# Attosecond Time Delay in Photoionization of Noble-Gas and Halogen Atoms
## 2 Overview of Theoretical Methods
### Inclusion of Collective Effects by Time-Dependent Local-Density Approximation  
If collective effects are important, for example, in low-energy ionization of heavy atoms, the single-electron picture is only good as a qualitative description. In the linear response regime, a TDLDA code provided in Refs. [57; 58] is employed to calculate the collective effects of an atom in the external dipole field \(z\). A change of charge density \(\delta\rho(\mathbf{r};\omega)\) occurs under the influence of the laser field, inducing a change, \(\delta V(\mathbf{r};\omega)\), in the potential. Formally, the dipole transition amplitude should be updated as  
\[\langle\psi_{\mathbf{k}}|z|\psi_{0}\rangle\rightarrow\langle\psi_{\mathbf{k}} |z+\delta V(\mathbf{r};\omega)|\psi_{0}\rangle\,, \tag{7}\]  
assuming the initial and final state wave functions are still calculated from the LDA potential Equation (5). Both the induced charge density and potential can be solved iteratively from the following equations  
\[\delta\rho(\mathbf{r};\omega) =\int\chi_{0}(\mathbf{r},\mathbf{r}^{\prime};\omega)\left[z^{ \prime}+\delta V(\mathbf{r}^{\prime};\omega)\right]d\mathbf{r}^{\prime}, \tag{8}\] \[\delta V(\mathbf{r};\omega) =\int\frac{\delta\rho(\mathbf{r}^{\prime};\omega)}{|\mathbf{r}- \mathbf{r}^{\prime}|}d\mathbf{r}^{\prime}+\left.\frac{\partial V_{XC}[\rho( \mathbf{r}^{\prime})]}{\partial\rho(\mathbf{r}^{\prime})}\right|_{\mathbf{r}^{ \prime}=\mathbf{r}}\delta\rho(\mathbf{r};\omega). \tag{9}\]  
In the above equation, \(\chi_{0}(\mathbf{r},\mathbf{r}^{\prime};\omega)\) is the susceptibility in the independent-electron approximation. It is constructed from the single-electron orbitals  
\[\chi_{0}(\mathbf{r},\mathbf{r}^{\prime};\omega)=\sum_{i}\psi_{i}^{*}(\mathbf{r })\psi_{i}(\mathbf{r}^{\prime})G(\mathbf{r},\mathbf{r}^{\prime};\varepsilon_{i} +\omega)+\sum_{i}\psi_{i}(\mathbf{r})\psi_{i}^{*}(\mathbf{r}^{\prime})G^{*}( \mathbf{r},\mathbf{r}^{\prime};\varepsilon_{i}-\omega), \tag{10}\]  
where \(i\) runs over the occupied states and \(G\) is the Green's function that satisfies the equation \[\left[\frac{1}{2}\
abla^{2}-V(\mathbf{r})+E\right]G(\mathbf{r},\mathbf{r}^{\prime}; E)=\delta(\mathbf{r}-\mathbf{r}^{\prime}). \tag{11}\]  
The Green's function can be expanded in the spherical basis:  
\[G(\mathbf{r},\mathbf{r}^{\prime};E)=\sum_{lm}G_{l}(r,r^{\prime};E)Y_{lm}^{*}( \Omega_{\mathbf{r}})Y_{lm}(\Omega_{\mathbf{r}^{\prime}}), \tag{12}\]  
where the radial part \(G_{l}(r,r^{\prime};E)\) satisfies the radial equation  
\[\left[\frac{1}{2}\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{ \partial}{\partial r}-V(r)-\frac{l(l+1)}{2r^{2}}+E\right]G_{l}(r,r^{\prime};E )=\frac{\delta(r-r^{\prime})}{r^{2}}. \tag{13}\]  
Letting \(j_{l}(r;E)\) and \(h_{l}(r;E)\) be the regular and irregular solution to the homogeneous version of Equation (13), one can construct the radial component of the Green's function as  
\[G_{l}(r,r^{\prime};E)=\frac{j_{l}(r_{<};E)h_{l}(r_{>};E)}{W[j_{l},h_{l}]}, \tag{14}\]  
where \(r_{<}\) (\(r_{>}\)) refers to the smaller (larger) one between \(r\) and \(r^{\prime}\). The Wronskian \(W[j_{l},h_{l}]\) is defined as  
\[W[j_{l},h_{l}]\equiv\frac{1}{2}r^{2}\left[j_{l}(r)\frac{dh_{l}(r)}{dr}-h_{l}(r )\frac{dj_{l}(r)}{dr}\right]_{r=a}, \tag{15}\]  
which is independent of an arbitrary distance \(a\). In practice, both the regular and irregular functions \(j_{l}(r;E)\) and \(h_{l}(r;E)\) are solved numerically. In summary, the collective effects of a many-electron atom are included as a frequency-dependent induced potential \(\delta V(\mathbf{r};\omega)\). It was shown in Ref.