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@ -114,16 +114,16 @@ For example, modeling core electron spectroscopy requires core ionization energi
Many-body perturbation theory can also be used to access optical excitation energies through the Bethe-Salpeter equation. \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} However, the accuracy is not yet satisfying for triplet excited states, where instabilities often occur. \cite{Bruneval_2015,Jacquemin_2017a,Jacquemin_2017b,Holzer_2018a}
Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require higher precision.
Unfortunately, defining a systematic way to go beyond $GW$ via the inclusion of vertex corrections has been demonstrated to be a tricky task. \cite{Baym_1961,Baym_1962,DeDominicis_1964a,DeDominicis_1964b,Bickers_1989a,Bickers_1989b,Bickers_1991,Hedin_1999,Bickers_2004,Shirley_1996,DelSol_1994,Schindlmayr_1998,Morris_2007,Shishkin_2007b,Romaniello_2009a,Romaniello_2012,Gruneis_2014,Hung_2017,Maggio_2017b,Mejuto-Zaera_2022}
For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasi-particle energies with respect to $GW$. \cite{Lewis_2019}
For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasiparticle energies with respect to $GW$. \cite{Lewis_2019}
We refer the reader to the recent review by Golze and co-workers (see Ref.~\onlinecite{Golze_2019}) for an extensive list of current challenges in many-body perturbation theory.
Many-body perturbation theory also suffers from the infamous intruder-state problem,\cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} where they manifest themselves as solutions of the quasi-particle equation with non-negligible spectral weights.
Many-body perturbation theory also suffers from the infamous intruder-state problem,\cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} where they manifest themselves as solutions of the quasiparticle equation with non-negligible spectral weights.
In some cases, this transfer of spectral weight makes it difficult to distinguish between a quasiparticle and a satellite.
These multiple solutions hinder the convergence of partially self-consistent schemes such as quasiparticle self-consistent $GW$ (qs$GW$) and eigenvalue-only self-consistent $GW$ (ev$GW$). \cite{Veril_2018,Forster_2021,Monino_2022}
The simpler one-shot $G_0W_0$ scheme is also impacted by these intruder states, leading to discontinuities in a variety of physical quantities including charged and neutral excitation energies, correlation and total energies.\cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
These convergence problems and discontinuities can even happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
In a recent study, Monino and Loos showed that the 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}
In a recent study, Monino and Loos showed that the discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasiparticle equation. \cite{Monino_2022}
Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, such as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} the present work investigates the application of the SRG formalism to many-body perturbation theory in its $GW$.
In particular, we focus here on the possibility of curing the qs$GW$ convergence problems using the SRG.
@ -160,7 +160,7 @@ This section starts by
\label{sec:gw}
%%%%%%%%%%%%%%%%%%%%%%
The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation which, within the $GW$ approximation, reads
The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasiparticle equation which, within the $GW$ approximation, reads
\begin{equation}
\label{eq:quasipart_eq}
\qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
@ -169,7 +169,8 @@ where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is (the corr
Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects.
Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently.
\ant{Note that $\bSig(\omega)$ is dynamical which implies that it depends on both the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.}
\titou{Note that $\bSig(\omega)$ is dynamical which implies that it depends on both the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.}
\PFL{I still don't like it.}
The matrix elements of $\bSig(\omega)$ have the following analytic expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
\begin{equation}
@ -181,56 +182,62 @@ The matrix elements of $\bSig(\omega)$ have the following analytic expression \c
where $\eta$ is a positive infinitesimal and the screened two-electron integrals are
\begin{equation}
\label{eq:GW_sERI}
W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu}+\bY_{\nu})_{ia},
W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty(\bX+\bY)_{ia,\nu},
\end{equation}
where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct RPA problem defined as
where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct (\ie without exchange) RPA problem defined as
\begin{equation}
\label{eq:full_dRPA}
\mqty( \bA & \bB \\ -\bA & -\bB ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ),
\mqty( \bA & \bB \\ -\bB & -\bA ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ),
\end{equation}
with
\begin{subequations}
\begin{align}
A_{ia,jb} & = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
A_{ia,jb} & = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj},
\\
B_{ia,jb} & = \eri{ij}{ab}.
B_{ia,jb} & = \eri{ij}{ab},
\end{align}
\end{subequations}
The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
The TDA case is discussed in Appendix \ref{sec:nonTDA}.
Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral excitations.
and
\begin{equation}
\braket{pq}{rs} = \iint \frac{\SO{p}(\bx_1) \SO{q}(\bx_2)\SO{r}(\bx_1) \SO{s}(\bx_2) }{\abs{\br_1 - \br_2}} d\bx_1 d\bx_2
\end{equation}
are two-electron integrals in the spin-orbital basis.
Because of the frequency dependence, fully solving the quasi-particle equation is a rather complicated task.
The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
\titou{In the Tamm-Dancoff approximation (TDA), which is discussed in Appendix \ref{sec:nonTDA}, one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$.}
Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral (single) excitations.
Because of the frequency dependence of the self-energy, fully solving the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task.
Hence, several approximate schemes have been developed to bypass self-consistency.
The most popular one is the one-shot (perturbative) scheme, known as $G_0W_0$, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
Assuming a HF starting point, this results in $K$ quasi-particle equations that read
The most popular strategy is the one-shot (perturbative) scheme, known as $G_0W_0$, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
Assuming a HF starting point, this results in $K$ quasiparticle equations that read
\begin{equation}
\label{eq:G0W0}
\epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
\end{equation}
where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
The previous equations are non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$ (where the index $s$ is numbering solutions).
The previous equations are non-linear with respect to $\omega$ and therefore have multiple solutions $\epsilon_{p,s}$ for a given $p$ (where the index $s$ is numbering solutions).
These solutions can be characterized by their spectral weight given by the renormalisation factor $Z_{p,s}$
\begin{equation}
\label{eq:renorm_factor}
0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
\end{equation}
The solution with the largest weight is referred to as the quasi-particle 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 quasi-particle is not well-defined.
The solution with the largest weight 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.
These 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.
Indeed, in Eq.~\eqref{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 can \ant{tune} the starting point to obtain the best one-shot energies possible, which is commonly done. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
Alternatively, one could solve this set of quasi-particle equations self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007a,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
The solutions $\epsilon_p$ are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equation are solved for $\omega$ again.
Therefore, one can ``tune'' the starting point to obtain the best one-shot energies possible, which is commonly done. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
Alternatively, one could solve this set of quasiparticle equations self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007a,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
The solutions $\epsilon_p$ are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equations are solved for $\omega$ again.
This procedure is iterated until convergence for $\epsilon_p$ is reached.
However, if one of the quasi-particle equations does not have a well-defined quasi-particle solution, reaching self-consistency can be quite difficult, if not impossible.
However, if one of the quasiparticle equations does not have a well-defined quasiparticle solution, reaching self-consistency can be quite difficult, if not impossible.
Even at convergence, the starting point dependence is not totally removed as the results still depend on the initial molecular orbitals. \cite{Marom_2012}
In order to update both the orbital energies and coefficients, one must consider the off-diagonal elements in $\bSig(\omega)$.
To take into account the off-diagonal elements without solving the dynamic quasi-particle equation [Eq.~\eqref{eq:quasipart_eq}], one can resort to the quasi-particle self-consistent (qs) $GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
To take into account the off-diagonal elements without solving the dynamic quasiparticle equation [Eq.~\eqref{eq:quasipart_eq}], one can resort to the quasiparticle self-consistent (qs) $GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
Then the qs$GW$ problem is solved using the usual HF algorithm with $\bF$ replaced by $\bF + \bSig^{\qs}$.
Various choices for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
\begin{equation}
@ -243,7 +250,7 @@ This will be done in the next sections.
Once again, in cases where multiple solutions have large spectral weights, qs$GW$ self-consistency can be difficult to reach.
Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasi-particle solution.
Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasiparticle solution.
If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
The satellites causing convergence problems are the so-called intruder states.
@ -253,10 +260,10 @@ Various other regularisers are possible and in particular one of us has shown th
But it would be more rigorous, and more instructive, to obtain this regulariser from first principles by applying the SRG formalism to many-body perturbation theory.
This is the aim of the rest of this work.
Applying the SRG to $GW$ could gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
Applying the SRG to $GW$ could gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle.
However, to do so one needs to identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
The way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version and the coupling terms will elegantly appear in the process.
The upfolded $GW$ quasi-particle equation is \cite{Bintrim_2021,Tolle_2022}
The upfolded $GW$ quasiparticle equation is \cite{Bintrim_2021,Tolle_2022}
\begin{equation}
\label{eq:GWlin}
\begin{pmatrix}
@ -277,7 +284,7 @@ The upfolded $GW$ quasi-particle equation is \cite{Bintrim_2021,Tolle_2022}
\end{pmatrix}
\boldsymbol{\epsilon},
\end{equation}
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
C^\text{2h1p}_{i\nu,j\mu} & = \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu},
@ -400,8 +407,8 @@ Then, the aim is to solve order by order the flow equation [see Eq.~\eqref{eq:fl
\bHod{1}(0) &= \begin{pmatrix} \bO & \bW \\ \bW^{\dagger} & \bO \end{pmatrix},
\end{align}
and $ \bHod{0}(0) = \bHd{1}(0) = \bO$, where we have defined the matrices $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness.
Once the analytical low-order perturbative expansions are known they can be inserted in Eq.~\eqref{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.
Once the analytical low-order perturbative expansions are known they can be inserted in Eq.~\eqref{eq:GWlin} before downfolding to obtain a renormalized quasiparticle equation.
In particular, in this manuscript, the focus will be on the second-order renormalized quasiparticle equation.
%///////////////////////////%
\subsection{Zeroth-order matrix elements}
@ -450,7 +457,7 @@ Note the close similarity of the first-order element expressions with the ones o
\subsection{Second-order matrix elements}
% ///////////////////////////%
The second-order renormalized quasi-particle equation is given by
The second-order renormalized quasiparticle equation is given by
\begin{equation}
\label{eq:GW_renorm}
\left( \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) \right) \bX = \omega \bX,
@ -461,7 +468,7 @@ with
\label{eq:srg_sigma}
\widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger}.
\end{align}
As can be readily seen above, $\bF^{(2)}$ is the only second-order block of the effective Hamiltonian contributing to the second-order SRG quasi-particle equation.
As can be readily seen above, $\bF^{(2)}$ is the only second-order block of the effective Hamiltonian contributing to the second-order SRG quasiparticle equation.
Collecting every second-order terms in the flow equation and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
\begin{multline}
\label{eq:diffeqF2}
@ -732,7 +739,7 @@ Within the TDA the renormalized matrix elements have the same $s$ dependence as
% The fact that the integrals are not screened in GF(2) manifests itself in the fact that the $\bC$ matrices are already diagonal.
% Applying the SRG formalism to this matrix is completely analog to the derivation exposed in the main text.
% We only give the analytical expressions of the matrix elements needed for the second-order SRG-GF(2) quasi-particle equations.
% We only give the analytical expressions of the matrix elements needed for the second-order SRG-GF(2) quasiparticle equations.
% \begin{equation}
% (V^\text{2h1p}_{p,ija})^{(1)}(s) = \frac{1}{\sqrt{2}}\aeri{pa}{ij} e^{- (\epsilon_p + \epsilon_a - \epsilon_i - \epsilon_j)^2 s}