## Introduction

All of quantum chemistry is based on the solution of the many-body Schrödinger equation. This solution consists of the eigenvalue (the energies typically obtained from *ab initio* calculations) and eigenfunction (the wave function, where **Ψ( r**,

*t*

**)·Ψ*(**= ρ

*r*)**(**,

*r**t*

**)**and is a quantum mechanical observable).

However, the majority of existing quantum chemistry calculation methodologies that map the important features of the potential energy surface (PES) place their main focus on the eigenvalues, for example, for time dependent (TD)/density functional theory (DFT), in the form of Kohn–Sham energies, and disregard a quantitative treatment of the eigenfunctions in the form of the total charge density distribution ρ**( r)**. As hugely successful as an energetics-driven approach to computational chemistry has been, there remain issues that need to be addressed, for example, a problem for DFT is the approximate exchange-correlation functional.1 The realization that the full Schrödinger equation solution consists of eigenfunctions as well as the energy eigenvalues leads to the understanding that to better investigate, for instance, the PES of structural isomers, a fully quantitative treatment of the eigenfunctions should also be included. Conventionally, the detail of the topology of conformers does not use quantum mechanics to guide computational searches to find isomeric conformers. In this short perspective, the author demonstrates that Bader's quantum theory of atoms in molecules (QTAIM)2 provides the basis for a search for the spanning set of isomeric topologies. QTAIM only requires the charge density ρ

**(**distributions that can be obtained from either theory, for example, TD-DFT, configuration interaction (Hartree Fock (HF)/

*r*)*cis*, complete active space self-consistent field (CASSCF), and multi-reference configuration interaction (MRCI)) or experiment, for example, X-ray diffraction3 and

^{1}H NMR spectroscopy data.4 QTAIM works by proving that the topological condition of zero-flux (

**∇**ρ(

**)·**

*r***n(**= 0) of the charge density ρ

*r*)**(**serves as the boundary condition for the application of Schwinger's principle of stationary action in the definition of an open system.

*r*)In this perspective, the author shows how the theory of elementary catastrophes has inspired a new non-Euclidean quantum mechanics-based geometry to describe the topology of molecules and clusters. We discuss some recent articles representative of current research that explore the relationship between the topology of the molecules and the phase space for both the molecular and the solid-state topologies as well as an alternative application of the Poincaré–Hopf relation; “molecular recognition.” The author's perspective on future directions is included in each subsection. The conclusions and outlook are then presented.