add TDA and non TDA equations

Manuscript/SRGGW.bib

@@ -6,7 +6,17 @@
 
 %% Saved with string encoding Unicode (UTF-8) 
 
-
+@misc{Tolle_2022,
+  title = {Exact Relationships between the {{GW}} Approximation and Equation-of-Motion Coupled-Cluster Theories through the Quasi-Boson Formalism},
+  author = {T{\"o}lle, Johannes and Chan, Garnet Kin-Lic},
+  year = {2022},
+  number = {arXiv:2212.08982},
+  eprint = {2212.08982},
+  eprinttype = {arxiv},
+  primaryclass = {cond-mat, physics:physics},
+  doi = {10.48550/arXiv.2212.08982},
+  archiveprefix = {arXiv}
+}
 
 @inbook{Bartlett_1986,
 	abstract = {A diagrammatic derivation of the coupled-cluster (CC) linear response equations for gradients is presented. MBPT approximations emerge as low-order iterations of the CC equations. In CC theory a knowledge of the change in cluster amplitudes with displacement is required, which would not be necessary if the coefficients were variationally optimum, as in the CI approach. However, it is shown that the CC linear response equations can be put in a form where there is no more difficulty in evaluating CC gradients than in the variational CI procedure. This offers a powerful approach for identifying critical points on energy surfaces and in evaluating other properties than the energy.},

Manuscript/SRGGW.tex

@@ -49,7 +49,7 @@ Here comes the abstract.
 \label{sec:intro}
 %=================================================================%
 
-One-body Green's functions provide a natural and elegant way to access charged excitations energies of a physical system. \cite{Martin_2016}
+One-body Green's functions provide a natural and elegant way to access charged excitations energies of a physical system. \cite{Martin_2016,Golze_2019}
 The one-body non-linear Hedin's equations give a recipe to obtain the exact interacting one-body Green's function and therefore the exact ionization potentials and electron affinities. \cite{Hedin_1965}
 Unfortunately, fully solving the Hedin's equations is out of reach and one must resort to approximations.
 In particular, the GW approximation, \cite{Hedin_1965} which has first been mainly used in the context of solids \cite{Strinati_1980,Strinati_1982,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely used for molecules as well \ant{ref?}, provides fairly accurate results for weakly correlated systems\cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2021}
@@ -217,32 +217,21 @@ which can be further developed to give the usual
   = \sum_{iv} \frac{W_{p,(i,v)} W_{q,(i,v)}}{\omega - \epsilon_i + \Omega_{v} - \ii \eta} 
   + \sum_{av} \frac{W_{p,(a,v)}W_{q,(a,v)}}{\omega - \epsilon_a - \Omega_{v} + \ii \eta},
 \end{equation}
-with the screened integral defined as
+with the screened integrals defined as
 \begin{equation}
   \label{eq:GW_sERI}
-  W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
+  W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia},
 \end{equation}
-where $\bX$ and $\bY$ are the matrix of eigenvectors of the direct particle-hole RPA problem defined as
+where $\bX$ is the matrix of eigenvectors of the particle-hole direct RPA (dRPA) problem in the Tamm-Dancoff approximation (TDA) defined as
 \begin{equation}
-  \begin{pmatrix}
-    \bA & \bB \\
-    - \bB & \bA \\
-  \end{pmatrix}
-  \begin{pmatrix}
-    \bX \\
-    \bY \\
-  \end{pmatrix} = \boldsymbol{\Omega}
-  \begin{pmatrix}
-    \bX \\
-    \bY \\
-  \end{pmatrix},
+  \bA \bX = \boldsymbol{\Omega} \bX,
 \end{equation}
 with
-\begin{align}
-  A^\dRPA_{ij,ab} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}, \\
-  B^\dRPA_{ij,ab} &= \eri{ij}{ab}.
-\end{align}
+\begin{equation}
+  A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
+\end{equation}
 $\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~(\ref{eq:GW_selfenergy}).
+The case of the non-TDA approximation is discussed in Appendix~\ref{sec:nonTDA}.
 
 Equations~(\ref{eq:GWlin}) and~(\ref{eq:GWnonlin}) have exactly the same solutions but one is linear and the other not.
 The price to pay for this linearity is that the size of the matrix in the former equation is $\mathcal{O}(K^3)$ while it is $\mathcal{O}(K)$ in the latter one.
@@ -506,6 +495,82 @@ The data that supports the findings of this study are available within the artic
 
 \appendix
 
+%%%%%%%%%%%%%%%%%%%%%%
+\section{Non-TDA $GW$ equations}
+\label{sec:nonTDA}
+%%%%%%%%%%%%%%%%%%%%%%
+
+The $GW$ self-energy without TDA is the same as in Eq.~(\ref{eq:GW_selfenergy}) but the screened integrals are now defined as
+\begin{equation}
+  \label{eq:GWnonTDA_sERI}
+  W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
+\end{equation}
+where $\bX$ and $\bY$ are the matrix of eigenvectors of the full particle-hole dRPA problem defined as
+\begin{equation}
+  \label{eq:full_dRPA}
+  \begin{pmatrix}
+    \bA & \bB \\
+    - \bB & \bA \\
+  \end{pmatrix}
+  \begin{pmatrix}
+    \bX \\
+    \bY \\
+  \end{pmatrix} = \boldsymbol{\Omega}
+  \begin{pmatrix}
+    \bX \\
+    \bY \\
+  \end{pmatrix},
+\end{equation}
+with
+\begin{align}
+  A_{ij,ab} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}, \\
+  B_{ij,ab} &= \eri{ij}{ab}.
+\end{align}
+$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues. Note that $\boldsymbol{\Omega}$ in this case has the same size as in the TDA because we consider only the positive excitations of the full dRPA problem.
+
+Defining an unfold version of this equation which 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}
+\begin{equation}
+  \label{eq:nonTDA_upfold}
+    \begin{pmatrix}
+    \bF				&	\bW^{\text{2h1p}}		&	\bW^{\text{2p1h}}	\\
+    (\bW^{\text{2h1p}})^{\mathrm{T}}	&	\bD^{\text{2h1p}}		&	\bO					\\
+    (\bW^{\text{2p1h}})^{\mathrm{T}}	&	\bO				&	\bD^{\text{2p1h}}	\\
+  \end{pmatrix}
+  \cdot
+  \begin{pmatrix}
+    \bX	\\
+    \bY^{\text{2h1p}}	\\
+    \bY^{\text{2p1h}}	\\
+  \end{pmatrix}
+  =
+  \begin{pmatrix}
+    \bX	\\
+    \bY^{\text{2h1p}}	\\
+    \bY^{\text{2p1h}}	\\
+  \end{pmatrix}
+  \cdot
+  \boldsymbol{\epsilon},
+\end{equation}
+which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~(\ref{eq:full_dRPA}).
+
+
+where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
+\begin{subequations}
+  \begin{align}
+    D^\text{2h1p}_{ija,klc} & = \delta_{ik}\delta_{jl} \delta_{ac} \qty[\epsilon_i - \Omega_{ja}] ,
+    \\
+    D^\text{2p1h}_{iab,kcd} & = \delta_{ik}\delta_{ac} \delta_{bd} \qty[\epsilon_a + \Omega_{ib}] ,
+  \end{align}
+\end{subequations}
+and the corresponding coupling blocks read
+\begin{align}
+  W^\text{2h1p}_{p,klc} & = \sum_{ia}\eri{pi}{ka} \qty( \bX_{lc} + \bY_{lc})_{ia} \\
+    W^\text{2p1h}_{p,kcd} & = \sum_{ia}\eri{pi}{ca} \qty( \bX_{kd} + \bY_{kd})_{ia}
+\end{align}
+
+Using the SRG on this matrix instead of Eq.~(\ref{eq:GWlin}) gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}.
+
 %%%%%%%%%%%%%%%%%%%%%%
 \section{GF(2) equations}
 \label{sec:GF2}