saving intro

Manuscript/ufGW.bib

@@ -1,13 +1,27 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2022-02-14 14:04:41 +0100 
+%% Created for Pierre-Francois Loos at 2022-02-18 09:40:08 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{DiSabatino_2021,
+	abstract = {Using the simple (symmetric) Hubbard dimer, we analyze some important features of the GW approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot GW method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard GW approximation and find, in particular, that 1) some neutral excitation energies become complex when the electron-electron interaction U increases, which can be traced back to the approximate nature of the GW quasiparticle energies; 2) the BSE formalism yields accurate correlation energies over a wide range of U when the trace (or plasmon) formula is employed; 3) the trace formula is sensitive to the occurrence of complex excitation energies (especially singlet), while the expression obtained from the adiabatic-connection fluctuation-dissipation theorem (ACFDT) is more stable (yet less accurate); 4) the trace formula has the correct behavior for weak (i.e., small U) interaction, unlike the ACFDT expression.},
+	author = {Di Sabatino, S. and Loos, P.-F. and Romaniello, P.},
+	date-added = {2022-02-15 09:53:24 +0100},
+	date-modified = {2022-02-15 09:54:07 +0100},
+	doi = {10.3389/fchem.2021.751054},
+	journal = {Front. Chem.},
+	pages = {751054},
+	title = {Scrutinizing GW-Based Methods Using the Hubbard Dimer},
+	volume = {9},
+	year = {2021},
+	bdsk-url-1 = {https://www.frontiersin.org/article/10.3389/fchem.2021.751054},
+	bdsk-url-2 = {https://doi.org/10.3389/fchem.2021.751054}}
+
 @article{Bannwarth_2020,
 	author = {Bannwarth,Christoph and Yu,Jimmy K. and Hohenstein,Edward G. and Mart{\'\i}nez,Todd J.},
 	date-added = {2022-02-14 14:04:04 +0100},
@@ -14063,9 +14077,8 @@
 @article{Gallandi_2016,
 	author = {Gallandi, Lukas and Marom, Noa and Rinke, Patrick and K{\"o}rzd{\"o}rfer, Thomas},
 	date-added = {2018-04-22 16:22:34 +0000},
-	date-modified = {2018-04-22 16:22:34 +0000},
+	date-modified = {2022-02-18 09:40:06 +0100},
 	doi = {10.1021/acs.jctc.5b00873},
-	issn = {1549-9618, 1549-9626},
 	journal = {J. Chem. Theory Comput.},
 	language = {en},
 	month = feb,

Manuscript/ufGW.tex

@@ -143,13 +143,25 @@ A simple and efficient regularization procedure is proposed to avoid such issues
 \section{Introduction}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-We consider {\GOWO} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes.
+The $GW$ approximation of many-body perturbation theory \cite{Hedin_1965,Martin_2016} allows to compute accurate charged excitation (\ie, ionization potentials and electron affinities) in solids and molecules. \cite{Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019}
+Its popularity in the molecular structure community is rapidly increasing \cite{Ke_2011,Bruneval_2012,Bruneval_2013,Bruneval_2015,Blase_2016,Bruneval_2016, Bruneval_2016a,Koval_2014,Hung_2016,Blase_2018,Boulanger_2014,Li_2017,Hung_2016,Hung_2017,vanSetten_2015,vanSetten_2018,vanSetten_2015, Maggio_2017,vanSetten_2018,Richard_2016,Gallandi_2016,Knight_2016,Dolgounitcheva_2016,Bruneval_2015,Krause_2015,Govoni_2018,Caruso_2016} thanks to its relatively low cost and somehow surprising accuracy for weakly-correlated systems. \cite{Bruneval_2021}
+
+In recent studies, \cite{Loos_2018b,Veril_2018,Berger_2021,DiSabatino_2021} we discovered that one can observe (unphysical) irregularities and/or discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, fundamental gap, total and correlation energies, as well as vertical excitation energies) even in the weakly correlated regime. 
+These issues were discovered in Ref.~\onlinecite{Loos_2018b} while studying a model two-electron system \cite{Seidl_2007,Loos_2009a,Loos_2009c} and they were further investigated in Ref.~\onlinecite{Veril_2018}, where we provided additional evidences and explanations of these undesirable features in real molecular systems.
+In particular, we show that each branch of the self-energy $\Sigma$ is associated with a distinct quasiparticle solution, and that each switch between solutions implies a significant discontinuity in the quasiparticle energy due to the transfer of weight between two solutions of the quasiparticle equation. \cite{Veril_2018}
+It was shown that these problems could be alleviated by using a static COHSEX self-energy \cite{Berger_2021} and by considering a fully self-consistent scheme. \cite{DiSabatino_2021}
+However, none of these solutions is completely satisfying as a static approximation of the self-energy can induce significant loss of accuracy and fully self-consistent calculations can be quite challenging terms of implementation and cost.
+
+In the present article, we provide further physical insights into the origin of these discontinuities by highlighting, in particular, the role of intruder states. 
+These new insights allow us to provide a cheap and efficient regularization scheme in order to avoid these issues.
+
+We consider {\GOWO} \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes.\cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Downfold: The non-linear $GW$ problem}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Here,  for the sake of simplicity, we consider a restricted Hartree-Fock (HF) starting point but the present analysis can be straightforwardly extended to a Kohn-Sham (KS) starting point.
+Here, for the sake of simplicity, we consider a restricted Hartree-Fock (HF) starting point but the present analysis can be straightforwardly extended to a Kohn-Sham (KS) starting point.
 Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$, the following (non-linear) quasiparticle equation
 \begin{equation}
 \label{eq:qp_eq}
@@ -173,26 +185,31 @@ with
 \begin{equation}
 	A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
 \end{equation}
-As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{}$ and their corresponding weight is given by the value of the so-called renormalization factor
+and 
 \begin{equation}
-	0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{}} ]^{-1} \le 1
+	\ERI{pq}{rs} = \iint \MO{p}(\br_1) \MO{q}(\br_1) \frac{1}{\abs{\br_1 - \br_2}} \MO{r}(\br_2) \MO{s}(\br_2) d\br_1 \dbr_2
+\end{equation}
+are two-electron integrals over the HF (spatial) orbitals $\MO{p}(\br)$. 
+As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW}$ and their corresponding weight is given by the value of the so-called renormalization factor
+\begin{equation}
+	0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1
 \end{equation} 
-In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{} \equiv \eps{p,s=0}{}$ has a large weight $Z_{} \equiv Z_{p,=0}$
+In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{\GW} \equiv \eps{p,s=0}{\GW}$ has a large weight $Z_{p} \equiv Z_{p,s=0}$.
 Note that we have the following important conservation rules
 \begin{align}
 	\sum_{s} Z_{p,s} & = 1
 	&
-	\sum_{s} Z_{p,s} \eps{p,s}{} & = \eps{p}{\HF}
+	\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF}
 \end{align}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Upfolding: the linear $GW$ problem}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The non-linear quasiparticle equation \eqref{eq:qp_eq} can be transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
-are upfolded from the 1h and 1p sectors.
+are upfolded from the 1h and 1p sectors. \cite{Bintrim_2021a}
 For each orbital $p$, this yields a linear eigenvalue problem of the form
 \begin{equation}
-	\bH^{(p)} \bc{}{(p,s)} = \eps{p,s}{} \bc{}{(p,s)}
+	\bH^{(p)} \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
 \end{equation}
 with
 \begin{equation}
@@ -214,11 +231,12 @@ where
 \end{align}
 and the corresponding coupling blocks read
 \begin{align}
-	V^\text{2h1p}_{p,kld} & = \sqrt{2} \ERI{pk}{cl}
+	V^\text{2h1p}_{p,klc} & = \sqrt{2} \ERI{pk}{cl}
 	&
-	V^\text{2p1h}_{p,cld} & = \sqrt{2} \ERI{pd}{kc}
+	V^\text{2p1h}_{p,kcd} & = \sqrt{2} \ERI{pd}{kc}
 \end{align}
 The size of this eigenvalue problem is $N = 1 + N^\text{2h1p} + N^\text{2p1h} = 1 + O^2 V + O V^2$, and this eigenvalue problem has to be solved for each orbital that one wants to correct.
+Note, however, that the block $\bC{}{\text{2h1p}}$ and $\bC{}{\text{2p1h}}$ do not need to be recomputed for each orbital.
 Because the renormalization factor corresponds to the projection of the vector $\bc{}{(p,s)}$ onto the reference space, the weight of a solution $(p,s)$ is given by the  the first coefficient of their corresponding eigenvector $\bc{}{(p,s)}$, \ie, 
 \begin{equation}
 	Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
@@ -275,6 +293,13 @@ Of course, the way of regularizing the self-energy is not unique but here we con
 
 This helps greatly convergence for (partially) self-consistent $GW$ methods.
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Concluding remarks}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Note that these issues do not only appear at the {\GOWO} level but also at the partially self-consistent levels such as ev$GW$ \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018} and qs$GW$. \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016}
+However, fully self-consistent $GW$ methods \cite{Stan_2006,Stan_2009,Rostgaard_2010,Caruso_2012,Caruso_2013,Caruso_2013a,Caruso_2013b,Koval_2014,Wilhelm_2018} where one considers not only the quasiparticle solutions but also the satellites at each iteration are free of these irregularities. \cite{DiSabatino_2021}
+The $T$-matrix-based formalism as well as second-order Green's function (or second Born) scheme \cite{SzaboBook,Casida_1989,Casida_1991,Stefanucci_2013,Ortiz_2013, Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Hirata_2015,Hirata_2017} exhibit the same problems.
+
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{
 This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}