Cohesive energy in ionic crystals

A satisfactory theory of the cohesive energy of ionic crystals can be based almost exclusively on Coulomb s law. If two particles i and j, having charges Zj and Zj, are placed a distance apart in vacuum, the energy of interaction between them is 

Consider a crystal, such as NaCl, in which the charges on the ions are equal and opposite in sign, z+ = —z. To calculate the cohesive energy of a cry stal that contains N ions of each sign, N ion pairs, we add the interaction of every ion with all the others. Since the 

The interaction energy of ion i with all of the others, j, is obtained by summing this expression. 

In every term of this sum, = Zj furthermore, each term in the sum is simply a number, determined by geometry, so the sum is a number that we write as Zj. Then 

Summing the energies of all the ions in the lattice yields the total energy of interaction U  

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The theorem has the important implication that intramolecular interactions can be calculated by the methods of classical electrostatics if the electronic wave function (or charge distribution) is correctly known. The one instance where it can be applied immediately is in the calculation of cohesive energies in ionic crystals. Taking NaCl as an example, the assumed complete ionization that defines a (Na+Cl-) crystal, also defines the charge distribution and the correct cohesive energy is calculated directly by the Madelung procedure. 

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