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Manuscript/ufGW.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-02-21 16:46:45 +0100
%% Created for Pierre-Francois Loos at 2022-02-21 21:38:32 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Shee_2021,
author = {Shee, James and Loipersberger, Matthias and Rettig, Adam and Lee, Joonho and Head-Gordon, Martin},
date-added = {2022-02-21 21:37:57 +0100},
date-modified = {2022-02-21 21:38:20 +0100},
doi = {10.1021/acs.jpclett.1c03468},
journal = {J. Phys. Chem. Lett.},
number = {50},
pages = {12084-12097},
title = {Regularized Second-Order M{\o}ller--Plesset Theory: A More Accurate Alternative to Conventional MP2 for Noncovalent Interactions and Transition Metal Thermochemistry for the Same Computational Cost},
volume = {12},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.1c03468}}
@article{Forster_2021,
abstract = {Low-order scaling GW implementations for molecules are usually restricted to approximations with diagonal self-energy. Here, we present an all-electron implementation of quasiparticle self-consistent GW for molecular systems. We use an efficient algorithm for the evaluation of the self-energy in imaginary time, from which a static non-local exchange-correlation potential is calculated via analytical continuation. By using a direct inversion of iterative subspace method, fast and stable convergence is achieved for almost all molecules in the GW100 database. Exceptions are systems which are associated with a breakdown of the single quasiparticle picture in the valence region. The implementation is proven to be starting point independent and good agreement of QP energies with other codes is observed. We demonstrate the computational efficiency of the new implementation by calculating the quasiparticle spectrum of a DNA oligomer with 1,220 electrons using a basis of 6,300 atomic orbitals in less than 4 days on a single compute node with 16 cores. We use then our implementation to study the dependence of quasiparticle energies of DNA oligomers consisting of adenine-thymine pairs on the oligomer size. The first ionization potential in vacuum decreases by nearly 1 electron volt and the electron affinity increases by 0.4 eV going from the smallest to the largest considered oligomer. This shows that the DNA environment stabilizes the hole/electron resulting from photoexcitation/photoattachment. Upon inclusion of the aqueous environment via a polarizable continuum model, the differences between the ionization potentials reduce to 130 meV, demonstrating that the solvent effectively compensates for the stabilizing effect of the DNA environment. The electron affinities of the different oligomers are almost identical in the aqueous environment.},
author = {F{\"o}rster, Arno and Visscher, Lucas},
date-added = {2022-02-21 21:12:20 +0100},
date-modified = {2022-02-21 21:12:37 +0100},
doi = {10.3389/fchem.2021.736591},
journal = {Front. Chem.},
title = {Low-Order Scaling Quasiparticle Self-Consistent GW for Molecules},
volume = {9},
year = {2021},
bdsk-url-1 = {https://www.frontiersin.org/article/10.3389/fchem.2021.736591},
bdsk-url-2 = {https://doi.org/10.3389/fchem.2021.736591}}
@misc{Riva_2022,
archiveprefix = {arXiv},
author = {Gabriele Riva and Timoth{\'e}e Audinet and Matthieu Vladaj and Pina Romaniello and J. Arjan Berger},

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Manuscript/ufGW.tex
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\begin{document}
\title{Unphysical Discontinuities in $GW$ Methods and the Role of Intruder States}
\title{Unphysical Discontinuities, Intruder States and Regularization in $GW$ Methods}
\author{Enzo \surname{Monino}}
\affiliation{\LCPQ}
@ -139,8 +139,8 @@ A simple and efficient regularization procedure is proposed to avoid such issues
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 growing \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}
The $GW$ approximation of many-body perturbation theory, \cite{Hedin_1965,Martin_2016} allows to compute accurate charged excitation (\ie, ionization potentials, electron affinities and fundamental gaps) in solids and molecules. \cite{Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019}
Its popularity in the molecular electronic structure community is rapidly growing \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 computational cost \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2021} and somehow surprising accuracy for weakly correlated systems. \cite{Korbel_2014,vanSetten_2015,Caruso_2016,Hung_2017,vanSetten_2018,Bruneval_2021}
The idea behind the $GW$ approximation is to recast the many-body problem into a set of non-linear one-body equations. The introduction of the self-energy $\Sigma$ links the non-interacting Green's function $G_0$ to its fully-interacting version $G$ via the following Dyson equation:
\begin{equation}
@ -176,7 +176,7 @@ These issues were discovered in Ref.~\onlinecite{Loos_2018b} while studying a mo
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, \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}
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}
In addition to obvious irregularities on potential energy surfaces that 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.
@ -185,8 +185,8 @@ In the present article, via an upfolding process of the non-linear $GW$ equation
Inspired by regularized electronic structure theories, \cite{Lee_2018a,Evangelista_2014b} these new insights allow us to propose a cheap and efficient regularization scheme in order to avoid these issues.
Here, we consider the one-shot {\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}
Moreover, we consider a restricted Hartree-Fock (HF) starting point but it can be straightforwardly extended to a Kohn-Sham (KS) starting point.
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 $i \to a$.
Moreover, we consider a restricted Hartree-Fock (HF) starting point but it can be straightforwardly extended to a Kohn-Sham starting point.
Throughout this article, $p$ and $q$ are general (spatial) orbitals, $i$, $j$, $k$, and $l$ denotes occupied orbitals, $a$, $b$, $c$, and $d$ are vacant orbitals, while $m$ labels single excitations $i \to a$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Downfolding: The non-linear $GW$ problem}
@ -234,7 +234,7 @@ Note that we have the following important conservation rules \cite{Martin_1959,B
&
\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF}
\end{align}
which physically shows that the mean-field solution of unit weight is ``scattered'' by the effect of correlation in many solutions with smaller weights.
which physically shows that the mean-field solution of unit weight is ``scattered'' by the effect of correlation in many solutions of smaller weights.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Upfolding: the linear $GW$ problem}
@ -295,7 +295,7 @@ One can see this downfolding process as the construction of a frequency-dependen
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 (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}.
In some situations, one (or several) 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.
@ -325,17 +325,16 @@ Similarly notations will be employed for the $(\Ne\pm1)$-electron configurations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Figure \ref{fig:H2} shows the evolution of the quasiparticle energy, energetically close-by satellites and their corresponding weights at the {\GOWO}@HF level as a function on the internuclear distance $\RHH$.
One can easily diagnose two problematic regions showing obvious discontinuities around $\RHH = \SI{0.5}{\angstrom}$ for the HOMO$+3$ ($p = 4$) and $\RHH = \SI{1.1}{\angstrom}$ for the HOMO$+2$ ($p = 3$).
One can easily diagnose two problematic regions showing obvious discontinuities around $\RHH = \SI{1.2}{\angstrom}$ for the HOMO$+2$ ($p = 3$) and $\RHH = \SI{0.5}{\angstrom}$ for the HOMO$+3$ ($p = 4$).
Let us first look more closely at the region around $\RHH = \SI{1.1}{\angstrom}$.
As one can see, an avoided crossing is formed between two $(\Ne+1)$-electron.
Inspection of their corresponding eigenvectors reveals that the determinants involved are the reference 1p determinant $\ket*{1\Bar{1}3}$ and an excited $(\Ne+1)$-electron of configuration $\ket*{12\Bar{2}}$.
Let us first look more closely at the region around $\RHH = \SI{1.2}{\angstrom}$.
As one can see, an avoided crossing is formed between two solutions of the quasiparticle equation.
Inspection of their corresponding eigenvectors reveals that the $(\Ne+1)$-electron determinants involved are the reference 1p determinant $\ket*{1\Bar{1}3}$ and an excited $(\Ne+1)$-electron of configuration $\ket*{12\Bar{2}}$ which becomes lower in energy than the reference determinant for $\RHH > \SI{1.2}{\angstrom}$.
The algorithm diabatically follows the reference determinant $\ket*{1\Bar{1}3}$, while it should be adiabatically switching to the $\ket*{12\Bar{2}}$ determinant which becomes lower in energy for $\RHH > \SI{1.1}{\angstrom}$.
This is not the $\Ne$-electron situation where one has to check if it is multireference, but for the $\Ne\pm1$-electron situations.
As similar scenario is at the play in the region around $\RHH = \SI{0.5}{\angstrom}$ but it now involves 3 electronic configurations: the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration $\ket*{1\Bar{?}?}$ and $\ket*{1\Bar{?}?}$.
These forms two avoided crossings in rapid successions, which involves two discontinuties in the energy surface.
These forms two avoided crossings in rapid successions, which involves two discontinuities in the energy surface.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introducing regularized $GW$ methods}
@ -354,15 +353,15 @@ 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 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}
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 Head-Gordon and coworkers within second-order M{\o}ller-Plesset theory. \cite{Lee_2018a,Shee_2021}
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{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}
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, hence avoiding intruder state problems. \cite{Li_2019a}
Of course, by construction, we have
\begin{equation}
\lim_{\eta\to0} \rSigC{p}(\omega;\eta) = \SigC{p}(\omega)
@ -398,7 +397,7 @@ This project has received funding from the European Research Council (ERC) under
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that supports the findings of this study are available within the article and its supplementary material.
The data that supports the findings of this study are available within the article.% and its supplementary material.
%%%%%%%%%%%%%%%%%%%%%%%%