\section{Better initial guess}
\label{sec:kkr-jlcdm}

% Slide PhD project flowchart % Section phd-project
\begin{frame}
  \frametitle{Vision: Electronic structure learning}
  % frametitle notes: PhD project flowchart
  \framesubtitle{as integrated, high-level multiscale workflows}
  \vspace*{0em}

  \includegraphics[width=1.0\textwidth]{../resources/fig/presentation-2023-02/atomistic-ml/classification-of-atomistic-ml_presentation-2023-02_02-emph-both_ktikz.pdf}

  \vspace*{2em}
  \begin{columns}[t]
    \hspace{1em}
    \begin{column}{0.6\linewidth}
      \begin{center}
        Better \enquote{initial guess}

        for fast SCF convergence
      \end{center}
    \end{column}
    \vrule{}
    \hspace{1em}
    \begin{column}{0.39\linewidth}
      \begin{center}
        Magnetic property prediction

        (ML-Exc) for spin dynamics
      \end{center}
    \end{column}
  \end{columns}
\end{frame}

% Slide AiiDA-KKR workflows % kkr-theory
\begin{frame}[plain,c]
  \frametitle{Project \enquote{Better initial guess}}
  \framesubtitle{\logoAiida{}-KKR workflows}
  \vspace*{0em}
  \begin{columns}[t]
    \vspace*{-1em}
    \begin{column}{0.3\linewidth}
      % {\footnotesize Single impurity}
      \begin{center}
        \includegraphics[width=0.95\linewidth]{../resources/fig/presentation-2023-03/ruess/ruess-aiida-kkr-paper-workflow-c.pdf}\footcite{russmannAiiDAKKRPluginIts2021}
      \end{center}
    \end{column}
    \begin{column}{0.5\linewidth}
      \vspace*{-2em}
      \begin{center}
        \includegraphics[width=0.75\textheight]{../resources/fig/aiida-kkr/graph//kkr_imp_wc.pdf}%
      \end{center}
      % \framezoom<1><2>(3.5cm,0.5cm)(4.5cm,3.25cm) % upper part, kkr_scf workflow
      % \framezoom<1><3>(4.5cm,3.5cm)(4.5cm,5.25cm) % lower part, kkr_imp workflow
  \end{column}
\end{columns}
\vspace*{-0em}
\end{frame}


% Slide dataset generation (single-impurity-database)
\begin{frame}[plain]
  \frametitle{Project \enquote{Better initial guess}}
  \framesubtitle{Data generation}
  % frametitle notes: Vimp-prediction qnd

  \begin{columns}[c]
    \begin{column}{0.45\linewidth}
      {\small
        \textbf{Data} 10'000 impurity embeddings into elemental crystals\vspace{0.5em}

      \textbf{Target} Electron potential difference \(\Delta
      V_{imp}(\vec{r})\)  \vspace{0.5em}
      }

      \includegraphics[width=1.0\linewidth]{../resources/fig/aiida-kkr-ml/da/conv_heatmap_scale-factor_count_linscale_annot.pdf}%

      \begin{center}
        {\footnotesize \textcolor{fzjgray50}{Dataset map. Rows: Element of host crystal,
            columns: impurity atom, color: num. calculations.}}
      \end{center}
    \end{column}

    \begin{column}{0.55\linewidth}
      \vspace*{-2em}
      \begin{columns}
        \begin{column}{0.25\linewidth}
          \hspace*{2.0em}
          % \begin{center}
            \includegraphics[width=1.0\linewidth]{../resources/fig/theo/kkr-impurity-region.png}%
          % \end{center}
        \end{column}
        \begin{column}{0.75\linewidth}
          \hspace*{2.5em}
          % \begin{center}
            \includegraphics[width=0.75\textwidth]{../resources/fig/jukkr/kkr-scf/kkr-scf_with-description.pdf}
          % \end{center}
        \end{column}
      \end{columns}


      \hspace*{0.0em}
      \includegraphics[width=1.0\linewidth]{../resources/fig/aiida-kkr-ml/da/potential_Hg_sf-1.0_logy-False_absval-False_zoom.png}

      \hspace*{0.0em}
      \includegraphics[width=1.0\linewidth]{../resources/fig/aiida-kkr-ml/da/potential_Hg_sf-1.0_logy-True.png}

      % \hspace*{-0.0em}
      \begin{center}
        {\footnotesize \textcolor{fzjgray50}{Spherical impurity potentials in first Voronoi cell of \ce{Hg}:\ce{X}
        embeddings,\\left upper to bottom right: \(V\), \(V\!-\!V^0\), \(|V|\), \(|V\!-\!V^0|\).}}
      \end{center}
    \end{column}
  \end{columns}
\end{frame}

% Slide Jacobi-Legendre framework
\begin{frame}[plain,c]
  \frametitle{The Jacobi-Legendre framework\footcite{dominaJacobiLegendreFrameworkMachine2024}}
  \framesubtitle{for electronic structure representation}
  \vspace*{0em}
  \begin{columns}[c]
    \begin{column}{0.68\linewidth}
      \begin{center}
      %   \includegraphics[width=1.0\linewidth]{../resources/fig/external/papers/dominaJacobiLegendreFrameworkMachine2024/processed/fig5.1-with-polynomials.png}%

      \includegraphics[width=0.9\linewidth]{../resources/fig/external/papers/dominaJacobiLegendreFrameworkMachine2024/originals/fig4.1.pdf}%
      \vspace*{0.0em}
      \includegraphics[width=0.9\linewidth]{../resources/fig/external/papers/dominaJacobiLegendreFrameworkMachine2024/processed/fig-5.1-first-row-only.png}%
      \end{center}
    \end{column}
    \vrule{}                    % vertical separator line
    \hspace*{1em}
    \begin{column}{0.32\linewidth}
      \vspace*{0em}

      \includegraphics[width=0.75\linewidth]{../resources/fig/external/papers/dominaJacobiLegendreFrameworkMachine2024/processed/fig6.2-300dpi-narrower-1.png}%
    \end{column}
  \end{columns}
  \vspace*{0em}
\end{frame}


% \begin{frame}[plain]
%   % \frametitle{Dataset analysis}
%   % \framesubtitle{}
%   \emph{Single-impurity database} \(\boldsymbol{\vert}\) Charge doping
%   \vspace{-1em}
%   \begin{center}
%     \includegraphics[width=0.88\textwidth]{../resources/fig/aiida-kkr-ml/da/conv_heatmap_charge-neutrality-imp_linscale.pdf}%
%   \end{center}
% \end{frame}

\begin{frame}[plain]
  \frametitle{Atom-based JLCDM}

  \begin{itemize}
    \item The full KKR potential is expanded in real-space Voronoi
      (Wigner-Seitz) cells around each atom and convoluted with shape functions


      \[
      V(\vec{r}) = \sum_L V_L(r) Y_L(\vec{r}) \quad \text{with} \quad V_{L L'}(r) = \sum_{L''} C_{L L' L''} V_{L''}(r)
    \]
    % side notes. C_L1L2 = Clebsch-Gordan, C_L1L2L3 = Gaunt coefficients

\[
V^n(\vec{r}) = V(\vec{r} + \vec{R}^n) \Theta(\vec{r}) \quad \text{with} \quad \Theta(\vec{r}) = \sum_L \Theta_L^n(r) Y_L(\vec{r})
\]
\item Idea: Expand the Jacobi-Legendre grid description at each atom, instead of
  each grid point.


\[
V(\vec{r}) = \sum_{i} \sum_{n} a_n \tilde{P}_{n}^{ig}
+\sum_{ij}\sum_{n_1,n2,L} a_{n_1,n_2,i} \bar{P}_{n_1}^{ig} \bar{P}_{n_2}^{ij} P_{L}^{ijg}
\]

\[
\text{Angular expansion}\quad \int P_L ( \hat{r}_{gi} \cdot \hat{r}_{gj} ) Y (
\hat{r}_{gi} ) \, \text{d} \hat{r} \longrightarrow \int P_L ( \hat{r}_{ij} \cdot \hat{r}_{ig} ) Y ( \hat{r}_{ig} ) \, \text{d} \hat{r}
\]
  \end{itemize}
\end{frame}



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