DOI: 10.1002/cphc.201300092 Domain-Averaged Exchange-Correlation Energies as a Physical Underpinning for Chemical Graphs M. García-Revilla, [a] E. Francisco, [a] Paul L. A. Popelier, [b] and Angel Martín Pendµs* [a] 1. Introduction At the core of all chemical disciplines lies the assumption of molecular structure. In most circumstances, molecules, or gen- erally molecular systems, are visualized as a set of nuclei linked by bonds. Provided that a set of nodes, together with a collec- tion of binary relationships among them, form a mathematical graph, it is not surprising that the development of chemical structure theory as formulated by KekulØ, Couper, Butlerov, and others (see ref. [1]) ran parallel to that of graph theory. Today, chemical graph theory is a rich discipline that has pro- vided a number of important insights, particularly in the devel- opment of useful indices for quantitative structure–activity re- lationships (QSAR [2] ). In a sense, chemistry has become the dis- cipline that studies any meaningful connectivity between sets of atoms, as well as their changes. Despite experimental and theoretical advances, there is no clear consensus about at which state two nodes of a molecular graph (MG) should share an edge, that is, when to consider them bonded or not. Frequently, one still relies either on em- pirical geometrical criteria (using interatomic distances, all kinds of atomic radii, and so forth) or on chemical intuition. On the other hand, standard theoretical proposals tend to use concepts, forged from molecular orbital theory, which are non- invariant upon orbital transformations and therefore arbitrary. Whether these issues should be considered closed or not in the absence of an unequivocal answer to the molecular graph key question depends, surprisingly, on taste. For many, includ- ing respected scholars (see ref. [3]), the ambiguity of the chem- ical bond concept is inherent to chemistry and enriches it. For others, this ambiguity simply hides a lack of understanding. One of the cleanest proposals that tackles this important issue stems from the quantum theory of atoms in molecules (QTAIM), proposed by Bader and co-workers. [4] This theory shows that standard non-relativistic quantum mechanics can be generalized to a 3D subsystems defined by the attraction basins of the electron density, 1(r). By examining numerically its gradient vector field, it was found that some of its critical points (CPs) lay between bonded atoms, and that the only two gradient paths emanating from these CPs ended up in the two bonded nuclei. These bond critical points (BCPs) and bond paths (BPs) reconstruct the accepted MGs of the vast majority of molecules, and provided the first algorithmic recipe to build MGs that did not depend neither on geometric databases nor on orbital arguments. Moreover, several scalars computed at the BCPs, such as electron density itself, 1 b , were shown to be correlated to bond strength, although each individual atomic pair had to be calibrated separately in order to transform 1 b into bond orders or bond energies. Nonetheless, orthodox identification of BCPs and BPs with chemical bonds is still em- pirical in nature, despite several attempts. [5] In the last few years, some have questioned the universal validity of this iden- tification and its on–off nature, for systems have been found where BCPs appear where chemists would not add an edge to the molecular graph or vice versa (see for instance refs. [6–13]). [a] Prof. M. García-Revilla, Prof. E. Francisco, Prof. A. Martín Pendµs Departamento de Qímica Física y Analítica Facultad de Química, Universidad de Oviedo 33006-Oviedo (Spain) Fax: (+ 34) 985103125 E-mail : angel@fluor.quimica.uniovi.es [b] P. L. A. Popelier Manchester Interdisciplinary Biocentre (MIB) 131 Princess Street, Manchester M1 7DN (Great Britain) and School of Chemistry, University of Manchester Oxford Road, Manchester M13 9PL (Great Britain) A novel solution to the problem of assigning a molecular graph to a collection of nuclei (i.e. how to draw a molecular structure) is presented. Molecules are universally understood as a set of nuclei linked by bonds, but establishing which nuclei are bonded and which are not is still an empirical matter. Our approach borrows techniques from quantum chemical topology, which showed for the first time the con- struction of chemical graphs from wave functions, shifting the focus on energetics. This new focus resolves issues surround- ing previous topological analyses, in which domain-averaged exchange-correlation energies (V xc ), quantities defined in real space between each possible atom pair, hold the key. Expo- nential decay of V xc in non-metallic systems as the intercenter distance increases guarantees a well-defined hierarchy for all possible V xc values in a molecule. Herein, we show that extract- ing the set of atom pairs that display the largest V xc values in the hierarchy is equivalent to retrieving the molecular graph itself. Notably, domain-averaged exchange-correlation energies are transferable, and they can be used to calculate bond strengths. Fine-grained details resulted to be related to simple stereoelectronic effects. These ideas are demonstrated in a set of simple pilot molecules.  2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2013, 14, 1211 – 1218 1211 CHEMPHYSCHEM ARTICLES