**2. Theory**

The SO process leading to the ejection of two electrons in the continuum can be exhibited graphically by an infinite sequence of diagrams presented in Figure 1a. Here, we use the following graphical symbols. A straight line with an arrow to the right represents the electron continuum states *k*, *f* whereas an arrow pointing to the left exhibits the holes in atomic shells *i*, *l*, *m*. Before photoabsorption, all the atomic shells are presumed to be fully occupied and thus closed. The atomic ground state (the "vacuum" state) contains neither holes nor excited electrons. The wavy line denotes the Coulomb interaction between the electrons. The dashed line represents an absorbed photon. A circle denotes the diagonal matrix element of the self-energy part of the Green's function Σ*i*. The SHGF self-energy is expanded graphically in more detail in Figure 1b.

**Figure 1.** (**a**) Schematic representation of the SO process leading to a two-electron continuum. The circle represents the self-energy part of the SHGF expanded in more detail in (**b**).

The corresponding DPI amplitude can be found by summing a geometric progression of terms in Figure 1a that leads to the following expression:

$$
\langle f \| M\_{\rm SO} \| i \rangle = \langle f \| r \| i \rangle \left( 1 - \frac{\Sigma\_i(\varepsilon)}{\varepsilon - \varepsilon\_i - i\delta} \right)^{-1} \frac{\langle ik \| \| L\_L \| \| lm \rangle}{\varepsilon - \varepsilon\_i - i\delta} \,. \tag{1}
$$

Here, *f ri*- is a dipole matrix element of the single-photon absorption and *ikULlm*- is a Coulomb matrix element, which includes a direct and exchange *l* ↔ *m* terms and involves a transfer of the angular momentum *L*. Both the dipole and Coulomb matrix elements are reduced to strip their angular momentum projections dependence. The symbol *ε* =  *f* − *ω* < 0 denotes an effective hole energy. In the absence of correlation, *ε* =  *i* is the orbital energy. Many-electron correlation adds a discrete spectrum of shake-up satellites and a shake-off continuum, which starts at the DPI threshold *E*∞ =  *l* +  *<sup>m</sup>*. The infinitesimally small *δ* → 0 in the energy denominator defines the pole bypass.

The imaginary part of Equation (1) gives rise to an additional phase of the DPI amplitude due to the final state correlation:

$$\arg M\_{\rm SO} = \arctan \frac{\mathrm{Im}\Sigma\_i(\varepsilon)}{\mathrm{Re}[\varepsilon - \varepsilon\_i - \Sigma\_i(\varepsilon)]} \approx \arctan \frac{\mathrm{Im}\Sigma\_i(\varepsilon)}{\mathrm{Re}[\varepsilon - \varepsilon\_i]} \,. \tag{2}$$

Here ImΣ*i* = *π*(2*<sup>L</sup>* + <sup>1</sup>)−<sup>1</sup>|*ikULlm*-|2 . The approximate equality is satisfied under the condition that |*ε* −  *i*||<sup>Σ</sup>*i*(*ε*)| . This condition defines the part of the double ionized continuum sufficiently far from the main photoelectron line.

The energy resolved single-differential cross-section (SDCS) is given by Equation (9) of [25]:

$$\frac{d\sigma\_i^{2+}}{d\epsilon\_f} = \sigma\_i^+ \, \frac{1}{\pi} \frac{\text{Im}\Sigma\_i(\varepsilon)}{|\varepsilon - \varepsilon\_i - \Sigma\_i(\varepsilon)|^2} \approx \sigma\_i^+ \, \frac{1}{\pi} \frac{\text{Im}\Sigma\_i(\varepsilon)}{|\varepsilon - \varepsilon\_i|^2} \,. \tag{3}$$

By solving this equation relative to ImΣ*i*, we can express the additional phase of the DPI amplitude due to the final state correlation in the following form:

$$\arg M\_{\rm SO} = \arctan \frac{\pi}{\sigma\_i^+} \frac{d \sigma\_i^{2+}}{d \varepsilon\_f} \left| \varepsilon - \varepsilon\_i \right| \,. \tag{4}$$

We note that all the quantities entering this expression are known from the experiment, which are the single photoionization cross-section of the primary photoelectron *σ*+*i* and the energy differential DPI cross-section *<sup>d</sup>σ*2+*i*/*<sup>d</sup> f* .

By integrating the SDCS Equation (3) over the fast photoelectron energy, we can obtain the double-to-single photoionization cross-section ratio:

$$R = \frac{\sigma^{2+}}{\sigma^{+}} = \frac{1}{\sigma^{+}} \int\_{0}^{\infty} \frac{d\sigma^{2+}}{d\varepsilon\_{f}} d\varepsilon\_{f} \,. \tag{5}$$

This ratio is known in He, and its isoelectronic sequence of ions [30]. It will serve as a convenient reference in Section 3.2.

Following [25], we can introduce the inverse SHGF

$$F(\varepsilon) = G^{-1}(\varepsilon) = \varepsilon - \varepsilon\_i - \Sigma\_i(\varepsilon) \tag{6}$$

Then the argumen<sup>t</sup> of the SO amplitude Equation (2) can be rewritten as

$$\arg M\_{\rm SO} = \arg G^{-1}(\varepsilon) = \arg F(\varepsilon) \tag{7}$$

This expression allows to use the integral rule presented by Equation (5) of [25], which relates the energies of the discrete shake-up satellites with the time delay:

$$\sum\_{k=0}^{\infty} (\varepsilon\_k - E\_k) = \frac{1}{\pi} \int\_{-\infty}^{E\_{\text{SO}}} \varepsilon \tau(\varepsilon) \, d\varepsilon \text{ , where } \tau(\varepsilon) = \frac{\partial}{\partial \varepsilon} \arg M\_{\text{SO}} = \frac{\partial}{\partial \varepsilon} \arg F(\varepsilon) \tag{8}$$

Here *Ek* =  *l* +  *m* −  *k* and *E*0 =  *i* are non-correlated energies of the shake-up excitations calculated from the HF orbital energies. Meanwhile, *εk* are the corresponding energies shifted by the final-state correlation and found as the poles of the SHGF. The integral time delay rule Equation (8) presents an analytical test for the numerical SO time delay values. This test will be conducted in Section 3.3.
