Data for "Electron-Density Morphology: A Structural Framework for Density-Informed Molecular and Materials Design"

While topology determines which interactions exist within a molecule and geometry specifies absolute properties, morphology bridges these levels by capturing how charge is distributed directionally within a given topological framework.

Using the Nudged Elastic Band (NEB) method with DFT calculations, we analyze bond critical points (bCPs) along minimum energy pathways for three classes of reactions: oxygen atom transfer (PhI/DMSO and PMe₃/DMSO systems), hydrogen atom transfer (propane to cysteine-, histidine-, and tyrosine-ligated heme iron models), and proton transfer (pyridine/acetic acid). The directional organization at each bCP is quantified using cone angles derived from the 3×3 Hessian of the electron density, yielding a scale-invariant "order parameter" <i>η</i> that characterizes the asymmetry of charge-transfer tubes linking reacting fragments.</p><p dir="ltr">Within a Landau theoretical framework, where system energy is expanded in terms of morphological order parameters, the transition state emerges as a stationary point (<i>dE/dη</i> = 0) that—at zero electrostatic bias—corresponds precisely to morphological symmetry (<i>η</i> = 0).