small changes

Manuscript/SRGGW.tex

@@ -132,7 +132,7 @@ In fact, these cases are related to the discontinuities and convergence problems
 One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
 Indeed, in Eq.~(\ref{eq:G0W0}) we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example.
 Therefore, one could try to optimize the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
-Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme. \cite{Shishkin_2006,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016}
+Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme. \cite{Shishkin_2007,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016}
 To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~(\ref{eq:G0W0}) and then this equation is solved for $\omega$ again until convergence is reached.
 However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach. 
 Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals. \cite{Marom_2012}
@@ -204,17 +204,17 @@ and the corresponding coupling blocks read
 \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}
+The usual $GW$ non-linear equation
 \begin{equation}
   \label{eq:GWnonlin}
   \left( \bF + \bSig(\omega) \right) \bX = \omega \bX,
 \end{equation}
-with
+can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021} which gives the following the expression for the self-energy
 \begin{align}
   \bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1}  (\bV^{\hhp})^{\mathsf{T}} \\
     &+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1}  (\bV^{\pph}\right)^{\mathsf{T}}, \notag
 \end{align}
-which can be further developed to give 
+which can be further developed as
 \begin{equation}
   \label{eq:GW_selfenergy}
   \Sigma_{pq}(\omega) 
@@ -310,7 +310,7 @@ Then, one can collect order by order the terms in Eq.~(\ref{eq:flowEquation}) an
 %%%%%%%%%%%%%%%%%%%%%% 
 
 Finally, the SRG formalism exposed above will be applied to $GW$.
-First, one needs to define the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
+The first step is to define the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
 As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal parts will be defined as
 \begin{align}
   \label{eq:diag_and_offdiag}
@@ -329,7 +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 order by order the flow equation [see Eq.~(\ref{eq:flowEquation})] knowing that the initial conditions are
+Then, the aim 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 \\
@@ -349,7 +349,7 @@ Then, the aim of this section is to solve order by order the flow equation [see
                                                 \end{pmatrix}   \notag 
 \end{align}
 where we have defined the matrices $\bC$ 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 renormalized quasi-particle equation.
+Once the analytical low-order perturbative expansions are known they can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalized quasi-particle equation.
 In particular, in this manuscript the focus will be on the second-order renormalized quasi-particle equation.
 
 %///////////////////////////%
@@ -373,26 +373,28 @@ where the $s$ dependence of $\bV^{(0)}$ and $\bV^{(0),\dagger}$  has been droppe
 $\bF^{(0)}$ 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 $\bW^{(0)}= \bV^{(0)} \bU$.
+where $\bW^{(0)}(s)= \bV^{(0)}(s) \bU$.
+The matrix elements of $\bU$ and $\bD^{(0)}$ are
+\begin{align}
+  U_{(p,v),(q,w)} &= \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.
 Note that the matrix $\bU$ is also used in the downfolding process of Eq.~(\ref{eq:GWlin}). \cite{Bintrim_2021}
-Due to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, these equations can be easily solved and give 
+
+Thanks to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, Eq.~\eqref{eq:eqdiffW0} can be easily solved and give 
 \begin{equation}
   W_{p,(q,v)}^{(0)}(s) = W_{p,(q,v)}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
 \end{equation}
 Due to the initial conditions  $\bV^{(0)}(0) = \bO$, we have $\bW^{(0)}(s)=\bO$ and therefore $\bV^{(0)}(s)=\bO=\bV^{(0)}(0) $.
-The two first equations of the system are trivial and finally, we have
+Therefore, the zeroth order Hamiltonian is
 \begin{equation}
-  \bH^{(0)}(s) = \bH^{(0)}(0)
+  \bH^{(0)}(s) = \bH^{(0)}(0),
 \end{equation}
-which shows that the zero-th order matrix elements are independent of $s$.
+\ie it is 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}
@@ -411,9 +413,9 @@ Once again the two first equations are easily solved
 and the first order coupling elements are given by (up to a multiplication by $\bU^{-1}$)
 \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}
+  &= W_{p,(q,v)}^{(1)}(0) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v)^2 s} \notag
 \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.
+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}) while 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}).
 
@@ -434,7 +436,7 @@ with
 \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 and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
+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{equation}
   \label{eq:diffeqF2}
   \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} .
@@ -443,35 +445,40 @@ This can be solved by simple integration along with the initial condition $\bF^{
 \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 $\Delta_{pqv} = \epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v$.
+with $\Delta_{prv} = \epsilon_p - \epsilon_r - \text{sign}(\epsilon_r-\epsilon_F)\Omega_v$.
 
 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) = \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}.
+  F_{pq}^{(2)}(\infty) =  \sum_{r,v}  \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),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 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}).
+Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction.
-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}).
+This transformation is done gradually starting from the states that have the largest denominators in Eq.~(\ref{eq:static_F2}).
-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.
+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}) which matrix elements read as
+\begin{equation}
+  \label{eq:static_F2}
+  \Sigma_{pq}^{\text{qs}GW}(\eta) =  \sum_{r,v}  \left( \frac{\Delta_{prv}}{\Delta_{prv}^2 + \eta^2} +\frac{\Delta_{qrv}}{\Delta_{qrv}^2 + \eta^2}  \right) W_{p,(r,v)} W^{\dagger}_{(r,v),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.
 
 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. \ant{ref fabien ?}
+Indeed, in qs$GW$ calculation using the symmetrized static form, 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{align}
   \label{eq:SRG_qsGW}
-  \Sigma_{pq}^{\text{SRG}}(s) &=  \epsilon_p \delta_{pq} + \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} \notag \\
+  \Sigma_{pq}^{\text{SRG}}(s) =  \frac{1}{2} \sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W_{q,(r,v)}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right)
-  &\times \left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s}   e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right) 
 \end{align}
-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 case.
-The fact that the $s\to\infty$ static limit does not well converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
+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.
 
 To conclude this section, we will discuss the case of discontinuities.
-Indeed, we have previously said that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
+Indeed, previously we mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
-So is it possible to remove discontinuities by using the SRG machinery developed above?
+So is it possible to use the SRG machinery developed above to remove discontinuities?
 In fact, not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
 However, doing a change of variable such that
 \begin{align}
@@ -483,7 +490,7 @@ In fact, the dynamic part after the change of variable is closely related to the
 %=================================================================%
 \section{Computational details}
 \label{sec:comp_det}
-% =================================================================%
+%=================================================================%
 
 The two qs$GW$ variants considered in this work have been implemented in an in-house program.
 The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}