Quantum-chemical quantification of molecular complexity and chirality

Abstract

We introduce a number of molecular complexity indexes on the grounds of quantization of notions known in the differential geometry of curves (specifically, electron trajectories). Mean lengths, L( Ψ) and L, reflect the general spatial extent of a many-electron system in a given state Ψ and, correspondingly, of the molecular structure as a whole. Analogously, mean curvatures, K( Ψ) and K, give a measure of the non-linearity of electron distribution. A mean torsion, τ( ψ), along with related indexes κ and κ f , presents a degree of molecular dissymmetry, i.e. a measure of chirality. In fact, all these indexes possess additivity properties and correct asymptotics under decomposition into separate parts or individual atoms. A topological approach to a one-electron hamiltonian through an adjacency matrix of the corresponding molecular graph is examined when calculating the above indexes. Some typical planar and nonplanar structures (Hückel-like systems, simple models of chiral allenes and chiral octahedral complexes with bidentate ligands) are studied.