\Section[
    \eng{Atomic dynamics}
    % \ger{}
]{ad}
\begin{formula}{hamiltonian}
    \desc{Electron Hamiltonian}{}{$\hat{T}$ \fRef{comp:est:kinetic_energy}, $\hat{V}$ \fRef{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 \fRef{comp:est:hamiltonian}, $\psi_\txe^i$ eigenstate $i$ of \fRef{comp:ad: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})}{$\Lambda_{ij}$ \fRef{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} \fRef{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{\fRef{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 \fRef{comp:ad: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 \fRef{comp:ad: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 \fRef{::approx}}{Product of single electronic and single nuclear state}{}
        \desc[german]{Ansatz für \fRef{::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{\fRef{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}{\fRef{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 \fRef{comp:est:dft:ks:equation}
                        \item Calculate density $n(\vecr)$
                        \item Repeat b-d until self consistent
                    \end{enumerate}
                    \item Calculate \fRef{:::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 \fRef{comp:ad:bo:surface} around every nucleus position $\vecR_I^0$}{$\Phi_{IJ}^{\mu\nu}$ \fRef{::force_constant_matrix}, $s$ displacement}
        \desc[german]{Harmonische Näherung}{Hesse matrix, Taylor Entwicklung der \fRef{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 \fRef{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{wavevector}, $\Phi$ \fRef{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}, $\mat{D}$ \fRef{::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 \fRef{comp:ad:bo:surface} is volume dependant.
            }}
        \end{formula}

        \begin{formula}{pertubative}
            \desc{Pertubative approaches}{}{}
            % \desc[german]{Störungs}{}{}
            \ttxt{\eng{
                Expand \fRef{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 Assumes fully classical nuclei
                \item Macroscropical observables from statistical ensembles 
                \item Number of points to consider does NOT scale with system size
                \item System evolves in time (\absRef{ehrenfest_theorem})
                \item Computes time-dependant observables
                \item Does not use \fRef{comp:ad:latvib:harmonic_approx} \Rightarrow Anharmonic effects included
            \end{itemize}
        }}
    \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 \fRef[Calculate forces]{comp:ad:opt:forces} from the \fRef{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 \fRef{::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}
