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Inquantum chemistry, thequantum theory of atoms in molecules (QTAIM), sometimes referred to asatoms in molecules (AIM), is a model of molecular andcondensed matter electronic systems (such ascrystals) in which the principal objects of molecular structure -atoms andbonds - are natural expressions of a system's observableelectron density distribution function. An electron density distribution of a molecule is aprobability distribution that describes the average manner in which the electronic charge is distributed throughout real space in the attractive field exerted by thenuclei. According to QTAIM, molecular structure is revealed by thestationary points of the electron density together with thegradient paths of the electron density that originate and terminate at these points.
QTAIM was primarily developed by ProfessorRichard Bader and his research group atMcMaster University over the course of decades, beginning with analyses of theoretically calculated electron densities of simple molecules in the early 1960s and culminating with analyses of both theoretically and experimentally measured electron densities of crystals in the 90s. The development of QTAIM was driven by the assumption that, since the concepts of atoms and bonds have been and continue to be so ubiquitously useful in interpreting, classifying, predicting and communicating chemistry, they should have a well-defined physical basis.
QTAIM recovers the central operational concepts of the molecular structure hypothesis, that of afunctional grouping of atoms with an additive and characteristic set of properties, together with a definition of the bonds that link the atoms and impart the structure. QTAIM defines chemical bonding and structure of a chemical system based on thetopology of the electron density. In addition to bonding, QTAIM allows the calculation of certain physical properties on a per-atom basis, by dividing space up into atomic volumes containing exactly one nucleus, which acts as a localattractor of the electron density. In QTAIM an atom is defined as aproper open system, i.e. a system that can shareenergy and electron density which is localized in the 3D space. The mathematical study of these features is usually referred to in the literature as charge densitytopology.
QTAIM rests on the fact that the dominant topological property of the vast majority of electron density distributions is the presence of strongmaxima that occur exclusively at the nuclei, certain pairs of which are linked together by ridges of electron density. In terms of an electron density distribution'sgradient vector field, this corresponds to a complete, non-overlapping partitioning of a molecule into three-dimensional basins (atoms) that are linked together by shared two-dimensional separatrices (interatomic surfaces). Within each interatomic surface, the electron density is a maximum at the corresponding internuclear saddle point, which also lies at the minimum of the ridge between corresponding pair of nuclei, the ridge being defined by the pair of gradient trajectories (bond path) originating at the saddle point and terminating at the nuclei. Because QTAIM atoms are always bounded by surfaces having zeroflux in the gradient vector field of the electron density, they have some unique quantum mechanical properties compared to other subsystem definitions. These include unique electronic kinetic energy, the satisfaction of an electronic virial theorem analogous to the molecular electronicvirial theorem, and some interesting variational properties. QTAIM has gradually become a method for addressing possible questions regarding chemical systems, in a variety of situations hardly handled before by any othermodel or theory inchemistry.[1][2][3][4]
QTAIM is applied to the description of certainorganic crystals with unusually short distances between neighboring molecules as observed byX-ray diffraction. For example in thecrystal structure of molecularchlorine, the experimental Cl...Cl distance between two molecules is 327 picometres, which is less than the sum of thevan der Waals radii of 350 picometres. In one QTAIM result, 12 bond paths start from each chlorine atom to other chlorine atoms including the other chlorine atom in the molecule. The theory also aims to explain the metallic properties ofmetallic hydrogen in much the same way.
The theory is also applied to so-calledhydrogen–hydrogen bonds[5] as they occur in molecules such asphenanthrene andchrysene. In these compounds, the distance between two ortho hydrogen atoms again is shorter than their van der Waals radii, and according toin silico experiments based on this theory, a bond path is identified between them. Both hydrogen atoms have identical electron density and areclosed shell and therefore they are very different from the so-calleddihydrogen bonds that are postulated for compounds such asH3NBH3, and also different from so-calledagostic interactions.
In mainstream chemistry descriptions, close proximity of two nonbonding atoms leads to destabilizingsteric repulsion but in QTAIM the observed hydrogen-hydrogen interactions are in fact stabilizing. It is well known that both kinkedphenanthrene andchrysene are around 6kcal/mol (25kJ/mol) more stable than their linearisomersanthracene andtetracene. One traditional explanation is given byClar's rule. QTAIM shows that a calculated stabilization of 8 kcal/mol (33 kJ/mol) for phenanthrene is the result of destabilization of the compound by 8 kcal/mol (33 kJ/mol) originating from electron transfer from carbon to hydrogen, offset by 12.1 kcal (51 kJ/mol) of stabilization due to a H...H bond path. The electron density at the critical point between the two hydrogen atoms is low (0.012 e) for phenanthrene. Another property of the bond path is its curvature.
Another molecule analyzed by QTAIM isbiphenyl. Its two phenyl ring planes are oriented at a 38° angle with respect to each other, with the planarmolecular geometry (resulting from a rotation around the central C-C bond) destabilized by 2.1 kcal/mol (8.8 kJ/mol) and the perpendicular one destabilized by 2.5 kcal/mol (10.5 kJ/mol). The classic explanations for this rotational barrier are steric repulsion between the ortho-hydrogen atoms (planar) and breaking ofdelocalization ofpi density over both rings (perpendicular).
In QTAIM, the energy increase on decreasing thedihedral angle from 38° to 0° is a summation of several factors. Destabilizing factors are the increase inbond length between the connecting carbon atoms (because they have to accommodate the approaching hydrogen atoms) and transfer of electronic charge from carbon to hydrogen. Stabilizing factors are increased delocalization of pi-electrons from one ring to the other and (the one that tips the balance) is a hydrogen–hydrogen bond between the ortho hydrogens.
QTAIM has also been applied to study the electron topology of solvated post-translational modifications to proteins. For example, covalent–bond force constants in a set of lysine-arginineadvanced glycation end-products were derived using electronic structure calculations, and then bond paths were used to illustrate differences in each of the appliedcomputational chemistry functionals.[6] Furthermore, QTAIM had been used to identify a bond-path network of hydrogen bonds betweenglucosepane and nearby water molecules.[7]
The hydrogen-hydrogen bond is not without its critics. According to one, the relative stability of phenanthrene compared to its isomers can be adequately explained by comparing resonance stabilizations.[8] Another critic argues that the stability of phenanthrene can be attributed to more effective pi-pi overlap in the central double bond; the existence of bond paths is not questioned but the stabilizing energy derived from them is.[9]
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