Hyperfine-mediated electric quadrupole shifts in Al$^+$ and In$^+$ ion clocks

We evaluate the electric quadrupole moments of the ${^1}\!S_0$ and ${^3}\!P_0$ clock states of $^{27}$Al$^+$ and $^{115}$In$^+$. To capture all dominant contributions, our analysis extends through third order of perturbation theory and includes hyperfine coupling of the electrons to both the magnetic dipole and electric quadrupole moments of the nucleus. For $^{27}$Al$^+$, a fortuitous cancellation leads to a suppressed frequency shift. This should allow for continued improvement of the clock without special techniques to control or cancel the shift, such as the averaging schemes that are critical to other optical ion clocks.