Extended Data Fig. 4: Hyperfine structure energy levels of HD⁺ in the two relevant rovibrational levels.

Shown are the values \({E}_{{{{\rm{spin}}}},p}^{({{{\rm{theor}}}})}/h\) for the ground vibrational level (v = 0, N = 0) (left) and \({E}_{{{{\rm{spin}}}},q}^{({{{\rm{theor}}}})}/h\) for the excited vibrational level \(({v}^{{\prime} }=5,{N}^{{\prime} }=1)\) (right) of the \({}^{2}{\Sigma }_{g}^{+}\) ground electronic state. The vertical axis is broken for better visualization. The numbers next to the arrows indicate the line numbers i(p, q), where p (q) denotes the particular lower (upper) spin state. In this work, line 12 and line 21 were measured. G₁ refers to the sum of electron and proton spin, G₂ to the sum of G₁ and deuteron spin, and F to the sum of G₂ and rotational angular momentum N. The ground rovibrational level has zero rotational angular momentum, N = 0, giving rise to 4 spin states with total angular momentum F = 0, 1, 2. The upper rovibrational level \(({v}^{{\prime} }=5,\,{N}^{{\prime} }=1)\) has 10 spin states, having values \({F}^{{\prime} }=0,\,1,\,2,\,3\). For both studied lines 12 and 21, the lower and upper hyperfine states have the same particle spin coupling \({G}_{1}={G}_{1}^{{\prime} }\) and \({G}_{2}={G}_{2}^{{\prime} }\). The line 12 measured here is analogous to line 12 in ref. ³ and to line 12 in ref. ⁴. The line 21 is analogous to line 16 in ref. ³. Spin states with total angular momentum F are (2F + 1)-fold degenerate in zero magnetic field. We denote the lines and corresponding transition frequencies in zero external fields by f_{line i}. i indicates the position in the list {f_spin,i} of the spin frequency values of all spin components in ascending order.