Second-order hyperfine correction to H, D, and 3 He energy levels

The complete second-order hyperfine-interaction correction is calculated for centroid energy levels of H, D, and 3 He atoms. For 3 He, the corrections of - 2 . 075 kHz and - 0 . 305 kHz beyond the leading hyperfine-mixing contribution are obtained for the 2 1 S and 23S states, respectively. These results shift the nuclear charge radii difference derived from the 3 He-4He isotope shift and largely resolve the previously reported disagreement between the muonic and electronic helium determinations [van der Werf et al., arXiv:2306.02333; Schuhmann et al., arXiv:2305.11679].