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\section{Random Facts}

This section is a collection of facts we discovered about xtp and ctp which should be included in the manual at some point, but lack proper background.


The cutoffs used should not exceed the dimension of the cell, at least for a non orthogonal unit cell. XQMM throws an error if your cutoff is larger than the box, but it does not take the extension of the molecule into accout, so often you may still have overlap.

XQMMM takes the segment coordinates from the MPS\textunderscore Files so be vary careful which MPS\textunderscore File you put in.


\item Use pewald3d, as a calculator. I am not sure what the rest is for. All still use the ewald options. The shape factor massively influences the results. For bulk systems "none" is the option of choice. Other options are "xyslab", "sphere", "cube" but I do not know what they do.
\item The induction cutoff should hardly ever exceed 3nm, because the calculation is expensive
\item If you want to use induction, you befor have to run ewdbgpol calculator and specify polar\textunderscore top.bgp in the options file for ewald. All the other parameters should be the same in ewdbpol and pewald3d


There is a wide range of different approximations for $GW$-BSE. Here I try to summarize common abbreviations and methods and explain what our code does. This is not complete and certainly has some mistakes in it. 

 \item COHSEX: RPA is only calculated for $\omega=0$. This amounts to $\varepsilon(\vctr{r},\vctr{r'},\omega)=\varepsilon(\vctr{r},\vctr{r'})$. This is also called the static approximation to $GW$
 \item Plasmon Pole model; RPA is calculated moslty twice, once at $\omega=0$ and $\omega=\omega_0$ , then these values are used to fit an analytic model to interpolate $\varepsilon(\vctr{r},\vctr{r'},\omega)$.
 \item Imaginary frequency integration, $\varepsilon$ is numerically integrated along the Imaginary frequency axis. This is done because $\varepsilon$ is much smoother along the Imaginary axis and thus requires less functional evaluations. Afterwards $\varepsilon$ is extended to real frequencies via analytic continuation.  

The calculation of $GW$. In VOTCA we also have the \textbf{shift} option. This is commonly called a scissor operator. This allows you to shift the unoccuppied KS-orbitals up in energy, making the resulting spectrum closer to the $GW$ spectrum. This is often a better starting point for the self-consistent evaluations.

 \item $G_0W_0$. $G$ and $\varepsilon$ are calculated once from DFT results. $W$ is evaluated once from that. Then energies $ \varepsilon_i^{GW}$ are corrected via:
   \varepsilon_i^{GW}=\varepsilon_i^{KS}+Z_i\bra{\phi^{KS}_i} \Sigma(\varepsilon_i^{KS})-V_{xc} \ket{\phi^{KS}_i}
 This is not implemented in VOTCA, because $GW_0$ is nearly as fast. 
 \item $GW_0$. $\varepsilon$ is calculated once from DFT results. $W$ is evaluated once from that. Then $\Sigma$ is calculated. The resulting energies are fed back into $\Sigma$, until self-consistency is achieved. This is denoted \textbf{fixed} in VOTCA.
   \varepsilon_i^{GW}=\varepsilon_i^{KS}+\bra{\phi^{KS}_i} \Sigma(\varepsilon_i^{GW})-V_{xc} \ket{\phi^{KS}_i}
 \item $scQPGW$. As $GW_0$, but after $\varepsilon_i^{GW}$ are converged, these energies are used to recalculate  $\varepsilon$ and then $W$, this is repeated until self-consistency is achieved. This is denoted \textbf{iterate} in VOTCA. This converges the ``eigenvalues'' of QP particles but along the $\ket{\phi^{GW}}$ states
 \item $scGW$. As $scQPGW$ but instead of correcting only the energies, the full $Sigma$ matrix is calculated and the eigenvalue problem for the QP is set up and solved and the whole system is then self-consistently solved in $\ket{\phi^{GW}}$ states and not in   $\ket{\phi^{KS}}$. This fully converges the eigenvalues and eigenvectors of the QP particles in the space spanned by $\ket{\phi^{KS}}$, e.g. $\ket{\phi^{GW}}$ are linear combinations of $\ket{\phi^{KS}}$. This is reported to be unstable because the vertex corrections are important now. This is at the moment implemented in VOTCA.


The calculations in BSE