Merge branch 'master' of https://git.irsamc.ups-tlse.fr/loos/SRGGW

Manuscript/SRGGW.bib

@@ -80,11 +80,9 @@
 	doi = {10.1021/acs.jctc.9b00353},
 	eprint = {https://doi.org/10.1021/acs.jctc.9b00353},
 	journal = {J. Chem. Theory Comput.},
-	note = {PMID: 31268704},
 	number = {8},
 	pages = {4399-4414},
 	title = {Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation},
-	url = {https://doi.org/10.1021/acs.jctc.9b00353},
 	volume = {15},
 	year = {2019},
 	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.9b00353}}
@@ -99,7 +97,6 @@
 	number = {11},
 	pages = {114111},
 	title = {Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics},
-	url = {https://doi.org/10.1063/5.0059362},
 	volume = {155},
 	year = {2021},
 	bdsk-url-1 = {https://doi.org/10.1063/5.0059362}}
@@ -111,11 +108,9 @@
 	doi = {10.1021/acs.jctc.1c00980},
 	eprint = {https://doi.org/10.1021/acs.jctc.1c00980},
 	journal = {J. Chem. Theory Comput.},
-	note = {PMID: 34839660},
 	number = {12},
 	pages = {7666-7681},
 	title = {Analytic Energy Gradients for the Driven Similarity Renormalization Group Multireference Second-Order Perturbation Theory},
-	url = {https://doi.org/10.1021/acs.jctc.1c00980},
 	volume = {17},
 	year = {2021},
 	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c00980}}
@@ -127,11 +122,9 @@
 	doi = {10.1021/acs.jctc.2c00966},
 	eprint = {https://doi.org/10.1021/acs.jctc.2c00966},
 	journal = {J. Chem. Theory Comput.},
-	note = {PMID: 36534617},
 	number = {1},
 	pages = {122-136},
 	title = {Assessment of State-Averaged Driven Similarity Renormalization Group on Vertical Excitation Energies: Optimal Flow Parameters and Applications to Nucleobases},
-	url = {https://doi.org/10.1021/acs.jctc.2c00966},
 	volume = {19},
 	year = {2023},
 	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00966}}
@@ -175,11 +168,9 @@
 	doi = {10.1021/acs.jctc.7b00586},
 	eprint = {https://doi.org/10.1021/acs.jctc.7b00586},
 	journal = {J. Chem. Theory Comput.},
-	note = {PMID: 28873298},
 	number = {10},
 	pages = {4765-4778},
 	title = {GW Vertex Corrected Calculations for Molecular Systems},
-	url = {https://doi.org/10.1021/acs.jctc.7b00586},
 	volume = {13},
 	year = {2017},
 	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.7b00586}}
@@ -397,11 +388,9 @@
 	doi = {10.1021/acs.jctc.2c00617},
 	eprint = {https://doi.org/10.1021/acs.jctc.2c00617},
 	journal = {J. Chem. Theory Comput.},
-	note = {PMID: 36322136},
 	number = {12},
 	pages = {7570-7585},
 	title = {Benchmark of GW Methods for Core-Level Binding Energies},
-	url = {https://doi.org/10.1021/acs.jctc.2c00617},
 	volume = {18},
 	year = {2022},
 	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00617}}
@@ -1474,7 +1463,7 @@
 	author = {Olsen, Jeppe and J{\o}rgensen, Poul and Helgaker, Trygve and Christiansen, Ove},
 	doi = {10.1063/1.481611},
 	issn = {0021-9606},
-	journal = {The Journal of Chemical Physics},
+	journal = {J. Chem. Phys.},
 	number = {22},
 	pages = {9736--9748},
 	title = {Divergence in {{M\o ller}}\textendash{{Plesset}} Theory: {{A}} Simple Explanation Based on a Two-State Model},
@@ -1487,7 +1476,7 @@
 	copyright = {\textcopyright{} 2001 American Institute of Physics.},
 	doi = {10.1063/1.1345510},
 	issn = {0021-9606},
-	journal = {The Journal of Chemical Physics},
+	journal = {J. Chem. Phys.},
 	number = {9},
 	pages = {3913},
 	title = {Identifying and Removing Intruder States in Multireference {{Mo}}/Ller\textendash{{Plesset}} Perturbation Theory},
@@ -1499,7 +1488,7 @@
 	author = {Battaglia, Stefano and Frans{\'e}n, Lina and Fdez. Galv{\'a}n, Ignacio and Lindh, Roland},
 	doi = {10.1021/acs.jctc.2c00368},
 	issn = {1549-9618},
-	journal = {Journal of Chemical Theory and Computation},
+        journal = {J. Chem. Theory Comput.},
 	number = {8},
 	pages = {4814--4825},
 	title = {Regularized {{CASPT2}}: An {{Intruder-State-Free Approach}}},
@@ -6637,7 +6626,7 @@
 	author = {Strinati, G. and Mattausch, H. J. and Hanke, W.},
 	date-modified = {2023-01-20 09:26:16 +0100},
 	doi = {10.1103/PhysRevB.25.2867},
-	journal = {Physical Review B},
+	journal = {Phys. Rev. B},
 	number = {4},
 	pages = {2867--2888},
 	title = {Dynamical Aspects of Correlation Corrections in a Covalent Crystal},
@@ -14957,7 +14946,7 @@
 	author = {Forsberg, Niclas and Malmqvist, Per-{\AA}ke},
 	doi = {10.1016/S0009-2614(97)00669-6},
 	issn = {0009-2614},
-	journal = {Chemical Physics Letters},
+	journal = {Chem. Phys. Lett.},
 	number = {1},
 	pages = {196--204},
 	title = {Multiconfiguration Perturbation Theory with Imaginary Level Shift},
@@ -16490,7 +16479,7 @@
 	author = {Shishkin, M. and Kresse, G.},
 	date-modified = {2023-01-30 15:40:02 +0100},
 	doi = {10.1103/PhysRevB.75.235102},
-	journal = {Physical Review B},
+	journal = {Phys. Rev. B},
 	number = {23},
 	pages = {235102},
 	title = {Self-Consistent \${{GW}}\$ Calculations for Semiconductors and Insulators},
@@ -16502,7 +16491,7 @@
 	author = {Wilhelm, Jan and Del Ben, Mauro and Hutter, J{\"u}rg},
 	doi = {10.1021/acs.jctc.6b00380},
 	issn = {1549-9618},
-	journal = {Journal of Chemical Theory and Computation},
+	journal = {J. Chem. Theory Comput.},
 	number = {8},
 	pages = {3623--3635},
 	title = {{{GW}} in the {{Gaussian}} and {{Plane Waves Scheme}} with {{Application}} to {{Linear Acenes}}},
@@ -16514,7 +16503,7 @@
 	author = {Gallandi, Lukas and K{\"o}rzd{\"o}rfer, Thomas},
 	doi = {10.1021/acs.jctc.5b00820},
 	issn = {1549-9618},
-	journal = {Journal of Chemical Theory and Computation},
+	journal = {J. Chem. Theory Comput.},
 	number = {11},
 	pages = {5391--5400},
 	title = {Long-{{Range Corrected DFT Meets GW}}: {{Vibrationally Resolved Photoelectron Spectra}} from {{First Principles}}},
@@ -16525,7 +16514,7 @@
 @article{Korzdorfer_2012,
 	author = {K{\"o}rzd{\"o}rfer, Thomas and Marom, Noa},
 	doi = {10.1103/PhysRevB.86.041110},
-	journal = {Physical Review B},
+	journal = {Phys. Rev. B},
 	number = {4},
 	pages = {041110},
 	title = {Strategy for Finding a Reliable Starting Point for \$\{\vphantom\}{{G}}\vphantom\{\}\_\{0\}\{\vphantom\}{{W}}\vphantom\{\}\_\{0\}\$ Demonstrated for Molecules},
@@ -16536,7 +16525,7 @@
 @article{Marom_2012,
 	author = {Marom, Noa and Caruso, Fabio and Ren, Xinguo and Hofmann, Oliver T. and K{\"o}rzd{\"o}rfer, Thomas and Chelikowsky, James R. and Rubio, Angel and Scheffler, Matthias and Rinke, Patrick},
 	doi = {10.1103/PhysRevB.86.245127},
-	journal = {Physical Review B},
+	journal = {Phys. Rev. B},
 	number = {24},
 	pages = {245127},
 	title = {Benchmark of \${{GW}}\$ Methods for Azabenzenes},

Manuscript/SRGGW.tex

@@ -212,7 +212,8 @@ and
 are bare two-electron integrals in the spin-orbital basis.
 
 The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problen defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
-In the Tamm-Dancoff approximation (TDA), which is discussed in Appendix \ref{sec:nonTDA}, one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$.
+In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$.
+\ant{The corresponding TDA screened two-electron integrals are computed using Eq.~(\ref{eq:GW_sERI}) with $\bY=0$.}
 
 Because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task. 
 Hence, several approximate schemes have been developed to bypass self-consistency.
@@ -255,7 +256,7 @@ which was first introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfg
 The corresponding matrix elements are
 \begin{equation}
   \label{eq:sym_qsGW}
-  \Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu}  \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}}  ) W_ {p,r\nu} W_{q,r\nu}.
+  \Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu}  \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2}  ) W_ {p,r\nu} W_{q,r\nu}.
 \end{equation}
 with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
 One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy. 
@@ -267,7 +268,7 @@ If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates b
 
 The satellites causing convergence problems are the above-mentioned intruder states.
 One can deal with them by introducing \textit{ad hoc} regularizers.
-\titou{The $\ii \eta$ term that is usually added in the denominators of the self-energy} [see Eq.~\eqref{eq:GW_selfenergy}] is similar to the usual imaginary-shift regularizer employed in various other theories affected by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}
+\ant{The $\ii\eta$ term in the denominators of Eq.~(\ref{eq:GW_selfenergy}), which stems from a regularization of the convolution to obtain $\Sigma$ and should theoretically be set to 0,\cite{Martin_2016}  is similar to the usual imaginary-shift regularizer employed in various other theories plagued by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}.}
 Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
 Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
 This is the central aim of the present work.
@@ -338,7 +339,8 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
 
 By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.
 However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
-A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version which elegantly highlights the coupling terms: \cite{Bintrim_2021,Tolle_2022}
+\ant{A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
+Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022} }
 \begin{equation}
   \label{eq:GWlin}
   \begin{pmatrix}
@@ -376,8 +378,9 @@ and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
 \end{align}
 The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021} 
 \begin{equation}
-\begin{split}
+  \begin{split}
-  \bSig(\omega) 
+    \label{eq:downfolded_sigma}
+  \bSig(\omega)
   & = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1}  (\bW^{\hhp})^\dag
   \\
   & + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1}  (\bW^{\pph})^\dag, 
@@ -431,7 +434,7 @@ where the supermatrices
 \end{align}
 \end{subequations}
 collect the 2h1p and 2p1h channels.
-Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:GWlin} before applying the downfolding process to obtain a renormalized version of the quasiparticle equation.
+Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma}\trashant{\eqref{eq:GWlin} before applying the downfolding process to obtain} to define a renormalized version of the quasiparticle equation.
 In particular, we focus here on the second-order renormalized quasiparticle equation.
 
 %///////////////////////////%
@@ -488,13 +491,13 @@ Equation \eqref{eq:F0_C0} implies
 \begin{align}
   \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
 \end{align}
-and, thanks to the \titou{diagonal structure of $\bF^{(0)}$} and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
+and, thanks to the \ant{diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point)} and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
 \begin{equation}
   W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
 \end{equation}
 At $s=0$ the elements  $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero.
 Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
-It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} \titou{in a different context} following a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
+\ant{It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} for the usual electronic Hamiltonian in the context of wave-function theory within a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).}
 
 %///////////////////////////%
 \subsection{Second-order matrix elements}
@@ -532,7 +535,7 @@ At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends
   F_{pq}^{(2)}(\infty) =  \sum_{r\nu}  \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}.
 \end{equation}
 Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
-\titou{Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}(s)$.}
+\ant{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.}
 This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
 %%% FIG 1 %%% 
@@ -540,7 +543,7 @@ This transformation is done gradually starting from the states that have the lar
   \centering
   \includegraphics[width=\linewidth]{fig1.pdf}
   \caption{
-    Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-$GW$ self-energy (right) for $s = 1/(2\eta^2)$.
+    Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2)$.
     \label{fig:fig1}}
 \end{figure*}
 %%% %%% %%% %%%
@@ -640,10 +643,8 @@ Then the accuracy of the IP yielded by the traditional and SRG schemes will be s
 
 This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set.
 Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
-The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation, a result which is now well understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
+\ant{The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result which is now well understood.} \cite{SzaboBook,Lewis_2019}
-\PFL{Check Szabo\&Ostlund, section on Koopman's theorem.}
 The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference.
-%The Neon atom is a well-behaved system and could be converged without regularisation parameter while for water $\eta$ was set to 0.01 to help convergence.
 
 Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IP as a function of the flow parameter (blue curve).
 At $s=0$, the IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
@@ -718,50 +719,50 @@ This project has received funding from the European Research Council (ERC) under
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The data that supports the findings of this study are available within the article.% and its supplementary material.
 
-\appendix
+% \appendix
 
-%%%%%%%%%%%%%%%%%%%%%%
+% %%%%%%%%%%%%%%%%%%%%%%
-\section{Non-TDA $GW$  and $GW_{\text{TDHF}}$ equations}
+% \section{Non-TDA $GW$  and $GW_{\text{TDHF}}$ equations}
-\label{sec:nonTDA}
+% \label{sec:nonTDA}
-%%%%%%%%%%%%%%%%%%%%%%
+% %%%%%%%%%%%%%%%%%%%%%%
 
-The matrix elements of the $GW$ self-energy within the TDA are the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
+% The matrix elements of the $GW$ self-energy within the TDA are the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
-\begin{equation}
+% \begin{equation}
-  \label{eq:GWnonTDA_sERI}
+%   \label{eq:GWnonTDA_sERI}
-  W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu})_{ia},
+%   W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu})_{ia},
-\end{equation}
+% \end{equation}
-where $\bX$ is the eigenvector matrix of the TDA particle-hole dRPA problem obtained by setting $\bB= \bO$ in Eq.~\eqref{eq:full_dRPA}, \ie
+% where $\bX$ is the eigenvector matrix of the TDA particle-hole dRPA problem obtained by setting $\bB= \bO$ in Eq.~\eqref{eq:full_dRPA}, \ie
-\begin{equation}
+% \begin{equation}
-  \label{eq:TDA_dRPA}
+%   \label{eq:TDA_dRPA}
-  \bA \bX = \bX \boldsymbol{\Omega}.
+%   \bA \bX = \bX \boldsymbol{\Omega}.
-\end{equation}
+% \end{equation}
 
-Defining an unfold version of this equation that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
+% Defining an unfold version of this equation that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
-However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
+% However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
-\begin{equation}
+% \begin{equation}
-  \label{eq:nonTDA_upfold}
+%   \label{eq:nonTDA_upfold}
-    \begin{pmatrix}
+%     \begin{pmatrix}
-      \bF				&	\bW^{\text{2h1p}}		&	\bW^{\text{2p1h}}	\\
+%       \bF				&	\bW^{\text{2h1p}}		&	\bW^{\text{2p1h}}	\\
-    (\bW^{\text{2h1p}})^{\mathrm{T}}	&	\bD^{\text{2h1p}}		&	\bO					\\
+%     (\bW^{\text{2h1p}})^{\mathrm{T}}	&	\bD^{\text{2h1p}}		&	\bO					\\
-    (\bW^{\text{2p1h}})^{\mathrm{T}}	&	\bO				&	\bD^{\text{2p1h}}	\\
+%     (\bW^{\text{2p1h}})^{\mathrm{T}}	&	\bO				&	\bD^{\text{2p1h}}	\\
-  \end{pmatrix}
+%   \end{pmatrix}
-  \cdot
+%   \cdot
-  \begin{pmatrix}
+%   \begin{pmatrix}
-    \bX	\\
+%     \bX	\\
-    \bY^{\text{2h1p}}	\\
+%     \bY^{\text{2h1p}}	\\
-    \bY^{\text{2p1h}}	\\
+%     \bY^{\text{2p1h}}	\\
-  \end{pmatrix}
+%   \end{pmatrix}
-  =
+%   =
-  \begin{pmatrix}
+%   \begin{pmatrix}
-    \bX	\\
+%     \bX	\\
-    \bY^{\text{2h1p}}	\\
+%     \bY^{\text{2h1p}}	\\
-    \bY^{\text{2p1h}}	\\
+%     \bY^{\text{2p1h}}	\\
-  \end{pmatrix}
+%   \end{pmatrix}
-  \cdot
+%   \cdot
-  \boldsymbol{\epsilon},
+%   \boldsymbol{\epsilon},
-\end{equation}
+% \end{equation}
-which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~\eqref{eq:full_dRPA}.
+% which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~\eqref{eq:full_dRPA}.
-Within the TDA the renormalized matrix elements have the same $s$ dependence as in the RPA case but the $s=0$ screened integrals $W_{p,q\nu}$ and eigenvalues $\Omega_\nu$ are replaced by the ones of Eq.~\eqref{eq:GWnonTDA_sERI} and \eqref{eq:TDA_dRPA}, respectively.
+% Within the TDA the renormalized matrix elements have the same $s$ dependence as in the RPA case but the $s=0$ screened integrals $W_{p,q\nu}$ and eigenvalues $\Omega_\nu$ are replaced by the ones of Eq.~\eqref{eq:GWnonTDA_sERI} and \eqref{eq:TDA_dRPA}, respectively.
 
 % %%%%%%%%%%%%%%%%%%%%%%
 % \section{GF(2) equations \ant{NOT SURE THAT WE KEEP IT}}