saving work

Manuscript/ufGW-SI.tex

@@ -48,7 +48,7 @@
 
 \begin{document}
 
-\title{Supporting Information for ``Unphysical Discontinuities, Intruder States and Regularization in $GW$ Methods''}
+\title{Supplementary Material for ``Unphysical Discontinuities, Intruder States and Regularization in $GW$ Methods''}
 
 
 \author{Enzo \surname{Monino}}

Manuscript/ufGW.tex

@@ -20,6 +20,7 @@
 \newcommand{\titou}[1]{\textcolor{red}{#1}}
 \newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}}
 \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
+\newcommand{\SupMat}{\textcolor{blue}{supplementary material}}
 
 \newcommand{\mc}{\multicolumn}
 \newcommand{\fnm}{\footnotemark}
@@ -374,6 +375,7 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
 	\caption{
 	\label{fig:H2reg}
 	Difference between regularized and non-regularized quasiparticle energies $\reps{p}{\GW} - \eps{p}{\GW}$ computed with $\alert{\kappa} = 1$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level.
+	\alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ are reported as {\SupMat}.}
 }
 \end{figure}	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -386,6 +388,7 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
 	\caption{
 	\label{fig:F2}
 	Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set.
+	\alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ are reported as {\SupMat}.}
 }
 \end{figure}	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -419,23 +422,22 @@ The most common and well-established way of regularizing $\Sigma$ is via the sim
 	f_\eta(\Delta) = (\Delta \pm \ii \eta)^{-1}
 \end{equation}
 (with $\eta > 0$), \cite{vanSetten_2013,Bruneval_2016a,Martin_2016,Duchemin_2020} a strategy somehow related to the imaginary shift used in multiconfigurational perturbation theory. \cite{Forsberg_1997}
-\alert{This type of broadening is customary in solid-state calculations, hence such regularization is naturally captured in many codes. \cite{Martin_2016}}
+\alert{Not that this type of broadening is customary in solid-state calculations, hence such regularization is naturally captured in many codes. \cite{Martin_2016}}
 In practice, an empirical value of $\eta$ around \SI{100}{\milli\eV} is suggested.
-Other choices are legitimate like the regularizers considered by Head-Gordon and coworkers within orbital-optimized second-order M{\o}ller-Plesset theory, which have the specificity of being energy-dependent. \cite{Lee_2018a,Shee_2021}
+Other choices are legitimate like the regularizers considered by Head-Gordon and coworkers within orbital-optimized second-order M{\o}ller-Plesset theory \alert{(MP2)}, which have the specificity of being energy-dependent. \cite{Lee_2018a,Shee_2021}
 In this context, the real version of the simple energy-independent regularizer \eqref{eq:simple_reg} has been shown to damage thermochemistry performance and was abandoned. \cite{Stuck_2013,Rostam_2017}
 
 Our investigations have shown that the following energy-dependent regularizer
 \begin{equation}
 \label{eq:srg_reg}
-	f_\kappa(\Delta) = \frac{1-e^{-2\Delta^2/\kappa^2}}{\Delta}
+	f_{\alert{\kappa}}(\Delta) = \frac{1-e^{-2\Delta^2/\alert{\kappa}^2}}{\Delta}
 \end{equation}
 derived from the (second-order) perturbative analysis of the similarity renormalization group (SRG) equations \cite{Wegner_1994,Glazek_1994,White_2002} by Evangelista \cite{Evangelista_2014} is particularly convenient and effective for our purposes.
 Increasing $\alert{\kappa}$ gradually integrates out states with denominators $\Delta$ larger than $\alert{\kappa}$ while the states with $\Delta \ll \alert{\kappa}$ are not decoupled from the reference space, hence avoiding intruder state problems. \cite{Li_2019a}
 
-
-Figure \ref{fig:H2reg_zoom} compares the non-regularized and regularized quasiparticle energies in the two regions of interest for various $\eta$ values.
+Figure \ref{fig:H2reg_zoom} compares the non-regularized and regularized quasiparticle energies in the two regions of interest \alert{for various $\eta$ and $\kappa$ values.}
 It clearly shows how the regularization of the $GW$ self-energy diabatically linked the two solutions to get rid of the discontinuities.
-However, this diabatization is more or less accurate depending on the value of $\eta$ and the actual form of the regularizer.
+However, this diabatization is more or less accurate depending on \alert{(i) the actual form of the regularizer, and (ii) the value of $\eta$ or $\kappa$.}
 
 Let us first discuss the simple energy-independent regularizer given by Eq.~\eqref{eq:simple_reg} (top panels of Fig.~\ref{fig:H2reg_zoom}).
 Mathematically, in order to link smoothly two solutions, the value of $\eta$ has to be large enough so that the singularity lying in the complex plane at the avoided crossing is moved to the real axis (see Ref.~\onlinecite{Marie_2021} and references therein).
@@ -445,21 +447,25 @@ For example, around $\RHH = \SI{1.1}{\angstrom}$ (top-left), a value of \SI{0.1}
 Note also that $\eta = \SI{0.1}{\hartree}$ is significantly larger than the suggested value of \SI{100}{\milli\eV} and if one uses smaller $\eta$ values, the regularization is clearly inefficient.
 
 Let us now discuss the SRG-based energy-dependent regularizer provided in Eq.~\eqref{eq:srg_reg} (bottom panels of Fig.~\ref{fig:H2reg_zoom}).
-For $\eta = \SI{10}{\hartree}$, the value is clearly too large inducing a large difference between the two sets of quasiparticle energies (purple curves).
-For $\eta = \SI{0.1}{\hartree}$, we have the opposite scenario where $\eta$ is too small and some irregularities remain (green curves).
-We have found that $\eta = \SI{1.0}{\hartree}$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions. 
+For $\alert{\kappa} = \SI{10}{\hartree}$, the value is clearly too large inducing a large difference between the two sets of quasiparticle energies (purple curves).
+For $\alert{\kappa} = \SI{0.1}{\hartree}$, we have the opposite scenario where $\alert{\kappa}$ is too small and some irregularities remain (green curves).
+We have found that $\alert{\kappa} = \SI{1.0}{\hartree}$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions. 
 Moreover, this value performs well in all scenarios that we have encountered.
-However, it can be certainly refined for specific applications. \cite{Shee_2021}
+However, it can be certainly refined for specific applications. 
+\alert{For example, in the case of regularized MP2 theory (where one relies on a similar energy-dependent regularizer), a value of $\kappa = 1.1$ have been found to be optimal for noncovalent interactions and transition metal thermochemistry. \cite{Shee_2021}}
 
-To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized (computed at $\eta = \SI{1.0}{\hartree}$ with the SRG-based regularizer) and non-regularized quasiparticle energies as functions of $\RHH$ for each orbital.
+To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized (computed at $\alert{\kappa} = \SI{1.0}{\hartree}$ with the SRG-based regularizer) and non-regularized quasiparticle energies as functions of $\RHH$ for each orbital.
 The principal observation is that, in the absence of intruder states, the regularization induces an error below \SI{10}{\milli\eV} for the HOMO ($p = 1$) and LUMO ($p = 2$), which is practically viable.
 Of course, in the troublesome regions ($p = 3$ and $p = 4$), the correction brought by the regularization procedure is larger (as it should) but it has the undeniable advantage to provide smooth curves.
+\alert{Similar graphs for $\kappa = 0.1$ and $\kappa = 10$ [and the simple regularizer given in Eq.~\eqref{eq:simple_reg}] are reported as {\SupMat}, where one clearly sees that the larger the value of $\kappa$, the larger the difference between regularized and non-regularizer quasiparticle energies.}
 
 As a final example, we report in Fig.~\ref{fig:F2} the ground-state potential energy surface of the \ce{F2} molecule obtained at various levels of theory with the cc-pVDZ basis.
 In particular, we compute, with and without regularization, the total energy at the Bethe-Salpeter equation (BSE) level \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} within the adiabatic connection fluctuation dissipation formalism \cite{Maggio_2016,Holzer_2018b,Loos_2020e} following the same protocol detailed in Ref.~\onlinecite{Loos_2020e}.
-These results are compared to high-level coupled-cluster calculations \cite{Purvis_1982,Christiansen_1995b} extracted from the same work.
-As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' around the equilibrium bond length and it is clear that the regularization scheme (black curve computed with $\eta = 1$) allows to smooth it out without significantly altering the overall accuracy.
+These results are compared to high-level coupled-cluster \alert{(CC)} calculations extracted from the same work: \alert{CC with singles and doubles (CCSD) \cite{Purvis_1982} and the non-perturbative third-order approximate CC method (CC3). \cite{Christiansen_1995b}}
+As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' around the equilibrium bond length and it is clear that the regularization scheme (black curve computed with $\alert{\kappa} = 1$) allows to smooth it out without significantly altering the overall accuracy.
 Moreover, while it is extremely challenging to perform self-consistent $GW$ calculations without regularization, it is now straightforward to compute the BSE@ev$GW$@HF potential energy surface (gray curve). 
+\alert{For the sake of completeness, a similar graph for $\kappa = 10$ is reported as {\SupMat}.
+Interestingly, for this rather large value of $\kappa$, the smooth BSE@{\GOWO}@HF and BSE@ev$GW$@HF curves are superposed, and of very similar quality as CCSD.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Concluding remarks}
@@ -470,6 +476,12 @@ A simple and efficient regularization procedure inspired by the similarity renor
 Moreover, this regularization of the self-energy significantly speeds up the convergence of (partially) self-consistent $GW$ methods.
 We hope that these new physical insights and technical developments will broaden the applicability of Green's function methods in the molecular electronic structure community and beyond.
 
+%%%%%%%%%%%%%%%%%%%%%%
+\section*{Supplementary Material}
+\label{sec:supmat}
+%%%%%%%%%%%%%%%%%%%%%%
+\alert{Included in the {\SupMat} are the raw data associated with each figure as well as additional figures showing the effect of the regularizer and its parameter, with, in particular, the difference between non-regularized and regularized quasiparticle energies for \ce{H2}, and the ground-state potential energy surface of \ce{F2} around its equilibrium geometry.}
+
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{
 The authors thank Pina Romaniello, Fabien Bruneval, and Xavier Blase for insightful discussions.