Spherically Averaged Exchange Potential

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 The Coulomb Potential is the main component of the potential, but there is another component that makes a significant contribution, especially as the distance from the origin of the atom increases. This is the exchange contribution to the potential. The Mattheiss prescription follows the method proposed by Slater which bases the value of the exchange potential on analytic calculations for a free electron gas. This is called the Xa approximation:

.

The value of the statistical exchange parameter a is, in general, 0.7 for all but the lightest atoms where it rises to about 0.78. A compilation of a values for up to Z=41 can be found from Karlheinz Schwarz, Phys. Rev. B 5 Num. 7, 2466 (1972).

The appropriate value of r(r) to use for the spherically averaged potential about a particular atom may be found as in the previous case of the Coulomb potential. Contributions to r(r) from neighboring atoms take the form

and are added together with the density of the atom of interest to find the total density.

The total spherically averaged potential about each atom is then taken to be

.

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Created: April 9, 1999 ---- Last Updated: April 12, 1999
By Mark D. Pauli