to find the values of nuclear octupole and hexadecapole moments from $C$ and $D$ we need electronic form-factors; these are also computed in Sec. \[Sec:electronic-reduced\]. We neglect contributions of one remaining HFS constant $E$ arising from the $2^5$-pole nuclear magnetic moment. This contribution is expected to be strongly suppressed compared to the contribution of the octupole moment (see Sec. \[sec:Discussion\]).

Calculations of electronic structure factors {#Sec:electronic-reduced}
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Dirac-Hartree-Fock calculations {#Sec:electronic-reduced-ml}
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$\mathrm{Yb^{+}}$ ion in the first excited state contains thirteen $4f$ electrons and two $6s$ electrons. In this section, we start our calculation of the electronic wave functions by employing the frozen core Dirac-Hartree-Fock (DHF) approximation. In this approximation, we compute the DHF orbitals of the $\mathrm{YbIII}\,([\mathrm{Xe}]4f^{14})$ core. Then the valence (outside the $\mathrm{[Xe]}4f^{14}$ core) orbitals are computed using the DHF potential of the core. The many-body wave function $\psi_{J\,,M}$ can be approximated as $$\begin{aligned}
\label{eq:multi-electron-wf}
&&{\ensuremath{|\psi_{J,\,M}\rangle}}\simeq \frac{1}{2} (-1)^{7/2-M}\times\nonumber\\
&&(\sum\limits_{m}(-1)^{m-1/2}{a_{6s_{1/2,m}}^{\dagger}a_{6s_{1/2,-m}}^{\dagger})\,a_{4f_{7/2,-M}}}{\ensuremath{|{0_c}\rangle}}\,,\end{aligned}$$where $a_{6s_{1/2},m}^{\dagger}$ are creation operators with magnetic quantum number $m$ equal to either $-1/2$ or $1/2$, $a_{4f_{7/2,\,M}}$ is an annihilation operator for the $4f_{7/2}$ orbital, and ${\ensuremath{|0_c\rangle}}$ represents the $\mathrm{[Xe]}4f^{14}$ core. The phase factor $(-1)^{7/2-M}$ is generated after moving the hole operator from the core state [@Johnson2007]. The two $6s_{1/2}$ orbitals 