Analysis of the Electronic Exchange in Atoms

The Hartree-Fock exchange and the statistical free-electron approximation are analyzed and compared for a number of atoms. The exchange energy is calculated in the two approximations---in both cases by using accurate Hartree-Fock orbitals---and split up into contributions from individual electronic shells and also into self-interaction and interelectronic parts. The results are conveniently expressed by the ratio of the various quantities in the two approximations. It is found that these parameters vary in a simple and regular way, which can be interpreted by means of an extended free-electron model. It is, e.g., found that the parameter of the total self-interaction is significantly larger than that of the total interelectronic exchange. This explains in a simple way the principal variation with the atomic number of the parameter in the so-called Xα method. The analysis is only intended to yield some insight into the exchange phenomenon, and no new---local or nonlocal---exchange potential is suggested.