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@ -49,33 +49,33 @@ Here comes the abstract.
\label{sec:intro} \label{sec:intro}
%=================================================================% %=================================================================%
One-body Green's functions provide a natural and elegant way to access the charged excitations energies of a physical system. \cite{Martin_2016,Golze_2019} One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{Martin_2016,Golze_2019}
The one-body non-linear Hedin's equations give a recipe to obtain the exact interacting one-body Green's function and therefore the exact ionization potentials and electron affinities. \cite{Hedin_1965} The non-linear Hedin's equations give a recipe to obtain the exact interacting one-body Green's function and therefore the exact ionization potentials and electron affinities. \cite{Hedin_1965}
Unfortunately, fully solving Hedin's equations is out of reach and one must resort to approximations. Unfortunately, fully solving Hedin's equations is out of reach and one must resort to approximations.
In particular, the $GW$ approximation, \cite{Hedin_1965} which has first been mainly used in the context of solids \cite{Strinati_1980,Strinati_1982,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely used for molecules as well \ant{ref?}, provides fairly accurate results for weakly correlated systems\cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2021} In particular, the $GW$ approximation, \cite{Hedin_1965} which has first been mainly used in the context of solids \cite{Strinati_1980,Strinati_1982,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely used for molecules as well \ant{ref?}, provides fairly accurate results for weakly correlated systems\cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2021}
The $GW$ approximation is an approximation for the self-energy $\Sigma$ which role is to relate the exact interacting Green's function $G$ to a non-interacting reference one $G_0$ through the Dyson equation The $GW$ method approximates the self-energy $\Sigma$ which relates the exact interacting Green's function $G$ to a non-interacting reference one $G_S$ through the Dyson equation
\begin{equation} \begin{equation}
\label{eq:dyson} \label{eq:dyson}
G = G_0 + G_0\Sigma G. G = G_S + G_S\Sigma G.
\end{equation} \end{equation}
The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken in account in the reference system. The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken into account in the reference system.
%Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation. %Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation.
Approximating $\Sigma$ as the first order term of its perturbation expansion with respect to the screened interaction $W$ gives the so-called $GW$ approximation. \cite{Hedin_1965, Martin_2016} Approximating $\Sigma$ as the first order term of its perturbation expansion with respect to the screened interaction $W$ gives the so-called $GW$ approximation. \cite{Hedin_1965, Martin_2016}
Alternatively one could choose to define $\Sigma$ as the $n$-th order expansion in terms of the bare Coulomb interaction leading to the GF($n$) class of approximations. \cite{Hirata_2015,Hirata_2017} Alternatively one could choose to define $\Sigma$ as the $n$-th order expansion in terms of the bare Coulomb interaction leading to the GF($n$) class of approximations. \cite{Hirata_2015,Hirata_2017}
The GF(2) approximation is also known as the second Born approximation. \ant{ref ?} The GF(2) approximation is also known as the second Born approximation. \ant{ref ?}
Despite a wide range of successes, many-body perturbation theory is not flawless. Despite a wide range of successes, many-body perturbation theory is not flawless.
It has been shown that a variety of physical quantities such as charged and neutral excitations energies or correlation and total energies computed within many-body perturbation theory exhibits some discontinuities. \cite{Veril_2018,Loos_2018b} It has been shown that a variety of physical quantities such as charged and neutral excitations energies or correlation and total energies computed within many-body perturbation theory exhibit some discontinuities. \cite{Veril_2018,Loos_2018b}
Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is thought to be valid. Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is thought to be valid.
These discontinuities are due to a transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022} These discontinuities are due to a transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022}
This is another occurrence of the infamous intruder-state problem. \cite{Roos_1995,Olsen_2000,Choe_2001} \ant{more ref} This is another occurrence of the infamous intruder-state problem. \cite{Roos_1995,Olsen_2000,Choe_2001} \ant{more ref}
In addition, systems for which two quasi-particle solutions have a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$. \cite{Forster_2021} In addition, systems for which two quasi-particle solutions have a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$. \cite{Forster_2021}
In a recent study, Monino and Loos showed that these discontinuities could be removed by introduction of a regularizer inspired by the similarity renormalisation group (SRG) in the quasi-particle equation. \cite{Monino_2022} In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasi-particle equation. \cite{Monino_2022}
Encouraged by this result, this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants. Encouraged by this result, this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants.
The SRG has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively. The SRG has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively.
This formalism has been been introduced in quantum chemistry by White \cite{White_2002} before being explored in more details by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,Li_2015, Li_2016, Li_2017, Li_2018, Li_2019a} This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more details by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,Li_2015, Li_2016, Li_2017, Li_2018, Li_2019a}
The SRG has also been successful in the context of nuclear theory, \cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016} see Ref.\onlinecite{Hergert_2016a} for a recent review in this field. \ant{Maybe search for recent papers of T. Duguet as well.} The SRG has also been successful in the context of nuclear theory, \cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016} see Ref.\onlinecite{Hergert_2016a} for a recent review in this field. \ant{Maybe search for recent papers of T. Duguet as well.}
The SRG transformation aims at decoupling a reference space from an external space while folding information about the coupling in the reference space. The SRG transformation aims at decoupling a reference space from an external space while folding information about the coupling in the reference space.
@ -83,12 +83,12 @@ This is often during such decoupling that intruder states appear. \ant{ref}
However, SRG is particularly well-suited to avoid them because the decoupling of each external configuration is inversely proportional to its energy difference with the reference space. However, SRG is particularly well-suited to avoid them because the decoupling of each external configuration is inversely proportional to its energy difference with the reference space.
Because intruder states have energies really close to the reference energies they will be the last ones decoupled. Because intruder states have energies really close to the reference energies they will be the last ones decoupled.
Therefore the SRG continuous transformation can be stopped once every external configurations except the intruder ones have been decoupled. Therefore the SRG continuous transformation can be stopped once every external configurations except the intruder ones have been decoupled.
Doing so, it gives a way to fold in information about the coupling in the reference space while avoiding intruder states. This provides a way to fold in information about the coupling in the reference space while avoiding intruder states.
The aim of this manuscript is to investigate whether SRG can treat the intruder-state problem in many-body perturbation theory as successfully as it has been in other fields. The aim of this manuscript is to investigate whether SRG can treat the intruder-state problem in many-body perturbation theory as successfully as it has been in other fields.
We begin by reviewing the $GW$ formalism in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}. We begin by reviewing the $GW$ formalism in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
Section~\ref{sec:theoretical} is concluded by a perturbative analysis of the SRG formalism applied to $GW$ (see Sec.~\ref{sec:srggw}). Section~\ref{sec:theoretical} is concluded by a perturbative analysis of the SRG formalism applied to $GW$ (see Sec.~\ref{sec:srggw}).
The computational details of our implementation are provided in Sec.~\ref{sec:comp_det} before turning to the results section. The computational details of our implementation are provided in Sec.~\ref{sec:comp_det} before turning to the results section.
This section starts by This section starts by
%=================================================================% %=================================================================%
@ -113,26 +113,26 @@ Note that $\bSig$ is dynamical, \ie it depends on both the eigenvalues $\epsilon
Because of this $\omega$ dependence, fully solving this equation is a rather complicated task, hence several approximate solving schemes has been developed. Because of this $\omega$ dependence, fully solving this equation is a rather complicated task, hence several approximate solving schemes has been developed.
The most popular one is probably the one-shot scheme, known as $G_0W_0$ if the self-energy is the $GW$ one, in which the off-diagonal elements of Eq.~(\ref{eq:quasipart_eq}) are neglected and the self-consistency is abandoned. The most popular one is probably the one-shot scheme, known as $G_0W_0$ if the self-energy is the $GW$ one, in which the off-diagonal elements of Eq.~(\ref{eq:quasipart_eq}) are neglected and the self-consistency is abandoned.
In this case, there are $K$ quasi-particle equations which read as In this case, there are $K$ quasi-particle equations that read
tr\begin{equation} tr\begin{equation}
\label{eq:G0W0} \label{eq:G0W0}
\epsilon_p^{\HF} + \Sigma_{p}(\omega) - \omega = 0, \epsilon_p^{\HF} + \Sigma_{p}(\omega) - \omega = 0,
\end{equation} \end{equation}
where $\Sigma_{p}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies. where $\Sigma_{p}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
The previous equation is non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$. The previous equation is non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$.
These solutions can be characterised by their spectral weight defined as the renormalisation factor $Z_{p,s}$ These solutions can be characterized by their spectral weight defined as the renormalization factor $Z_{p,s}$
\begin{equation} \begin{equation}
\label{eq:renorm_factor} \label{eq:renorm_factor}
0 \leq Z_{p,s} = \left[ 1 - \pdv{\Sigma_{p}(\omega)}{\omega}\bigg|_{\omega=\epsilon_{p,s}} \right]^{-1} \leq 1. 0 \leq Z_{p,s} = \left[ 1 - \pdv{\Sigma_{p}(\omega)}{\omega}\bigg|_{\omega=\epsilon_{p,s}} \right]^{-1} \leq 1.
\end{equation} \end{equation}
The solution with the largest weight is referred to as the quasi-particle solution while the others are known as satellites or shake-up solutions. The solution with the largest weight is referred to as the quasi-particle solution while the others are known as satellites or shake-up solutions.
However, in some cases Eq.~(\ref{eq:G0W0}) can have two (or more) solutions with similar weights and the quasi-particle solution is not well-defined. However, in some cases, Eq.~(\ref{eq:G0W0}) can have two (or more) solutions with similar weights and the quasi-particle solution is not well-defined.
In fact, these cases are related to the discontinuities and convergence problems discussed earlier because the additional solutions with large weights are the previously mentioned intruder states. In fact, these cases are related to the discontinuities and convergence problems discussed earlier because the additional solutions with large weights are the previously mentioned intruder states.
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence. One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
Indeed, in Eq.~(\ref{eq:G0W0}) we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example. Indeed, in Eq.~(\ref{eq:G0W0}) we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example.
Therefore, one could try to optimise the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016} Therefore, one could try to optimize the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
Alternatively, one could solve this equation self-consistently leading to the eigenvalue only self-consistent scheme. \cite{Shishkin_2006,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016} Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme. \cite{Shishkin_2006,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016}
To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~(\ref{eq:G0W0}) and then this equation is solved for $\omega$ again until convergence is reached. To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~(\ref{eq:G0W0}) and then this equation is solved for $\omega$ again until convergence is reached.
However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach. However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach.
Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals. \cite{Marom_2012} Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals. \cite{Marom_2012}
@ -140,7 +140,7 @@ Even if self-consistency has been reached, the starting point dependence has not
To update both the energies and the molecular orbitals, one needs to take into account the off-diagonal elements in Eq.~(\ref{eq:quasipart_eq}). To update both the energies and the molecular orbitals, one needs to take into account the off-diagonal elements in Eq.~(\ref{eq:quasipart_eq}).
To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$. To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$. The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$.
Various choice for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation Various choices for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
\begin{equation} \begin{equation}
\label{eq:sym_qsgw} \label{eq:sym_qsgw}
\Sigma_{pq}^\qs = \frac{1}{2}\Re\left(\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) \right). \Sigma_{pq}^\qs = \frac{1}{2}\Re\left(\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) \right).
@ -148,10 +148,10 @@ Various choice for $\bSig^\qs$ are possible but the most popular one is the foll
This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017} This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
One of the main results of this manuscript is the derivation from first principles of an alternative static Hermitian form, this will be done in the next section. One of the main results of this manuscript is the derivation from first principles of an alternative static Hermitian form, this will be done in the next section.
In this case as well self-consistency can be difficult to reach in cases where multiple solutions have large spectral weights. In this case, as well self-consistency can be difficult to reach in cases where multiple solutions have large spectral weights.
Multiple solutions arise due to the $\omega$ dependence of the self-energy. Multiple solutions arise due to the $\omega$ dependence of the self-energy.
Therefore, by suppressing this dependence the static qs approximation relies on the fact that there is one well-defined quasi-particle solution. Therefore, by suppressing this dependence the static qs approximation relies on the fact that there is one well-defined quasi-particle solution.
If it is not the case, the qs scheme will oscillates between the solutions with large weights. \cite{Forster_2021} If it is not the case, the qs scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
Convergence problems arise when a shake-up configuration has an energy similar to the associated quasi-particle solution, this satellite is the so-called intruder state. Convergence problems arise when a shake-up configuration has an energy similar to the associated quasi-particle solution, this satellite is the so-called intruder state.
The intruder state problem can be dealt with by introducing \textit{ad hoc} regularizers. The intruder state problem can be dealt with by introducing \textit{ad hoc} regularizers.
@ -163,7 +163,7 @@ This is the aim of this work.
Therefore if we apply it, the SRG would gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle. Therefore if we apply it, the SRG would gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
However, to do so one needs to identify the coupling terms in Eq.~(\ref{eq:quasipart_eq}), which is not straightforward. However, to do so one needs to identify the coupling terms in Eq.~(\ref{eq:quasipart_eq}), which is not straightforward.
The way around this problem is to transform Eq.~(\ref{eq:quasipart_eq}) to its upfolded version and the coupling terms will elegantly appear in the process. The way around this problem is to transform Eq.~(\ref{eq:quasipart_eq}) to its upfolded version and the coupling terms will elegantly appear in the process.
From now on, we will restrict ourselves to the $GW$ in the Tamm-Dancoff approximation (TDA) case but the same derivation could be done for the non-TDA $GW$ and GF(2) self-energies. From now on, we will restrict ourselves to the $GW$ in the Tamm-Dancoff approximation (TDA) case but the same derivation could be done for the non-TDA $GW$ and GF(2) self-energies.
The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively. The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively.
The upfolded $GW$ quasi-particle equation is the following The upfolded $GW$ quasi-particle equation is the following
\begin{equation} \begin{equation}
@ -237,18 +237,18 @@ with
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~(\ref{eq:GW_selfenergy}). $\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~(\ref{eq:GW_selfenergy}).
Equations~(\ref{eq:GWlin}) and~(\ref{eq:GWnonlin}) have exactly the same solutions but one is linear and the other not. Equations~(\ref{eq:GWlin}) and~(\ref{eq:GWnonlin}) have exactly the same solutions but one is linear and the other not.
The price to pay for this linearity is that the size of the matrix in the former equation is $\mathcal{O}(K^3)$ while it is $\mathcal{O}(K)$ in the latter one. The price to pay for this linearity is that the size of the matrix in the former equation is $\order{K^3}$ while it is $\order{K}$ in the latter one.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case). We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case).
As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $ V^\text{2p1h}$ are coupling the 1h and 1p configuration to the dressed 2h1p and 2p1h configurations. As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $ V^\text{2p1h}$ are coupling the 1h and 1p configuration to the dressed 2h1p and 2p1h configurations.
Therefore, these blocks will be the target of our SRG transformation but before going in more details we will review the SRG formalism. Therefore, these blocks will be the target of our SRG transformation but before going into more detail we will review the SRG formalism.
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
\subsection{The similarity renormalization group} \subsection{The similarity renormalization group}
\label{sec:srg} \label{sec:srg}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
The similarity renormalization group method aims at continuously transforming an Hamiltonian to a diagonal form, or more often to a block-diagonal form. The similarity renormalization group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
Therefore, the transformed Hamiltonian Therefore, the transformed Hamiltonian
\begin{equation} \begin{equation}
\label{eq:SRG_Ham} \label{eq:SRG_Ham}
@ -265,13 +265,13 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
\begin{equation} \begin{equation}
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s). \boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
\end{equation} \end{equation}
To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce approximation for $\boldsymbol{\eta}(s)$. To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce an approximation for $\boldsymbol{\eta}(s)$.
Before defining such an approximation, we need to define what are the blocks to suppress to obtain a block-diagonal Hamiltonian. Before defining such an approximation, we need to define what are the blocks to suppress to obtain a block-diagonal Hamiltonian.
The Hamiltonian is separated in two parts as The Hamiltonian is separated into two parts as
\begin{equation} \begin{equation}
\bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}}, \bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}},
\end{equation} \end{equation}
and by definition we have the following condition on $\bH^\text{od}$ and, by definition, we have the following condition on $\bH^\text{od}$
\begin{equation} \begin{equation}
\bH^\text{od}(s=\infty) = \boldsymbol{0}. \bH^\text{od}(s=\infty) = \boldsymbol{0}.
\end{equation} \end{equation}
@ -305,12 +305,12 @@ Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as w
Then, one can collect order by order the terms in Eq.~(\ref{eq:flowEquation}) and solve analytically the low-order differential equations. Then, one can collect order by order the terms in Eq.~(\ref{eq:flowEquation}) and solve analytically the low-order differential equations.
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
\subsection{Renormalised GW} \subsection{Renormalized GW}
\label{sec:srggw} \label{sec:srggw}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
Finally, the SRG formalism exposed above will be applied to $GW$. Finally, the SRG formalism exposed above will be applied to $GW$.
First, one needs to defined the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian. First, one needs to define the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal parts will be defined as As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal parts will be defined as
\begin{align} \begin{align}
\label{eq:diag_and_offdiag} \label{eq:diag_and_offdiag}
@ -348,15 +348,15 @@ Then, the aim of this section is to solve order by order the flow equation [see
\bV{}{\dagger} & \bO \notag \bV{}{\dagger} & \bO \notag
\end{pmatrix} \notag \end{pmatrix} \notag
\end{align} \end{align}
where we have defined the matrix $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness. where we have defined the matrices $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness.
Then, the perturbative expansions can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalised quasi-particle equation. Then, the perturbative expansions can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalized quasi-particle equation.
In particular, in this manuscript the focus will be on the second-order renormalized quasi-particle equation. In particular, in this manuscript the focus will be on the second-order renormalized quasi-particle equation.
%///////////////////////////% %///////////////////////////%
\subsubsection{Zero-th order matrix elements} \subsubsection{Zero-th order matrix elements}
%///////////////////////////% %///////////////////////////%
There is only one zero-th order term in the right hand side of the flow equation There is only one zeroth order term in the right-hand side of the flow equation
\begin{equation} \begin{equation}
\dv{\bH^{(0)}}{s} = \comm{\comm{\bHd{0}}{\bHod{0}}}{\bH^{(0)}}, \dv{\bH^{(0)}}{s} = \comm{\comm{\bHd{0}}{\bHod{0}}}{\bH^{(0)}},
\end{equation} \end{equation}
@ -382,7 +382,7 @@ Due to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, these equations ca
W_{p,(q,v)}^{(0)}(s) = W_{p,(q,v)}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s} W_{p,(q,v)}^{(0)}(s) = W_{p,(q,v)}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
\end{equation} \end{equation}
Due to the initial conditions $\bV^{(0)}(0) = \bO$, we have $\bW^{(0)}(s)=\bO$ and therefore $\bV^{(0)}(s)=\bO=\bV^{(0)}(0) $. Due to the initial conditions $\bV^{(0)}(0) = \bO$, we have $\bW^{(0)}(s)=\bO$ and therefore $\bV^{(0)}(s)=\bO=\bV^{(0)}(0) $.
The two first equations of the system are trivial and finally we have The two first equations of the system are trivial and finally, we have
\begin{equation} \begin{equation}
\bH^{(0)}(s) = \bH^{(0)}(0) \bH^{(0)}(s) = \bH^{(0)}(0)
\end{equation} \end{equation}
@ -421,7 +421,7 @@ Note the close similarity of the first-order element expressions with the ones o
\subsubsection{Second-order matrix elements} \subsubsection{Second-order matrix elements}
% ///////////////////////////% % ///////////////////////////%
The second-order renormalised quasi-particle equation is given by The second-order renormalized quasi-particle equation is given by
\begin{equation} \begin{equation}
\label{eq:GW_renorm} \label{eq:GW_renorm}
\left( \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) \right) \bX = \omega \bX \left( \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) \right) \bX = \omega \bX
@ -456,7 +456,7 @@ Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~(\ref{eq:GW_reno
Yet, both are closely related as they share the same diagonal terms. Yet, both are closely related as they share the same diagonal terms.
Also, note that the hermiticity is naturally enforced in the SRG static approximation as opposed to the symmetrized case. Also, note that the hermiticity is naturally enforced in the SRG static approximation as opposed to the symmetrized case.
However, as will be discussed in more details in the results section, the convergence of the qs$GW$ scheme using $\widetilde{\bF}(\infty)$ is very poor. However, as will be discussed in more detail in the results section, the convergence of the qs$GW$ scheme using $\widetilde{\bF}(\infty)$ is very poor.
This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero. This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
Indeed, in qs$GW$ calculation using the symmetrized static form, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable. \ant{ref fabien ?} Indeed, in qs$GW$ calculation using the symmetrized static form, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable. \ant{ref fabien ?}
Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
@ -485,10 +485,10 @@ In fact, the dynamic part after the change of variable is closely related to the
\label{sec:comp_det} \label{sec:comp_det}
% =================================================================% % =================================================================%
The two qs$GW$ variants considered in this work has been implemented in a in-house program. The two qs$GW$ variants considered in this work have been implemented in an in-house program.
The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016} The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
The geometry have been optimized at the CC3 level in the aug-cc-pvtz basis set without frozen core using the CFOUR program. The geometries have been optimized at the CC3 level in the aug-cc-pvtz basis set without frozen core using the CFOUR program.
The reference CCSD(T) IP energies have obtained using default parameters of Gaussian 16. The reference CCSD(T) IP energies have been obtained using default parameters of Gaussian 16.
This means that the cations used an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted formalism. This means that the cations used an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted formalism.
%=================================================================% %=================================================================%
@ -527,7 +527,7 @@ The $GW$ self-energy without TDA is the same as in Eq.~(\ref{eq:GW_selfenergy})
\label{eq:GWnonTDA_sERI} \label{eq:GWnonTDA_sERI}
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia}, W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
\end{equation} \end{equation}
where $\bX$ and $\bY$ are the matrix of eigenvectors of the full particle-hole dRPA problem defined as where $\bX$ and $\bY$ are the eigenvector matrices of the full particle-hole dRPA problem defined as
\begin{equation} \begin{equation}
\label{eq:full_dRPA} \label{eq:full_dRPA}
\begin{pmatrix} \begin{pmatrix}
@ -550,7 +550,7 @@ with
\end{align} \end{align}
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues. Note that $\boldsymbol{\Omega}$ in this case has the same size as in the TDA because we consider only the positive excitations of the full dRPA problem. $\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues. Note that $\boldsymbol{\Omega}$ in this case has the same size as in the TDA because we consider only the positive excitations of the full dRPA problem.
Defining an unfold version of this equation which does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}). Defining an unfold version of this equation that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022} However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
\begin{equation} \begin{equation}
\label{eq:nonTDA_upfold} \label{eq:nonTDA_upfold}