396 lines
24 KiB
TeX
396 lines
24 KiB
TeX
\Section[
|
|
\eng{Electronic structure theory}
|
|
% \ger{}
|
|
]{est}
|
|
|
|
\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[
|
|
\eng{Tight-binding}
|
|
\ger{Modell der stark gebundenen Elektronen / Tight-binding}
|
|
]{tb}
|
|
\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[
|
|
\eng{Density functional theory (DFT)}
|
|
\ger{Dichtefunktionaltheorie (DFT)}
|
|
]{dft}
|
|
\abbrLink{dft}{DFT}
|
|
\Subsubsection[
|
|
\eng{Hartree-Fock}
|
|
\ger{Hartree-Fock}
|
|
]{hf}
|
|
\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[
|
|
\eng{Hohenberg-Kohn Theorems}
|
|
\ger{Hohenberg-Kohn Theoreme}
|
|
]{hk}
|
|
\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[
|
|
\eng{Kohn-Sham DFT}
|
|
\ger{Kohn-Sham DFT}
|
|
]{ks}
|
|
\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[
|
|
\eng{Exchange-Correlation functionals}
|
|
\ger{Exchange-Correlation Funktionale}
|
|
]{xc}
|
|
\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}
|
|
|
|
\begin{bigformula}{comparison}
|
|
\desc{Comparison of DFT functionals}{}{}
|
|
\desc[german]{Vergleich von DFT Funktionalen}{}{}
|
|
% \begin{tabular}{l|c}
|
|
% \fRef[Hartree-Fock]{comp:est:dft:hf:potential} & only exchange, no correlation \Rightarrow upper bound of GS energy \\
|
|
% \abbrRef{lda} & understimates e repulsion \Rightarrow Overbinding \\
|
|
% \abbrRef{gga} & underestimate band gap \\
|
|
% hybrid & underestimate band gap
|
|
% \end{tabular}
|
|
|
|
\TODO{HFtotal energy: upper boundary for GS density $n$}
|
|
|
|
\newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
|
|
% \begin{tabular}{|P{0.15\textwidth}|P{0.2\textwidth}|P{0.1\textwidth}|P{0.2\textwidth}|P{0.1\textwidth}|P{0.1\textwidth}|P{0.15\textwidth}|}
|
|
% \hline
|
|
% \textbf{Method} & \textbf{Description} & \textbf{Mean Absolute Error (eV)} & \textbf{Band Gap Accuracy} & \textbf{Computational Cost} & \textbf{Usage} & \textbf{Other Notes} \\
|
|
% \hline
|
|
% Hartree-Fock (HF) &
|
|
% $E_C \sim E_C^{HF\text{theory}}$
|
|
% $E_X \sim E_X^{FOCK}$
|
|
% & 3.1 (Underbinding) & \tabitem no SIE \tabitem correct long-range behaviour \tabitem nonlinear chemical potential (missing DD) \tabitem positive correlation effects & High & Reference for exact exchange, useful for small molecules. & Self-interaction free, but lacks correlation. \\
|
|
% \hline
|
|
% Local Density Approximation (LDA) &
|
|
% $E_x \sim n(r)$
|
|
% $E_c \sim n(r)$
|
|
% & 1.3 (Overbinding) & \tabitem SIE \tabitem wrong long-range behaviour \tabitem nonlinear chemical potential (missing DD) & Low & Basic solids and metallic systems, where accuracy is not critical. & Simple and computationally cheap. \\
|
|
% \hline
|
|
% Generalised Gradient Approximation (GGA) &
|
|
% $E_x \sim n(r), \nabla n(r)$
|
|
% $E_c \sim n(r), \nabla n(r)$
|
|
% & 0.3 (Mostly overbinding) & \tabitem SIE \tabitem wrong long-range behaviour \tabitem nonlinear chemical potential (missing DD) & Moderate & More accurate for molecules and chemical bonding studies. & Better than LDA for chemical bonding. \\
|
|
% \hline
|
|
% Hybrid Functionals &
|
|
% $E_x = E_x^{GGA}$
|
|
% $E_x = (1-\alpha)E_x^{GGA} + \alpha E_X^{FOCK}$
|
|
% \tabitem Add expensive non-local Fock term to reduce self-interaction
|
|
% & Lower than GGA (Improved balance) & \tabitem reduced SIE \tabitem wrong long-range behaviour \tabitem nonlinear chemical potential (missing DD) & Higher & Molecular chemistry, solid-state physics requiring better accuracy. & Balances accuracy and cost. \\
|
|
% \hline
|
|
% Range-Separated Hybrid (RSH) &
|
|
% $E_x = E_x^{GGA}$
|
|
% $E_{X,SR} = (1-\alpha)E_x^{GGA} + \alpha E_X^{FOCK}$
|
|
% $E_{X,LR} = E_x^{GGA}$
|
|
% \tabitem Mix-in expensive Fock term only for short-range interactions $\rightarrow$ since for LR the Coulomb interaction gets screening in dielectric substances ($\epsilon > 1$), such as crystalline materials.
|
|
% & Lower than Hybrid (Even better balance) & \tabitem reduced SIE \tabitem wrong long-range behaviour \tabitem nonlinear chemical potential (missing DD) & Very High & Semiconductors, materials with screened Coulomb interactions. & Used for dielectric materials. \\
|
|
% \hline
|
|
% Optimally Tuned RSH (OT-RSH) &
|
|
% $E_x = E_x^{GGA}$
|
|
% $E_{X,SR} = E_x^{GGA}$ and $E_X^{FOCK}$
|
|
% $E_{X,LR} = E_x^{GGA}$ and $E_X^{FOCK}$
|
|
% \tabitem More advanced tuning between Fock and GGA. So that set also have the correct asymptotic behaviour of $1/r$ (Coulomb e.g. Fock) instead of $e^{-r}$ from GGA
|
|
% & Lowest & \tabitem reduced SIE \tabitem better long-range behaviour \tabitem /+ better chemical potential - they include non-multiplicative, orbital dependent terms. Hence, in principle they allow for including a DD. & Extremely High & Precise calculations for band gap predictions and electronic properties. & Most flexible but computationally expensive. \\
|
|
% \hline
|
|
% \end{tabular}
|
|
|
|
% \begin{tabularx}{\textwidth}{lXlllll}
|
|
% \toprule
|
|
% \textbf{Method} & \textbf{Description} & \textbf{Mean Absolute Error (eV)} & \textbf{Band Gap Accuracy} & \textbf{Computational Cost} & \textbf{Usage} & \textbf{Other Notes} \\
|
|
% \midrule
|
|
% Hartree-Fock (HF) & $E_C \sim E_C^{HF\text{theory}}$ $E_X \sim E_X^{FOCK}$ & 3.1 (Underbinding) & Overestimates
|
|
% \tabitem no SIE
|
|
% \tabitem correct long-range behaviour
|
|
% \tabitem nonlinear chemical potential (missing DD)
|
|
% \tabitem positive correlation effects
|
|
% & High & Reference for exact exchange, useful for small molecules. & Self-interaction free, but lacks correlation. \\
|
|
% \midrule
|
|
% Local Density Approximation (LDA) &
|
|
% $E_x \sim n(r)$
|
|
% $E_c \sim n(r)$
|
|
% & 1.3 (Overbinding) & Underestimates
|
|
% \tabitem SIE
|
|
% \tabitem wrong long-range behaviour
|
|
% \tabitem nonlinear chemical potential (missing DD)
|
|
% & Low & Basic solids and metallic systems, where accuracy is not critical. & Simple and computationally cheap. \\
|
|
% \midrule
|
|
% Generalised Gradient Approximation (GGA) &
|
|
% $E_x \sim n(r), \nabla n(r)$
|
|
% $E_c \sim n(r), \nabla n(r)$
|
|
% & 0.3 (Mostly overbinding) & Improved over LDA
|
|
% \tabitem SIE
|
|
% \tabitem wrong long-range behaviour
|
|
% \tabitem nonlinear chemical potential (missing DD)
|
|
% & Moderate & More accurate for molecules and chemical bonding studies. & Better than LDA for chemical bonding. \\
|
|
% \midrule
|
|
% Hybrid Functionals &
|
|
% $E_x = E_x^{GGA}$
|
|
% $E_x = (1-\alpha)E_x^{GGA} + \alpha E_X^{FOCK}$
|
|
% \tabitem Add expensive non-local Fock term to reduce self-interaction
|
|
% & Lower than GGA (Improved balance) & Better than GGA
|
|
% \tabitem reduced SIE
|
|
% \tabitem wrong long-range behaviour
|
|
% \tabitem nonlinear chemical potential (missing DD)
|
|
% & Higher & Molecular chemistry, solid-state physics requiring better accuracy. & Balances accuracy and cost. \\
|
|
% \midrule
|
|
% Range-Separated Hybrid (RSH) &
|
|
% $E_x = E_x^{GGA}$
|
|
% $E_{X,SR} = (1-\alpha)E_x^{GGA} + \alpha E_X^{FOCK}$
|
|
% $E_{X,LR} = E_x^{GGA}$
|
|
% \tabitem Mix-in expensive Fock term only for short-range interactions $\rightarrow$ since for LR the Coulomb interaction gets screening in dielectric substances ($\epsilon > 1$), such as crystalline materials.
|
|
% & Lower than Hybrid (Even better balance) & Strongly underestimates
|
|
% \tabitem reduced SIE
|
|
% \tabitem wrong long-range behaviour
|
|
% \tabitem nonlinear chemical potential (missing DD)
|
|
% & Very High & Semiconductors, materials with screened Coulomb interactions. & Used for dielectric materials. \\
|
|
% \midrule
|
|
% Optimally Tuned RSH (OT-RSH) &
|
|
% $E_x = E_x^{GGA}$
|
|
% $E_{X,SR} = E_x^{GGA}$ and $E_X^{FOCK}$
|
|
% $E_{X,LR} = E_x^{GGA}$ and $E_X^{FOCK}$
|
|
% \tabitem More advanced tuning between Fock and GGA. So that set also have the correct asymptotic behaviour of $1/r$ (Coulomb e.g. Fock) instead of $e^{-r}$ from GGA
|
|
% & Lowest & Most accurate
|
|
% \tabitem reduced SIE
|
|
% \tabitem better long-range behaviour
|
|
% \tabitem /+ better chemical potential - they include non-multiplicative, orbital dependent terms. Hence, in principle they allow for including a DD.
|
|
% & Extremely High & Precise calculations for band gap predictions and electronic properties. & Most flexible but computationally expensive. \\
|
|
% \bottomrule
|
|
% \end{tabularx}
|
|
\end{bigformula}
|
|
|
|
\Subsubsection[
|
|
\eng{Basis sets}
|
|
\ger{Basis-Sets}
|
|
]{basis}
|
|
\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[
|
|
\eng{Pseudo-Potential method}
|
|
\ger{Pseudopotentialmethode}
|
|
]{pseudo}
|
|
\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}
|