The Atoms in Molecules or Atoms-in-Molecules or Quantum Theory of Atoms in Molecules (Qtaim) approach is a quantum chemical model that characterizes the chemical bonding of a system based on the topology of the quantum charge 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. Developed by Professor Richard Bader since the early 60's, during the past decades QTAIM has gradually become a theory for addressing possible questions regarding chemical systems, in a variety of situations hardly handled before by any other model or theory in Chemistry ^[1]. 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. Each atom acts as a local attractor of the electron density, and therefore it can be defined in terms of the local curvatures of the electron density. The mathematical study of these features is usually referred in the literature as charge density topology. Nevertheless, the term topology is used in a different sense in Mathematics. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. ... A chemical bond is the physical process responsible for the attractive interactions between atoms and molecules, and that which confers stability to diatomic and polyatomic chemical compounds. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Charge density is the amount of electric charge per unit volume. ... Richard F. W. Bader is a quantum chemist, noted for his work on the Atoms in Molecules approach. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... Look up model in Wiktionary, the free dictionary. ... For other uses, see Chemistry (disambiguation). ... Properties For other meanings of Atom, see Atom (disambiguation). ... Electron density is the measure of the probability of an electron being present at a specific location. ... The space we live in is three-dimensional space. ... In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

According to the theorems of QTAIM, the molecular structure is given by the stationary points of the electron density. In mathematics, theory is used informally to refer to a body of knowledge about mathematics. ... Stationary points (red pluses) and inflection points (green circles). ...

Main results

The major conclusions of the Freakshow approach are:

• A molecule can be uniquely divided into a set of atomic volumes. These volumes are divided by a series of surfaces through which which the gradient vector field of the electron density has no flux. Atomic properties such as atomic charge, dipole moment, and energies can be calculated by integrating their corresponding operators over the atomic volume.
• Two atoms are bonded if their atomic volumes share a common interatomic surface, and there is a (3, −1) critical point on this surface. A critical point is defined as a point in space where the gradient is zero. A (3, −1) critical point is defined as a critical point at which two of the eigenvalues of the Hessian matrix at the critical point are negative, while the other eigenvalue is positive. In other words, a bonding critical point is a first-order saddle point in the electron density scalar field. A bond path the line along which the electron density is a maximum with respect to a neighboring line. Along the associated virial path the potential energy is maximally stabilizing.
• The interatomic bonds are classified as either closed shell or shared, if the Laplacian of the electron density at the critical point is positive or negative, respectively.
• Geometric bond strain can be gauged by examining the deviation of the bonding critical point from the interatomic axis between the two atoms. A large deviation implies larger bond strain.

In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In mathematics, the Hessian matrix is the square matrix of second order partial derivatives of a function. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

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 micrometres which is less than the sum of the van der Waals radii of 260 micrometres. 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. For other uses, see Crystal (disambiguation). ... X-ray crystallography is a technique in crystallography in which the pattern produced by the diffraction of x-rays through the closely spaced lattice of atoms in a crystal is recorded and then analyzed to reveal the nature of that lattice. ... Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... General Name, symbol, number chlorine, Cl, 17 Chemical series halogens Group, period, block 17, 3, p Appearance yellowish green Standard atomic weight 35. ... The van der Waals radius of an atom is the radius of an imaginary hard sphere which can be used to model the atom for many purposes. ... Metallic hydrogen results when hydrogen is sufficiently compressed and undergoes a phase change; it is an example of degenerate matter. ...

The theory is also applied to so-called hydrogen-hydrogen bonds ^[2] 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. Phenanthrene is a polycyclic aromatic hydrocarbon composed of three fused benzene rings--as the above formula shows. ... Chrysene is a polycyclic aromatic hydrocarbon (PAH) with the molecular formula C18H12 that consists of four fused benzene rings. ... In Silico is a full length artist album by Deepsky View From a Stairway Jareths Church The Mansion World (Deepskys Trippin In Unknown Territory Mix) Ride Three Sheets to the Wind Atia Metro Smile Cosmic Dancer (2002 remix) Until the End of the World Let Me Live Categories... Example of a sodium electron shell model An electron shell, also known as a main energy level, is a group of atomic orbitals with the same value of the principal quantum number n. ... In chemistry, a dihydrogen bond is a kind of hydrogen bond, an interaction between a metal hydride bond and an OH or NH group or another proton donor. ... The word agostic was first coined for organometallic chemistry by Malcolm Green and Maurice Brookhart to describe a weak bonding interaction between a transition metal and a C-H bond on a carbon atom that is held close to the metal center through an additional Ï€- or σ-bond. ...

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/mole 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/mole is the result of destabilization of the compound by 8 kcal/mol originating from electron transfer from carbon to hydrogen, offset by 12.1 kcal 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. The steric effect of tri-(tert-butyl)amine makes electrophilic reactions, like forming the tetraalkylammonium cation, difficult. ... kcal is an abbreviation for kilocalorie, which is equivalent to 1 Calorie, or about 4. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... In chemistry, isomers are molecules with the same chemical formula and often with the same kinds of chemical bonds between atoms, but in which the atoms are arranged differently (analogous to a chemical anagram). ... In chemistry, anthracene is a solid polycyclic aromatic hydrocarbon consisting of three benzene rings derived from coal-tar. ... Tetracene, also called naphthacene and 2,3-benzanthracene, is a polycyclic aromatic hydrocarbon. ... An illustration of typical polycyclic aromatic hydrocarbons. ...

Another molecule studied in Qtaim is biphenyl. Its two phenyl rings are oriented in an 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 and the perpendicular one destabilized by 2.5 kcal/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). Image File history File links HydrogenhydrogenbondingE.svg‎ Hydrogen hydrogen bonding in phenanthrene File historyClick on a date/time to view the file as it appeared at that time. ... Biphenyl (or Diphenyl or 1,1-Biphenyl or lemonene) is a solid compound that forms colourless to yellowish crystals. ... Geometry of the water molecule Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule, inferred from the spectroscopic studies of the compound. ... In chemistry, delocalized electrons are electrons in a molecule that do not belong to a single atom or a covalent bond. ... Electron atomic and molecular orbitals, showing a Pi-bond at the bottom right of the picture In chemistry, pi bonds (Ï€ bonds) are chemical bonds of the covalent type, where two lobes of one involved electron orbital overlap two lobes of the other involved electron orbital. ...

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. In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... In molecular geometry, bond length or bond distance is the distance between two bonded atoms in a molecule. ...

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 ^[3]. Another critic ^[4] 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.

External links

• Atoms in Molecules page at McMaster University

References

1. ^ Bader, Richard (1994). Atoms in Molecules: A Quantum Theory. USA: Oxford University Press. ISBN 978-0198558651. 
2. ^ Hydrogen - Hydrogen Bonding: A Stabilizing Interaction in Molecules and Crystals Cherif F. Matta, Jesus Hernandez-Trujillo, Ting-Hua Tang, Richard F. W. Bader Chem. Eur. J. 2003, 9, 1940 ± 1951 doi:10.1002/chem.200204626
3. ^ Molecular Recognition in Organic Crystals: Directed Intermolecular Bonds or Nonlocalized Bonding? Jack D. Dunitz and Angelo Gavezzotti Angew. Chem. Int. Ed. 2005, 44, 1766 – 1787 doi:10.1002/anie.200460157
4. ^ Polycyclic Benzenoids: Why Kinked is More Stable than Straight Jordi Poater, Ruud Visser, Miquel Sola, F. Matthias Bickelhaupt J. Org. Chem. 2007, 72, 1134-1142 doi:jo061637p

• Hydrogen Bonding without Borders: An Atoms-in-Molecules Perspective, R. Parthasarathi, V. Subramanian, and N. Sathyamurthy, J. Phys. Chem. A; 2006; 110(10) pp 3349 - 3351; (Letter)[DOI: 10.1021/jp060571z]
  http://pubs3.acs.org/acs/journals/doilookup?in_doi=10.1021/jp060571z]