regularizer

Manuscript/ufGW.bib

@@ -1,13 +1,42 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2022-02-21 11:30:33 +0100 
+%% Created for Pierre-Francois Loos at 2022-02-21 14:28:44 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@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.},
+	date-added = {2022-02-21 14:27:55 +0100},
+	date-modified = {2022-02-21 14:28:27 +0100},
+	doi = {10.1146/annurev-physchem-042018-052416},
+	journal = {Annu. Rev. Phys. Chem.},
+	number = {1},
+	pages = {245-273},
+	title = {Multireference Theories of Electron Correlation Based on the Driven Similarity Renormalization Group},
+	volume = {70},
+	year = {2019},
+	bdsk-url-1 = {https://doi.org/10.1146/annurev-physchem-042018-052416}}
+
+@article{Forsberg_1997,
+	abstract = {In multiconfigurational perturbation theory, so-called intruders may cause singularities in the potential energy functions, at geometries where an energy denominator becomes zero. When the singularities are weak, they may be successfully removed by level shift techniques. When applied to excited states, a small shift merely moves the singularity. A large shift may cause new divergencies, and too large shifts are unacceptable since the potential function is affected in regions further away from the singularities. This Letter presents an alternative which may be regarded as an imaginary shift. The singularities are not moved, but disappear completely. They are replaced by a small distortion of the potential function. Applications to the N2 ground state, its A3/gEu+ state, and the Cr2 ground state show that the distortion caused by this procedure is small.},
+	author = {Niclas Forsberg and Per-{\AA}ke Malmqvist},
+	date-added = {2022-02-21 14:05:35 +0100},
+	date-modified = {2022-02-21 14:05:55 +0100},
+	doi = {https://doi.org/10.1016/S0009-2614(97)00669-6},
+	journal = {Chem. Phys. Lett.},
+	number = {1},
+	pages = {196-204},
+	title = {Multiconfiguration perturbation theory with imaginary level shift},
+	volume = {274},
+	year = {1997},
+	bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0009261497006696},
+	bdsk-url-2 = {https://doi.org/10.1016/S0009-2614(97)00669-6}}
+
 @article{Golze_2018,
 	author = {Golze, Dorothea and Wilhelm, Jan and van Setten, Michiel J. and Rinke, Patrick},
 	date-added = {2022-02-21 11:30:10 +0100},

Manuscript/ufGW.tex

@@ -108,7 +108,7 @@
 \newcommand{\EB}{E_B}
 
 \newcommand{\RHH}{R_{\ce{H-H}}}
-
+\newcommand{\ii}{\mathrm{i}}
 
 % addresses
 \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
@@ -171,12 +171,14 @@ Electron correlation is then explicitly incorporated into one-body quantities vi
 %\end{subequations}
 %where $v$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\boldsymbol{r}_1,t_1)$.
 
-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. 
+In recent studies, \cite{Loos_2018b,Veril_2018,Loos_2020e,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 showed 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}
-Multiple solution issues in $GW$ appears frequently, especially for orbitals that are energetically far from the Fermi level, such as in core ionized states. \cite{Golze_2018,Golze_2020}
+Multiple solution issues in $GW$ appears frequently, \cite{vanSetten_2015,Maggio_2017,Duchemin_2020} especially for orbitals that are energetically far from the Fermi level, such as in core ionized states. \cite{Golze_2018,Golze_2020}
 
-It was shown that these problems could be alleviated by using a static Coulomb-hole plus screened-exchange (COHSEX) \cite{Hedin_1965,Hybertsen_1986,Hedin_1999,Bruneval_2006} self-energy \cite{Berger_2021} or by considering a fully self-consistent scheme. \cite{DiSabatino_2021}
+In addition to obvious irregularities on potential energy surfaces which hampers the accurate determination of properties such as equilibrium bond lengths and harmonic vibrational frequencies, \cite{Loos_2020e,Berger_2021} one direct consequence of these discontinuities is the difficulty to converge (partially) self-consistent $GW$ calculations as the self-consistent procedure jumps erratically from one solution to the other even if convergence accelerator techniques such as DIIS are employed. \cite{Pulay_1980,Pulay_1982,Veril_2018}
+
+It was shown that these problems can be tamed by using a static Coulomb-hole plus screened-exchange (COHSEX) \cite{Hedin_1965,Hybertsen_1986,Hedin_1999,Bruneval_2006} self-energy \cite{Berger_2021} or by considering a fully self-consistent $GW$ scheme, \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 solution but also the satellites at each iteration. \cite{DiSabatino_2021}
 However, none of these solutions is completely satisfying as a static approximation of the self-energy can induce significant loss in accuracy and fully self-consistent calculations can be quite challenging in terms of implementation and cost.
 
 In the present article, via an upfolding process of the non-linear $GW$ equation, \cite{Bintrim_2021a} we provide further physical insights into the origin of these discontinuities by highlighting, in particular, the role of intruder states.
@@ -195,13 +197,14 @@ Within the {\GOWO} approximation, in order to obtain the quasiparticle energies
 \label{eq:qp_eq}
 	\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
 \end{equation}
-where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy
+where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy is constituted by a hole (h) and a particle (p) term, as follows
 \begin{equation}
+\label{eq:SigC}
 	\SigC{p}(\omega) 
-	= \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}}
+	= \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}
 \end{equation}
-is constituted by a hole (h) and a particle (p) term.
+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
@@ -303,7 +306,7 @@ 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.
-We denote as $\ket*{1\Bar{1}}$ the $\Ne$-electron ground-state determinant where the orbital 1 is occupied by one spin-up and spin-down electron.
+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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -335,38 +338,56 @@ These forms two avoided crossings in rapid successions, which involves two disco
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Introducing regularized $GW$ methods}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-One direct consequences of such irregularities is the difficulty to converge (partially) self-consistent $GW$ methods as, depending on the iteration, the self-consistent procedure will jump erratically from one solution to the other (even if DIIS is employed).
+One way to alleviate these issues and to massively improve the convergence properties of self-consistent $GW$ calculations is to resort to a regularization of the self-energy without altering too much the quasiparticle energies.
 
-One way to hamper such issues and to massively improve convergence is to resort to regularization of the $GW$ self-energy.
-Of course, the way of regularizing the self-energy is not unique but here we consider three different ways directly imported from MP2 theory.
-We show that it removes discontinuities, speed up convergence without altering the quasiparticle energy.
-
-The regularized self-energy is 
+From a general perspective, a regularized $GW$ self-energy reads 
 \begin{equation}
 \begin{split}
-	\rSigC{p}(\omega;\kappa) 
-	& = \sum_{im} 2\ERI{pi}{m}^2 f_\kappa(\omega - \eps{i}{\HF} + \Om{m}{\RPA})
+	\rSigC{p}(\omega;\eta) 
+	& = \sum_{im} 2\ERI{pi}{m}^2 f_\eta(\omega - \eps{i}{\HF} + \Om{m}{\RPA})
 	\\
-	& + \sum_{am} 2\ERI{pa}{m}^2 f_\kappa(\omega - \eps{a}{\HF} - \Om{m}{\RPA})
+	& + \sum_{am} 2\ERI{pa}{m}^2 f_\eta(\omega - \eps{a}{\HF} - \Om{m}{\RPA})
 \end{split}
 \end{equation}
-where various choices for the ``regularizer'' $f_\kappa$ are possible.
-Our investigation have shown that
+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}
+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
 \begin{equation}
-	f_\kappa(\Delta) = \frac{1-e^{-2\kappa\Delta^2}}{\Delta}
+	f_\eta(\Delta) = \frac{1-e^{-2\Delta^2/\eta^2}}{\Delta}
 \end{equation}
-is a very convenient form and has been derived from the flow equation within driven-similarity renormalization group \cite{Evangelista_2014}
+derived from the (second-order) perturbative analysis of the similarity renormalization group equations \cite{Evangelista_2014,Li_2019a} is particularly convenient and effective in the present context.
+%Decreasing $\eta$ gradually decouples states with small denominator in the self-energy (which is exactly our purpose).
 Of course, by construction, we have
 \begin{equation}
-	\lim_{s\to\infty} \rSigC{p}(\omega;s) = \SigC{p}(\omega) 
+	\lim_{\eta\to0} \rSigC{p}(\omega;\eta) = \SigC{p}(\omega) 
 \end{equation}
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%	FIGURE 2
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{figure*}
+%	\includegraphics[width=\linewidth]{fig1a}
+%	\includegraphics[width=\linewidth]{fig1b}
+	\caption{
+	\label{fig:H2_reg}
+	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.
+}
+\end{figure*}	
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+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}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-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.
+
+%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{