Finite-field calculation of the polarizabilities and hyperpolarizabilities of Al$^{+}$

Abstract: In this study, accurate static dipole polarizability and hyperpolarizability are calculated for Al$^+$ ground state 3s$^{2}$ $^{1}$S$_{0}$ and excited state $3s3p$ $^{3}$P$_{J}$ with $J$=0, 1, 2. The finite-field computations use energies obtained with the relativistic configuration interaction approach and the relativistic coupled-cluster approach. Excellent agreement with previously recommended values is found for the dipole polarizability of Al$^{+}$ ground state 3s$^{2}$ $^{1}$S$_{0}$ and excited state $3s3p$ $^{3}$P$_{0}$ as well as the hyperpolarizability of the ground state 3s$^{2}$ $^{1}$S$_{0}$. The recommended values of the dipole polarizability of the Al$^{+}$ $3s3p$ $^{3}$P$_{1}$ and $^{3}$P$_{2}$ and the hyperpolarizability of Al$^{+}$ $3s3p$ $^{3}$P$_{0}$, $^{3}$P$_{1}$, and $^{3}$P$_{2}$ are also given. The impacts of the relativity and spin-orbit coupling are elucidated by analyzing the angular momentum dependence of the dipole polarizability and the hyperpolarizability and comparing the fully and scalar relativistic calculated data. It is shown that the impact of the relativity and spin-orbit coupling are small for the dipole polarizability but become significant for the hyperpolarizability. Finally, the black-body radiation shifts contributed by the dipole polarizability and hyperpolarizability respectively are evaluated for transitions of Al$^{+}$ 3s$^{2}$ $^{1}$S$_{0}$ to $3s3p$ $^{3}$P$_{J}$ with $J$=0, 1, 2.