Exchange- and Correlation-Energy Calculations in Finite Systems

In the local-effective-potential approximation of the Kohn-Sham self-consistent scheme applied to an atomic system, it was reported that while the calculated exchange energies for various atoms were accurate to within 10%, the correlation energies of atoms were too large by a factor of 2. A model system is set up to study the reason why the exchange energy estimates are so much better than those of the correlation energy even though they were treated on the same footing. It was found that, in using expressions derived from an infinite electron gas system to study finite systems, the major source of error comes from the fact that the low-lying levels of a finite system are discrete and have finite spacings. The replacement of summation by integration in the perturbation diagrams results in an overestimate of correlation energy, and a small error in the exchange energy in a finite system like an atom.