is used as an optical frequency standard [@Huntemann2016], where hyperfine structure within these states allows control of systematics. Some of the current best limits on the time variation of fundamental constants are based on precision measurements between specific hyperfine components of this E3 transition [@Godun2014; @Huntemann2014b].

As the experimental progress with this species continues to achieve higher accuracy and precision [@Huntemann2016; @Sanner2018], theoretical work is needed in parallel with these improvements to understand contributions to systematics. Ref. [@Beloy2008a] has shown how to calculate the second-order energy correction due to hyperfine interaction for the alkali-atoms in the first excited state, and gives the theoretical basis for higher-order terms calculated in this paper. The values of hyperfine constants $A$ and $B$ for $^{171}\mathrm{Yb}^{+}$ and $^{173}\mathrm{Yb}^{+}$ in the ${}^2F_{7/2}^o$ state are also given in Ref. [@Dzuba2016a], and are used as a comparison to our values.

The paper is organized as follows. In Sec. \[Sec:General-theory\], we review the theory of hyperfine structure. Based on this general theory, we derive the first- and second-order corrections to the HFS of $^{173}\mathrm{Yb^{+}}$ in the first excited state. In Sec. \[Sec:electronic-reduced-ml\], we compute $^{173}\mathrm{Yb^{+}}$ electronic-structure factors required for extracting nuclear moments. We discuss the importance of correlation effects 