Quantum Metrics for Continuous Shape Measures of Molecules

The field of continuous molecular shape and symmetry (dis)similarity quantifiers habitually called measures (specifically continuous shape measures - CShM or continuous symmetry measures - CSM) is obfuscated by the combinatorial numerical algorithms used in the field which restricts the applicability to the molecules containing up to twenty equivalent atoms. In the present paper we analyze this problem using various tools of classical probability theory as well as of one-particle and many-particle quantum mechanics. Applying these allows us to lift the combinatorial restriction and to identify the adequate renumbering of atoms (vertices) without considering all N! permutations of an N-vertex set so that in the end purely geometric molecular shape (dis)similarity quantifier can be defined. Developed methods can be easily implemented in the relevant computer code.