  The present work is a generalisation of the dynamical screening factor
presented in [1] to consider an atom located at an arbitrary position within
the fullerene. A more elaborated investigation into the case where the atom is
located at the centre is performed and compared with quantum mechanical
calculations for dynamical screening factor of Ar@C$_{60}$ [2] and Mg@C$_{60}$
[3]. The $\pi$ and $\sigma$ plasmons of the fullerene are accounted for in a
modified screening factor to improve correspondence with the quantum
calculations. The spatial dependence of the screening factor was explored with
Ar@C$_{60}$ and Ar@C$_{240}$ and found to depend significantly on the radial
distance of the atom from the centre of the fullerene. A spatial averaging of
the screening factor is presented.
