and dipole-quadrupole terms, there are three possible $J'$ values, while for quadrupole-quadrupole term, there are five possible $J'$ values. Among the ${\ensuremath{|\gamma{'}{J'}\rangle}}$ intermediate states, the dominant contribution comes from the configuration $4f^{13}(^2\!F^{o})6s^2$ with $J=5/2$. The electronic matrix elements of other possible intermediate states are small enough to be neglected, or the energy denominators are large. With the single intermediate state fixed, we rewrite Eq. (\[eq:2nd-order\]) as $$\begin{aligned}
W_F^{(2)}&\approx &C_{\mu{\mu}}[J',F]\times\eta_{\mu{\mu}}[\gamma{'}J']+ \nonumber \\
&& C_{\mu{Q}}[J',F]\times\eta_{\mu{Q}}[\gamma'J'] + \label{Eq:W2-individual} \\
&&C_{QQ}[J',F]\times\eta_{QQ}[\gamma{'}J']\,,\nonumber\end{aligned}$$ where the angular factors are $$\begin{aligned}
C_{{\mu\mu}}[J{'},F]&=&{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&1\end{Bmatrix}}}^2\,,\nonumber\\
C_{{\mu{Q}}}[J{'},F]&=&{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&1\end{Bmatrix}}}{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&2\end{Bmatrix}}}\,,\label{eq:2nd-order-coeff}\\
C_{{{Q}{Q}}}[J{'},F]&=&{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&2\end{Bmatrix}}}^2 \,.\nonumber\end{aligned}$$ Adding the first-order, Eq. (\[Eq:W1-individual\]) and second-order, Eq (\[Eq:W2-individual\]), corrections for individual hyperfine levels, we arrive at $$\begin{aligned}
 \label{Eq:W12-individual}
    W_6^{(1+2)}&=&W_6^{(1)}+0\times{\eta_{\mu\mu}}+0\times{\eta_{\mu{Q}}}+0\times\eta_{QQ}\,,\nonumber\\    
    W_5^{(1+2)}&=&W_5^{(1 )}+\frac{1}{98}\eta_{\mu\mu}+\frac{\sqrt\frac{5}{6}}{98}\eta_{\mu{Q}}+\frac{5}{588}\eta_{QQ}\,,\nonumber\\
    W_4^{(1+2)}&=&W_4^{(1)}+\frac{11}{882}\eta_{\mu\mu}+0\times\eta_{\mu{Q}}+0\times\eta_{QQ}\,,\nonumber\\
    W_3^{(1+2)}&=&W_3^{(1)}+\frac{1}{98}\eta_{\mu\mu}-\frac{\sqrt{\frac{2}{15}}}{49}{\eta_{\mu{Q}}}+\frac{4}{735}\eta_{QQ}\,,\\
    W_2^{(1+2)}&=&W_2^{(1)}+\frac{3}{490}\eta_{\mu\mu}-\frac{\sqrt{\frac{3}{10}}}{70}\eta_{\mu{Q}}+\frac{1}{100}\eta_{QQ}\,,\nonumber\\
    W_1^{(1+
    2)}&=&W_1^{(1)}+\frac{1}{441}\eta_{\mu\mu}-\frac{1}{{49}\sqrt{30}}{\eta_{\mu{Q}}}+\frac{3}{490}\eta_{QQ}\,.\nonumber
 \end{aligned}$$

Experimentally relevant quantities are the HFS energy intervals $\Delta{W}_F = W_{F+1} - W_F$. Explicitly,

$$\begin{aligned}
 \label{eq:DeltaW12}  
\Delta{W}_5^{(1+2)}&=&6 A+\frac{18 }{35}B+\frac{144 }{35}C+\frac{48 }{7}D-\frac{1}{98}\eta_{\mu\mu} -\frac{1}{98} \sqrt{\frac{5}{6}} \eta_{\mu{Q}}-\frac{5}{588}\eta_{{Q}{Q}} \,, \nonumber\\
\Delta{W}_4^{(1+2)}&=&5 A+\frac{1}{28}B-\frac{31 }{7}C-\frac{125 }{7}D-\frac{\eta_{\mu\mu}}{441} +\frac{1}{98} \sqrt{\frac{5}{6}} \eta_{\mu{Q}}+\frac{5}{588}\eta_{{Q}{Q}}\,,\nonumber\\
\Delta{W}_3^{(1+2)}&=&4 A-\frac{8 }{35}B-\frac{64 }{35}C+\frac{128 }{7}D+\frac{\eta_{\mu\mu}}{441}+\frac{1}{49} \sqrt{\frac{2}{15}} \eta_{\mu{Q}} -\frac{4}{735}\eta_{{Q}{Q}} \,,\\
\Delta{W}_2^{(1+2)}&=&3 A-\frac{9 }{28}B+\frac{99 }{35}C+\frac{33 }{7}D+\frac{1}{245} \eta_{\mu\mu}+(\frac{1}{70} \sqrt{\frac{3}{10}} -\frac{1}{49} \sqrt{\frac{2}{15}}) \eta_{\mu{Q}}-\frac{67}{14700}\eta_{{Q}{Q}}\,,\nonumber\\
\Delta{W}_1^{(1+2)}&=&2 A-\frac{2 }{7}B+\frac{176 }{35}C-\frac{176 }{7}D+\frac{17 }{4410}\eta_{\mu\mu} +(\frac{1}{49 \sqrt{30}}-\frac{1}{70} \sqrt{\frac{3}{10}}) \eta_{\mu{Q}}+\frac{19}{4900}\eta_{{Q}{Q}}\,.\nonumber
 \end{aligned}$$

To determine the HFS constants $A,B,C$, and $D$ from experimental measurements of $\Delta{W}_F$, in Sec. \[Sec:electronic-reduced\] we compute the second-order corrections. Further, 