User:Thompsonc14/sandbox

The Quantum Theory of Atoms in Molecules (QTAIM) is a model of molecular and condensed matter electronic systems (such as crystals) in which the principal objects of molecular structure - atoms and bonds - are natural expressions of a system's observable electron density distribution function. An electron density distribution of a molecule is a probability distribution that describes the average manner in which the electronic charge is distributed throughout real space in the attractive field exerted by the nuclei. According to QTAIM, molecular structure is revealed by the stationary points of the electron density together with the gradient paths of the electron density that originate and terminate at these points.

QTAIM recovers the central operational concepts of the molecular structure hypothesis, that of a functional 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 the topology of the electron density. In addition to bonding, AIM 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 local attractor of the electron density. In QTAIM an atom is defined as a proper open system, i.e. a system that can share energy and electron density, which is localized in the 3D space. The mathematical study of these features is usually referred in the literature as charge density topology.

QTAIM rests on the fact that the dominant topological property of the vast majority of electron density distributions is the presence of strong maxima 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's gradient 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 zero flux in the gradient vector field of the electron density, they have some unique quantum mechanical properties compared to other subsystem definitions, including unique electronic kinetic energy, the satisfaction of an electronic virial theorem analogous to the molecular electronic virial 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 other model or theory in Chemistry

History
In the 19th century molecular structure hypothesis was the leading theory of experimental chemistry. This hypothesis was the main way of organizing observations taken from experimental chemistry, and was still used for a long period of time. This hypothesis, however, became obsolete, because it failed to relate to quantum mechanics, which is the physics behind the motions of nuclei and electrons in atoms and bonds. It was believed by some that the hypothesis of molecular structure was beyond its theoretical definition despite its usefulness.

The new formed theory of Richard Feynman and Julian Schwinger helps us to discover what an atom is in a molecule, and its properties. The new theory helps to describe the observed topology of the distribution of electronic charge, which shows a unique partitioning of the total system into a set of bounded regions. The characteristics of the molecule are described by its structure and substituents. This relationship between characteristics and structure allows for a greater understanding of quantum mechanics, which can be used to predict other properties in molecules.

This new resulting theory of quantum mechanics combines the concepts of the molecular structure hypothesis and the definition of bonds that link atoms and make the structure. The new theory is able to quantify the bonds that link each atom and provide a meaningful understanding to the concepts of chemistry. Thus, enabling further applications of theory. This will allow experiments to be performed electronically rather than in the laboratory, and thus being more efficient in all aspects of the process.

The theory of atoms in molecules allows others to have one of the most pertinent observations in chemistry, that of substituents with a very specific set of properties.

The late Professor Richard Bader and research group at McMaster University created the Quantum Theory of Atoms in Molecules over several years in the 1960's. The research began by analyzing theoretical calculated electron densities of structurally simple molecules and comparing these results with the 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.

Applications
QTAIM is applied to the description of certain organic crystals with unusually short distances between neighboring molecules as observed by X-ray diffraction. For example in the crystal structure of molecular chlorine the experimental Cl...Cl distance between two molecules is 327 picometres which is less than the sum of the van 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 of metallic hydrogen in much the same way. The theory is also applied to so-called hydrogen-hydrogen bonds as they occur in molecules such as phenanthrene and chrysene. In these compounds the distance between two ortho hydrogen atoms again is shorter than their van der Waals radii and according to in silico experiments based on this theory, a bond path is identified between them. Both hydrogen atoms have identical electron density and are closed shell and therefore they are very different from the so-called dihydrogen bonds which are postulated for compounds such as (CH3)2NHBH3 and also different from so-called agostic interactions. In mainstream chemistry close proximity of two nonbonding atoms leads to destabilizing steric repulsion but in QTAIM the observed hydrogen hydrogen interactions are in fact stabilizing. It is well known that both kinked phenanthrene and chrysene are around 6 kcal/mol (25 kJ/mol) more stable than their linear isomers anthracene and tetracene. One traditional explanation is given by Clar's rule. QTAIM shows that a calculated stabilization for phenanthrene by 8 kcal/mol (33 kJ/mol) 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 studied in QTAIM is biphenyl. Its two phenyl rings are oriented at a 38° angle with respect to each other with the planar molecular geometry (encountered in 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 rotation barrier are steric repulsion between the ortho-hydrogen atoms (planar) and breaking of delocalization of pi density over both rings (perpendicular).

In QTAIM the energy increase on decreasing the dihedral angle from 38° to 0° is a summation of several factors. Destabilizing factors are the increase in bond 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. The 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. 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 are not questioned but the stabilizing energy derived from it is.

Inductive Effect
Inductive effect can be easily studied through QTAIM specifically using the transmission of charge through a molecule based on electron density movement. In QTAIM, a molecule is divided into a number of atomic basins using the zero-flux surface determined by the electron density. Atomic properties, such as energy and charge can be found through integration of the atomic basins. The sum of all of the atomic properties results in the property of the molecule. Critical points in can be used to identify structural features, such as the bond critical point (BCP), which is the identifying characteristic of a bonding interaction between two atoms.

Topology Effect
The use of QTAIM allows us to create topological maps of the electron probability densities in molecules. These maps can be used to predict reactivity and geometric structure of atoms in molecules. Each topological feature whether a maximum, a minimum or a saddle has an associated critical point where the first derivative of the function is equal to 0. The critical points at which the gradient of the electron density vanishes define bonds, rings, and cages. The rank of a critical point, denoted by ω, is the number of nonzero curvatures, its signature, denoted by σ, is the sum of their algebraic signs. The critical point is labeled by giving the duo of values (ω, σ). The majority have charged distributions for molecules at or in the neighborhood of energetically stable nuclear configurations are all of rank three. A critical point with ω less than 3 is said to be degenerate. A degenerate critical point is unstable in the sense that a small change in ρ(r) caused by a nuclear displacement causes it either to vanish or to bifurcate into a number of nondegenerate, i.e., stable, critical points with ω = 3. Some possible rank three critical points are: (3,-3) All curvatures of ρ(r) at the critical point are negative and ρ(r) is a local maximum at rc. (3,-1) Two curvatures are negative and ρ(r) is a maximum at rc in the plane deﬁned by the two associated axes. The third curvature is positive and ρ(r) is a minimum at rc along the axis perpendicular to this plane. (3,+1) Two curvatures are positive and ρ(r) is a minimum at       rc in the plane deﬁned by the two associated axes. The third curvature is negative and ρ(r) is a maximum at       rc along the axis perpendicular to this plane. (3,+3) All curvatures are positive and ρ(r) is a local minimum at rc.