Comparison of Several Exchange Potentials for Electrons in theCu+Ion

Self-consistent calculations of the ${\mathrm{Cu}}^{+}$ ion have been carried out using five different methods of approximating the Hartree-Fock exchange. These calculations have been compared with Hartree's ${\mathrm{Cu}}^{+}$ calculation to test the accuracy of the various approximations and to interpret their interrelations. The best results were obtained from two quite different methods. The first, suggested by Liberman with modifications which we have introduced, uses a different local exchange potential for each orbital and gives a very good approximation to the Hartree-Fock method, but with considerable computational difficulty. The second amounts to multiplying the local potential proportional to the ⅓ power of the electronic charge density, suggested by the senior author in 1951, by a constant factor $\alpha$ chosen to minimize the total energy. This second method is much simpler to apply than the first and gives very nearly as good orbitals, as well as a very good total energy, but gives poor one-electron energies for the x-ray levels. The reasons for the different results are discussed. The latter method, which has been empirically arrived at by a number of the workers in the energy-band field, is probably the most useful one for practical calculation.