Atoms in molecules

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 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 ^[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.

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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 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.

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.

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.

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 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 (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 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.

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]