Formal approach to deriving analytically asymptotic formulas for static polarizabilities of atoms and ions in Rydberg states

On the basis of analytical expressions in the Fues model potential approach for the second-order Stark effect on single-electron Rydberg states in atoms and ions, general equations are derived for coefficients of polynomials in powers of the principal quantum number n in asymptotic presentations of static scalar and tensor dipole polarizabilities. The power dependence for polarizabilities of isolated Rydberg states | nl > at n >> 1 scales as n(7), in contrast with that of polarizabilities for hydrogenic states, degenerate in the orbital quantum number l <= n -1, which scales as n(6). This difference is demonstrated analytically in the asymptotic dependencies of the second-order matrix elements, determining the Stark shifts of the isolated and degenerate states. Analytical equations for polynomial coefficients use the data on quantum defects determined from the level energies of corresponding series of states. Numerical values of coefficients are presented for S-, P-and D-series of Rydberg states in neutral atoms of alkali-metal elements and helium in comparison with existing data of the literature. The analytical equations are also used for determining numerical values of coefficients in asymptotic polynomials for polarizabilities of Rydberg states in positive ions of alkaline-earth-metal elements.