\Part[
    \eng{Computational Physics}
    \ger{Computergestützte Physik}
]{cmp}
\Section[
    \eng{Quantum many-body physics}
    \ger{Quanten-Vielteilchenphysik}
]{mb}
\TODO{TODO}
    \Subsection[
        \eng{Importance sampling}
        \ger{Importance sampling / Stichprobenentnahme nach Wichtigkeit}
    ]{importance_sampling}
    \TODO{Monte Carlo}

    \Subsection[
        \eng{Matrix product states}
        \ger{Matrix Produktzustände}
    ]{mps}



\Section[
    \eng{Electronic structure theory}
    % \ger{}
]{elsth}
    \begin{formula}{hamiltonian}
        \desc{Electronic structure Hamiltonian}{}{$\hat{T}$ kinetic energy, $\hat{V}$ electrostatic potential, $\txe$ electrons, $\txn$ nucleons}
        % \desc[german]{}{}{}
        \eq{
            \hat{H} &= \hat{T}_\txe + \hat{T}_\txn + V_{\e \leftrightarrow \e} + V_{\n \leftrightarrow \e} + V_{\n \leftrightarrow \n} \\
            \shortintertext{with}
            \hat{T}_i &= -\sum_{n=1}^{N_i} \frac{\hbar^2}{2 m_i} \vec{\nabla}^2_n \\
            \hat{V}_{i \leftrightarrow j} &= -\sum_{k,l} \frac{Z_i Z_j \e^2}{\abs{\vecr_k - \vecr_l}}
        }
    \end{formula}
    \begin{formula}{mean_field}
        \desc{Mean field approximation}{Replaces 2-particle operator by 1-particle operator}{Example for Coulumb interaction between many electrons}
        \desc[german]{Molekularfeldnäherung}{Ersetzt 2-Teilchen Operator durch 1-Teilchen Operator}{Beispiel für Coulumb Wechselwirkung zwischen Elektronen}
        \eq{
            \frac{1}{2}\sum_{i\neq j} \frac{e^2}{\abs{\vec{r}_i - \vec{r}_j}} \approx \sum_{i} V_\text{eff}(\vec{r}_i)
        }
    \end{formula}


\Subsection[
    \eng{Tight-binding}
    \ger{Tight-binding}
]{tb}


\Subsection[
    \eng{Density functional theory (DFT)}
    \ger{Dichtefunktionaltheorie (DFT)}
]{dft}
    \Subsubsection[
        \eng{Hartree-Fock}
        \ger{Hartree-Fock}
    ]{hf}
        \begin{ttext}
            \eng{
                \begin{itemize}
                    \item \fqEqRef{comp:misc:mean_field} theory
                    \item Self-interaction free: Self interaction is cancelled out by the Fock-term
                \end{itemize}
            }
        \end{ttext}
        \begin{formula}{equation}
            \desc{Hartree-Fock equation}{}{
                $\varphi_\xi$ single particle wavefunction of $\xi$th orbital,
                $\hat{T}$ kinetic electron energy,
                $\hat{V}_{\text{en}}$ electron-nucleus attraction,
                $\hat{V}_{\text{HF}}$ \fqEqRef{comp:dft:hf:potential},
            }
            \desc[german]{Hartree-Fock Gleichung}{}{
                $\varphi_\xi$ ein-Teilchen Wellenfunktion des $\xi$-ten Orbitals,
                $\hat{T}$ kinetische Energie der Elektronen,
                $\hat{V}_{\text{en}}$ Electron-Kern Anziehung,
                $\hat{V}_{\text{HF}}$ \fqEqRef{comp:dft:hf:potential}
            }
            \eq{
                \left(\hat{T} + \hat{V}_{\text{en}} + \hat{V}_{\text{HF}}^\xi\right)\varphi_\xi(x) = \epsilon_\xi \varphi_\xi(x)
            }
        \end{formula}
        \begin{formula}{potential}
            \desc{Hartree-Fock potential}{}{}
            \desc[german]{Hartree Fock Potential}{}{}
            \eq{
                V_{\text{HF}}^\xi(\vecr) = 
                \sum_{\vartheta} \int \d x' 
                \frac{e^2}{\abs{\vecr - \vecr'}}
                \left(
                    \underbrace{\abs{\varphi_\xi(x')}^2}_{\text{Hartree-Term}}
                    - \underbrace{\frac{\varphi_{\vartheta}^*(x') \varphi_{\xi}(x') \varphi_{\vartheta}(x)}{\varphi_\xi(x)}}_{\text{Fock-Term}}
                \right)
            }
        \end{formula}
        \begin{formula}{scf}
            \desc{Self-consistend field cycle}{}{}
            % \desc[german]{}{}{}
            \ttxt{
                \eng{
                    \begin{enumerate}
                        \item Initial guess for $\psi$
                        \item Solve SG for each particle
                        \item Make new guess for $\psi$
                    \end{enumerate}
                }
            }
        \end{formula}

\Section[
    \eng{Atomic dynamics}
    % \ger{}
]{ad}
    \Subsection[
        \eng{Kohn-Sham}
        \ger{Kohn-Sham}
    ]{ks}
    \TODO{TODO}

    \Subsection[
        \eng{Born-Oppenheimer Approximation}
        \ger{Born-Oppenheimer Näherung}
    ]{bo}
        \TODO{TODO, BO surface}

    \Subsection[
        \eng{Molecular Dynamics}
        \ger{Molekulardynamik}
    ]{md}
        \begin{ttext}
            \eng{Statistical method}
            
        \end{ttext}

        \TODO{ab-initio MD, force-field MD}



\Section[
    \eng{Gradient descent}
    \ger{Gradientenverfahren}
]{gd}
    \TODO{TODO}



