add some refs

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Antoine Marie 2022-12-13 15:08:08 +01:00
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2 changed files with 80 additions and 23 deletions

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@ -15730,21 +15730,6 @@
year = {2012}, year = {2012},
bdsk-url-1 = {https://dx.doi.org/10.1063/1.4718428}} 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, @article{Bruneval_2016a,
author = {Bruneval, Fabien}, author = {Bruneval, Fabien},
doi = {10.1063/1.4972003}, doi = {10.1063/1.4972003},
@ -15972,6 +15957,77 @@
doi = {10.1088/1361-648X/aa7803} 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, @article{vanSchilfgaarde_2006,
author = {{van Schilfgaarde}, M. and Kotani, Takao and Faleev, S.}, author = {{van Schilfgaarde}, M. and Kotani, Takao and Faleev, S.},
date-modified = {2018-04-14 07:31:33 +0000}, 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
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence. 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. 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} 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. 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. 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 This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
\begin{align} \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 \\ 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} \end{align}
At $s=0$, this second-order correction is null and for $s\to\infty$ it tends towards the following static limit At $s=0$, this second-order correction is null and for $s\to\infty$ it tends towards the following static limit
\begin{equation} \begin{equation}
\label{eq:static_F2} \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} \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. 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. 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. 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} \begin{align}
\label{eq:SRG_qsGW} \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) &= \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) &\times \left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right)
\end{align} \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 $\tilde{\bF}$. 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. Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.