SRG GW and GF2 ready to be implemented (modulo some typos...)
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@ -551,11 +551,11 @@ In the following, upper case indices correspond to the 2h1p and 2p1h sectors whi
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\end{align}
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The differential equation for the coupling blocks can be solved in the GF(2) case because in this case $\bC{}{}$ is diagonal (see Appendix~\ref{sec:diagC}).
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However, in the general case this matrix differential equation is not trivial to solve.
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In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we only need to solve the above equations for $\bF{}{} = \epsilon_p$.
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In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we only need to solve the above equations for $\bF{}{} = \epsilon_r$.
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\begin{align}
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\label{eq:matrixdiffeq}
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\dv{\bV{}{(1)}}{s} &= \bV{}{(1)} (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \\
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\dv{\bV{}{(1),\dagger}}{s} &= (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger}
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\dv{\bV{r}{(1)}}{s} &= \bV{}{(1)} (2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2) \\
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\dv{\bV{r}{(1),\dagger}}{s} &= (2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger}
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\end{align}
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\begin{itemize}
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@ -577,7 +577,7 @@ In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we
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\end{equation}
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\item \textbf{GW}
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In order to solve the matrix differential equation Eq.~(\ref{eq:matrixdiffeq}), we need to diagonalize $2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2$ which is a polynomial in $\bC{}{(0)}$ so the polynomial admits the same eigenvectors $\bC{}{(0)}$.
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In order to solve the matrix differential equation Eq.~(\ref{eq:matrixdiffeq}), we need to diagonalize $2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2$ which is a polynomial in $\bC{}{(0)}$ so the polynomial admits the same eigenvectors $\bC{}{(0)}$.
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The elements of the matrix $\bC{}{(0)}$ in the GW case are given in Eq.~(\ref{eq:GW_unfolded}) and can be written equivalently as
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\begin{align}
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\label{eq:GW_unfolded}
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@ -594,23 +594,23 @@ In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we
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\begin{align}
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\Omega_{(i,v)} &= \epsilon_i - \Omega_v & \Omega_{(a,v)} &= \epsilon_a + \Omega_v
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\end{align}
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Therefore the eigenvalues of $2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2$ are $2 \epsilon_p \Omega_{(q,v)} - \epsilon_p^2 - \Omega_{q,v}^2 = -(\epsilon_p - \Omega_{(q,v)})^2$.
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Therefore the eigenvalues of $2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2$ are $2 \epsilon_r \Omega_{(q,v)} - \epsilon_r^2 - \Omega_{q,v}^2 = -(\epsilon_r - \Omega_{(q,v)})^2$.
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And finally the analytical expressions for the GW coupling blocks at first order are
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\begin{align}
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\bV{}{\hhp,(1)}(s) &= \bU \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s})\bU^{-1}\bV{}{\hhp,(1)}(0) \\
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\bV{}{\pph,(1)}(s) &= \bU \text{diag}(e^{-(\epsilon_p - \Omega_{(a,v)})^2s})\bU^{-1}\bV{}{\pph,(1)}(0)
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\bV{r}{\hhp,(1),\dagger}(s) &= \bU \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s})\bU^{-1}\bV{r}{\hhp,(1),\dagger}(0) \\
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\bV{r}{\pph,(1),\dagger}(s) &= \bU \text{diag}(e^{-(\epsilon_r - \Omega_{(a,v)})^2s})\bU^{-1}\bV{r}{\pph,(1),\dagger}(0)
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\end{align}
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Therefore the downfolded SRG self-energy is
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\begin{align}
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&\bSig(\omega)^{\hhp} = \bV{}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s}) \\
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&\text{diag}(\frac{1}{\omega - \epsilon_i + \Omega^{(v)}}) \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)}(0))^{\mathsf{T}} \notag \\
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&= \bV{}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(\frac{e^{-2(\epsilon_p - \Omega_{(i,v)})^2s}}{\omega - \epsilon_i + \Omega^{(v)}}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)}(0))^{\mathsf{T}} \notag
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&\bSig(\omega)^{\hhp}_r = \bV{r}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s}) \\
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&\text{diag}(\frac{1}{\omega - \epsilon_i + \Omega^{(v)}}) \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s}) (\bU^{\hhp})^{-1} (\bV{r}{\hhp,(1)}(0))^{\mathsf{T}} \notag \\
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&= \bV{r}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(\frac{e^{-2(\epsilon_r - \Omega_{(i,v)})^2s}}{\omega - \epsilon_i + \Omega^{(v)}}) (\bU^{\hhp})^{-1} (\bV{r}{\hhp,(1)}(0))^{\mathsf{T}} \notag
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\end{align}
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Renaming the index $p$ as $r$ and using a similar matrix product as in Appendix~\ref{sec:downfolding}, we finally obtain
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Using a similar matrix product as in Appendix~\ref{sec:downfolding}, we finally obtain
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\begin{align}
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\label{eq:SRGGW_selfenergy}
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\Sigma(\omega)_{pq} &= \sum_{(i,v)} \frac{M_{ip}^{(v)}M_{iq}^{(v)}}{\omega - \epsilon_i + \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(i,v)})^2s} \notag \\
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(\Sigma(\omega)_r^{\GW})_{pq} &= \sum_{(i,v)} \frac{M_{ip}^{(v)}M_{iq}^{(v)}}{\omega - \epsilon_i + \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(i,v)})^2s} \notag \\
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&+ \sum_{(a,v)} \frac{M_{ap}^{(v)}M_{aq}^{(v)}}{\omega - \epsilon_a - \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(a,v)})^2s}
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\end{align}
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where the $\pph$ part has been obtained by an analog derivation.
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@ -621,7 +621,7 @@ In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we
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A compeltely analog derivation gives
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\begin{align}
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\label{eq:SRGGF2_selfenergy}
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\Sigma_{pq}^{GF(2)}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} e^{-2(\epsilon_r - \Delta_{ij}^a)^2s}\notag \\
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(\Sigma_r^{\GF2})_{pq}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} e^{-2(\epsilon_r - \Delta_{ij}^a)^2s}\notag \\
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&+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} e^{-2(\epsilon_r - \Delta_{i}^{ab})^2s} \notag
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\end{align}
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@ -643,6 +643,57 @@ In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we
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The two first equations can be solved by simple integrations.
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The two last equations admit the same solutions as the first order coupling blocks differential equations with different initial conditions.
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We focus on $\bF{}{(2)}$ because it is the only second order block contributing to the second order quasiparticle equation.
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We are still in the case where $\bF{}{(2)} = \epsilon_r^{(2)}$
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\begin{align}
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\dv{\epsilon_r^{(2)}}{s} &= \epsilon_r^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\epsilon_r^{(0)} - 2 \bV{}{(1)}\bC{}{(0)}\bV{}{(1),\dagger} \\
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&= 2\bV{}{(1)} \left(\epsilon_r^{(0)} \mathbb{1} - \bC{}{(0)} \right) \bV{}{(1),\dagger} \notag \\
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&= 2\bV{}{\hhp,(1)} \left(\epsilon_r^{(0)} \mathbb{1} - \bC{}{\hhp,(0)} \right) \bV{}{\hhp,(1),\dagger} \notag \\
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&+ 2\bV{}{\pph,(1)} \left(\epsilon_r^{(0)} \mathbb{1} - \bC{}{\pph,(0)} \right) \bV{}{\pph,(1),\dagger} \notag
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\end{align}
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We focus on the $\hhp$ part
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\begin{align}
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&= 2\bV{}{\hhp,(1)}(0) \bU e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1} \left(\epsilon_r^{(0)} \mathbb{1} - \bC{}{\hhp,(0)} \right) \bU \notag \\
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&\times e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1}\bV{}{\hhp,(1),\dagger}(0) \notag \\
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&= 2\bV{}{\hhp,(1)}(0) \bU e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} (\epsilon_r^{(0)} \mathbb{1} - \boldsymbol{\Omega}) e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1}\bV{}{\hhp,(1),\dagger}(0) \notag \\
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\end{align}
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\begin{itemize}
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\item \textbf{GW}
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In the GW case this evaluates to
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\begin{align}
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&= 2 \sum_{(i,v)} M_{ir}^{(v)}M_{ir}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(i,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag
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\end{align}
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The other part is analog and we obtain
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\begin{align}
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\dv{\epsilon_r^{(2)}}{s} &= 2 \sum_{(i,v)} M_{ir}^{(v)}M_{ir}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(i,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag \\
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&+ 2 \sum_{(a,v)} M_{ar}^{(v)}M_{ar}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(a,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(a,v)})^2s} \notag
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\end{align}
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which can be integrated as
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\begin{align}
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\epsilon_r^{(2)}(s) &= - \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(i,v)}} e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag \\
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&- \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(a,v)}} e^{-2(\epsilon_r^{(0)} - \Omega_{(a,v)})^2s} + \text{Cte} \notag
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\end{align}
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The constant is determined as
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\begin{equation}
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\epsilon_r^{(2)}(0) = 0 = - \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(i,v)}} - \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(a,v)}} + \text{Cte} \notag
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\end{equation}
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Which finally gives
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\begin{align}
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\epsilon_r^{(2)}(s) &= \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r - \Omega_{(i,v)}} \left(1 - e^{-2(\epsilon_r - \Omega_{(i,v)})^2s}\right) \notag \\
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&+ \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r - \Omega_{(a,v)}} \left(1 - e^{-2(\epsilon_r - \Omega_{(a,v)})^2s}\right) \notag
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\end{align}
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\item \textbf{GF(2)}
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The expression for the GF(2) case is
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\begin{align}
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\epsilon_r^{(2)}(s) &= \sum _{ija} \frac{\aeri{ra}{ij}^2}{\epsilon_r ^{(0)}- \Delta_{ij}^a} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{ij}^a)^2s}\right) \notag \\
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&+ \sum _{iab} \frac{\aeri{ri}{ab}^2}{\epsilon_r^{(0)} - \Delta_{i}^{ab}} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{i}^{ab})^2s}\right) \notag
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\end{align}
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\end{itemize}
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% =================================================================%
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\section{Towards second quantized effective Hamiltonians for MBPT?}
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\label{sec:second_quant_mbpt}
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