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We developed recently [A. Fert\'e, et al., J. Phys. Chem. Lett. 11, 4359(2020)] a method to compute single site double core hole (ssDCH or K$^{-2}$)spectra. We refer to that method as NOTA+CIPSI. In the present paper thismethod is applied to the O K$^{-2}$ spectrum of the CO$_2$ molecule, and we usethis as an example to discuss in detail its convergence properties. Using thisapproach, a theoretical spectra in excellent agreement with the experimentalone is obtained. Thanks to a thorough interpretation of the shake-up statesresponsible for the main satellite peaks and with the help of a comparison withthe O K$^{-2}$ spectrum of CO, we can highlight the clear signature of the twonon equivalent carbon oxygen bonds in the oxygen ssDCH CO$_2$ dication.

Specific chemical bond relaxation unravelled by analysis of shake-up satellites in the oxygen single site double core hole spectrum of CO$_2$