\Section{est}
    \desc{Electronic structure theory}{}{}
    % \desc[german]{}{}{}


\begin{formula}{kinetic_energy}
    \desc{Kinetic energy}{of species $i$}{$i$ = nucleons/electrons, $N$ number of particles, $m$ \qtyRef{mass}}
    \desc[german]{Kinetische Energie}{von Spezies $i$}{$i$ = Nukleonen/Elektronen, $N$ Teilchenzahl, $m$ \qtyRef{mass}}
    \eq{\hat{T}_i &= -\sum_{n=1}^{N_i} \frac{\hbar^2}{2 m_i} \vec{\nabla}^2_n}
\end{formula}
\begin{formula}{potential_energy}
    \desc{Electrostatic potential}{between species $i$ and $j$}{$i,j$ = nucleons/electrons, $r$ particle position, $Z_i$ charge of species $i$, \ConstRef{charge}}
    \desc[german]{Elektrostatisches Potential}{zwischen Spezies $i$ und $j$}{}
    \eq{\hat{V}_{i \leftrightarrow j} &= -\sum_{k,l} \frac{Z_i Z_j e^2}{\abs{\vecr_k - \vecr_l}}}
\end{formula}
\begin{formula}{hamiltonian}
    \desc{Electronic structure Hamiltonian}{}{$\hat{T}$ \fRef{comp:est:kinetic_energy}, $\hat{V}$ \fRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
    \eq{\hat{H} &= \hat{T}_\txe + \hat{T}_\txn + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe} + V_{\txn \leftrightarrow \txn}}
\end{formula}
\begin{formula}{mean_field}
    \desc{Mean field approximation}{Replaces 2-particle operator by 1-particle operator}{Example for Coulomb interaction between many electrons}
    \desc[german]{Molekularfeldnäherung}{Ersetzt 2-Teilchen Operator durch 1-Teilchen Operator}{Beispiel für Coulomb 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{tb}
    \desc{Tight-binding}{}{}
    \desc[german]{Modell der stark gebundenen Elektronen / Tight-binding}{}{}
    \begin{formula}{assumptions}
        \desc{Assumptions}{}{}
        \desc[german]{Annahmen}{}{}
        \ttxt{
            \eng{
                \begin{itemize}
                    \item Atomic wave functions are localized \Rightarrow Small overlap, interaction cutoff
                \end{itemize}
            }
        }
    \end{formula}
    \begin{formula}{hamiltonian}
        \desc{Tight-binding Hamiltonian}{in second quantized form}{$\hat{a}_i^\dagger$, $\hat{a}_i$ \GT{creation_annihilation_ops} create/destory an electron on site $i$, $\epsilon_i$ on-site energy, $t_{i,j}$ hopping amplitude, usually $\epsilon$ and $t$ are determined from experiments or other methods}
        \desc[german]{Tight-binding Hamiltonian}{in zweiter Quantisierung}{$\hat{a}_i^\dagger$, $\hat{a}_i$ \GT{creation_annihilation_ops} erzeugen/vernichten ein Elektron auf Platz $i$, $\epsilon_i$ on-site Energie, $t_{i,j}$ hopping Amplitude, meist werden $\epsilon$ und $t$ aus experimentellen Daten oder anderen Methoden bestimmt}
        \eq{\hat{H} = \sum_i \epsilon_i \hat{a}_i^\dagger \hat{a}_i - \sum_{i,j} t_{i,j} \left(\hat{a}_i^\dagger \hat{a}_j + \hat{a}_j^\dagger \hat{a}_i\right)}
    \end{formula}
    


\Subsection{dft}
    \desc{Density functional theory (DFT)}{}{}
    \desc[german]{Dichtefunktionaltheorie (DFT)}{}{}
    \abbrLink{dft}{DFT}
    \Subsubsection{hf}
        \desc{Hartree-Fock}{}{}
        \desc[german]{Hartree-Fock}{}{}
        \begin{formula}{description}
            \desc{Description}{}{}
            \desc[german]{Beschreibung}{}{}
            \begin{ttext}
                \eng{
                    \begin{itemize}
                        \item Assumes wave functions are \fRef{qm:other:slater_det} \Rightarrow Approximation
                        \item \fRef{comp:est:mean_field} theory obeying the Pauli principle
                        \item Self-interaction free: Self interaction is cancelled out by the Fock-term
                    \end{itemize}
                }
            \end{ttext}
        \end{formula}
        \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,
                $h\hat{V}_{\text{HF}}$ \fRef{comp:est:dft:hf:potential},
                $x = \vecr,\sigma$ position and spin 
            }
            \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,
                $\hat{V}_{\text{HF}}$ \fRef{comp:est:dft:hf:potential},
                $x = \vecr,\sigma$ Position and Spin 
            }
            \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-consistent field cycle}{}{}
            % \desc[german]{}{}{}
            \ttxt{
                \eng{
                    \begin{enumerate}
                        \item Initial guess for $\varphi$
                        \item Solve SG for each particle
                        \item Make new guess for $\varphi$
                    \end{enumerate}
                }
            }
        \end{formula}

    \Subsubsection{hk}
        \desc{Hohenberg-Kohn Theorems}{}{}
        \desc[german]{Hohenberg-Kohn Theoreme}{}{}
        \begin{formula}{hk1}
            \desc{Hohenberg-Kohn theorem (HK1)}{}{}
            \desc[german]{Hohenberg-Kohn Theorem (HK1)}{}{}
            \ttxt{
                \eng{For any system of interacting electrons, the ground state electron density $n(\vecr)$ determines $\hat{V}_\text{ext}$ uniquely up to a trivial constant. }
                \ger{Die Elektronendichte des Grundzustandes $n(\vecr)$ bestimmt ein einzigartiges $\hat{V}_{\text{ext}}$ eines Systems aus interagierenden Elektronen bis auf eine Konstante.}
            }
        \end{formula}
        \begin{formula}{hk2}
            \desc{Hohenberg-Kohn theorem (HK2)}{}{}
            \desc[german]{Hohenberg-Kohn Theorem (HK2)}{}{}
            \ttxt{
                \eng{Given the energy functional $E[n(\vecr)]$, the ground state density and energy can be obtained variationally. The density that minimizes the total energy is the exact ground state density. }
                \ger{Für ein Energiefunktional $E[n(\vecr)]$ kann die Grundzustandsdichte und Energie durch systematische Variation bestimmt werden. Die Dichte, welche die Gesamtenergie minimiert ist die exakte Grundzustandsichte. }
            }
        \end{formula}

        \begin{formula}{density}
            \desc{Ground state electron density}{}{}
            \desc[german]{Grundzustandselektronendichte}{}{}
            \eq{n(\vecr) = \Braket{\psi_0|\sum_{i=1}^N \delta(\vecr-\vecr_i)|\psi_0}}
        \end{formula}

    \Subsubsection{ks}
        \desc{Kohn-Sham DFT}{}{}
        \desc[german]{Kohn-Sham DFT}{}{}
        \abbrLink{ksdft}{KS-DFT}
        \begin{formula}{map}
            \desc{Kohn-Sham map}{}{}
            \desc[german]{Kohn-Sham Map}{}{}
            \ttxt{
                \eng{Maps fully interacting system of electrons to a system of non-interacting electrons with the same ground state density $n^\prime(\vecr) = n(\vecr)$}
            }
            \eq{n(\vecr) = \sum_{i=1}^N \abs{\phi_i(\vecr)}^2}
        \end{formula}
        \begin{formula}{functional}
            \desc{Kohn-Sham functional}{}{$T_\text{KS}$ kinetic enery, $V_\text{ext}$ external potential, $E_\txH$ \fRef[Hartree term]{comp:est:dft:hf:potential}, $E_\text{XC}$ \fRef{comp:est:dft:xc:xc}}
            \desc[german]{Kohn-Sham Funktional}{}{}
            \eq{E_\text{KS}[n(\vecr)] = T_\text{KS}[n(\vecr)] + V_\text{ext}[n(\vecr)] + E_\text{H}[n(\vecr)] + E_\text{XC}[n(\vecr)] }
        \end{formula}

        \begin{formula}{equation}
            \desc{Kohn-Sham equation}{Exact single particle \abbrRef{schroedinger_equation} (though often exact $E_\text{XC}$ is not known)\\ Solving it uses up a large portion of supercomputer resources}{$\phi_i^\text{KS}$ KS orbitals, $\int\d^3r v_\text{ext}(\vecr)n(\vecr)=V_\text{ext}[n(\vecr)]$}
            \desc[german]{Kohn-Sham Gleichung}{Exakte Einteilchen-\abbrRef{schroedinger_equation} (allerdings ist das exakte $E_\text{XC}$ oft nicht bekannt)\\ Die Lösung der Gleichung macht einen großen Teil der Supercomputer Ressourcen aus}{}
            \begin{multline}
                \biggr\{
                    -\frac{\hbar^2\nabla^2}{2m} 
                    + v_\text{ext}(\vecr)
                    + e^2 \int\d^3 \vecr^\prime \frac{n(\vecr^\prime)}{\abs{\vecr-\vecr^\prime}} \\
                    + \pdv{E_\txX[n(\vecr)]}{n(\vecr)} 
                    + \pdv{E_\txC[n(\vecr)]}{n(\vecr)}
                \biggr\} \phi_i^\text{KS}(\vecr) =\\
                = \epsilon_i^\text{KS} \phi_i^\text{KS}(\vecr)
            \end{multline}
        \end{formula}
        \begin{formula}{scf}
            \desc{Self-consistent field cycle for Kohn-Sham}{}{}
            % \desc[german]{}{}{}
            \ttxt{
                \itemsep=\parsep
                \eng{
                    \begin{enumerate}
                        \item Initial guess for $n(\vecr)$
                        \item Calculate effective potential $V_\text{eff}$
                        \item Solve \fRef{comp:est:dft:ks:equation}
                        \item Calculate density $n(\vecr)$
                        \item Repeat 2-4 until self consistent
                    \end{enumerate}
                }
            }
        \end{formula}

        \Subsubsection{xc}
            \desc{Exchange-Correlation functionals}{}{}
            \desc[german]{Exchange-Correlation Funktionale}{}{}
            \begin{formula}{xc}
                \desc{Exchange-Correlation functional}{}{}
                \desc[german]{Exchange-Correlation Funktional}{}{}
                \eq{ E_\text{XC}[n(\vecr)] = \Braket{\hat{T}} - T_\text{KS}[n(\vecr)] + \Braket{\hat{V}_\text{int}} - E_\txH[n(\vecr)] }
                \ttxt{\eng{
                    Accounts for:
                    \begin{itemize}
                            \item Kinetic energy difference between interaction and non-interacting system
                            \item Exchange energy due to Pauli principle
                            \item Correlation energy due to many-body Coulomb interaction (not accounted for in mean field Hartree term $E_\txH$)
                    \end{itemize}
                }}
            \end{formula}
            \begin{formula}{lda}
                \desc{Local density approximation (LDA)}{Simplest DFT functionals}{$\epsilon_\txX$ calculated exchange energy from \fRef[HEG model]{comp:qmb:models:heg}, $\epsilon_\txC$ correlation energy calculated with \fRef{comp:qmb:methods:qmonte-carlo}}
                \desc[german]{}{}{}
                \abbrLabel{LDA}
                \eq{E_\text{XC}^\text{LDA}[n(\vecr)] = \int \d^3r\,n(r) \Big[\epsilon_\txX[n(\vecr)] + \epsilon_\txC[n(\vecr)]\Big]}
            \end{formula}

            \begin{formula}{gga}
                \desc{Generalized gradient approximation (GGA)}{}{$\epsilon_\txX$ calculated exchange energy from \fRef[HEG model]{comp:qmb:models:heg}, $F_\text{XC}$ function containing exchange-correlation energy dependency on $n$ and $\Grad n$}
                \desc[german]{}{}{}
                \abbrLabel{GGA}
                \eq{E_\text{XC}^\text{GGA}[n(\vecr)] = \int \d^3r\,n(r) \epsilon_\txX[n(\vecr)]\,F_\text{XC}[n(\vecr), \Grad n(\vecr)]}
            \end{formula}

            \begin{formula}{hybrid}
                \desc{Hybrid functionals}{}{}
                \desc[german]{Hybride Funktionale}{}{$\alpha$ mixing paramter, $E_\txX$ exchange energy, $E_\txC$ correlation energy}
                \eq{\alpha E_\txX^\text{HF} + (1-\alpha) E_\txX^\text{GGA} + E_\txC^\text{GGA}}
                \ttxt{\eng{
                    Include \fRef[Fock term]{comp:est:dft:hf:potential} (exact exchange) in other functional, like \abbrRef{gga}. Computationally expensive
                }}
            \end{formula}


            \begin{formula}{range-separated-hybrid}
                \desc{Range separated hyrid functionals (RSH)}{Here HSE as example}{$\alpha$ mixing paramter, $E_\txX$ exchange energy, $E_\txC$ correlation energy}
                % \desc[german]{}{}{}
                \begin{gather}
                    \frac{1}{r} = \frac{\erf(\omega r)}{r} + \frac{\erfc{\omega r}}{r} \\
                    E_\text{XC}^\text{HSE} = \alpha E_\text{X,SR}^\text{HF}(\omega) + (1-\alpha)E_\text{X,SR}^\text{GGA}(\omega) + E_\text{X,LR}^\text{GGA}(\omega) + E_\txC^\text{GGA}
                \end{gather}
                \separateEntries
                \ttxt{\eng{
                        Use \abbrRef{gga} and \fRef[Fock]{comp:est:dft:hf:potential} exchange for short ranges (SR) and only \abbrRef{GGA} for long ranges (LR).
                        \abbrRef{GGA} correlation is always used. Useful when dielectric screening reduces long range interactions, saves computational cost.
                }}
            \end{formula}   

    
    \Subsubsection{basis}
        \desc{Basis sets}{}{}
        \desc[german]{Basis-Sets}{}{}
        \begin{formula}{plane_wave}
            \desc{Plane wave basis}{Plane wave ansatz in \fRef{comp:est:dft:ks:equation}\\Good for periodic structures, allows computation parallelization over a sample points in the \abbrRef{brillouin_zone}}{}
            \desc[german]{Ebene Wellen als Basis}{}{}
            \eq{\sum_{\vecG^\prime} \left[\frac{\hbar^2 \abs{\vecG+\veck}^2}{2m} \delta_{\vecG,\vecG^\prime} + V_\text{eff}(\vecG-\vecG^\prime)\right] c_{i,\veck,\vecG^\prime} = \epsilon_{i,\veck} c_{i,\veck,\vecG}}
        \end{formula}
        \begin{formula}{plane_wave_cutoff}
            \desc{Plane wave cutoff}{Number of plane waves included in the calculation must be finite}{}
            % \desc[german]{}{}{}
            \eq{E_\text{cutoff} = \frac{\hbar^2 \abs{\veck+\vecG}^2}{2m}}
        \end{formula}

    \Subsubsection{pseudo}
        \desc{Pseudo-Potential method}{}{}
        \desc[german]{Pseudopotentialmethode}{}{}
        \begin{formula}{ansatz}
            \desc{Ansatz}{}{}
            \desc[german]{Ansatz}{}{}
            \ttxt{\eng{
                Core electrons are absorbed into the potential since they do not contribute much to interesting properties.
            }}
        \end{formula}
