The relativistic effects on the dipole polarizabilities and hyperpolarizabilities are considered for different kinds of energy levels in hydrogen- and helium-like atoms. The relativistic Coulomb Green's function is used for calculating the susceptibilities of the ground-state hydrogen up to terms of order (α Z)10. Both relativistic and interelectronic corrections are determined for the ground state of helium. The formulas are given for polarizability and hyperpolarizability in the relativistic "screened-charge" approximation. The anticrossing of the triplet 3PJ states with zero magnetic quantum number is studied on the basis of perturbation theory for degenerate states. General expressions are given for the dipole matrix elements, up to the fourth order in field strength, within the basis of close fine-structure substates with equal angular momenta L and different total momenta J. The calculation of the higher order matrix elements is carried out with the use of the Green's function in the model potential approximation. PACS Nos.: 31.10Dk, 31.15Ar, 31.30Jv, 32.10-f, 31.25Eb
Relativistic Coulomb problem: Analytic upper bounds on energy levels