Gauge-invariance method for accurate atomic-physics calculations: Application to relativistic polarizabilities

An accurate method for variational calculations of atomic properties is constructed using the principle that the results must be invariant under gauge transformations. The technique is used with multiexponential basis sets and provides very fast convergence. The method is applied to the relativistic calculation of the dipole polarizability in hydrogenic ions. The results obtained with 21 relativistic Slater-type basis functions are accurate to 13 digits for nuclear charges from Z=1 to Z=115, therefore overcoming the large loss of accuracy for large values of Z, which is typical of single-exponential basis sets. The results are compared with previous calculations and the coefficients of a relativistic expansion of the polarizability in powers of (${\alpha }_{\mathrm{FS}\mathrm{Z}{)}^2}$ up to (${\alpha }_{\mathrm{FS}\mathrm{Z}{)}^8}$ are obtained, where ${\alpha }_{\mathrm{FS}}$ is the fine-structure constant.