Application of the correlated B-spline basis functions to the leading relativistic and QED corrections of helium

Abstract

B-spline functions have been widely used in computational atomic physics. Different from the traditional B-spline basis (a simple product of two B-splines), the recently developed correlated B-spline basis functions(C-BSBF), in which the interelectronic coordinate $r_{12}$ is included explicitly, have greatly improved the computational accuracy of polarizability [S. J. Yang \textit{et al}., Phys. Rev. A \textbf{95}, 062505 (2017)] and bethe logarithm [ S. J. Yang \textit{et al}., Phys. Rev. A \textbf{100}, 042509 (2019)] for singlet states of helium. Here, we report the extension of the C-BSBF to the leading relativistic and QED correction calculations for energy levels of the $1{}^{1}S$, $2{}^{1}S$, $2{}^{3}S$, and $3{}^{3}S$ states of helium. The relativistic kinetic term $p_1^4$, contact potential ${\delta }^3(r_1)$, ${\delta }^3(r_{12})$ and Araki-Sucher correction $⟨1/r_{12}^3⟩$ are calculated by using the global operator method, in which $r_{12}^n$ and $r_{12}^n\ln r_{12}$ involved are calculated with the generalization of Laplace's expansions. The obtained values for the ground state are $\delta E_{rel}/{\alpha }^2=-$1.951 754 7(2) and $\delta E_{QED}/{\alpha }^3=$57.288 165(2), consistent with previous results, which opens the possibility of calculating higher-order relativistic and QED effects using the C-BSBF.