**3. Results**

## *3.1. Electron-Lithium Excitation*

In the seminal publications of Bederson [43,44], the idea of perfect scattering experiments was introduced. In such experiments, the maximum amount of the underlying scattering information is measured. In the case of P-state excitation, the angular correlation parameters discussed above, or the equivalent Stokes parameters [45], which may depend on total electron spin [46], complement the differential cross sections in providing the extra experimental information to compare with the theory. In the following three figures, we examine e-Li excitation of the 2P state by presenting spin-dependent and -weighted results for the differential cross sections (DCS) and the three Stokes parameters. For each total electron spin *S*, at every scattering angle, there are two independent scattering (complex) amplitudes resulting in four independent theoretical predictions. While experiment [8] is available only for the spin-weighted parameters, it can be thought of as perfect for the spin-weighted case.

In addition to presenting the CCC calculations, which are convergen<sup>t</sup> in the treatment of both the target discrete and the continuous spectrum, the CC calculations are also presented. These are convergen<sup>t</sup> in the treatment of just the discrete spectrum; hence, the differences between CCC and CC highlight the importance of the inclusion of the target continuum on the transition of interest.

In Figure 1, we present the results for 7 eV e-Li excitation of the 2P state. We see that there is a substantial dependence of the results on the total electron spin *S*. The importance of the continuum is quite small at this energy. The agreemen<sup>t</sup> of the calculations with the experiment of Karaganov et al. [8] is quite extraordinary. Note that this experiment utilises the superelastic technique, which yields much better statistics than the traditional electron–photon coincidence experiments [1,2].

**Figure 1.** Differential cross section and (reduced) Stokes parameters for 7 eV electron-lithium 2P excitation. CCC results are convergen<sup>t</sup> in the treatment of the Li discrete and continuous spectrum, whereas CC results are convergen<sup>t</sup> in the treatment of the discrete spectrum only. Experiment is due to Karaganov et al. [8].

The results for 14 eV e-Li excitation of the 2P state are presented in Figure 2. This time we see substantial differences between the CCC and CC calculations, though mostly for the *S* = 0 DCS and Stokes parameters. These differences are much less visible when compared with the experiment for the spin-weighted Stokes parameters, where the agreemen<sup>t</sup> is again outstanding.

Lastly, the 22 eV e-Li excitation of the 2P state is presented in Figure 3. At this energy the inclusion of the target continuum is clearly important for *S* = 0, though this is less visible for the spin-weighted parameters. The agreemen<sup>t</sup> with the experiment is outstanding once more. It was such good agreement, first reported by [6], that others seriously questioned the accuracy of the corresponding e-H 2P excitation measurements.

**Figure 2.** Same as for Figure 1, except for 14 eV.

**Figure 3.** Same as for Figure 1, except for 22 eV.

## *3.2. Double Photoionisation*

The knowledge of the scattering *T*-matrix allows expression of the two-electron dipole matrix element in the form of an integral [47]:

$$\begin{split} \langle \Psi\_{j}^{(-)}(\mathbf{k}) | \mathcal{D} | \Psi\_{0} \rangle &= \langle \mathbf{k}^{(-)} \Phi\_{\mathbf{j}} | \mathcal{D} | \Psi\_{0} \rangle \\ &+ \sum\_{i} \sum\_{i} d^{3} \mathbf{x} \frac{\langle \mathbf{k}^{(-)} \Phi\_{\mathbf{j}} | T | \Phi\_{\mathbf{i}} \mathbf{x}^{(+)} \rangle \langle \mathbf{x}^{(+)} \Phi\_{\mathbf{i}} | \mathcal{D} | \Psi\_{0} \rangle}{E - \varepsilon\_{\mathbf{x}} - \varepsilon\_{i} + i0}. \end{split} \tag{6}$$

Here, the bare dipole matrix element *k*(−)*φj*|D| <sup>Ψ</sup>0- is calculated between the twoelectron initial state Ψ0 and the final channels *k*(−) *b φj*|. The initial state is expressed in the form of the multiconfiguration Hartree-Fock (MCHF) expansion

$$\Psi\_0(\boldsymbol{r}\_1, \boldsymbol{r}\_2) = \sum\_{l=0}^{l\_{\text{max}}} \sum\_{m=n}^{n\_{\text{max}}} \mathbb{C}\_{ml} \left| \psi\_{ml}(\boldsymbol{r}\_1) \,\psi\_{ml}(\boldsymbol{r}\_2) : \, ^1\mathbb{S} \right|. \tag{7}$$

The MCHF orbitals *ψml*(*r*) are found in the frozen 1*s*<sup>2</sup> core in the case of Li− (*n* = 2), while this core is absent in the case of H−(*n* = 1). Only diagonal *ml*<sup>2</sup> terms are included in expansion (7), as is always the case for the closed-shell MCHF ground state. The coefficients *Cml* in the MCHF expansion (7) are found by using the multiconfiguration Dirac-Fock computer code [48].

The cross section of a two-electron transition, measured as a function of the photon energy *ω* and corresponding to a particular state *j* of the remaining bound electron, is given by [49]:

$$\left|\sigma\_{\rangle}(\omega)\right\rangle = \frac{4\pi^2}{\omega\varepsilon} \sum\_{m\_{\parallel}}^{\text{f}} d^3k \left|\langle\Psi\_{\text{j}}^{(-)}(\mathbf{k})\left|\mathcal{D}\right|\Psi\_0\right\rangle|^2 \delta(\omega - E + E\_0) \,. \tag{8}$$

Here, *c* 137 is the speed of light in atomic units, while other fundamental constants are set to *e* = *m* = *h*¯ = 1. The final channels are separated into single and double ionization according to the energy  *j*, which is positive for the double ionized channels and negative for the singly ionized channels.

The phase of the ionization amplitude

$$\delta\_{\rangle}(\mathbf{k}) = \arg \langle \Psi\_{\rangle}^{(-)}(\mathbf{k}) \left| \mathcal{D} \right| \Psi\_{0} \rangle \,, \ \tau\_{\rangle}(\mathbf{k}) = \partial \delta\_{\rangle}(\mathbf{k}) / \partial \mathbf{E} \tag{9}$$

is used to calculate the photoemission time delay *<sup>τ</sup>j*(*k*) in the given emission direction as the phase derivative with respect to the photoelectron energy *E* = *k*2/2 (see Equation (S10) of [37]).

Our numerical results for photodetachment of H− and Li− are displayed in Figures 4 and 5, respectively. We select these two targets because the threshold laws manifest themselves particularly clearly in negative ions, which have a very simple spectrum and can support only a limited number of discrete excited states in comparison to an infinite number in their neutral atomic counterparts.

The photodetachment cross section (8) as well as the phase and the time delay (9) of H− near the *n* = 2 threshold are presented in Figure 4. The top panel displays the partial photodetachment cross sections in various channels leaving the H atom in the ground 1*s* and excited 2*s* or 2*p* states. The total cross section is compared with the experiment [50]. The sharp resonance below the *n* = 2 threshold is due to autodetachment of a two-electron bound state into the ground state of the H atom (Feshbach resonance). We see that all four autodetachment channels have very similar cross sections. The time delay in the *n* = 2 channels is strongly angular-dependent. Near the opening of the *n* = 2 channels, it reaches very large negative values.

Figure 5 displays the set of Li− results analogous to that of H− shown in Figure 4. The photodetachment cross section exhibited in the top panel of Figure 5 displays a very clear threshold cusp prescribed by the Wigner threshold law [51], which suppresses all the partial waves in the photoelectron wave packet in the newly opened channel except the *s*-wave.

**Figure 4.** (Color online) (**a**) The cross section (8), (**b**) the phase of the transition amplitude, and (**c**) the time delay (9) of H− in various photodetachment channels near the *n* = 2 threshold. The experiment [50] is displayed in the top panel.

In the Wigner theory, the cross section near the opening of a new channel can be parameterized as

$$
\sigma(E) = \sigma \mathbf{0} - 2A|E - E\_0|^{1/2} \times \begin{cases}
\sin^2 \delta\_0 & \text{for} \quad E > E\_0 \\
\sin \delta\_0 \cos \delta\_0 & \text{for} \quad E < E\_0
\end{cases} \tag{10}
$$

The scattering phase *δ*0 is rather small in H<sup>−</sup>, as the inelastic threshold is 3/4 Ry above the photodetachment threshold. In contrast, in Li− , the inelastic 2*p* threshold is only 2 eV above the photodetachment threshold, and the phase *δ*0 is rather large. The Wigner Formula (10) predicts the falling cross section above the threshold, while it is always rising below the threshold, hence the cusp formation and the strong dominance of the *s*-wave in the inelastic channel of Li− . This dominance reduces dramatically the angular dependence of the phase and time delay. The latter is only weakly angular dependent in comparison with the H− ion.

**Figure 5.** (Color online) (**a**) The cross section (8), (**b**) the phase of the transition amplitude, and (**c**) the time delay (9) of H− in various photodetachment channels near the *n* = 2 threshold. Experimental cross section results [52,53] are displayed in the top panel.

#### *3.3. Atomic, Molecular and Optical Science Gateway*

The CCC calculations presented here may be reproduced in a matter of minutes using the Atomic, Molecular and Optical Science (AMOS) Gateway https://amosgateway.org (accessed on 4 February 2022), which is accessible to anyone who is able to authenticate via their own institution. The gateway is a sustainable community-oriented platform for enabling AMOS applications as a service for the AMOS community with intuitive interfaces. Several other AMOS codes are either available, or being made available, via this interface [42].

The gateway-accessible CCC code has been installed and compiled for efficient execution on the machines of the Extreme Science and Engineering Discovery Environment (XSEDE). The code has inbuilt hybrid MPI and OpenMP parallelization, as well as GPU acceleration. The latter has only been implemented recently [54] and continues to be a focus of development. Sample e-Li scattering inputs for the presented calculations are available. The gateway is constantly under development and we expect that more user-friendly interfaces will be built in due course.
