correcting zeroth order subsections + saving work

Manuscript/SRGGW.tex

@@ -352,7 +352,7 @@ which satisfied the following condition \cite{Kehrein_2006}
 \end{equation}
 This implies that the matrix elements of the off-diagonal part decrease in a monotonic way throughout the transformation.
 Moreover, the coupling coefficients associated with the highest-energy determinants are removed first as we shall evidence in the perturbative analysis below.
-The main drawback of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \ant{ref}
+The main drawback of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \cite{Hergert_2016a}
 However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions so we will not be affected by this problem. \cite{Evangelista_2014,Hergert_2016}
 
 Let us now perform the perturbative analysis of the SRG equations.
@@ -418,47 +418,14 @@ Once the analytical low-order perturbative expansions are known they can be inse
 In particular, in this manuscript, the focus will be on the second-order renormalized quasi-particle equation.
 
 %///////////////////////////%
-\subsection{Zeroth-order matrix elements \ANT{This subsection is false, I'll rewrite it soon}}
-%///////////////////////////%
+\subsection{Zeroth-order matrix elements}
+% ///////////////////////////%
 
-There is only one zeroth order term in the right-hand side of the flow equation
+The choice of the Wegner generator associated with the form of the flow equation [see Eq.~(\ref{eq:flowEquation})] implies that the off-diagonal corrections are of order $\order{\lambda}$ while the correction to the diagonal blocks are at least $\order{\lambda^2}$.
+Therefore, the zeroth order Hamiltonian is independent of $s$ and we have
 \begin{equation}
-  \dv{\bH^{(0)}}{s} = \comm{\comm{\bHd{0}}{\bHod{0}}}{\bH^{(0)}},
+  \bH^{(0)}(s) = \bH^{(0)}(0).
 \end{equation}
-and performing the block matrix products gives the following system of equations 
-\begin{subequations}
-  \begin{align}
-    \dv{\bF^{(0)}}{s}  &= \bO \\
-    \dv{\bC^{(0)}}{s}  &= \bO \\
-    \dv{\bW^{(0),\dagger}}{s} &= 2 \bC^{(0)}\bW^{(0),\dagger}\bF^{(0)} - \bW^{(0),\dagger}(\bF^{(0)})^2 - (\bC^{(0)})^2\bW^{(0),\dagger} \\
-    \dv{\bW^{(0)}}{s} &= 2 \bF^{(0)}\bW^{(0)}\bC^{(0)} - (\bF^{(0)})^2\bW^{(0)} - \bW^{(0)}(\bC^{(0)})^2 
-  \end{align}
-\end{subequations}
-where the $s$ dependence of $\bW^{(0)}$ and $\bW^{(0),\dagger}$  has been dropped in the last two equations.
-$\bF^{(0)}$ and $\bC^{(0)}$ do not depend on $s$ as a consequence of the first two equations.
-% The last equation can be solved by introducing $\bU$ the matrix that diagonalizes $\bC^{(0)} = \bU \bD^{(0)} \bU^{-1}$ such that the differential equation for $\bV^{(0)}$ becomes
-% \begin{equation}
-%   \label{eq:eqdiffW0}
-%   \dv{\bW^{(0)}}{s} = 2 \bF^{(0)}\bW^{(0)} \bD^{(0)} - (\bF^{(0)})^2\bW^{(0)} - \bW^{(0)} (\bD^{(0)})^2
-% \end{equation}
-% where $\bW^{(0)}(s)= \bV^{(0)}(s) \bU$.
-% The matrix elements of $\bU$ and $\bD^{(0)}$ are
-% \begin{align}
-%   U_{p\nu,q\mu} &= \delta_{pq} \bX_{\nu\mu} \\
-%   D_{p\nu,q\mu}^{(0)} &= \left(\epsilon_p + \sgn(\epsilon_p-\epsilon_\text{F})\Omega_\nu\right)\delta_{pq}\delta_{\nu\mu}
-% \end{align}
-% where $\epsilon_\text{F}$ is the Fermi level.
-% Note that the matrix $\bU$ is also used in the downfolding process of Eq.~\eqref{eq:GWlin}. \cite{Bintrim_2021}
-
-Thanks to the diagonal structure of $\bF^{(0)}$ and $\bC^{(0)}$, the last equation can be easily solved and give 
-\begin{equation}
-  W_{p,q\nu}^{(0)}(s) = W_{p,q\nu}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{q\nu,q\nu}^{(0)})^2 s}
-\end{equation}
-The initial condition $\bW^{(0)}(0) = \bO$ implies $\bW^{(0)}(s)=\bO$ and therefore the zeroth order Hamiltonian is
-\begin{equation}
-  \bH^{(0)}(s) = \bH^{(0)}(0),
-\end{equation}
-\ie it is independent of $s$.
 
 %///////////////////////////%
 \subsection{First-order matrix elements}
@@ -468,17 +435,27 @@ Knowing that $\bHod{0}(s)=\bO$, the first order flow equation is
 \begin{equation}
   \dv{\bH^{(1)}}{s} = \comm{\comm{\bHd{0}}{\bHod{1}}}{\bHd{0}}
 \end{equation}
-which gives the same system of equations as in the previous subsection except that $\bW^{(0)}$ and $\bW^{(0),\dagger}$ should be replaced by $\bW^{(1)}$ and $\bW^{(1),\dagger}$.
-
-Once again the two first equations are easily solved
+which gives the following system of equations
+\ANT{Do you know a cleaner way to write this system? The vertical spaces are too large...}
+\begin{subequations}
+  \begin{align}
+    \dv{\bF^{(0)}}{s}&=\bO   &  \dv{\bC^{(0)}}{s}&=\bO
+  \end{align}
+  \begin{multline}
+    \dv{\bW^{(1),\dagger}}{s}{(s)} = 2 \bC^{(0)}\bW^{(1),\dagger}(s)\bF^{(0)} - \bW^{(1),\dagger}(s)(\bF^{(0)})^2 \\ - (\bC^{(0)})^2\bW^{(1),\dagger}(s)
+  \end{multline}
+  \begin{multline}
+    \dv{\bW^{(1)}}{s}{(s)} = 2 \bF^{(0)}\bW^{(1)}(s)\bC^{(0)} - (\bF^{(0)})^2\bW^{(1)}(s) \\ - \bW^{(1)}(s)(\bC^{(0)})^2.
+  \end{multline}
+\end{subequations}
+The two first equations imply
 \begin{align}
   \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO.
 \end{align}
-and the first order coupling elements are given by (up to a multiplication by $\bU^{-1}$)
-\begin{align}
-  W_{p,q\nu}^{(1)}(s) &= W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{q\nu,q\nu}^{(0)})^2 s} \\
-  &= W_{p,q\nu}^{(1)}(0) e^{- [\epsilon_p - \epsilon_q - \sgn(\epsilon_q-\epsilon_F)\Omega_\nu]^2 s} \notag
-\end{align}
+and thanks to the diagonal structure of $\bF^{(0)}$ and $\bC^{(0)}$ the differential equations for the coupling elements are easily solved and give
+ \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 two-electron screened integrals defined in Eq.~\eqref{eq:GW_sERI} while for $s\to\infty$ they go to zero.
 Therefore, $W_{p,q\nu}^{(1)}(s)$ are renormalized two-electrons screened integrals.
 Note the close similarity of the first-order element expressions with the ones of Evangelista in Ref.~\onlinecite{Evangelista_2014b} obtained in a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
@@ -490,15 +467,14 @@ Note the close similarity of the first-order element expressions with the ones o
 The second-order renormalized quasi-particle equation is given by
 \begin{equation}
   \label{eq:GW_renorm}
-  \left( \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) \right) \bX = \omega \bX 
+  \left( \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) \right) \bX = \omega \bX,
 \end{equation}
 with
 \begin{align}
-  \widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s)\\
+  \widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s),\\
   \label{eq:srg_sigma}
-  \widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1}  (\bV^{(1)}(s))^{\dagger}
+  \widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1}  (\bV^{(1)}(s))^{\dagger}.
 \end{align}
-
 As can be readily seen above, $\bF^{(2)}$ is the only second-order block of the effective Hamiltonian contributing to the second-order SRG quasi-particle equation.
 Collecting every second-order terms in the flow equation and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
 \begin{multline}
@@ -518,27 +494,27 @@ At $s=0$, this second-order correction is null while for $s\to\infty$ it tends t
   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^{\dagger}_{r\nu,q}.
 \end{equation}
 Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
-Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction.
+Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into 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}.
-y
-Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~\eqref{eq:GW_renorm} defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} which matrix elements read as
+
+Interestingly, the static limit, \ie the $s\to\infty$ limit, of Eq.~\eqref{eq:GW_renorm} defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} which matrix elements read as
 \begin{equation}
   \label{eq:sym_qsGW}
-    \Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu}  \left( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2}  \right) W_{p,r\nu} W^{\dagger}_{r\nu,q}.
+  \Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu}  \left( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2}  \right) W_{p,r\nu} W^{\dagger}_{r\nu,q}.
 \end{equation}
-Yet, both approximation are closely related as they share the same diagonal terms when $\eta=0$.
-Also, note that the SRG static approximation is naturally Hermitian as opposed to the symmetrized case where it is enforced by symmetrization.
+This alternative static form will be refered to as SRG-qs$GW$ in the following. 
+Both approximations are closely related as they share the same diagonal terms when $\eta=0$.
+Also, note that the SRG static approximation is naturally Hermitian as opposed to the usual case where it is enforced by symmetrization.
 
 However, as will be discussed in more detail in the results section, the convergence of the qs$GW$ scheme using $\widetilde{\bF}(\infty)$ is very poor.
 This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
-Indeed, in qs$GW$ calculation using the symmetrized static form, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable.
+Indeed, in traditional qs$GW$ calculation increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable.
 Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
-
 \begin{multline}
   \label{eq:SRG_qsGW}
   \Sigma_{pq}^{\text{SRG}}(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} \times \\ \left(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}\right)  
 \end{multline}
-which depends on one regularising parameter $s$ analogously to $\eta$ in the usual case.
+which depends on one regularising parameter $s$ analogously to $\eta$ in the usual qs$GW$.
 The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
 Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.
 
@@ -659,8 +635,6 @@ Finally, the Beryllium oxyde will be studied as a prototypical example of a mole
 The test set considered in this study is composed of the GW20 set of molecules introduced by Lewis and Berkelbach. \cite{Lewis_2019}
 This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set.
 We also added the MgO and O3 molecules which are part of GW100 and are known to be difficult to converged for qs$GW$. \cite{vanSetten_2015,Forster_2021}
-In addition, we considered the Quest 1 and 2 sets which is made of small and medium size organic molecules.
-
 
 %=================================================================%
 \section{Conclusion}