386 lines
19 KiB
TeX
386 lines
19 KiB
TeX
\Section[
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\eng{Atomic dynamics}
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% \ger{}
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]{ad}
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\begin{formula}{hamiltonian}
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\desc{Electron Hamiltonian}{}{$\hat{T}$ \fRef{comp:est:kinetic_energy}, $\hat{V}$ \fRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
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\desc[german]{Hamiltonian der Elektronen}{}{}
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\eq{\hat{H}_\txe = \hat{T}_\txe + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe}}
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\end{formula}
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\begin{formula}{ansatz}
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\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}
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\desc[german]{Wellenfunktion Ansatz}{}{}
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\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)}
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\end{formula}
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\begin{formula}{equation}
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\desc{Equation}{}{}
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% \desc[german]{}{}{}
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\eq{
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\label{eq:\fqname}
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\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)
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}
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\end{formula}
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\begin{formula}{coupling_operator}
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\desc{Exact nonadiabtic coupling operator}{Electron-phonon couplings / electron-vibrational couplings}{$\psi^i_\txe$ electronic states, $\vecR$ nucleus position, $M$ nucleus \qtyRef{mass}}
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% \desc[german]{}{}{}
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\begin{multline}
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\Lambda_{ij} = \int \d^3r (\psi_\txe^j)^* \left(-\sum_I \frac{\hbar^2\nabla_{\vecR_I}^2}{2M_I}\right) \psi_\txe^i \\
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+ \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})
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\end{multline}
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\end{formula}
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\Subsection[
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\eng{Born-Oppenheimer Approximation}
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\ger{Born-Oppenheimer Näherung}
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]{bo}
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\begin{formula}{adiabatic_approx}
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\desc{Adiabatic approximation}{Electronic configuration remains the same when atoms move (\absRef{adiabatic_theorem})}{$\Lambda_{ij}$ \fRef{comp:ad:coupling_operator}}
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\desc[german]{Adiabatische Näherung}{Elektronenkonfiguration bleibt gleich bei Bewegung der Atome gleichl (\absRef{adiabatic_theorem})}{}
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\eq{\Lambda_{ij} = 0 \quad \text{\GT{for} } i\neq j}
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\end{formula}
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\begin{formula}{approx}
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\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 $E_\txe^j$}
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\desc[german]{Born-Oppenheimer Näherung}{Elektronen werden nicht durch die Bewegung der Atome beeinflusst}{}
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\begin{gather}
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\Lambda_{ij} = 0
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% \shortintertext{\fRef{comp:ad:bo:equation} \Rightarrow}
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\left[E_e^i\big(\{\vecR\}\big) + \hat{T}_\txn - E^n\right]c^{ni}\big(\{\vecR\}\big) = 0
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\end{gather}
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\end{formula}
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\begin{formula}{surface}
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\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}}
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\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}}
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\begin{gather}
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V_\text{BO}\big(\{\vecR\}\big) = E_\txe^0\big(\{\vecR\}\big) \\
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M_I \ddot{\vecR}_I(t) = - \Grad_{\vecR_I} V_\text{BO}\big(\{\vecR(t)\}\big)
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\end{gather}
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\end{formula}
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\begin{formula}{ansatz}
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\desc{Ansatz for \fRef{::approx}}{Product of single electronic and single nuclear state}{}
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\desc[german]{Ansatz für \fRef{::approx}}{Produkt aus einem einzelnen elektronischen Zustand und einem Nukleus-Zustand}{}
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\eq{
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\psi_\text{BO} = c^{n0} \big(\{\vecR\}\big) \,\psi_\txe^0 \big(\{\vecr,\sigma\},\{\vecR\}\big)
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}
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\end{formula}
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\begin{formula}{limitations}
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\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}
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\desc[german]{Limitationen}{}{}
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\ttxt{
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\eng{
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\begin{itemize}
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\item Nuclei velocities must be small and electron energy state differences large
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\item Nuclei need spin for effects like spin-orbit coupling
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\item Nonadiabitc effects in photochemistry, proteins
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\end{itemize}
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Valid when Massey parameter $\xi \gg 1$
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}
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}
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\eq{
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\xi = \frac{\tau_\txn}{\tau_\txe} = \frac{L \Delta E}{\hbar \abs{\dot{\vecR}}}
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}
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\end{formula}
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\Subsection[
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\eng{Structure optimization}
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\ger{Strukturoptimierung}
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]{opt}
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\begin{formula}{forces}
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\desc{Forces}{}{}
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\desc[german]{Kräfte}{}{}
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\eq{
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\vec{F}_I = -\Grad_{\vecR_I} E
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\explOverEq{\fRef{qm:se:hellmann_feynmann}}
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-\Braket{\psi(\vecR_I) | \left(\Grad_{\vecR_I} \hat{H}(\vecR_I)\right) | \psi(\vecR)}
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}
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\end{formula}
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\begin{formula}{ionic_cycle}
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\desc{Ionic cycle}{\fRef{comp:est:dft:ks:scf} for geometry optimization}{}
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\desc[german]{}{}{}
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\ttxt{
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\eng{
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\begin{enumerate}
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\item Initial guess for $n(\vecr)$
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\begin{enumerate}
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\item Calculate effective potential $V_\text{eff}$
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\item Solve \fRef{comp:est:dft:ks:equation}
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\item Calculate density $n(\vecr)$
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\item Repeat b-d until self consistent
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\end{enumerate}
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\item Calculate \fRef{:::forces}
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\item If $F\neq0$, get new geometry by interpolating $R$ and restart
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\end{enumerate}
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}
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}
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\end{formula}
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\begin{formula}{transformation}
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\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}
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\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}
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\eq{R_\alpha(\ten{\epsilon}_{\alpha\beta}) = \sum_\beta \big(\delta_{\alpha\beta} + \ten{\epsilon}_{\alpha\beta}\big)R_\beta(0)}
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\end{formula}
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\begin{formula}{stress_tensor}
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\desc{Stress tensor}{}{$\Omega$ unit cell volume, \ten{\epsilon} \qtyRef{strain} tensor}
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\desc[german]{Spannungstensor}{}{}
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\eq{\ten{\sigma}_{\alpha,\beta} = \frac{1}{\Omega} \pdv{E_\text{total}}{\ten{\epsilon}_{\alpha\beta}}_{\ten{\epsilon}=0}}
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\end{formula}
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\begin{formula}{pulay_stress}
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\desc{Pulay stress}{}{}
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\desc[german]{Pulay-Spannung}{}{}
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\eq{
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N_\text{PW} \propto E_\text{cut}^\frac{3}{2} \propto \abs{\vec{G}_\text{max}}^3
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}
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\ttxt{\eng{
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Number of plane waves $N_\text{PW}$ depends on $E_\text{cut}$.
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If $G$ changes during optimization, $N_\text{PW}$ may change, thus the basis set can change.
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This typically leads to too small volumes.
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}}
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\end{formula}
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\Subsection[
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\eng{Lattice vibrations}
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\ger{Gitterschwingungen}
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]{latvib}
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\begin{formula}{force_constant_matrix}
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\desc{Force constant matrix}{}{}
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% \desc[german]{}{}{}
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\eq{\Phi_{IJ}^{\mu\nu} = \pdv{V(\{\vecR\})}{R_I^\mu,R_J^\nu}_{\{\vecR_I\}=\{\vecR_I^0\}}}
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\end{formula}
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\begin{formula}{harmonic_approx}
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\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}
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\desc[german]{Harmonische Näherung}{Hesse matrix, Taylor Entwicklung der \fRef{comp:ad:bo:surface} in zweiter Oddnung um Atomposition $\vecR_I^0$}{}
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\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} }
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\end{formula}
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% solving difficult becaus we need to calculate (3N)^2 derivatives, Hellmann-Feynman cant be applied directly
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% -> DFPT
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% finite-difference method
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\Subsubsection[
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\eng{Finite difference method}
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% \ger{}
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]{fin_diff}
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\begin{formula}{approx}
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\desc{Approximation}{Assume forces in equilibrium structure vanish}{$\Delta s$ displacement of atom $J$}
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% \desc[german]{}{}{}
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\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}}
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\end{formula}
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\begin{formula}{dynamical_matrix}
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\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{wave_vector}, $\Phi$ \fRef{comp:ad:latvib:force_constant_matrix}, $M$ \qtyRef{mass}}
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% \desc[german]{}{}{}
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\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})}}
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\end{formula}
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\begin{formula}{eigenvalue_equation}
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\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}}
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\desc[german]{Eigenwertgleichung}{}{}
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\eq{\omega^2 \vecc(\vecq) = \mat{D}(\vecq) \vecc(\vecq) }
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\end{formula}
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\Subsubsection[
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\eng{Anharmonic approaches}
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\ger{Anharmonische Ansätze}
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]{anharmonic}
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\begin{formula}{qha}
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\desc{Quasi-harmonic approximation}{}{}
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\desc[german]{}{}{}
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\ttxt{\eng{
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Include thermal expansion by assuming \fRef{comp:ad:bo:surface} is volume dependant.
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}}
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\end{formula}
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\begin{formula}{pertubative}
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\desc{Pertubative approaches}{}{}
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% \desc[german]{Störungs}{}{}
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\ttxt{\eng{
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Expand \fRef{comp:ad:latvib:force_constant_matrix} to third order.
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}}
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\end{formula}
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\Subsection[
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\eng{Molecular Dynamics}
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\ger{Molekulardynamik}
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]{md} \abbrLink{md}{MD}
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\begin{formula}{desc}
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\desc{Description}{}{}
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\desc[german]{Beschreibung}{}{}
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\ttxt{\eng{
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\begin{itemize}
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\item Exact (within previous approximations) approach to treat anharmonic effects in materials.
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\item Computes time-dependant observables.
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\item Assumes fully classical nuclei.
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\item Macroscropical observables from statistical ensembles
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\item System evolves in time (ehrenfest). Number of points to consider does NOT scale with system size.
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\item Exact because time dependance is studied explicitly, not via harmonic approx.
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\end{itemize}
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\TODO{cleanup}
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}}
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\end{formula}
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\begin{formula}{procedure}
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\desc{MD simulation procedure}{}{}
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\desc[german]{Ablauf von MD Simulationen}{}{}
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\ttxt{\eng{
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\begin{enumerate}
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\item Initialize with optimized geometry, interaction potential, ensemble, integration scheme, temperature/pressure control
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\item Equilibrate to desired temperature/pressure (eg with statistical starting velocities)
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\item Production run, run MD long enough to calculate desired observables
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\end{enumerate}
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}}
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\end{formula}
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\Subsubsection[
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\eng{Ab-initio molecular dynamics}
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\ger{Ab-initio molecular dynamics}
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]{ab-initio}
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\begin{formula}{bomd}
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\abbrLabel{BOMD}
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\desc{Born-Oppenheimer MD (BOMD)}{}{}
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\desc[german]{Born-Oppenheimer MD (BOMD)}{}{}
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\ttxt{\eng{
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\begin{enumerate}
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\item Calculate electronic ground state of current nucleui configuration $\{\vecR(t)\}$ with \abbrRef{ksdft}
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\item \fRef[Calculate forces]{comp:ad:opt:forces} from the \fRef{comp:ad:bo:surface}
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\item Update positions and velocities
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\end{enumerate}
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\begin{itemize}
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\gooditem "ab-inito" - no empirical information required
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\baditem Many expensive \abbrRef{dft} calculations
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\end{itemize}
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}}
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\end{formula}
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\begin{formula}{cpmd}
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\desc{Car-Parrinello MD (CPMD)}{}{$\mu$ electron orbital mass, $\varphi_i$ \abbrRef{ksdft} eigenststate, $\lambda_{ij}$ Lagrange multiplier}
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\desc[german]{Car-Parrinello MD (CPMD)}{}{}
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\ttxt{\eng{
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Evolve electronic wave function $\varphi$ (adiabatically) along with the nuclei \Rightarrow only one full \abbrRef{ksdft}
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}}
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\begin{gather}
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M_I \odv[2]{\vecR_I}{t} = -\Grad_{\vecR_I} E[\{\varphi_i\},\{\vecR_I\}] \\
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% not using pdv because of comma in parens
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% E[\{\varphi_i\}\{\vecR_I\}] = \Braket{\psi_0|H_\text{el}^\text{KS}|\psi_0}
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\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)
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\end{gather}
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\end{formula}
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\Subsubsection[
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\eng{Force-field MD}
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\ger{Force-field MD}
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]{ff}
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\begin{formula}{ffmd}
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\desc{Force field MD (FFMD)}{}{}
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% \desc[german]{}{}{}
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\ttxt{\eng{
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\begin{itemize}
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\item Use empirical interaction potential instead of electronic structure
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\baditem Force fields need to be fitted for specific material \Rightarrow not transferable
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\gooditem Faster than \abbrRef{bomd}
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\item Example: \absRef[Lennard-Jones]{lennard_jones}
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\end{itemize}
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}}
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\end{formula}
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\Subsubsection[
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\eng{Integration schemes}
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% \ger{}
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]{scheme}
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\begin{ttext}
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\eng{Procedures for updating positions and velocities to obey the equations of motion.}
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\end{ttext}
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\begin{formula}{euler}
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\desc{Euler method}{First-order procedure for solving \abbrRef{ode}s with a given initial value.\\Taylor expansion of $\vecR/\vecv (t+\Delta t)$}{}
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\desc[german]{Euler-Verfahren}{Prozedur um gewöhnliche DGLs mit Anfangsbedingungen in erster Ordnung zu lösen.\\Taylor Entwicklung von $\vecR/\vecv (t+\Delta t)$}{}
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\eq{
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\vecR(t+\Delta t) &= \vecR(t) + \vecv(t) \Delta t + \Order{\Delta t^2} \\
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\vecv(t+\Delta t) &= \vecv(t) + \veca(t) \Delta t + \Order{\Delta t^2}
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}
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\ttxt{\eng{
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Cumulative error scales linearly $\Order{\Delta t}$. Not time reversible.
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}}
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\end{formula}
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\begin{formula}{verlet}
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\desc{Verlet integration}{Preverses time reversibility, does not require velocity updates. Integration in 2nd order}{}
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\desc[german]{Verlet-Algorithmus}{Zeitumkehr-symmetrisch. Interation in zweiter Ordnung}{}
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\eq{
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\vecR(t+\Delta t) = 2\vecR(t) -\vecR(t-\Delta t) + \veca(t) \Delta t^2 + \Order{\Delta t^4}
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}
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\end{formula}
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\begin{formula}{velocity-verlet}
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\desc{Velocity-Verlet integration}{}{}
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% \desc[german]{}{}{}
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\eq{
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\vecR(t+\Delta t) &= \vecR(t) + \vecv(t)\Delta t + \frac{1}{2} \veca(t) \Delta t^2 + \Order{\Delta t^4} \\
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\vecv(t+\Delta t) &= \vecv(t) + \frac{\veca(t) + \veca(t+\Delta t)}{2} \Delta t + \Order{\Delta t^4}
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}
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\end{formula}
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\begin{formula}{leapfrog}
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\desc{Leapfrog}{Integration in 2nd order}{}
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\desc[german]{Leapfrog}{Integration in zweiter Ordnung}{}
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\eq{
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x_{i+1} &= x_i + v_{i+1/2} \Delta t_i \\
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v_{i+1/2} &= v_{i-1/2} + a_{i} \Delta t_i
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}
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\end{formula}
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\Subsubsection[
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\eng{Thermostats and barostats}
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\ger{Thermostate und Barostate}
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]{stats}
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\begin{formula}{velocity_rescaling}
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\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}}
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% \desc[german]{}{}{}
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\eq{
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\Delta T(t) &= T_0 - T(t) \\
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&= \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} \\
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&= (\lambda^2 - 1) T(t)
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}
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\eq{\lambda = \sqrt{\frac{T_0}{T(t)}}}
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\end{formula}
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\begin{formula}{berendsen}
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\desc{Berendsen thermostat}{Does not obey \absRef{c_ensemble} but efficiently brings system to target temperature}{}
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% \desc[german]{}{}{}
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\eq{\odv{T}{t} = \frac{T_0-T}{\tau}}
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\end{formula}
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\begin{formula}{nose-hoover}
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\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}
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\desc[german]{Nosé-Hoover Thermostat}{}{}
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\begin{gather}
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\d\tilde{t} = \tilde{s}\d t \\
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\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}
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\end{gather}
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\end{formula}
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\Subsubsection[
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\eng{Calculating observables}
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\ger{Berechnung von Observablen}
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]{obs}
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\begin{formula}{spectral_density}
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\desc{Spectral density}{Wiener-Khinchin theorem\\\absRef{fourier_transform} of \absRef{autocorrelation}}{$C$ \absRef{autocorrelation}}
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\desc[german]{Spektraldichte}{Wiener-Khinchin Theorem\\\absRef{fourier_transform} of \absRef{autocorrelation}}{}
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\eq{S(\omega) = \int_{-\infty}^\infty \d\tau C(\tau) \e^{-\I\omega t} }
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\end{formula}
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\begin{formula}{vdos} \abbrLabel{VDOS}
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\desc{Vibrational density of states (VDOS)}{}{$S_{v_i}$ velocity \fRef{::spectral_density} of particle $I$}
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\desc[german]{Vibrationszustandsdicht (VDOS)}{}{}
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\eq{g(\omega) \sim \sum_{I=1}^N M_I S_{v_I}(\omega)}
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\end{formula}
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