Structural and chemical properties of the building block of silica nanowires, (SiO_2)_6, are investigated with the theory of atoms and molecules (QTAIM). Twenty-five conformers have been analyzed, ten of which have not been reported before. We extend the silica (SiO_2) _6 topology phase space using QTAIM; the Poincaré-Hopf topological sum rules are applied and used to identify the spanning set of topologies, and this includes finding eight new distinct topologies that satisfy the Poincaré-Hopf relation. A simple phase diagram of the solutions of the Poincaré-Hopf relation is created with the aid of a new classification scheme to determine the boundary between topological stability and instability. Sum rules are then found to be applicable to any set of isomers. We determine that O-O bonding interactions exist for the silica (SiO_2)_6 conformers in regions where the energy surface is flattest. In addition, we identify unstable local minima in the topology of the charge density in order to further compare conformer instabilities. We quantify the dimensionality of a molecule using the Poincaré-Hopf relation instead of Euclidean geometry. This quantum topological definition of geometry shows that the four most energetically stable (SiO_2)_6 conformers are quantified as two-dimensional within the new quantum topology.
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