Precision calculation of fine structure in helium and Li⁺

The fine structure constant α can be extracted from high-precision spectroscopy of the 2³P_J fine structure splittings in helium and light helium-like ions. In this work, the 2³P_J fine structure splittings of helium and Li⁺ ion are calculated, including relativistic and QED corrections of order mα⁴, mα⁴(m/M), mα⁵, mα⁵(m/M), and Douglas-Kroll operators of mα⁶ and mα⁶(m/M), which provide an independent verification for the previous calculations performed by Drake Can. J. Phys. 80 1195 (2002) and by Pachucki and Yerokhin Phys. Rev. A 79 062516 (2009); Phys. Rev. Lett. 104 070403 (2010); Can. J. Phys. 89 1139 (2011). The results of the three groups agree with each other.