Isotope shifts of the $1s^{2}2s^2$(${}^1{S}_0$) $\to$ $1s^{2}2p^2$(${}^1{S}_0$) transition in the doubly ionized carbon ion ${\mathrm{C}}^{2+}$

Highly accurate quantum mechanical calculations are performed for the $1s^{2}2s^2$ (${}^1{S}_0$) $\to$ $1s^{2}2p^2$ (${}^1{S}_0$) transition energy in the isotopomers of ${\mathrm{C}}^{2+}$ ion to determine the isotope shifts. Explicitly correlated Gaussian functions and a variational approach that explicitly includes the nuclear motion are employed in the calculations. The leading relativistic and quantum electrodynamics corrections to the transition energy are also calculated using the perturbation theory with the nonrelativistic wave function as the zero-order approximation. It is determined that the ${}^{12}{\mathrm{C}}^{2+}$ transitions energy, which is obtained from the calculations to be 182 519.031 cm${}^{-1}$ (vs the experimental value of 182 519.88 cm${}^{-1}$, an excellent sub-wave-number agreement) up-shifts by 1.755 cm${}^{-1}$ for ${}^{13}{\mathrm{C}}^{2+}$ and by additional 1.498 cm${}^{-1}$ for ${}^{14}{\mathrm{C}}^{2+}$. Those shifts are sufficiently large to be measured experimentally.