add some refs

Manuscript/SRGGW.bib

@@ -15730,21 +15730,6 @@
 	year = {2012},
 	bdsk-url-1 = {https://dx.doi.org/10.1063/1.4718428}}
 
-@article{Bruneval_2013,
-	author = {Bruneval, Fabien and Marques, Miguel A. L.},
-	doi = {10.1021/ct300835h},
-	file = {/Users/loos/Zotero/storage/6ERH93TH/Bruneval_2012b.pdf},
-	issn = {1549-9618, 1549-9626},
-	journal = {J. Chem. Theory Comput.},
-	language = {en},
-	month = jan,
-	number = {1},
-	pages = {324--329},
-	title = {Benchmarking the {{Starting Points}} of the {{{\emph{GW}}}} {{Approximation}} for {{Molecules}}},
-	volume = {9},
-	year = {2013},
-	bdsk-url-1 = {https://dx.doi.org/10.1021/ct300835h}}
-
 @article{Bruneval_2016a,
 	author = {Bruneval, Fabien},
 	doi = {10.1063/1.4972003},
@@ -15972,6 +15957,77 @@
   doi = {10.1088/1361-648X/aa7803}
 }
 
+@article{Bruneval_2013,
+  title = {Benchmarking the {{Starting Points}} of the {{GW Approximation}} for {{Molecules}}},
+  author = {Bruneval, Fabien and Marques, Miguel A. L.},
+  year = {2013},
+  journal = {Journal of Chemical Theory and Computation},
+  volume = {9},
+  number = {1},
+  pages = {324--329},
+  issn = {1549-9618},
+  doi = {10.1021/ct300835h}
+}
+
+@article{Caruso_2016,
+  title = {Benchmark of {{GW Approaches}} for the {{GW100 Test Set}}},
+  author = {Caruso, Fabio and Dauth, Matthias and {van Setten}, Michiel J. and Rinke, Patrick},
+  year = {2016},
+  journal = {Journal of Chemical Theory and Computation},
+  volume = {12},
+  number = {10},
+  pages = {5076--5087},
+  issn = {1549-9618},
+  doi = {10.1021/acs.jctc.6b00774}
+}
+
+@article{Gallandi_2015,
+  title = {Long-{{Range Corrected DFT Meets GW}}: {{Vibrationally Resolved Photoelectron Spectra}} from {{First Principles}}},
+  author = {Gallandi, Lukas and K{\"o}rzd{\"o}rfer, Thomas},
+  year = {2015},
+  journal = {Journal of Chemical Theory and Computation},
+  volume = {11},
+  number = {11},
+  pages = {5391--5400},
+  issn = {1549-9618},
+  doi = {10.1021/acs.jctc.5b00820}
+}
+
+@article{Gallandi_2016,
+  title = {Accurate {{Ionization Potentials}} and {{Electron Affinities}} of {{Acceptor Molecules II}}: {{Non-Empirically Tuned Long-Range Corrected Hybrid Functionals}}},
+  author = {Gallandi, Lukas and Marom, Noa and Rinke, Patrick and K{\"o}rzd{\"o}rfer, Thomas},
+  year = {2016},
+  journal = {Journal of Chemical Theory and Computation},
+  volume = {12},
+  number = {2},
+  pages = {605--614},
+  issn = {1549-9618},
+  doi = {10.1021/acs.jctc.5b00873}
+}
+
+@article{Korzdorfer_2012,
+  title = {Strategy for Finding a Reliable Starting Point for \$\{\vphantom\}{{G}}\vphantom\{\}\_\{0\}\{\vphantom\}{{W}}\vphantom\{\}\_\{0\}\$ Demonstrated for Molecules},
+  author = {K{\"o}rzd{\"o}rfer, Thomas and Marom, Noa},
+  year = {2012},
+  journal = {Physical Review B},
+  volume = {86},
+  number = {4},
+  pages = {041110},
+  doi = {10.1103/PhysRevB.86.041110}
+}
+
+@article{Marom_2012,
+  title = {Benchmark of \${{GW}}\$ Methods for Azabenzenes},
+  author = {Marom, Noa and Caruso, Fabio and Ren, Xinguo and Hofmann, Oliver T. and K{\"o}rzd{\"o}rfer, Thomas and Chelikowsky, James R. and Rubio, Angel and Scheffler, Matthias and Rinke, Patrick},
+  year = {2012},
+  journal = {Physical Review B},
+  volume = {86},
+  number = {24},
+  pages = {245127},
+  doi = {10.1103/PhysRevB.86.245127}
+}
+
+
 @article{vanSchilfgaarde_2006,
 	author = {{van Schilfgaarde}, M. and Kotani, Takao and Faleev, S.},
 	date-modified = {2018-04-14 07:31:33 +0000},

Manuscript/SRGGW.tex

@@ -130,7 +130,7 @@ 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 optimise the starting point to obtain the best one-shot energies possible. \ant{add ref}
+Therefore, one could try to optimise 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{Kaplan_2016}
 To do so one use the energy of the quasi-particle solution of the previous iteration to build Eq.~(\ref{eq:G0W0}) and then solves 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. 
@@ -439,27 +439,28 @@ Collecting every second-order terms and performing the block matrix products res
 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) 
+                  &\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}
   \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) = \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}.
 \end{equation}
-Therefore, the SRG flows 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}).
+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}).
-Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~(\ref{eq:GW_renorm}) defines an alternative qsGW approximation to the one defined by Eq.~(\ref{eq:sym_qsgw}).
+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.
 
-However, as will be discussed in more details in the results section, the convergence of the qs$GW$ scheme using $\tilde{\bF}(\infty)$ is very poor.
+However, as will be discussed in more details 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.
-Therefore, we will define the SRG-qs$GW$ as
+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 ?}
+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 \\
   &\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 the parameter $s$ analogously to the $eta$ 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 $\tilde{\bF}$.
 Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.