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Cover_Letter/CoverLetter.tex
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@ -22,7 +22,7 @@ This contribution has never been submitted in total nor in parts to any other jo
In the present contribution, we apply the similarity renormalization group (SRG) approach to the well-known $GW$ approximation of many-body perturbation theory. In the present contribution, we apply the similarity renormalization group (SRG) approach to the well-known $GW$ approximation of many-body perturbation theory.
We show that the SRG transformation allows us to derive, from first principles, a new static and hermitian expression for the self-energy that can be directly employed in self-consistent $GW$ calculations. We show that the SRG transformation allows us to derive, from first principles, a new static and hermitian expression for the self-energy that can be directly employed in self-consistent $GW$ calculations.
As shown on a large set of molecules, the resulting SRG-based regularized self-energy significantly accelerates the convergence of $GW$ calculations and slightly improves the overall accuracy. As shown on a large set of molecules, the resulting SRG-based regularized self-energy significantly accelerates the convergence of $GW$ calculations and improves the overall accuracy.
We hope that these new technical developments will broaden the applicability of Green’s function methods in the molecular electronic structure community and beyond. We hope that these new technical developments will broaden the applicability of Green’s function methods in the molecular electronic structure community and beyond.
Because of the novelty of this work and its potential impact in quantum chemistry and condensed matter physics, we expect it to be of interest to a wide audience within the chemistry and physics communities. Because of the novelty of this work and its potential impact in quantum chemistry and condensed matter physics, we expect it to be of interest to a wide audience within the chemistry and physics communities.

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Manuscript/SRGGW.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2023-03-10 09:19:55 +0100 %% Created for Pierre-Francois Loos at 2023-03-10 11:18:15 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -307,11 +307,10 @@
@inbook{Bickers_2004, @inbook{Bickers_2004,
abstract = {Self-consistent field techniques for the many-electron problem are examined using the modern formalism of functional methods. Baym-Kadanoff, or $\Phi$-derivable, approximations are introduced first. After a brief review of functional integration results, the connection between conventional mean-field theory and higher-order Baym-Kadanoff approximations is established through the concept of the action functional. The $\Phi$-derivability criterion for thermodynamic consistency is discussed, along with the calculation of free-energy derivatives. Parquet, or crossing-symmetric, approximations are introduced next. The principal advantages of the parquet approach and its relationship to Baym-Kadanoff theory are outlined. A linear eigenvalue equation is derived to study instabilities of the electronic normal state within Baym-Kadanoff or parquet theory. Finally, numerical techniques for the solution of self-consistent field approximations are reviewed, with particular emphasis on renormalization group methods for frequency and momentum space.}, abstract = {Self-consistent field techniques for the many-electron problem are examined using the modern formalism of functional methods. Baym-Kadanoff, or $\Phi$-derivable, approximations are introduced first. After a brief review of functional integration results, the connection between conventional mean-field theory and higher-order Baym-Kadanoff approximations is established through the concept of the action functional. The $\Phi$-derivability criterion for thermodynamic consistency is discussed, along with the calculation of free-energy derivatives. Parquet, or crossing-symmetric, approximations are introduced next. The principal advantages of the parquet approach and its relationship to Baym-Kadanoff theory are outlined. A linear eigenvalue equation is derived to study instabilities of the electronic normal state within Baym-Kadanoff or parquet theory. Finally, numerical techniques for the solution of self-consistent field approximations are reviewed, with particular emphasis on renormalization group methods for frequency and momentum space.},
address = {New York, NY},
author = {Bickers, N. E.}, author = {Bickers, N. E.},
booktitle = {Theoretical Methods for Strongly Correlated Electrons}, booktitle = {Theoretical Methods for Strongly Correlated Electrons},
date-added = {2023-01-30 14:19:12 +0100}, date-added = {2023-01-30 14:19:12 +0100},
date-modified = {2023-01-30 14:19:12 +0100}, date-modified = {2023-03-10 11:18:04 +0100},
doi = {10.1007/0-387-21717-7_6}, doi = {10.1007/0-387-21717-7_6},
editor = {S{\'e}n{\'e}chal, David and Tremblay, Andr{\'e}-Marie and Bourbonnais, Claude}, editor = {S{\'e}n{\'e}chal, David and Tremblay, Andr{\'e}-Marie and Bourbonnais, Claude},
isbn = {978-0-387-21717-8}, isbn = {978-0-387-21717-8},
@ -15327,11 +15326,12 @@
@article{Lee_2018a, @article{Lee_2018a,
author = {J. Lee and M. Head-Gordon}, author = {J. Lee and M. Head-Gordon},
date-added = {2018-09-01 12:02:40 +0200}, date-added = {2018-09-01 12:02:40 +0200},
date-modified = {2020-12-09 09:44:44 +0100}, date-modified = {2023-03-10 11:16:08 +0100},
doi = {10.1021/acs.jctc.8b00731}, doi = {10.1021/acs.jctc.8b00731},
journal = {J. Chem. Theory Comput.}, journal = {J. Chem. Theory Comput.},
pages = {ASAP article}, pages = {5203--5219},
title = {Regularized Orbital-Optimized Second-Order M{\o}ller--Plesset Perturbation Theory: A Reliable Fifth-Order-Scaling Electron Correlation Model with Orbital Energy Dependent Regularizers}, title = {Regularized Orbital-Optimized Second-Order M{\o}ller--Plesset Perturbation Theory: A Reliable Fifth-Order-Scaling Electron Correlation Model with Orbital Energy Dependent Regularizers},
volume = {14},
year = {2018}, year = {2018},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.8b00731}} bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.8b00731}}

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Manuscript/SRGGW.tex
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@ -1,5 +1,5 @@
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace,ulem} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace}
\usepackage[version=4]{mhchem} \usepackage[version=4]{mhchem}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -232,7 +232,7 @@ These solutions can be characterized by their spectral weight given by the renor
The solution with the largest weight $Z_p \equiv Z_{p,z=0}$ is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions). The solution with the largest weight $Z_p \equiv Z_{p,z=0}$ is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions).
However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined. However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined.
One obvious drawback of the one-shot scheme mentioned above is its starting point dependence. One obvious drawback of the one-shot scheme mentioned above is its starting-point dependence.
Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead. Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead.
As commonly done, one can even ``tune'' the starting point to obtain the best possible one-shot $GW$ quasiparticle energies. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016,Gallandi_2016} As commonly done, one can even ``tune'' the starting point to obtain the best possible one-shot $GW$ quasiparticle energies. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016,Gallandi_2016}
@ -290,8 +290,8 @@ This transformation can be performed continuously via a unitary matrix $\bU(s)$,
\label{eq:SRG_Ham} \label{eq:SRG_Ham}
\bH(s) = \bU(s) \, \bH \, \bU^\dag(s), \bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
\end{equation} \end{equation}
\ant{where the flow parameter $s$ controls the extent of the decoupling and is related to an energy cutoff $\Lambda=s^{-1/2}$. where the flow parameter $s$ controls the extent of the decoupling and is related to an energy cutoff $\Lambda=s^{-1/2}$.
For a given value of $s$, only states with energy denominators smaller than $1/\Lambda$ will be decoupled from the reference space, hence avoiding potential intruders.} For a given value of $s$, only states with energy difference (with respect to the reference space) greater than $\Lambda$ are decoupled from the reference space, hence avoiding potential intruders.
By definition, the boundary conditions are $\bH(s=0) = \bH$ [or $\bU(s=0) = \bI$] and $\bH^\text{od}(s=\infty) = \bO$. By definition, the boundary conditions are $\bH(s=0) = \bH$ [or $\bU(s=0) = \bI$] and $\bH^\text{od}(s=\infty) = \bO$.
An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.~\eqref{eq:SRG_Ham} with respect to $s$, yielding the flow equation An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.~\eqref{eq:SRG_Ham} with respect to $s$, yielding the flow equation
@ -518,11 +518,11 @@ The second-order renormalized quasiparticle equation is given by
% \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) ] \bX = \omega \bX, % \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) ] \bX = \omega \bX,
\qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx), \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
\end{equation} \end{equation}
with a regularized Fock matrix of the form with a renormalized Fock matrix of the form
\begin{equation} \begin{equation}
\widetilde{\bF}(s) = \bF^{(0)}+\bF^{(2)}(s), \widetilde{\bF}(s) = \bF^{(0)}+\bF^{(2)}(s),
\end{equation} \end{equation}
and a regularized dynamical self-energy and a renormalized dynamical self-energy
\begin{equation} \begin{equation}
\label{eq:srg_sigma} \label{eq:srg_sigma}
\widetilde{\bSig}(\omega; s) = \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger}, \widetilde{\bSig}(\omega; s) = \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger},
@ -570,14 +570,14 @@ For $s\to\infty$, it tends towards the following static limit
\label{eq:static_F2} \label{eq:static_F2}
\lim_{s\to\infty} \widetilde{\bF}(s) \lim_{s\to\infty} \widetilde{\bF}(s)
= \epsilon_p \delta_{pq} = \epsilon_p \delta_{pq}
+ \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu}. + \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu},
\end{equation} \end{equation}
while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie, while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
\begin{equation} \begin{equation}
\lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = \bO. \lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = \bO.
\end{equation} \end{equation}
Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$. Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
As illustrated in Fig.~\ref{fig:flow} (magenta curve), this transformation is done gradually starting from the states that have \ant{the largest denominators} in Eq.~\eqref{eq:static_F2}. As illustrated in Fig.~\ref{fig:flow} (magenta curve), this transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{pr}^{\nu} e^{-(\Delta_{pr}^{\nu})^2 s} \approx 0$, meaning that the state is decoupled from the 1h and 1p configurations, while, for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{pr}^{\nu}(s) \approx W_{pr}^{\nu}$, that is, the state remains coupled. For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{pr}^{\nu} e^{-(\Delta_{pr}^{\nu})^2 s} \approx 0$, meaning that the state is decoupled from the 1h and 1p configurations, while, for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{pr}^{\nu}(s) \approx W_{pr}^{\nu}$, that is, the state remains coupled.
@ -615,7 +615,7 @@ Indeed, the fact that SRG-qs$GW$ calculations do not always converge in the larg
Therefore, one should use a value of $s$ large enough to include as many states as possible but small enough to avoid intruder states. Therefore, one should use a value of $s$ large enough to include as many states as possible but small enough to avoid intruder states.
It is instructive to examine the functional form of both regularizing functions (see Fig.~\ref{fig:plot}). It is instructive to examine the functional form of both regularizing functions (see Fig.~\ref{fig:plot}).
These have been plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first-order Taylor expansion around $(x,y) = (0,0)$ of both functional forms is equal. These have been plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/(2\eta^2)$ such that the first-order Taylor expansion around $(x,y) = (0,0)$ of both functional forms is equal.
One can observe that the SRG-qs$GW$ surface is much smoother than its qs$GW$ counterpart. One can observe that the SRG-qs$GW$ surface is much smoother than its qs$GW$ counterpart.
This is due to the fact that the SRG-qs$GW$ functional at $\eta=0$, $f^{\SRGqsGW}(x,y;0)$, has fewer irregularities. This is due to the fact that the SRG-qs$GW$ functional at $\eta=0$, $f^{\SRGqsGW}(x,y;0)$, has fewer irregularities.
In fact, there is a single singularity at $x=y=0$. In fact, there is a single singularity at $x=y=0$.
@ -656,9 +656,9 @@ However, in order to perform black-box comparisons, these parameters have been f
The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations (where we eschew linearizing the quasiparticle equation) while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set. The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculations (where we eschew linearizing the quasiparticle equation) while, for the qs$GW$ calculations, $\eta$ has been chosen as the largest value where one successfully converges the 50 systems composing the test set.
The various $GW$-based sets of values are compared with a set of reference values computed at the $\Delta$CCSD(T) level with the same basis set. The various $GW$-based sets of values are compared with a set of reference values computed at the $\Delta$CCSD(T) level with the same basis set.
The $\Delta$CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using \textsc{gaussian 16} \cite{g16} with default parameters within the restricted and unrestricted HF formalism for the neutral and charged species, respectively. The $\Delta$CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using \textsc{gaussian 16} \cite{g16} (with default parameters) within the restricted and unrestricted formalism for the neutral and charged species, respectively.
The numerical data associated with this study are reported in the {\SupInf}. All the numerical data associated with this study are reported in the {\SupInf}.
%=================================================================% %=================================================================%
\section{Results} \section{Results}