working on notations

Manuscript/SRGGW.tex

@@ -202,6 +202,8 @@ and the corresponding coupling blocks read
   &
     V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
 \end{align}
+Throughout the manuscript $p,q,r,s$ indices are used for general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
+The indices $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
 The usual $GW$ non-linear equation can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021}
 \begin{equation}
   \label{eq:GWnonlin}
@@ -327,7 +329,7 @@ As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal part
   \end{pmatrix}
 \end{align}
 where we have omitted the $s$ dependence of the matrix elements for the sake of brevity.
-Then, the aim of this section is to solve analytically the flow equation [see Eq.~(\ref{eq:flowEquation})] order by order knowing that the initial conditions are
+Then, the aim of this section is to solve order by order the flow equation [see Eq.~(\ref{eq:flowEquation})] knowing that the initial conditions are
 \begin{align}
   \bHd{0}(0) &=   \begin{pmatrix}
                     \bF{}{} & \bO \\
@@ -346,7 +348,7 @@ Then, the aim of this section is to solve analytically the flow equation [see Eq
                                                   \bV{}{\dagger} & \bO \notag
                                                 \end{pmatrix}   \notag 
 \end{align}
-where we have defined $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness.
+where we have defined the matrix $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness.
 Then, the perturbative expansions can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalised quasi-particle equation.
 In particular, in this manuscript the focus will be on the second-order renormalized quasi-particle equation.
 
@@ -358,7 +360,7 @@ There is only one zero-th order term in the right hand side of the flow equation
 \begin{equation}
   \dv{\bH^{(0)}}{s} = \comm{\comm{\bHd{0}}{\bHod{0}}}{\bH^{(0)}},
 \end{equation}
-and performing the block matrix products gives the following system of equations
+and performing the block matrix products gives the following system of equations 
 \begin{subequations}
   \begin{align}
     \dv{\bF^{(0)}}{s}  &= \bO \\
@@ -367,6 +369,8 @@ and performing the block matrix products gives the following system of equations
     \dv{\bV^{(0)}}{s} &= 2 \bF^{(0)}\bV^{(0)}\bC^{(0)} - (\bF^{(0)})^2\bV^{(0)} - \bV^{(0)}(\bC^{(0)})^2 
   \end{align}
 \end{subequations}
+where the $s$ dependence of $\bV^{(0)}$ and $\bV^{(0),\dagger}$  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}
   \dv{\bW^{(0)}}{s} = 2 \bF^{(0)}\bW^{(0)} \bD^{(0)} - (\bF^{(0)})^2\bW^{(0)} - \bW^{(0)} (\bD^{(0)})^2
@@ -383,6 +387,12 @@ The two first equations of the system are trivial and finally we have
   \bH^{(0)}(s) = \bH^{(0)}(0)
 \end{equation}
 which shows that the zero-th order matrix elements are independent of $s$.
+The matrix elements of $\bU$ and $\bD$ are
+\begin{align}
+  U_{(p,v),(q,w)}^{(0)} &= \delta_{pq} \bX_{v,w} \\
+  D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_F)\Omega_v\right)\delta_{pq}\delta_{vw}
+\end{align}
+where $\epsilon_F$ is the Fermi level.
 
 %///////////////////////////%
 \subsubsection{First order matrix elements}
@@ -399,9 +409,10 @@ Once again the two first equations are easily solved
   \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{equation}
-  W_{p,(q,v)}^{(1)}(s) = W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
-\end{equation}
+\begin{align}
+  W_{p,(q,v)}^{(1)}(s) &= W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s} \\
+  W_{p,(q,v)}^{(1)}(s) &= \left( \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia} \right) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v)^2 s}
+\end{align}
 Note that at $s=0$ the elements  $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~(\ref{eq:GW_sERI}) and that for $s\to\infty$ they go to zero.
 Therefore, $W_{p,(q,v)}^{(1)}(s)$ 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}).
@@ -418,6 +429,7 @@ The second-order renormalised quasi-particle equation is given by
 with
 \begin{align}
   \widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s)\\
+  \label{eq:srg_sigma}
   \widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \mathbb{1} - \bC^{(0)}\right)^{-1}  (\bV^{(1)}(s))^{\dagger}
 \end{align}
 
@@ -428,16 +440,18 @@ Collecting every second-order terms and performing the block matrix products res
   \dv{\bF^{(2)}}{s}  = \bF^{(0)}\bV^{(1)}\bV^{(1),\dagger} + \bV^{(1)}\bV^{(1),\dagger}\bF^{(0)} - 2 \bV^{(1)}\bC^{(0)}\bV^{(1),\dagger} .
 \end{equation}
 This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
-\begin{align}
-  F_{pq}^{(2)}(s) &= \sum_{r,v} \frac{\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W_{p,(r,v)} \notag \\
-                  &\times W^{\dagger}_{(r,v),q}\left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s}   e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right).
-\end{align}
-At $s=0$, this second-order correction is null and for $s\to\infty$ it tends towards the following static limit
+\begin{equation}
+  F_{pq}^{(2)}(s) = \sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right).
+\end{equation}
+with $\Delta_{pqv} = \epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v$.
+
+At $s=0$, this second-order correction is null while for $s\to\infty$ it tends towards the following static limit
 \begin{equation}
   \label{eq:static_F2}
-  F_{pq}^{(2)}(\infty) = \sum_{r,v} \frac{\left(\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)\right) W_{p,(r,v)} W_{q,(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2}.
+  F_{pq}^{(2)}(\infty) = \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}.
 \end{equation}
-Therefore, the SRG flow gradually transforms the dynamic degrees of freedom in $\bSig(\omega)$ in static ones, starting from the ones that have the largest denominators in Eq.~(\ref{eq:static_F2}).
+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 gradually transforms the dynamic degrees of freedom of $\bSig(\omega)$ in static ones, starting from the ones that have the largest denominators in Eq.~(\ref{eq:static_F2}).
 Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~(\ref{eq:GW_renorm}) defines an alternative qs$GW$ approximation to the one defined by Eq.~(\ref{eq:sym_qsgw}).
 Yet, both are closely related as they share the same diagonal terms.
 Also, note that the hermiticity is naturally enforced in the SRG static approximation as opposed to the symmetrized case.