\Section[
    \eng{Atomic dynamics}
    % \ger{}
]{ad}
\begin{formula}{hamiltonian}
    \desc{Electron Hamiltonian}{}{$\hat{T}$ \fqEqRef{comp:est:kinetic_energy}, $\hat{V}$ \fqEqRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
    \desc[german]{Hamiltonian der Elektronen}{}{}
    \eq{\hat{H}_\txe = \hat{T}_\txe + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe}}
\end{formula}
\begin{formula}{ansatz}
    \desc{Wave function ansatz}{}{$\psi_\text{en}^n$ eigenstate $n$ of \fqEqRef{comp:est:hamiltonian}, $\psi_\txe^i$ eigenstate $i$ of \fqEqRef{comp:ad:bo:hamiltonian}, $\vecr,\vecR$ electron/nucleus positions, $\sigma$ electron spin, $c^{ni}$ coefficients}
    \desc[german]{Wellenfunktion Ansatz}{}{}
    \eq{\psi_\text{en}^n\big(\{\vecr,\sigma\},\{\vecR\}\big) = \sum_i c^{ni}\big(\{\vecR\}\big)\, \psi_\txe^i\big(\{\vecr,\sigma\},\{\vecR\}\big)}
\end{formula}
\begin{formula}{equation}
    \desc{Equation}{}{}
    % \desc[german]{}{}{}
    \eq{
        \label{eq:\fqname}
        \left[E_\txe^j\big(\{\vecR\}\big) + \hat{T}_\txn + V_{\txn \leftrightarrow \txn} - E^n \right]c^{nj} = -\sum_i \Lambda_{ij} c^{ni}\big(\{\vecR\}\big)
    }
\end{formula}
\begin{formula}{coupling_operator}
    \desc{Exact nonadiabtic coupling operator}{Electron-phonon couplings / electron-vibrational couplings}{$\psi^i_\txe$ electronic states, $\vecR$ nucleus position, $M$ nucleus \qtyRef{mass}}
    % \desc[german]{}{}{}
    \begin{multline}
        \Lambda_{ij} = \int \d^3r (\psi_\txe^j)^* \left(-\sum_I \frac{\hbar^2\nabla_{\vecR_I}^2}{2M_I}\right) \psi_\txe^i \\
        + \sum_I \frac{1}{M_I} \int\d^3r \left[(\psi_\txe^j)^* (-i\hbar\nabla_{\vecR_I})\psi_\txe^i\right](-i\hbar\nabla_{\vecR_I})
    \end{multline}
\end{formula}

\Subsection[
    \eng{Born-Oppenheimer Approximation}
    \ger{Born-Oppenheimer Näherung}
]{bo}
    \begin{formula}{adiabatic_approx}
        \desc{Adiabatic approximation}{Electronic configuration remains the same when atoms move (\absRef{adiabatic_theorem})}{$\Lambda_{ij}$ \fqEqRef{comp:ad:coupling_operator}} 
        \desc[german]{Adiabatische Näherung}{Elektronenkonfiguration bleibt gleich bei Bewegung der Atome gleichl (\absRef{adiabatic_theorem})}{}
        \eq{\Lambda_{ij} = 0 \quad \text{\GT{for} } i\neq j}
    \end{formula}
    \begin{formula}{approx}
        \desc{Born-Oppenheimer approximation}{Electrons are not influenced by the movement of the atoms}{\GT{see} \fqEqRef{comp:ad:equation}, $V_{\txn \leftrightarrow \txn} = \const$ absorbed into $E_\txe^j$}
        \desc[german]{Born-Oppenheimer Näherung}{Elektronen werden nicht durch die Bewegung der Atome beeinflusst}{}
        \begin{gather}
            \Lambda_{ij} = 0
            \shortintertext{\fqEqRef{comp:ad:bo:equation} \Rightarrow}
            \left[E_e^i\big(\{\vecR\}\big) + \hat{T}_\txn - E^n\right]c^{ni}\big(\{\vecR\}\big) = 0
        \end{gather}
    \end{formula}
    \begin{formula}{surface}
        \desc{Born-Oppenheimer surface}{Potential energy surface (PES)\\ The nuclei follow Newtons equations of motion on the BO surface if the system is in the electronic ground state}{$E_\txe^0, \psi_\txe^0$ lowest eigenvalue/eigenstate of \fqEqRef{comp:ad:bo:hamiltonian}}
        \desc[german]{Born-Oppenheimer Potentialhyperfläche}{Die Nukleonen Newtons klassichen Bewegungsgleichungen auf der BO Hyperfläche wenn das System im elektronischen Grundzustand ist}{$E_\txe^0, \psi_\txe^0$ niedrigster Eigenwert/Eigenzustand vom \fqEqRef{comp:ad:bo:hamiltonian}}
        \begin{gather}
            V_\text{BO}\big(\{\vecR\}\big) = E_\txe^0\big(\{\vecR\}\big) \\
            M_I \ddot{\vecR}_I(t) = - \Grad_{\vecR_I} V_\text{BO}\big(\{\vecR(t)\}\big)
        \end{gather}
    \end{formula}
    \begin{formula}{ansatz}
        \desc{Ansatz for \secEqRef{approx}}{Product of single electronic and single nuclear state}{}
        \desc[german]{Ansatz für \secEqRef{approx}}{Produkt aus einem einzelnen elektronischen Zustand und einem Nukleus-Zustand}{}
        \eq{
            \psi_\text{BO} = c^{n0} \big(\{\vecR\}\big) \,\psi_\txe^0 \big(\{\vecr,\sigma\},\{\vecR\}\big)
        }
    \end{formula}

    \begin{formula}{limitations}
        \desc{Limitations}{}{$\tau$ passage of time for electrons/nuclei, $L$ characteristic length scale of atomic dynamics, $\dot{\vec{R}}$ nuclear velocity, $\Delta E$ difference between two electronic states}
        \desc[german]{Limitationen}{}{}
        \ttxt{
            \eng{ 
                \begin{itemize}
                    \item Nuclei velocities must be small and electron energy state differences large
                    \item Nuclei need spin for effects like spin-orbit coupling
                    \item Nonadiabitc effects in photochemistry, proteins
                \end{itemize}
                Valid when Massey parameter $\xi \gg 1$
            }
        }
        \eq{
            \xi = \frac{\tau_\txn}{\tau_\txe} = \frac{L \Delta E}{\hbar \abs{\dot{\vecR}}}
        }
    \end{formula}   

\Subsection[
    \eng{Structure optimization}
    \ger{Strukturoptimierung}
]{opt}
    \begin{formula}{forces}
        \desc{Forces}{}{}
        \desc[german]{Kräfte}{}{}
        \eq{\vec{F}_I = -\Grad_{\vecR_I} E \explOverEq{\fqEqRef{qm:se:hellmann_feynmann}} -\Braket{\psi(\vecR_I) | \left(\Grad_{\vecR_I} \hat{H}(\vecR_I)\right) | \psi(\vecR) }}
    \end{formula}
    \begin{formula}{ionic_cycle}
        \desc{Ionic cycle}{\fqEqRef{comp:est:dft:ks:scf} for geometry optimization}{}
        \desc[german]{}{}{}
        \ttxt{
            \eng{
                \begin{enumerate}
                    \item Initial guess for $n(\vecr)$
                    \begin{enumerate}
                        \item Calculate effective potential $V_\text{eff}$
                        \item Solve \fqEqRef{comp:est:dft:ks:equation}
                        \item Calculate density $n(\vecr)$
                        \item Repeat b-d until self consistent
                    \end{enumerate}
                    \item Calculate \secEqRef{forces}
                    \item If $F\neq0$, get new geometry by interpolating $R$ and restart
                \end{enumerate}
            }
        }
    \end{formula}

    \begin{formula}{transformation}
        \desc{Transformation of atomic positions under stress}{}{$\alpha,\beta=1,2,3$ position components, $R$ position, $R(0)$ zero-strain position, $\ten{\epsilon}$ \qtyRef{strain} tensor}
        \desc[german]{Transformation der Atompositionen unter Spannung}{}{$\alpha,\beta=1,2,3$ Positionskomponenten, $R$ Position, $R(0)$ Position ohne Dehnung, $\ten{\epsilon}$ \qtyRef{strain} Tensor}
        \eq{R_\alpha(\ten{\epsilon}_{\alpha\beta}) = \sum_\beta \big(\delta_{\alpha\beta} + \ten{\epsilon}_{\alpha\beta}\big)R_\beta(0)}
    \end{formula}

    \begin{formula}{stress_tensor}
        \desc{Stress tensor}{}{$\Omega$ unit cell volume, \ten{\epsilon} \qtyRef{strain} tensor}
        \desc[german]{Spannungstensor}{}{}
        \eq{\ten{\sigma}_{\alpha,\beta} = \frac{1}{\Omega} \pdv{E_\text{total}}{\ten{\epsilon}_{\alpha\beta}}_{\ten{\epsilon}=0}}
    \end{formula}

    \begin{formula}{pulay_stress}
        \desc{Pulay stress}{}{}
        \desc[german]{Pulay-Spannung}{}{}
        \eq{
            N_\text{PW} \propto E_\text{cut}^\frac{3}{2} \propto \abs{\vec{G}_\text{max}}^3
        }
        \ttxt{\eng{
            Number of plane waves $N_\text{PW}$ depends on $E_\text{cut}$.
            If $G$ changes during optimization, $N_\text{PW}$ may change, thus the basis set can change.
            This typically leads to too small volumes.
        }}
    \end{formula}

\Subsection[
    \eng{Lattice vibrations}
    \ger{Gitterschwingungen}
]{latvib}
    \begin{formula}{force_constant_matrix}
        \desc{Force constant matrix}{}{}
        % \desc[german]{}{}{}
        \eq{\Phi_{IJ}^{\mu\nu} = \pdv{V(\{\vecR\})}{R_I^\mu,R_J^\nu}_{\{\vecR_I\}=\{\vecR_I^0\}}}
    \end{formula}

    \begin{formula}{harmonic_approx}
        \desc{Harmonic approximation}{Hessian matrix, 2nd order Taylor expansion of the \fqEqRef{comp:ad:bo:surface} around every nucleus position $\vecR_I^0$}{$\Phi_{IJ}^{\mu\nu}$ \secEqRef{force_constant_matrix}, $s$ displacement}
        \desc[german]{Harmonische Näherung}{Hesse matrix, Taylor Entwicklung der \fqEqRef{comp:ad:bo:surface} in zweiter Oddnung um Atomposition $\vecR_I^0$}{}
        \eq{ V^\text{BO}(\{\vecR_I\}) \approx V^\text{BO}(\{\vecR_I^0\}) + \frac{1}{2} \sum_{I,J}^N \sum_{\mu,\nu}^3 s_I^\mu s_J^\nu \Phi_{IJ}^{\mu\nu} }
    \end{formula}
    
    % solving difficult becaus we need to calculate (3N)^2 derivatives, Hellmann-Feynman cant be applied directly
    % -> DFPT

    % finite-difference method
    \Subsubsection[
        \eng{Finite difference method}
        % \ger{}
    ]{fin_diff}

        \begin{formula}{approx}
            \desc{Approximation}{Assume forces in equilibrium structure vanish}{$\Delta s$ displacement of atom $J$}
            % \desc[german]{}{}{}
            \eq{\Phi_{IJ}^{\mu\nu} \approx \frac{\vecF_I^\mu(\vecR_1^0, \dots, \vecR_J^0+\Delta s_J^\nu,\dots, \vecR_N^0)}{\Delta s_J^\nu}}
        \end{formula}
        \begin{formula}{dynamical_matrix}
            \desc{Dynamical matrix}{Mass reduced \absRef[fourier transform]{fourier_transform} of the \fqEqRef{comp:ad:latvib:force_constant_matrix}}{$\vec{L}$ vector from origin to unit cell $n$, $\alpha/\beta$ atom index in th unit cell, $\vecq$ \qtyRef{wave_vector}, $\Phi$ \fqEqRef{comp:ad:latvib:force_constant_matrix}, $M$ \qtyRef{mass}}
            % \desc[german]{}{}{}
            \eq{D_{\alpha\beta}^{\mu\nu} = \frac{1}{\sqrt{M_\alpha M_\beta}} \sum_{n^\prime} \Phi_{\alpha\beta}^{\mu\nu}(n-n^\prime) \e^{\I \vec{q}(\vec{L}_n - \vec{L}_{n^\prime})}}
        \end{formula}

        \begin{formula}{eigenvalue_equation}
            \desc{Eigenvalue equation}{For a periodic crystal, reduces number of equations from $3N_p\times N$ to $3N_p$. Eigenvalues represent phonon band structure.}{$N_p$ number of atoms per unit cell, $\vecc$ displacement amplitudes, $\vecq$ \qtyRef{wave_vector}, $\mat{D}$ \secEqRef{dynamical_matrix}}
            \desc[german]{Eigenwertgleichung}{}{}
            \eq{\omega^2 \vecc(\vecq) = \mat{D}(\vecq) \vecc(\vecq) }
        \end{formula}

    \Subsubsection[
        \eng{Anharmonic approaches}
        \ger{Anharmonische Ansätze}
    ]{anharmonic}

        \begin{formula}{qha}
            \desc{Quasi-harmonic approximation}{}{}
            \desc[german]{}{}{}
            \ttxt{\eng{
                Include thermal expansion by assuming \fqEqRef{comp:ad:bo:surface} is volume dependant.
            }}
        \end{formula}

        \begin{formula}{pertubative}
            \desc{Pertubative approaches}{}{}
            % \desc[german]{Störungs}{}{}
            \ttxt{\eng{
                Expand \fqEqRef{comp:ad:latvib:force_constant_matrix} to third order.
            }}
        \end{formula}

    


\Subsection[
    \eng{Molecular Dynamics}
    \ger{Molekulardynamik}
]{md} \abbrLink{md}{MD}
    \begin{formula}{desc}
        \desc{Description}{}{}
        \desc[german]{Beschreibung}{}{}
        \ttxt{\eng{
            \begin{itemize}
                \item Exact (within previous approximations) approach to treat anharmonic effects in materials.
                \item Computes time-dependant observables.
                \item Assumes fully classical nuclei.
                \item Macroscropical observables from statistical ensembles 
                \item System evolves in time (ehrenfest). Number of points to consider does NOT scale with system size.
                \item Exact because time dependance is studied explicitly, not via harmonic approx.
            \end{itemize}
            \TODO{cleanup}
        }}
    \end{formula}

    \begin{formula}{procedure}
        \desc{MD simulation procedure}{}{}
        \desc[german]{Ablauf von MD Simulationen}{}{}
        \ttxt{\eng{
            \begin{enumerate}
                \item Initialize with optimized geometry, interaction potential, ensemble, integration scheme, temperature/pressure control
                \item Equilibrate to desired temperature/pressure (eg with statistical starting velocities)
                \item Production run, run MD long enough to calculate desired observables
            \end{enumerate}
        }}
    \end{formula}

    \Subsubsection[
        \eng{Ab-initio molecular dynamics}
        \ger{Ab-initio molecular dynamics}
    ]{ab-initio}
        \begin{formula}{bomd}
            \abbrLabel{BOMD}
            \desc{Born-Oppenheimer MD (BOMD)}{}{}
            \desc[german]{Born-Oppenheimer MD (BOMD)}{}{}
            \ttxt{\eng{
                \begin{enumerate}
                    \item Calculate electronic ground state of current nucleui configuration $\{\vecR(t)\}$ with \abbrRef{ksdft}
                    \item \hyperref[f:comp:ad:opt:forces]{Calculate forces} from the \fqEqRef{comp:ad:bo:surface} 
                    \item  Update positions and velocities
                \end{enumerate}
                \begin{itemize}
                    \gooditem "ab-inito" - no empirical information required
                    \baditem Many expensive \abbrRef{dft} calculations
                \end{itemize}
            }}
        \end{formula}

        \begin{formula}{cpmd}
            \desc{Car-Parrinello MD (CPMD)}{}{$\mu$ electron orbital mass, $\varphi_i$ \abbrRef{ksdft} eigenststate, $\lambda_{ij}$ Lagrange multiplier}
            \desc[german]{Car-Parrinello MD (CPMD)}{}{}
            \ttxt{\eng{
                Evolve electronic wave function $\varphi$ (adiabatically) along with the nuclei \Rightarrow only one full \abbrRef{ksdft} 
            }}
            \begin{gather}
                M_I \odv[2]{\vecR_I}{t} = -\Grad_{\vecR_I} E[\{\varphi_i\},\{\vecR_I\}] \\
                % not using pdv because of comma in parens
                % E[\{\varphi_i\}\{\vecR_I\}] = \Braket{\psi_0|H_\text{el}^\text{KS}|\psi_0}
                \mu \odv[2]{\varphi_i(\vecr,t)}{t} = - \frac{\partial}{\partial\varphi_i^*(\vecr,t)} E[\{\varphi_i\},\{\vecR_I\}] + \sum_j \lambda_{ij} \varphi_j(\vecr,t)
            \end{gather}
        \end{formula}

    \Subsubsection[
        \eng{Force-field MD}
        \ger{Force-field MD}
    ]{ff}

        \begin{formula}{ffmd}
            \desc{Force field MD (FFMD)}{}{}
            % \desc[german]{}{}{}
            \ttxt{\eng{
                \begin{itemize}
                    \item Use empirical interaction potential instead of electronic structure
                    \baditem Force fields need to be fitted for specific material \Rightarrow not transferable 
                    \gooditem Faster than \abbrRef{bomd}
                    \item Example: \absRef[Lennard-Jones]{lennard_jones}
                \end{itemize}
            }}
        \end{formula}



    \Subsubsection[
        \eng{Integration schemes}
        % \ger{}
    ]{scheme}
        \begin{ttext} 
            \eng{Procedures for updating positions and velocities to obey the equations of motion.}
        \end{ttext}
        
        \begin{formula}{euler}
            \desc{Euler method}{First-order procedure for solving \abbrRef{ode}s with a given initial value.\\Taylor expansion of $\vecR/\vecv (t+\Delta t)$}{}
            \desc[german]{Euler-Verfahren}{Prozedur um gewöhnliche DGLs mit Anfangsbedingungen in erster Ordnung zu lösen.\\Taylor Entwicklung von $\vecR/\vecv (t+\Delta t)$}{}
            \eq{
                \vecR(t+\Delta t) &= \vecR(t) + \vecv(t) \Delta t + \Order{\Delta t^2} \\
                \vecv(t+\Delta t) &= \vecv(t) + \veca(t) \Delta t + \Order{\Delta t^2}
            }
            \ttxt{\eng{
                Cumulative error scales linearly $\Order{\Delta t}$. Not time reversible.
            }}
        \end{formula}

        \begin{formula}{verlet}
            \desc{Verlet integration}{Preverses time reversibility, does not require velocity updates. Integration in 2nd order}{}
            \desc[german]{Verlet-Algorithmus}{Zeitumkehr-symmetrisch. Interation in zweiter Ordnung}{}
            \eq{
                \vecR(t+\Delta t) = 2\vecR(t) -\vecR(t-\Delta t) + \veca(t) \Delta t^2 + \Order{\Delta t^4}
            }
        \end{formula}
        
        \begin{formula}{velocity-verlet}
            \desc{Velocity-Verlet integration}{}{}
            % \desc[german]{}{}{}
            \eq{
                \vecR(t+\Delta t) &= \vecR(t) + \vecv(t)\Delta t + \frac{1}{2} \veca(t) \Delta t^2 + \Order{\Delta t^4} \\
                \vecv(t+\Delta t) &= \vecv(t) + \frac{\veca(t) + \veca(t+\Delta t)}{2} \Delta t + \Order{\Delta t^4} 
            }
        \end{formula}
        
        \begin{formula}{leapfrog}
            \desc{Leapfrog}{Integration in 2nd order}{}
            \desc[german]{Leapfrog}{Integration in zweiter Ordnung}{}
            \eq{
                x_{i+1} &= x_i + v_{i+1/2} \Delta t_i \\
                v_{i+1/2} &= v_{i-1/2} + a_{i} \Delta t_i
            }
        \end{formula}

    \Subsubsection[
        \eng{Thermostats and barostats}
        \ger{Thermostate und Barostate}
    ]{stats}
        \begin{formula}{velocity_rescaling}
            \desc{Velocity rescaling}{Thermostat, keep temperature at $T_0$ by rescaling velocities. Does not allow temperature fluctuations and thus does not obey the \absRef{c_ensemble}}{$T$ target \qtyRef{temperature}, $M$ \qtyRef{mass} of nucleon $I$, $\vecv$ \qtyRef{velocity}, $f$ number of degrees of freedom, $\lambda$ velocity scaling parameter, \ConstRef{boltzmann}}
            % \desc[german]{}{}{}
            \eq{
                \Delta T(t) &= T_0 - T(t) \\
                            &= \sum_I^N \frac{M_I\,(\lambda \vecv_I(t))^2}{f\kB} - \sum_I^N \frac{M_I\,\vecv_I(t)^2}{f\kB} \\
                            &= (\lambda^2 - 1) T(t)
            }
            \eq{\lambda = \sqrt{\frac{T_0}{T(t)}}}
        \end{formula}

        \begin{formula}{berendsen}
            \desc{Berendsen thermostat}{Does not obey \absRef{c_ensemble} but efficiently brings system to target temperature}{}
            % \desc[german]{}{}{}
            \eq{\odv{T}{t} = \frac{T_0-T}{\tau}}
        \end{formula}

        \begin{formula}{nose-hoover}
            \desc{Nosé-Hoover thermostat}{Control the temperature with by time stretching with an associated mass.\\Compliant with \absRef{c_ensemble}}{$s$ scaling factor, $Q$ associated "mass", $\mathcal{L}$ \absRef{lagrangian}, $g$ degrees of freedom}
            \desc[german]{Nosé-Hoover Thermostat}{}{}
            \begin{gather}
                \d\tilde{t} = \tilde{s}\d t \\
                \mathcal{L} = \sum_{I=1}^N \frac{1}{2} M_I \tilde{s}^2 v_i^2 - V(\tilde{\vecR}_1, \ldots, \tilde{\vecR}_I, \ldots, \tilde{\vecR}_N) + \frac{1}{2} Q \dot{\tilde{s}}^2 - g \kB T_0 \ln \tilde{s}
            \end{gather}
        \end{formula}

    \Subsubsection[
        \eng{Calculating observables}
        \ger{Berechnung von Observablen}
    ]{obs}
        \begin{formula}{spectral_density}
            \desc{Spectral density}{Wiener-Khinchin theorem\\\absRef{fourier_transform} of \absRef{autocorrelation}}{$C$ \absRef{autocorrelation}}
            \desc[german]{Spektraldichte}{Wiener-Khinchin Theorem\\\absRef{fourier_transform} of \absRef{autocorrelation}}{}
            \eq{S(\omega) = \int_{-\infty}^\infty \d\tau C(\tau) \e^{-\I\omega t} }
        \end{formula}

        \begin{formula}{vdos} \abbrLabel{VDOS}
            \desc{Vibrational density of states (VDOS)}{}{$S_{v_i}$ velocity \secEqRef{spectral_density} of particle $I$}
            \desc[german]{Vibrationszustandsdicht (VDOS)}{}{}
            \eq{g(\omega) \sim \sum_{I=1}^N M_I S_{v_I}(\omega)}
        \end{formula}
