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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 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., Comment: 16 pages, 7 figures, 7 tables

The velocity imaging photoionisation coincidence (VIPCO) technique is applied to the photoionisation of CO into the inner-valence 4 2Σ+ state of CO+, and the molecular frame photoelectron angular distributions are measured on the high energy side of the known σ* shape resonance. The angular distributions at 40.8 and 48.4 eV show that partial waves with the angular momentum l⩽3 provide the dominant contribution to both parallel and perpendicular transitions, and the contributions from the l4 partial waves are relatively negligible. By restricting the partial wave expansion to l⩽3, eight and four sets of dipole matrix elements and phase shift difference are extracted from the measured angular distributions for parallel and perpendicular transitions, respectively. Although it is not possible to choose unambiguously among these sets, combined enhancement among the l⩽3 partial waves for the parallel transition is obvious at the shape resonance.

Auger decay of an inner shell hole is an efficient way to create multiply charged ions in the gas phase. We illustrate this with the example of the argon 2s decay, and show that multi-electron coincidence spectroscopy between the 2s photoelectron and all released Auger electrons leads to a complete reconstruction of the Ar 2s decay cascade. Spectra of the intermediate and final Ar(n+) states are obtained and are compared with a theoretical model.