add some refs
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@ -15730,21 +15730,6 @@
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year = {2012},
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bdsk-url-1 = {https://dx.doi.org/10.1063/1.4718428}}
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@article{Bruneval_2013,
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author = {Bruneval, Fabien and Marques, Miguel A. L.},
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doi = {10.1021/ct300835h},
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file = {/Users/loos/Zotero/storage/6ERH93TH/Bruneval_2012b.pdf},
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issn = {1549-9618, 1549-9626},
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journal = {J. Chem. Theory Comput.},
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language = {en},
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month = jan,
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number = {1},
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pages = {324--329},
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title = {Benchmarking the {{Starting Points}} of the {{{\emph{GW}}}} {{Approximation}} for {{Molecules}}},
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volume = {9},
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year = {2013},
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bdsk-url-1 = {https://dx.doi.org/10.1021/ct300835h}}
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@article{Bruneval_2016a,
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author = {Bruneval, Fabien},
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doi = {10.1063/1.4972003},
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@ -15972,6 +15957,77 @@
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doi = {10.1088/1361-648X/aa7803}
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}
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@article{Bruneval_2013,
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title = {Benchmarking the {{Starting Points}} of the {{GW Approximation}} for {{Molecules}}},
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author = {Bruneval, Fabien and Marques, Miguel A. L.},
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year = {2013},
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journal = {Journal of Chemical Theory and Computation},
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volume = {9},
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number = {1},
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pages = {324--329},
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issn = {1549-9618},
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doi = {10.1021/ct300835h}
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}
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@article{Caruso_2016,
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title = {Benchmark of {{GW Approaches}} for the {{GW100 Test Set}}},
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author = {Caruso, Fabio and Dauth, Matthias and {van Setten}, Michiel J. and Rinke, Patrick},
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year = {2016},
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journal = {Journal of Chemical Theory and Computation},
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volume = {12},
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number = {10},
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pages = {5076--5087},
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issn = {1549-9618},
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doi = {10.1021/acs.jctc.6b00774}
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}
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@article{Gallandi_2015,
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title = {Long-{{Range Corrected DFT Meets GW}}: {{Vibrationally Resolved Photoelectron Spectra}} from {{First Principles}}},
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author = {Gallandi, Lukas and K{\"o}rzd{\"o}rfer, Thomas},
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year = {2015},
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journal = {Journal of Chemical Theory and Computation},
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volume = {11},
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number = {11},
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pages = {5391--5400},
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issn = {1549-9618},
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doi = {10.1021/acs.jctc.5b00820}
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}
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@article{Gallandi_2016,
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title = {Accurate {{Ionization Potentials}} and {{Electron Affinities}} of {{Acceptor Molecules II}}: {{Non-Empirically Tuned Long-Range Corrected Hybrid Functionals}}},
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author = {Gallandi, Lukas and Marom, Noa and Rinke, Patrick and K{\"o}rzd{\"o}rfer, Thomas},
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year = {2016},
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journal = {Journal of Chemical Theory and Computation},
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volume = {12},
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number = {2},
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pages = {605--614},
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issn = {1549-9618},
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doi = {10.1021/acs.jctc.5b00873}
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}
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@article{Korzdorfer_2012,
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title = {Strategy for Finding a Reliable Starting Point for \$\{\vphantom\}{{G}}\vphantom\{\}\_\{0\}\{\vphantom\}{{W}}\vphantom\{\}\_\{0\}\$ Demonstrated for Molecules},
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author = {K{\"o}rzd{\"o}rfer, Thomas and Marom, Noa},
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year = {2012},
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journal = {Physical Review B},
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volume = {86},
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number = {4},
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pages = {041110},
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doi = {10.1103/PhysRevB.86.041110}
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}
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@article{Marom_2012,
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title = {Benchmark of \${{GW}}\$ Methods for Azabenzenes},
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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},
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year = {2012},
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journal = {Physical Review B},
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volume = {86},
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number = {24},
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pages = {245127},
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doi = {10.1103/PhysRevB.86.245127}
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}
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@article{vanSchilfgaarde_2006,
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author = {{van Schilfgaarde}, M. and Kotani, Takao and Faleev, S.},
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date-modified = {2018-04-14 07:31:33 +0000},
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@ -130,7 +130,7 @@ In fact, these cases are related to the discontinuities and convergence problems
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One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
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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.
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Therefore, one could try to optimise the starting point to obtain the best one-shot energies possible. \ant{add ref}
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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}
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Alternatively, one could solve this equation self-consistently leading to the eigenvalue only self-consistent scheme. \cite{Kaplan_2016}
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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.
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However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach.
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@ -439,27 +439,28 @@ Collecting every second-order terms and performing the block matrix products res
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This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
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\begin{align}
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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 \\
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&\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)
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&\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).
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\end{align}
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At $s=0$, this second-order correction is null and for $s\to\infty$ it tends towards the following static limit
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\begin{equation}
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\label{eq:static_F2}
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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}
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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}.
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\end{equation}
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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}).
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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}).
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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}).
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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}).
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Yet, both are closely related as they share the same diagonal terms.
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Also, note that the hermiticity is naturally enforced in the SRG static approximation as opposed to the symmetrized case.
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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.
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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.
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This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
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Therefore, we will define the SRG-qs$GW$ as
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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 ?}
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Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
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\begin{align}
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\label{eq:SRG_qsGW}
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\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 \\
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&\times \left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right)
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\end{align}
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which depends on the parameter $s$ analogously to the $eta$ in the usual case.
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which depends on one regularising parameter $s$ analogously to $eta$ in the usual case.
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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 $\tilde{\bF}$.
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Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.
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