Atoms in Molecules
Encyclopedia
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 bond
ing 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 1960s, 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
. 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 curvature
s 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
.
According to the theorems of QTAIM, the molecular structure is given by the stationary points of the electron density.
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 bond
s which are postulated for compounds such as (CH3)2NHBH3 and also different from so-called agostic interaction
s.
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 isomer
s 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.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
chemical
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
model that characterizes the chemical bond
Chemical bond
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...
ing of a system based on the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
of the quantum charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
. 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
Richard Bader
Richard F. W. Bader is a quantum chemist, noted for his work on the Atoms in Molecules approach. This approach explains chemistry, particularly established chemical models such as the Lewis structure, in terms of the topology of the electron density function in three-dimensional space.He was born...
since the early 1960s, during the past decades QTAIM has gradually become a theory
Theory
The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...
for addressing possible questions regarding chemical systems, in a variety of situations hardly handled before by any other model
Scientific modelling
Scientific modelling is the process of generating abstract, conceptual, graphical and/or mathematical models. Science offers a growing collection of methods, techniques and theory about all kinds of specialized scientific modelling...
or theory in Chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
. In QTAIM an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
is defined as a proper open system, i.e. a system that can share energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
and electron density
Electron density
Electron density is the measure of the probability of an electron being present at a specific location.In molecules, regions of electron density are usually found around the atom, and its bonds...
, which is localized in the 3D
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
space. Each atom acts as a local attractor
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
of the electron density, and therefore it can be defined in terms of the local curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
s 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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
.
According to the theorems of QTAIM, the molecular structure is given by the stationary points of the electron density.
Main results
The major conclusions of the AIM 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 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 matrixHessian matrixIn mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
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 is 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.
Applications
QTAIM is applied to the description of certain organic crystalsCrystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
with unusually short distances between neighboring molecules as observed by X-ray diffraction. For example in the crystal structure
Crystal structure
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
of molecular chlorine
Chlorine
Chlorine is the chemical element with atomic number 17 and symbol Cl. It is the second lightest halogen, found in the periodic table in group 17. The element forms diatomic molecules under standard conditions, called dichlorine...
the experimental Cl...Cl distance between two molecules is 327 picometres which is less than the sum of the van der Waals radii
Van der Waals radius
The van der Waals radius, r, of an atom is the radius of an imaginary hard sphere which can be used to model the atom for many purposes. It is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics, as he was the first to recognise that atoms had a finite size and to...
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
Metallic hydrogen
Metallic hydrogen is a state of hydrogen which results when it is sufficiently compressed and undergoes a phase transition; it is an example of degenerate matter. Solid metallic hydrogen is predicted to consist of a crystal lattice of hydrogen nuclei , with a spacing which is significantly smaller...
in much the same way.
The theory is also applied to so-called hydrogen-hydrogen bonds as they occur in molecules such as phenanthrene
Phenanthrene
Phenanthrene is a polycyclic aromatic hydrocarbon composed of three fused benzene rings. The name phenanthrene is a composite of phenyl and anthracene. In its pure form, it is found in cigarette smoke and is a known irritant, photosensitizing skin to light...
and chrysene
Chrysene
Chrysene is a polycyclic aromatic hydrocarbon with the molecular formula that consists of four fused benzene rings. It is a natural constituent of coal tar, from which it was first isolated and characterized...
. In these compounds the distance between two ortho hydrogen atoms again is shorter than their van der Waals radii and according to in silico
In silico
In silico is an expression used to mean "performed on computer or via computer simulation." The phrase was coined in 1989 as an analogy to the Latin phrases in vivo and in vitro which are commonly used in biology and refer to experiments done in living organisms and outside of living organisms,...
experiments based on this theory, a bond path is identified between them. Both hydrogen atoms have identical electron density and are closed shell
Electron shell
An electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" , followed by the "2 shell" , then the "3 shell" , and so on further and further from the nucleus. The shell letters K,L,M,.....
and therefore they are very different from the so-called dihydrogen bond
Dihydrogen bond
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 first example of this phenomenon is credited to Brown and Heseltine. They observed intense absorptions in the IR bands at 3300 and...
s which are postulated for compounds such as (CH3)2NHBH3 and also different from so-called agostic interaction
Agostic interaction
Agostic interaction is a term in organometallic chemistry for the interaction of a coordinately-unsaturated transition metal with a C-H bond, when the two electrons involved in the C-H bond enter the empty d-orbital of a transition metal, resulting in a three-center two-electron bond. Many...
s.
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
Calorie
The calorie is a pre-SI metric unit of energy. It was first defined by Nicolas Clément in 1824 as a unit of heat, entering French and English dictionaries between 1841 and 1867. In most fields its use is archaic, having been replaced by the SI unit of energy, the joule...
/mol
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...
(25 kJ/mol) more stable than their linear isomer
Isomer
In chemistry, isomers are compounds with the same molecular formula but different structural formulas. Isomers do not necessarily share similar properties, unless they also have the same functional groups. There are many different classes of isomers, like stereoisomers, enantiomers, geometrical...
s anthracene
Anthracene
Anthracene is a solid polycyclic aromatic hydrocarbon consisting of three fused benzene rings. It is a component of coal-tar. Anthracene is used in the production of the red dye alizarin and other dyes...
and tetracene
Tetracene
Tetracene, also called naphthacene, is a polycyclic aromatic hydrocarbon. It has the appearance of a pale orange powder. Tetracene is the four-ringed member of the series of acenes, the previous one being anthracene and the next one being pentacene.Tetracene is a molecular organic semiconductor,...
. 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.
Biphenyl
Biphenyl is an organic compound that forms colorless crystals. It has a distinctively pleasant smell. Biphenyl is an aromatic hydrocarbon with a molecular formula 2...
. Its two phenyl rings are oriented at a 38° angle with respect to each other with the planar molecular geometry
Molecular geometry
Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...
(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
Dihedral angle
In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...
from 38° to 0° is a summation of several factors. Destabilizing factors are the increase in bond length
Bond length
- Explanation :Bond length is related to bond order, when more electrons participate in bond formation the bond will get shorter. Bond length is also inversely related to bond strength and the bond dissociation energy, as a stronger bond will be shorter...
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.
External links
- Atoms in Molecules page at McMaster University
- Bader Charge Analysis, freeware program for Bader and Voronoi charge analyses for a charge density on a grid computed by virtually any ab initio code