Finding the ultra-narrow $^3\!P_2 \rightarrow \, ^3\!P_0$ electric quadrupole transition in Ni$^{12+}$ ion for an optical clock

The Ni$^{12+}$ ion features an electronic transition with a natural width of only 8 mHz, allowing for a highly stable optical clock. We predict that the energy of this strongly forbidden $3s^2 3p^4\, ^3\!P_2 \rightarrow 3s^2 3p^4 \, ^3\!P_0$ electric quadrupole transition is 20081(10) cm$^{-1}$. For this, we use both a hybrid approach combining configuration interaction (CI) with coupled-cluster (CC) method and a pure CI calculation for the complete 16-electron system, ensuring convergence. The resulting very small theoretical uncertainty of only 0.05\% allowed us to find the transition experimentally in a few hours, yielding an energy of 20078.984(10) cm$^{-1}$. This level of agreement for a 16-electron system is unprecedented and qualifies our method for future calculations of many other complex atomic systems. While paving the way for a high-precision optical clock based on Ni$^{12+}$, our theory and code development will also enable better predictions for other highly charged ions and other complex atomic systems.