From aeb254af01c6fe5d5a78c418c90e1d230f0096ac Mon Sep 17 00:00:00 2001 From: Antoine MARIE Date: Tue, 13 Dec 2022 15:08:08 +0100 Subject: [PATCH] add some refs --- Manuscript/SRGGW.bib | 86 ++++++++++++++++++++++++++++++++++++-------- Manuscript/SRGGW.tex | 17 ++++----- 2 files changed, 80 insertions(+), 23 deletions(-) diff --git a/Manuscript/SRGGW.bib b/Manuscript/SRGGW.bib index 51b68ab..04cc462 100644 --- a/Manuscript/SRGGW.bib +++ b/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}, diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 0b5ce35..807f5a4 100644 --- a/Manuscript/SRGGW.tex +++ b/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}). -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}). +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 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.