Electrostatic Bonding Forces between Atoms
This Demonstration examines bonding forces by placing a charge between two separated ions. The contour represents the curve where the repulsive and attractive forces are counterbalanced by the charge. Changing the charge ratio lets you examine heteronuclear () to ionic () species.
This Demonstration examines the electronic bonding forces between two ions A and B separated by a distance along the axis. In the absence of any electrons, the repulsive force between the ions acting along the A-B axis would be
Now, suppose we put a single charge, , at some point P in the - plane. We then have two attractive forces and between the ions and the point charge. If we consider the component of along the A-B bond axis, we can define a bonding force,
where and denote the angles PAB and PBA, respectively. In a molecule, the bonding force is then computed by taking the expectation value over the electronic state ; however, we can gain some insight into bonding by simply looking at the functional form of . For this we follow the discussion in Barry, Rice, and Ross  that in turn is based upon ideas introduced by Kajans and Berlin . This is also discussed in the classic book by Hirschfelder, Curtiss, and Bird .
First, whenever the charge is located between the two ions, there is an attractive force, since and , hence . Likewise, if any charge is placed "behind" either ion, there is a destabilizing force, since . The curve where is then the demarcation between the bonding and anti-bonding regions.
The boundary surface given by
where and depends on only one physical parameter, the charge ratio between the two ions. By varying this parameter you can see how bonding occurs between different ionic species. corresponds to any homonuclear diatomic species, while various heteronuclear species correspond to . For example, corresponds to NaCl and corresponds to HCl.
References  R. S. Berry, S. A. Rice, and J. Ross, Physical Chemistry, New York: Wiley, 1980.  K. Fajans and T. Berlin, "Quantization of Molecules, Inter- and Intramolecular Forces," Phys. Rev. 63(7-8), 1943 pp. 309–312.  J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, New York: Wiley, 1954.