Extended Data Table 1 Spin Hamiltonian coefficients, spin-structure frequencies and spin-frequency derivatives

From: Precise test of quantum electrodynamics and determination of fundamental constants with HD+ ions

  1. \({ {\mathcal E} }_{k}^{{\prime} }\) (\({ {\mathcal E} }_{k}\)) are the updated coefficients of the spin Hamiltonian35 of the upper (lower) rotational level, in MHz. \({f}_{{\rm{spin}},i}^{({\rm{theor}})}\) are theoretical spin frequencies in MHz. γ are the dimensionless sensitivities of the spin frequencies to the spin Hamiltonian coefficients. \({\gamma }_{i,k}^{{\prime} }=\partial {f}_{{\rm{spin}},i}^{({\rm{theor}})}/\partial { {\mathcal E} }_{k}^{{\prime} }\) refer to the upper state and \({\gamma }_{i,k}=-\partial {f}_{{\rm{spin}},i}^{({\rm{theor}})}/\partial { {\mathcal E} }_{k}\) to the lower state. The entries for line 19 are decimal representations of rational values (see equation (6) in ref. 37). Note that because of the tracelessness of the spin Hamiltonian36, \({\sum }_{i}{d}_{i}{\gamma }_{i,k}=0\) and \({\sum }_{i}{d}_{i}^{{\prime} }{\gamma }_{i,k}^{{\prime} }=0\), where di = (2F(i) + 1)/36 and \({d}_{i}^{{\prime} }\) = (2F′(i) + 1)/36 are the degeneracies of the respective spin states, and the sum is over the ten favoured transitions i = 12, …, 21.
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