For systematics that are unique to the ${}^2\!\!\;F_{7/2}^o$ state, the largest is anticipated to be the energy shifts from the electronic electric quadrupole interacting with static electric field gradients in the trap. The diagonal contributions to the shifts are be given by $$\begin{split}
E^{(e.q.)}_{F,M_F}= -e \sum_\mu T_{2,\mu}(\nabla \mathbf{E}) \,\langle \gamma{IJF}|T_{2,-\mu}({{\mbox{\boldmath$ {\Theta} $}}})|\gamma IJF\rangle\\
= - e \,T_{2,0}(\nabla \mathbf{E})\,\frac{2(3M_F^2 - F(F+1))}{\sqrt{(2F+3)(2F+2)(2F+1)2F(2F-1)}}\\
\times (-1)^{I+J+F} (2F+1)
\frac{\begin{Bmatrix}
    J & F & I \\
    F & J & 2
    \end{Bmatrix}}{
    {\ensuremath{\begin{pmatrix}J&2&J\\#4&0&J\end{pmatrix}}}}\Theta(\gamma J) ,
\end{split}\label{Eq:DiagonalQuad}$$ where the quadrupole moment has been measured to be $\Theta({}^2\!\!\;F_{7/2}^o) = -0.041(5) a_\mathrm{o}^2$ [@Huntemann2012]. These contribute sub-Hz shifts for an electric field gradient of $1\mbox{ kV}/\mbox{cm}^2$, which is significantly larger than the gradient in our current trap. There are also potentially, off-resonant shifts due to the Paul trap’s radiofrequency drive if pairs of states happen to be split by a frequency near the rf drive, in which case the rf drive frequency may need to be changed. We are therefore not aware of any barriers to achieving a precision of $1 \, \mathrm{ Hz}$ for this measurement.

Discussion {#sec:Discussion}
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Equations (\[eq:DeltaW12\]) provide the relationship between the 5 quantities that will be measured experimentally (the $\Delta W^{(1+2)}_F$) and the 7 