Generation of Helical States - Breaking of Symmetries, Curie's Principle, and Excited States.

Herein, we deeply detail for the very first time mathematical concepts behind the generation of helical molecular orbitals (MOs) for linear chains of atoms. We first give a definition of helical MOs and we provide an index measuring how far a given helical states is from a perfect helical distribution. Structural properties of helical distribution for twisted -cumulene and cumulene version of Möbius systems are given.

Relating the molecular topology and local geometry: Haddon's pyramidalization angle and the Gaussian curvature.

The pyramidalization angle and spherical curvature are well-known quantities used to characterize the local geometry of a molecule and to provide a measure of regio-chemical activity of molecules. In this paper, we give a self-contained presentation of these two concepts and discuss their limitations. These limitations can bypass, thanks to the introduction of the notions of angular defect and discrete Gauss curvature coming from discrete differential geometry.