The dielectric function of the uniform electron gas: approximation to the Hartree-Fock ladder series

The equation of motion of the density fluctuation Green function is cast into a form from which an integral equation for the full Hartree-Fock proper polarisation function Pi (q omega ) is extracted. An iterative solution procedure which can be used to write down approximate 'all order' solutions is presented. It is shown that even the lowest order iteration satisfies the full Hartree-Fock compressibility sum rule exactly. An approximate procedure to calculate the fully iterated all order dynamic polarisability Pi (q omega ) is presented. The small r s behaviour of Pi (q,0) yields a local field which is in excellent agreement with that of the first-order HF result, and provides estimates of the second-order HF function Pi 2 (q,0) which has not been calculated directly, as yet. Further, it is shown that if the Coulomb potential is replaced by a certain type of Fermi surface average (FSA), the resulting very simple polarisation function Pi FSA (q omega ) provides a surprisingly good approximation to Pi (q omega ).