High-precision atomic clocks with highly charged ions: Nuclear-spin-zero $f^{12}$-shell ions

Optical atomic clocks using highly charged ions hold an intriguing promise of metrology at the 19th significant figure. Here we study transitions within the $4f^{12}$ ground-state electronic configuration of highly charged ions. We consider isotopes lacking hyperfine structure and show that the detrimental effects of coupling of electronic quadrupole moments to the gradients of a trapping electric field can be effectively reduced by using specially chosen virtual clock transitions. The estimated systematic fractional clock accuracy is shown to be below ${10}^{-19}$.