Finite-field calculation of static polarizabilities and hyperpolarizabilities of ${\text{In}}^{+}$ and Sr

The dipole polarizabilities, dipole hyperpolarizabilities, quadrupole moments, and quadrupole polarizabilities of the $5s^2{}^{1}S$ and $5s5p{}^3{P}^o$ states of ${\mathrm{In}}^{+}$ and Sr are calculated by using the finite-field method. The electron correlation effect and the basis set convergence are investigated in the relativistic coupled-cluster and configuration interaction calculations in order to obtain polarizabilities of high accuracy. Comparative study of the fully and scalar relativistic calculations reveals the effect of the spin-orbit coupling on the dipole polarizabilities of ${\mathrm{In}}^{+}$ and Sr. The blackbody-radiation shifts of the clock transition $5s^2{}^1{S}_0-5s5fp{}^3{P}_0^o$ due to the dipole polarizabilities, quadrupole polarizabilities, and dipole hyperpolarizabilities are evaluated to be 0.017, $8.33\times {10}^{-10}$, and $1.93\times {10}^{-17}$ Hz for ${\mathrm{In}}^{+}$ and 2.09 and $5.82\times {10}^{-8}$, and $1.69\times {10}^{-15}$ Hz for Sr. The blackbody-radiation shifts from the quadrupole polarizabilities and dipole hyperpolarizabilities are significantly less than that from the dipole polarizabilities and therefore can be safely omitted for the quoted ${10}^{-18}$ uncertainty of the optical frequency standard of ${\mathrm{In}}^{+}$ and Sr.