**2. Theory**

The details of the implementation of the CCC theory to electron scattering on quasi one-electron targets, such as the alkalis, have been given by Bray [18]. Briefly, the coreelectron wave-functions *ψj* of target *T* are solved for by utilizing the Self-consistent-Field Hartree-Fock (SCHF) equations [16]

$$\left(K + V^{\rm HF} - \varepsilon\_{\dot{j}}\right)\psi\_{\dot{j}}(\mathbf{r}) = 0, \qquad \forall\_{\dot{j}} \in T,\tag{1}$$

where

$$\begin{split} V^{\text{HF}} \psi\_{\vec{\jmath}}(\mathbf{r}) &= \quad \left( -\frac{Z}{r} + 2 \sum\_{\forall \boldsymbol{r} \in \boldsymbol{T}} \int d^3 r' \frac{|\psi\_{\vec{\jmath}'}(\mathbf{r}')|^2}{|\mathbf{r} - \mathbf{r}'|} \right) \psi\_{\vec{\jmath}}(\mathbf{r}) \\ &- \sum\_{\forall \boldsymbol{r}' \in \boldsymbol{T}} \int d^3 r' \frac{\psi\_{\vec{\jmath}'}^\*(\mathbf{r}') \psi\_{\vec{\jmath}}(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} \psi\_{\vec{\jmath}'}(\mathbf{r}) . \end{split} \tag{2}$$

The core-electron wave-functions are then used to define the Frozen-Core Hartree-Fock (FCHF) potential *V*FC as

$$\begin{split} V^{\rm FC} \phi\_{\vec{\gamma}}(\mathbf{r}) &= \quad \left( -\frac{Z}{r} + 2 \sum\_{\forall j' \in \mathcal{C}} \int d^3 r' \frac{|\psi\_{\vec{\gamma}'}(\mathbf{r}')|^2}{|\mathbf{r} - \mathbf{r}'|} \right) \phi\_{\vec{\gamma}}(\mathbf{r}) \\ &- \sum\_{\forall j' \in \mathcal{C}} \int d^3 r' \frac{\psi\_{\vec{\gamma}'}^\*(\mathbf{r}') \phi\_{\vec{\gamma}}(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} \psi\_{\vec{\gamma}'}(\mathbf{r}), \end{split} \tag{3}$$

where the notation *C* indicates the set of frozen-core wave-functions. The target wavefunctions are then obtained from the effective one-electron Hamiltonian *K* + *V*FC via

$$\left(K + V^{\text{FC}} - \epsilon\_{\dot{\rangle}}\right)\phi\_{\dot{\rangle}}(\mathbf{r}) = 0. \tag{4}$$

The eigenstates can be obtained directly [17] or via diagonalization in some suitable basis. In the CCC method we do both, with the utilization of the Laguerre basis, which yields negative-energy eigenstates and a discretization of the target continuum.

With all of the potentials *V* defined, the close-coupling equations are formed in momentum space as coupled Lippmann-Schwinger equations directly for the transition amplitudes *kf φf* |*TS*|*φiki*-, as if the problem was a three-body one [4,18]

$$
\langle \mathbf{k}\_f \Phi\_f | T\_S | \phi\_i \mathbf{k}\_i \rangle = \langle \mathbf{k}\_f \Phi\_f | V\_S | \phi\_i \mathbf{k}\_i \rangle + \sum\_{n=1}^{N} \int d^3 k \frac{\langle \mathbf{k}\_f \Phi\_f | V\_S | \phi\_n \mathbf{k} \rangle \langle \mathbf{k} \phi\_n | T\_S | \phi\_i \mathbf{k}\_i \rangle}{E^{(+)} - \epsilon\_n - k^2/2}, \tag{5}$$

where *S* is the total electron spin, and *N* is the number of Laguerre-based target states. We check for convergence in the required *kf φf* |*TS*|*φiki*- by simply increasing *N*.
