Atoms in Molecules

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The Atoms in Molecules or Atoms-in-Molecules 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. 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 flux of the gradient of the electron density is zero. 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.
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.