fig1

Manuscript/ufGW.bib

@@ -1,13 +1,66 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2022-02-21 14:28:44 +0100 
+%% Created for Pierre-Francois Loos at 2022-02-21 16:46:45 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@misc{Riva_2022,
+	archiveprefix = {arXiv},
+	author = {Gabriele Riva and Timoth{\'e}e Audinet and Matthieu Vladaj and Pina Romaniello and J. Arjan Berger},
+	date-added = {2022-02-21 15:51:38 +0100},
+	date-modified = {2022-02-21 15:51:44 +0100},
+	eprint = {2110.05623},
+	primaryclass = {cond-mat.str-el},
+	title = {Photoemission spectral functions from the three-body Green's function},
+	year = {2022}}
+
+@article{Wegner_1994,
+	author = {Franz Wegner},
+	date-added = {2022-02-21 15:45:42 +0100},
+	date-modified = {2022-02-21 15:47:47 +0100},
+	doi = {10.1002/andp.19945060203},
+	journal = {Ann. Phys. Leipzig},
+	pages = {77},
+	title = {Flow-equations for Hamiltonians},
+	volume = {3},
+	year = {1994},
+	bdsk-url-1 = {https://doi.org/10.1002/andp.19945060203}}
+
+@article{Glazek_1994,
+	author = {Glazek, Stanislaw D. and Wilson, Kenneth G.},
+	date-added = {2022-02-21 15:44:27 +0100},
+	date-modified = {2022-02-21 15:44:35 +0100},
+	doi = {10.1103/PhysRevD.49.4214},
+	issue = {8},
+	journal = {Phys. Rev. D},
+	month = {Apr},
+	numpages = {0},
+	pages = {4214--4218},
+	publisher = {American Physical Society},
+	title = {Perturbative renormalization group for Hamiltonians},
+	url = {https://link.aps.org/doi/10.1103/PhysRevD.49.4214},
+	volume = {49},
+	year = {1994},
+	bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevD.49.4214},
+	bdsk-url-2 = {https://doi.org/10.1103/PhysRevD.49.4214}}
+
+@article{White_2002,
+	author = {White,Steven R.},
+	date-added = {2022-02-21 15:42:00 +0100},
+	date-modified = {2022-02-21 15:42:14 +0100},
+	doi = {10.1063/1.1508370},
+	journal = {J. Chem. Phys.},
+	number = {16},
+	pages = {7472-7482},
+	title = {Numerical canonical transformation approach to quantum many-body problems},
+	volume = {117},
+	year = {2002},
+	bdsk-url-1 = {https://doi.org/10.1063/1.1508370}}
+
 @article{Li_2019a,
 	abstract = { The driven similarity renormalization group (DSRG) provides an alternative way to address the intruder state problem in quantum chemistry. In this review, we discuss recent developments of multireference methods based on the DSRG. We provide a pedagogical introduction to the DSRG and its various extensions and discuss its formal properties in great detail. In addition, we report several illustrative applications of the DSRG to molecular systems. },
 	author = {Li, Chenyang and Evangelista, Francesco A.},
@@ -5368,15 +5421,12 @@
 @article{Wilhelm_2018,
 	author = {Wilhelm, Jan and Golze, Dorothea and Talirz, Leopold and Hutter, J{\"u}rg and Pignedoli, Carlo A.},
 	date-added = {2020-05-18 21:40:28 +0200},
-	date-modified = {2020-05-18 21:40:28 +0200},
+	date-modified = {2022-02-21 16:46:11 +0100},
 	doi = {10.1021/acs.jpclett.7b02740},
-	eprint = {https://doi.org/10.1021/acs.jpclett.7b02740},
-	journal = {The Journal of Physical Chemistry Letters},
-	note = {PMID: 29280376},
+	journal = {J. Phys. Chem. Lett.},
 	number = {2},
 	pages = {306-312},
 	title = {Toward GW Calculations on Thousands of Atoms},
-	url = {https://doi.org/10.1021/acs.jpclett.7b02740},
 	volume = {9},
 	year = {2018},
 	bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.7b02740}}

Manuscript/ufGW.tex

@@ -189,7 +189,7 @@ Moreover, we consider a restricted Hartree-Fock (HF) starting point but it can b
 Throughout this article, $p$ and $q$ are general (spatial) orbitals, $i$, $j$, $k$, and $l$ denotes occupied orbitals, $a$, $b$, $c$, and $d$ are (unoccupied) virtual orbitals, while $m$ labels single excitations $i \to a$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Downfold: The non-linear $GW$ problem}
+\section{Downfolding: The non-linear $GW$ problem}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 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
@@ -201,10 +201,9 @@ where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of
 \begin{equation}
 \label{eq:SigC}
 	\SigC{p}(\omega) 
-	= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - \ii \eta}
-	+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + \ii \eta}
+	= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA}}
+	+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}}
 \end{equation}
-and $\eta$ is a positive infinitesimal that is set to zero in the following.
 Within the Tamm-Dancoff approximation (that we enforce here for the sake of simplicity), the screened two-electron integrals are given by
 \begin{equation}
 	\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
@@ -228,7 +227,7 @@ As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW
 \label{eq:Z}
 	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) $\eps{p}{\GW} \equiv \eps{p,s=0}{\GW}$ has a large weight $Z_{p} \equiv Z_{p,s=0}$.
+In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{\GW}$ has a large weight $Z_{p}$.
 Note that we have the following important conservation rules \cite{Martin_1959,Baym_1961,Baym_1962}
 \begin{align}
 	\sum_{s} Z_{p,s} & = 1
@@ -241,7 +240,7 @@ which physically shows that the mean-field solution of unit weight is ``scattere
 \section{Upfolding: the linear $GW$ problem}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The non-linear quasiparticle equation \eqref{eq:qp_eq} can be \textit{exactly} transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
-are upfolded from the 1h and 1p sectors. \cite{Bintrim_2021a}
+are upfolded from the 1h and 1p sectors. \cite{Bintrim_2021a,Riva_2022}
 For each orbital $p$, this yields a linear eigenvalue problem of the form
 \begin{equation}
 	\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
@@ -295,8 +294,8 @@ where $\bI$ is the identity matrix.
 One can see this downfolding process as the construction of a frequency-dependent effective Hamiltonian where the internal space is composed by a single determinant of the 1h or 1p sector and the external (or outer) space by all the 2h1p and 2p1h configurations. \cite{Dvorak_2019a,Dvorak_2019b,Bintrim_2021a} 
 The main mathematical difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the internal and external components of the eigenvectors associated with each quasiparticle and satellite, and not only their projection in the reference space as shown by Eq.~\eqref{eq:Z_proj}.
 
-The element $\eps{p}{\HF}$ of $\bH^{(p)}$ [see Eq.~\eqref{eq:Hp}] corresponds to the relative energy of the $(\Ne\pm1)$-electron reference determinant (compared to the $\Ne$-electron HF determinant) while the eigenvalues of the blocks $\bC{}{\text{2h1p}}$ and $\bC{}{\text{2p1h}}$ provide an estimate of the relative energy of the 2h1p and 2p1h determinants.
-In some situations, one of these determinants from the external space may become of similar energy than the reference determinant.
+The element $\eps{p}{\HF}$ of $\bH^{(p)}$ [see Eq.~\eqref{eq:Hp}] corresponds to the (approximate) relative energy of the $(\Ne\pm1)$-electron reference determinant (compared to the $\Ne$-electron HF determinant) while the eigenvalues of the blocks $\bC{}{\text{2h1p}}$ and $\bC{}{\text{2p1h}}$, which are $\eps{i}{\HF} - \Om{m}{\RPA}$ and $\eps{a}{\HF} + \Om{m}{\RPA}$ respectively, provide an estimate of the relative energy of the 2h1p and 2p1h determinants.
+In some situations, one of these determinants from the external space may become of similar energy than the reference determinant, resulting in a vanishing denominator in the self-energy \eqref{eq:SigC}.
 Hence, these two diabatic electronic configurations may cross and form an avoided crossing, and this outer-space determinant may be labeled as an intruder state.
 As we shall see below, discontinuities, which are ubiquitous in molecular systems, arise in such scenarios.
 
@@ -306,6 +305,7 @@ As we shall see below, discontinuities, which are ubiquitous in molecular system
 
 In order to illustrate the appearance and the origin of these multiple solutions, we consider the hydrogen molecule in the 6-31G basis set which corresponds to a two-electron system with four spatial orbitals (one occupied and three virtuals).
 This example was already considered in our previous work \cite{Veril_2018} but here we provide further insights on the origin of the appearances of these discontinuities.
+The downfolded and upfolded {\GOWO} schemes have been implemented in the electronic structure package QuAcK \cite{QuAcK} which is freely available at \url{https://github.com/pfloos/QuAcK}.
 We denote as $\ket*{1\Bar{1}}$ the $\Ne$-electron ground-state determinant where the orbital 1 is occupied by one spin-up and one spin-down electron.
 Similarly notations will be employed for the $(\Ne\pm1)$-electron configurations.
 
@@ -313,11 +313,13 @@ Similarly notations will be employed for the $(\Ne\pm1)$-electron configurations
 %	FIGURE 1
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure*}
-%	\includegraphics[width=\linewidth]{fig1a}
-%	\includegraphics[width=\linewidth]{fig1b}
+	\includegraphics[width=0.47\linewidth]{fig1a}
+	\hspace{0.05\linewidth}
+	\includegraphics[width=0.47\linewidth]{fig1b}
 	\caption{
 	\label{fig:H2}
-	Quasiparticle energies (left), correlation part of the self-energy (center) and renormalization factor (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) for various orbitals of \ce{H2} at the {\GOWO}@HF/6-31G level.
+	Selection of quasiparticle and satellite energies $\eps{p,s}{\GW}$ (solid lines) and their renormalization factor $Z_{p,s}$ (dashed lines) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) for the HOMO$+2$ ($p=3$) and HOMO$+3$ ($p=4$) orbitals of \ce{H2} at the {\GOWO}@HF/6-31G level.
+	The quasiparticle solution (which corresponds to the solution with the largest weight) is represented as a thicker line.
 }
 \end{figure*}	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -352,14 +354,14 @@ From a general perspective, a regularized $GW$ self-energy reads
 where various choices for the ``regularizer'' $f_\eta$ are possible.
 The main purpose of $f_\eta$ is to ensure that $\rSigC{p}(\omega;\eta)$ remains finite even if one of the denominators goes to zero.
 
-The most common way to regularize $\Sigma$ is by increasing the value of $\eta$ in Eq.~\eqref{eq:SigC}, which consists in the simple regularizer $f_\eta(\Delta) = (\Delta \pm \eta)^{-1}$, a strategy somehow related to the imaginary shift used in multiconfigurational perturbation theory. \cite{Forsberg_1997}
+The most common and well=established way of regularizing $\Sigma$ is via the simple regularizer $f_\eta(\Delta) = (\Delta \pm \eta)^{-1}$, a strategy somehow related to the imaginary shift used in multiconfigurational perturbation theory. \cite{Forsberg_1997}
 Other choices are legitimate like the regularizers considered by Lee \textit{et al.} within second-order M{\o}ller-Plesset theory. \cite{Lee_2018a}
 
-Our investigation has shown that the following regularizer
+Our investigations have shown that the following regularizer
 \begin{equation}
 	f_\eta(\Delta) = \frac{1-e^{-2\Delta^2/\eta^2}}{\Delta}
 \end{equation}
-derived from the (second-order) perturbative analysis of the similarity renormalization group equations \cite{Evangelista_2014} is particularly convenient and effective in the present context.
+derived from the (second-order) perturbative analysis of the similarity renormalization group equations \cite{Wegner_1994,Glazek_1994,White_2002,Evangelista_2014} is particularly convenient and effective in the present context.
 Increasing $\eta$ gradually integrates out states with denominators $\Delta$ larger than $\eta$ while the states with $\Delta \ll \eta$ are not decoupled from the reference space (which is exactly our purpose), hence avoiding intruder state problems. \cite{Li_2019a}
 Of course, by construction, we have
 \begin{equation}
@@ -381,8 +383,6 @@ Of course, by construction, we have
 
 Figure \ref{fig:H2_reg} evidences how the regularization of the $GW$ self-energy diabatically linked the two solutions to get rid of the discontinuities.
 
-
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Concluding remarks}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -391,9 +391,16 @@ Figure \ref{fig:H2_reg} evidences how the regularization of the $GW$ self-energy
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{
+The authors thank Pina Romaniello for insightful discussions.
 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).}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Data availability statement}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The data that supports the findings of this study are available within the article and its supplementary material.
+
+
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 \bibliography{ufGW}
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