of the atoms in the system relative to the fitting plane. It can be seen from the examples that this representation is very useful in graphically exhibiting molecular planarity. The methods proposed in this work have been implemented in our open-source analysis code Multiwfn, which can be freely obtained via http://sobereva.com/multiwfn . The use is very simple and rich file formats are supported as input file.