Relativistic corrections to the electric multipole polarisability and the shielding factor of a hydrogen-like ion

The exact values of the first relativistic corrections to the electric 2 l -pole polarisability alpha l and the shielding factor beta l of a hydrogen-like ion are calculated using the Dirac equation. It is shown that the explicit solution of the perturbed Dirac equation is not necessary to calculate the exact values of the corrections. The results for l=1 (dipole) and for l=2 (quadrupole) are: alpha 1 =(a 3 /Z 4 ( 9 / 2 - 14 / 3 ( alpha Z) 2 ), beta 1 =1/Z; alpha 2 =(3a 5 /Z 6 )(5- 293 / 40 ( alpha Z) 2 ), beta 2 =(1/3Z)(1- 2 / 5 ( alpha Z) 2 ). The corrections to the higher-multipole polarisabilities and shielding factors are also calculated.