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As a typical one-neutron halo nucleus, $^{11}$Be holds unique significance in atomic and nuclear physics research. The nucleus comprises a tightly bound $^{10}$Be core and a loosely bound valence neutron, forming an exotic nuclear configuration that exhibits remarkable differences in both magnetic and charge radii compared to conventional nuclei, thereby establishing a unique platform for investigating nuclear-electron interactions. This study focuses on the helium-like $^{11}$Be$^{2+}$ ion, employing the relativistic configuration interaction (RCI) method combined with high-order $B$-spline basis functions to systematically calculate the energies and wavefunctions of the $n\,^{3}\!S_1$ and $n\,^{3}\!P_{0,1,2}$ states up to principal quantum number $n=8$. By directly incorporating the nuclear mass shift operator $H_M$ into the Dirac-Coulomb-Breit (DCB) Hamiltonian, this work achieves a comprehensive treatment of relativistic effects, Breit interactions, and nuclear mass corrections for $^{11}$Be$^{2+}$. The results demonstrate that the energies of states with $n\leq5$ converge to eight significant digits, showing excellent agreement with existing NRQED values, such as $-9.298\,711\,91(5)$ a.u. for the $2\,^{3}\!S_1$ state. The nuclear mass corrections are on the order of $10^{-4}$ a.u. and decrease with increasing principal quantum number.
Using the high-precision wavefunctions, the electric dipole oscillator strengths for $k\,^3\!S_1 \rightarrow m\,^3\!P_{0,1,2}$ transitions ($k \leq 5$, $m \leq 8$) were determined, with results for low-lying excited states ($m\leq4$) accurate to six significant digits, providing reliable data for evaluating transition probabilities and radiative lifetimes. Furthermore, the dynamic electric dipole polarizabilities of the $n'\,^3\!S_1$ ($n' \leq 5$) states were calculated via the sum-over-states method. The static polarizabilities exhibit a significant increase with principal quantum number. For the $J=1$ state, the difference in polarizability between the magnetic sublevels $M_J=0$ and $M_J=\pm1$ is three times the tensor polarizability. In the calculation of dynamic polarizabilities, the precision reaches $10^{-6}$ in non-resonant regions, whereas achieving the same accuracy near resonance requires higher energy precision. These high-precision computational results provide crucial theoretical foundations and key input parameters for evaluating Stark shifts in high-precision measurements, simulating light-matter interactions, and investigating single-neutron halo nuclear structures. -
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