  A theoretical description of time correlation functions for electron
properties in the presence of a positive ion of charge number Z is given. The
simplest case of an electron gas distorted by a single ion is considered. A
semi-classical representation with a regularized electron - ion potential is
used to obtain a linear kinetic theory that is asymptotically exact at short
times. This Markovian approximation includes all initial (equilibrium) electron
- electron and electron - ion correlations through renormalized pair
potentials. The kinetic theory is solved in terms of single particle
trajectories of the electron - ion potential and a dielectric function for the
inhomogeneous electron gas. The results are illustrated by a calculation of the
autocorrelation function for the electron field at the ion. The dependence on
charge number Z is shown to be dominated by the bound states of the effective
electron - ion potential. On this basis, a very simple practical representation
of the trajectories is proposed and shown to be accurate over a wide range
including strong electron - ion coupling. This simple representation is then
used for a brief analysis of the dielectric function for the inhomogeneous
electron gas.
