The Analysis of Electron Densities: From Basics to Emergent Applications

The electron density determines all properties of a system of nuclei and electrons. It is both computable and observable. Its topology allows gaining insight into the mechanisms of bonding and other phenomena in a way that is complementary to and beyond that available from the molecular orbital picture and the formal oxidation state (FOS) formalism. The ability to derive mechanistic insight from electron density is also important with methods where orbitals are not available, such as orbital-free density functional theory (OF-DFT). While density topology-based analyses such as QTAIM (quantum theory of atoms-in-molecules) have been widely used, novel, vector-based techniques recently emerged such as next-generation (NG) QTAIM. Density-dependent quantities are also actively used in machine learning (ML)-based methods, in particular, for ML DFT functional development, including machine-learnt kinetic energy functionals. We review QTAIM and its recent extensions such as NG-QTAIM and localization-delocalization matrices (LDM) and their uses in the analysis of bonding, conformations, mechanisms of redox reactions excitations, as well as ultrafast phenomena. We review recent research showing that direct density analysis can circumvent certain pitfalls of the FOS formalism, in particular in the description of anionic redox, and of the widely used (spherically) projected density of states analysis. We discuss uses of density-based quantities for the construction of DFT functionals and prospects of applications of analyses of density topology to get mechanistic insight with OF-DFT and recently developed time-dependent OF-DFT.