& J
    \end{Bmatrix}\langle \gamma{J}||T^e_k||\gamma{J}\rangle\langle I||T^{n}_k||I\rangle\,,
\end{split}\label{Eq:1st-order-ME}$$ where $I$ is the nuclear spin, $J$ is the total electronic angular momentum, $F$ is the grand total angular momentum $\bm{F}=\bm{J} +\bm{I}$, and $\gamma$ stands for remaining quantum numbers.

The first-order energy corrections are conventionally expressed as linear combinations of HFS constants $A$, $B$, $C$, $D$...($k$=1,2,3,4...). The first four constants are defined as [@Beloy2008a] $$\begin{aligned}
\label{eq:hpf-constants}
     A&=&\frac{1}{IJ}\langle{T_1^n}\rangle_{I}\langle{T_1^e}\rangle_{J}=\frac{1}{IJ}\mu\langle{T_1^e}\rangle_J\,,\nonumber     \\
     B&=&4\langle{T_2^n}\rangle_I\langle{T_2^e}\rangle{_J}=2Q\langle{T_2^e}\rangle_{J}\,,\\ 
     C&=&\langle{T_3^n}\rangle_I\langle{T_3^e}\rangle{_J}=-\Omega\langle{T_3^e}\rangle{_J}\,,\nonumber\\
     D&=&\langle{T_4^n}\rangle{_I}\langle{T_4^e}\rangle{_J}=\Pi{\langle{T_4^e}}\rangle{_J} \,.\nonumber 
 \end{aligned}$$ Here, the stretched matrix element $\langle T_k^e \rangle_J$ is defined as $\langle T_k^e \rangle_J ={\ensuremath{\begin{pmatrix}J&k&J\\#4&0&J\end{pmatrix}}}{\ensuremath{\langle \gamma{J}|}}|T_{k}^e|{\ensuremath{|\gamma{J}\rangle}}$. Nuclear stretched matrix elements are proportional to the nuclear moments: $\langle{T_1^n}\rangle_I=\mu$, $\langle{T_2^n}\rangle_I=Q/2$, $\langle{T_3^n}\rangle_I=-\Omega$, and $\langle{T_4^n}\rangle_I=\Pi$.

The second-order energy correction due to hyperfine interaction reads

$$\begin{aligned}
\label{eq:2nd-order-general}
         W^{(2)}_{F}&=&\sum_{\gamma{'}J^{'}}\frac{\langle\gamma{IJFM_F}|H_{\mathrm{HFI}}|\gamma^{'}IJ'FM_F\rangle\langle\gamma^{'}IJ'FM_F|H_{\mathrm{HFI}}|\gamma{IJFM_F}\rangle}{E_{\gamma{J}}-E_{\gamma{'}J'}}.
    \end{aligned}$$

This equation reduces to $$\label{eq:2nd-order}
    \begin{split}
        W_F^{(2)} = \sum\limits_{\gamma'J'}\frac{1}{E_{\gamma{J}}-E_{\gamma{'}J'}}\sum\limits_{k_1,k2}{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&k_1\end{Bmatrix}}}{\ensuremath{\begin{Bmatrix}I&J&F\\#4&I&k_2\end{Bmatrix}}} \times \\
        {\ensuremath{\langle I|}} |T_{k_1}^n|{\ensuremath{|I\rangle}}{\ensuremath{\langle I|}} |T_{k_2}^n|{\ensuremath{|I\rangle}}{\ensuremath{\langle \gamma{J}|}}|T_{k_1}^e|{\ensuremath{|\gamma'J'\rangle}}{\ensuremath{\langle \gamma J|}}|T_{k_2}^e| {\ensuremath{|\gamma' J'\rangle}},
    \end{split}$$ where primed quantities refer to intermediate states; $E_{\gamma{J}}$ and $E_{\gamma{'}J{'}}$ are the HFI-unperturbed energy levels.

Based on the general theory, in the next section we investigate the hyperfine structure of $^{173}\mathrm{Yb^{+}}$ in the first excited state.

Hyperfine structure of $\mathrm{Yb}^{+}$ in the first excited state {#Sec:HFI-YbII}
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The first excited state of $\mathrm{Yb^{+}}$ has the electronic configuration $4f^{13}(^2\!F^o)6s^2$ with electronic angular momentum $J$ equal to 