  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.

Chemical Physics