*3.1. Computation Details*

We use the ATOM suite of programs developed by Miron Amusia and co-workers [31]. The SCFHF and FCHF computer codes calculate the atomic ground and excited states in the self-consistent and frozen-core Hartree–Fock (HF) approximations, respectively. Then the Coulomb matrix elements are evaluated, and the SHGF and its self-energy are found. The latter are used to calculate the SDCS, the *R* ratio and the time delay associated with the SO process.

We consider the two types of the SO process. In the first type, all the hole states *i*, *l*, *m* are confined to the same *ns* shell. The fast primary photoelectron is ejected from this shell into the *p*-continuum, whereas the slow electron is shaken off into the *s*-continuum. Such an intra-shell SO process takes place in the outer valence shell of the He, Be and Mg atoms as well as the H− ion. The second inter-shell type of the SO process accompanies ionization of the sub-valent *ns* shells of noble gas atoms Ne, Ar and Xe. While the primary hole *i* is made in the *ns* shell, the secondary holes *l*, *m* are made in the outermost *np* shell. The primary fast photoelectron is ejected in the *p*-wave, whereas the secondary electron is shaken off primarily into the *d*-wave.

#### *3.2. Energy Resolved DPI Cross-Sections*

In Figure 2, we exhibit the energy resolved DPI cross section of helium at the excess energies above the DPI threshold *E* = 100, 450 and 720 eV (from left to right). The SDCS Equation (3) is symmetrized by adding the two energy distributions of the slow and fast photoelectrons, thus giving it a characteristic *U*-shape. The present shake-off calculations using Equation (3) are compared with various reference data indicated in the figure caption. The integrated cross-sections of single photoionization *σ*+1*s* are used as tabulated in [32]. We can observe in Figure 2 that the SO mechanism is becoming gradually dominant as the photon energy grows. This is particularly true for a highly asymmetric energy sharing between the photoelectrons. Under this kinematics, the primary photoelectron takes nearly all the photon energy and is ejected in the dipole channel as a *p*-wave. At the same time, the secondary SO photoelectron is very slow and is ejected almost isotropically as an *s*-wave in the intra-shell SO process. It is this characteristic photoelectron angular distribution that was observed experimentally in He at *E* = 450 eV [7].

**Figure 2.** The energy resolved DPI cross-section of helium *<sup>d</sup>σ*2<sup>+</sup>/*dE*) (in bn/eV) at the excess energies above the DPI threshold *E* = 100, 450 and 720 eV (from left to right). The present shake-off calculations using Equation (3) are compared with the following reference data. At *E* = 100 eV, the relative measurement [33] is normalized to the TDCC calculation [34] and shown along with an analogous CCC calculation [35]. At *E* = 450 eV, the relative experiment [7] is normalized to the CCC calculation from the same reference. At *E* = 720 eV, the TDCC [36] and the CCC [11] calculations are shown.

Meanwhile, the equal energy sharing between the photoelectrons is affected by a competing KO process and deviates from the present SO predictions. This deviation, which is strongest at *E* = 100 eV, can also be seen at higher photon energies near the mid-point of the photoelectron energy distribution.

The double-to-single ratios as calculated by the SO model using Equation (5) are presented in Table 1. The SO ratio in He is equal to 1.44%, which is very close to the CCC ratio of 1.67% [30]. For other targets, this comparison is less accurate, especially for the H− ion, which is twice overestimated, while Be and Mg are 50% underestimated.

**Table 1.** Asymtptoc double-to-single cross-section ratios in various targets as calculated by the SO model using Equation (5) and compared with earlier CCC calculations.

