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B-spline functions have been widely used in computational atomic physics.Different from the traditional B-spline basis (a simple product of twoB-splines), the recently developed correlated B-spline basis functions(C-BSBF),in which the interelectronic coordinate $r_{12}$ is included explicitly, havegreatly 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)] forsinglet states of helium. Here, we report the extension of the C-BSBF to theleading relativistic and QED correction calculations for energy levels of the$1\,^1S$, $2\,^1S$, $2\,^3S$, and $3\,^3S$ states of helium. The relativistickinetic term $p_{1}^{4}$, contact potential $\delta^{3}(r_{1})$,$\delta^{3}(r_{12})$ and Araki-Sucher correction $\langle 1/r_{12}^{3} \rangle$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 ofLaplace's expansions. The obtained values for the ground state are $\deltaE_{rel}/\alpha ^{2}=-$1.951 754 7(2) and $\delta E_{QED}/\alpha^{3}=$57.288165(2), consistent with previous results, which opens the possibility ofcalculating higher-order relativistic and QED effects using the C-BSBF.

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