Revisiting the foundations of the quantum theory of atoms in molecules: Some open problems

In a series of papers in the last ten years, various aspects of the mathematical foundations of the quantum theory of atoms in molecules have been considered by this author and his coworkers in some details. Although these considerations answered part of the questions raised by some critics on the mathematical foundations of the quantum theory of atoms in molecules, however, new mathematical problems also emerged during these studies that were reviewed elsewhere [Int. J. Quantum Chem. 111, 4497 (2011)]. Beyond mathematical subtleties of the formalism that were the original motivation for initial exchanges and disputes, the questions raised by critics had a constructive effect and prompted the author to propose a novel extension of the theory, now called the multi-component quantum theory of atoms in molecules [Theor. Chem. Acc. 132, 1365 (2013)]. Taking this background into account, in this paper a new set of open problems is put forward that the author believes proper answers to these questions, may open new doors for future theoretical developments of the quantum theory of atoms in molecules. Accordingly, rather than emphasizing on the rigorous mathematical formulation, the practical motivations behind proposing these questions are discussed in detail and the relevant literature are reviewed while when possible, evidence and routes to answers are also provided. The author hopes that proposing these open questions as a compact package may motivate more mathematically oriented people to participate in future developments of the quantum theory of atoms in molecules and its multi-component version.