change of notations

Manuscript/SRGGW.tex

@@ -184,13 +184,13 @@ The matrix elements of $\bSig(\omega)$ have the following closed-form expression
 \begin{equation}
   \label{eq:GW_selfenergy}
   \Sigma_{pq}(\omega) 
-  = \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} 
+  = \sum_{i\nu} \frac{W_{pi}^{\nu} W_{qi}^{\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} 
-  + \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
+  + \sum_{a\nu} \frac{W_{pa}^{\nu} W_{qa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
 \end{equation}
 where $\eta$ is a positive infinitesimal and the screened two-electron integrals are
 \begin{equation}
   \label{eq:GW_sERI}
-  W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty(\bX+\bY)_{ia,\nu},
+  W_{pq}^{\nu} = \sum_{ia}\eri{pi}{qa}\qty(\bX+\bY)_{ia}^{\nu},
 \end{equation}
 with $\bX$ and $\bY$ the components of the eigenvectors of the direct (\ie without exchange) RPA problem defined as 
 \begin{equation}
@@ -253,9 +253,11 @@ 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}^{\qsGW} = \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}.
+  \Sigma_{pq}^{\qsGW} 
+  = \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_ {pr}^{\nu} W_{qr}^{\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).
+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. 
 
 Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
@@ -371,9 +373,9 @@ where the 2h1p and 2p1h matrix elements are
 \end{subequations}
 and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
 \begin{align}
-  W^\text{2h1p}_{p,i\nu} & = W_{p,i\nu},
+  W^\text{2h1p}_{p,i\nu} & = W_{pi}^{\nu},
   &
-    W^\text{2p1h}_{p,a\nu} & = W_{p,a\nu}.
+    W^\text{2p1h}_{p,a\nu} & = W_{pa}^{\nu}.
 \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}
@@ -492,10 +494,13 @@ Equation \eqref{eq:F0_C0} implies
 \end{align}
 and, thanks to the 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) \titou{e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}}
+  W_{pq}^{\nu(1)}(s) = W_{pq}^{\nu} e^{-(\Delta_{pq}^{\nu})^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.
+At $s=0$, $W_{pq}^{\nu(1)}(s)$ reduces to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while,
-Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
+\begin{equation}
+	\lim_{s\to\infty} W_{pq}^{\nu(1)}(s) = 0.
+\end{equation}
+Therefore, $W_{pq}^{\nu(1)}(s)$ 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} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}).
 
 %///////////////////////////%
@@ -522,8 +527,8 @@ with elements
   \label{eq:SRG-GW_selfenergy}
   \begin{split}
     \widetilde{\bSig}_{pq}(\omega; s) 
-  &= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu}} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\ 
+  &= \sum_{i\nu} \frac{W_{pi}^{\nu} W_{qi}^{\nu}}{\omega - \epsilon_i + \Omega_{\nu}} e^{-\qty[(\Delta_{pi}^{\nu})^2 + (\Delta_{qi}^{\nu})^2 ] s} \\ 
-  &+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}.
+  &+ \sum_{a\nu} \frac{W_{pa}^{\nu} W_{qa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-\qty[(\Delta_{pa}^{\nu})^2 + (\Delta_{qa}^{\nu})^2 ] s}.
   \end{split}
 \end{equation}
 
@@ -535,8 +540,10 @@ Collecting every second-order term in the flow equation and performing the block
 \end{multline}
 which can be solved by simple integration along with the initial condition $\bF^{(2)}(0)=\bO$ to yield
 \begin{multline}
-  F_{pq}^{(2)}(s) =  \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}  \\
+  F_{pq}^{(2)}(s) 
-  \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}].
+  = \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu}  
+  \\
+  \times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s}].
 \end{multline}
 
 %%% FIG 1 %%% 
@@ -556,7 +563,9 @@ At $s=0$, the second-order correction vanishes, hence giving
 For $s\to\infty$, it tends towards the following static limit
 \begin{equation}
   \label{eq:static_F2}
-  \lim_{s\to\infty} \widetilde{\bF}(s) = \epsilon_p \delta_{pq} + \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}.
+  \lim_{s\to\infty} \widetilde{\bF}(s) 
+  = \epsilon_p \delta_{pq} 
+  + \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu}.
 \end{equation}
 while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
 \begin{equation}
@@ -565,7 +574,7 @@ while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends t
 Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
 As illustrated in Fig.~\ref{fig:flow}, this transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
-For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{p,r\nu} e^{-(\Delta_{pr\nu})^2 s} \approx 0$, meaning that the state is decoupled, while, for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{p,r\nu}(s) \approx W_{p,r\nu}$, that is, the state remains coupled.
+For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{pr}^{\nu} e^{-(\Delta_{pr}^{\nu})^2 s} \approx 0$, meaning that the state is decoupled from the 1h and 1p configurations, while, for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{pr}^{\nu}(s) \approx W_{pr}^{\nu}$, that is, the state remains coupled.
 
 %%% FIG 2 %%% 
 \begin{figure*}
@@ -587,9 +596,9 @@ This yields a $s$-dependent static self-energy which matrix elements read
 \begin{multline}
   \label{eq:SRG_qsGW}
    \Sigma_{pq}^{\SRGqsGW}(s) 
-   = \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}  
+   = \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu}  
    \\
-   \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ].
+   \times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s} ].
 \end{multline}
 Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is naturally hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization.
 Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator.
@@ -691,7 +700,7 @@ This behavior as a function of $s$ can be understood by applying matrix perturba
 Through second order in the coupling block, the principal IP is 
 \begin{equation}
   \label{eq:2nd_order_IP}
-  \text{IP} \approx - \epsilon_\text{h} - \sum_{i\nu} \frac{W_{\text{h},i\nu}^2}{\epsilon_\text{h} - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{W_{\text{h},a\nu}^2}{\epsilon_\text{h} - \epsilon_a - \Omega_\nu} 
+  \text{IP} \approx - \epsilon_\text{h} - \sum_{i\nu} \frac{(W_{\text{h}}^{i\nu})^2}{\epsilon_\text{h} - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{(W_{\text{h}}^{a\nu})^2}{\epsilon_\text{h} - \epsilon_a - \Omega_\nu} 
 \end{equation}
 where $\text{h}$ is the index of the highest occupied molecular orbital (HOMO).
 The first term of the right-hand side of Eq.~\eqref{eq:2nd_order_IP} is the zeroth-order IP and the two following terms originate from the 2h1p and 2p1h coupling, respectively.