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

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 this method is applied to the O K$^{-2}$ spectrum of the CO$_2$ molecule, and we use this as an example to discuss in detail its convergence properties. Using this approach, a theoretical spectra in excellent agreement with the experimental one is obtained. Thanks to a thorough interpretation of the shake-up states responsible for the main satellite peaks and with the help of a comparison with the O K$^{-2}$ spectrum of CO, we can highlight the clear signature of the two non equivalent carbon oxygen bonds in the oxygen ssDCH CO$_2$ dication.