Complementarity of QTAIM and MO theory in the study of bonding in donor–acceptor complexes

Abstract

The quantum theory of an atom in a molecule, QTAIM, provides chemists with a choice of how to interpret, understand and predict the observations of experimental chemistry. They may continue with the use of subjective orbital models and associated energy and charge partitioning schemes, or they may combine the classification and ordering of electronic states obtained from molecular orbital (MO) theory with QTAIM to obtain unique physical answers to chemical questions. MO theory provides a prediction and understanding of the electronic structure of atoms and molecules. The eminently useful models that spring from this theory, the ligand field description of the metal complexes for example, set the stage for the application of quantum mechanics to complete our understanding of the properties predicted by wave functions obtained from theory. The paper demonstrates the complementary nature of MO theory and QTAIM, together with the means of bridging the two approaches in a discussion of the bonding in the carbonyl complexes of Cr, Fe and Ni and the metallocene complexes of Fe, Al + and Ge. Particular emphasis is placed on the atomic expectation value of the exchange operator that determines the number of electron pairs exchanged between bonded atoms, obviating the need for the terms ‘covalency’ and ‘resonance’. One side of the dichotomy in approaches, one based on physics the other eschewing it, is illustrated by a quotation from the abstract of a talk presented by Roald Hoffmann at a symposium of the American Chemical Society on ‘Contemporary Aspects of Chemical Bonding’, Sept. 2003: “And yet the concept of a chem bond, so essential to chem., and with a venerable history, has a life, generating controversy and incredible interest. Even if we can’t reduce it to physics. … Push the concept to its limits, accept that a bond will be a bond by some criteria, maybe not by others, respect chem tradition, have fun with the richness of something that cannot be defined clearly, and spare us the hype”. The abstract laid the groundwork for the message that a chemical bond lies not only beyond the domain of physics but is incapable of precise physical understanding. Whether physics does or does not offer a definition of a chemical bond, it does set out the necessary and sufficient conditions for two atoms to be chemically bonded to one another, conditions summarised by Slater in terms of the two theorems that are pivotal to the physics of bonding: “… both the virial theorem and Feynman's theorem are exact consequences of wave mechanics. Both interpretations agree in pointing to the existence of the overlap charge density as the essential feature in the attraction between the atoms”. The attraction leads to a decrease in the potential energy and to the existence of a potential well which, if of a depth greater than that of the zero point energy, guarantees the system possesses an equilibrium structure wherein no Feynman forces act on the nuclei and wherein the bonding can be characterised by all of the criteria set forth by Hoffmann for the chemical bond; length, energy, force constant etc. So this article makes a modest proposal. Reserve the name and concept of a bond for use by those who believe that chemistry lies beyond the scope of physics (in spite of their reliance on molecular orbitals and valence bond structures), and reserve the concept of bonding for use by those intent on the pursuit and understanding of chemistry using the tools of quantum mechanics.