are coupled so that the $6s^2$ valence shell has zero value of angular momentum. Using Wick’s theorem, we write the matrix element (\[Eq:single-electron-operator\]) in the multi-electron state as an expectation value in the hole orbital (see Appendix [\[app:2nd-quantization\]]{} for derivation) $${\ensuremath{\langle \psi_{J,\,M}|}} T^e_{k,\,\mu} {\ensuremath{|\psi_{J,\,M}\rangle}}  = -{\ensuremath{\langle \phi_{J,\,-M}|}}t_{k,\,\mu}^e {\ensuremath{|\phi_{J,\,-M}\rangle}},\label{eq:multi-electron-state}$$ where ${\ensuremath{|\phi_{J,\,-M}\rangle}}$ represents the $4f_{7/2}$ hole orbital with $J$ and $-M$ being the electron’s angular momentum and magnetic quantum number. The electronic tensors $T_{k,\,\mu}^e$ are given by Eq. (\[Eq:HF-general\]). In Appendix [\[app:2nd-quantization\]]{}, we show that the reduced matrix elements are related as $$\begin{aligned}
\label{eq:relation_multi_hole}
{\ensuremath{\langle \psi_{J}|}}|T^e_{k,\mu}|{\ensuremath{|\psi_{J}\rangle}}=(-1)^{k+1}{\ensuremath{\langle \phi_{J}|}}|t_k^e|{\ensuremath{|\phi_{J}\rangle}}\,,\end{aligned}$$ with reduced matrix elements specified in Appendix \[app:matrix-element\]. The transition from a multi-electron state to the single-electron hole orbital greatly simplifies our calculation since it only requires the one-electron $4f_{7/2}$ orbital, which can be easily obtained self-consistently with the DHF method. Our computed values of the first- and second-order hyperfine constants are listed in the first row of Table \[Table:HFS\].

[cccccccc]{} Method & $A$ & $B$& $C/{\Omega}$ & $D/{\Pi}$ & $\eta_{\mu\mu}$& $\eta_{\mu{Q}}$ &$\eta_{QQ}$\
\
DHF & $-239$ & $-5330$ &  $4.53\times 10^{-4} $ & $2.00\times{10^{-4}}$ & $-1.50\times{10^{-2}}$ & $-0.112$ & $-0.209$\
DHF (GRASP) & $-252$ & $-5622$ &  $4.83 \times 10^{-4}$ & $2.25\times{10^{-4}}$ & $-1.91\times{10^{-2}}$ 