One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{CsanakBook,FetterBook,Martin_2016,Golze_2019}
The non-linear Hedin's equations consist of a closed set of equations leading to the exact interacting one-body Green's function and, therefore, the total energy, density, ionization potentials, electron affinities, as well as spectral functions, without the explicit knowledge of the wave functions associated with the neutral and charged states of the system. \cite{Hedin_1965}
Unfortunately, solving exactly Hedin's equations is usually out of reach and one must resort to approximations.
In particular, the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} which has been first introduced in the context of solids \cite{Strinati_1980,Strinati_1982a,Strinati_1982b,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely applied to molecular systems, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Li_2017,Li_2019,Li_2020,Li_2021,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022} yields accurate charged excitation energies for weakly correlated systems \cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a relatively low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
The $GW$ method approximates the self-energy $\Sigma$ which relates the exact interacting Green's function $G$ to a non-interacting reference version $G_0$ through a Dyson equation of the form
%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 perturbative expansion with respect to the screened Coulomb potential $W$ yields the so-called $GW$ approximation \cite{Hedin_1965,Martin_2016}
Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017}
The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013}
Despite a wide range of successes, many-body perturbation theory is not flawless. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017,Duchemin_2020}
\ant{ For example, modelling core electron spectroscopy requires core ionisation energies which have been proved to be challenging for routine $GW$ calculations. \cite{Golze_2018}
Many-body perturbation theory can also be used to access optical excitation energies through the Bethe-Salpeter. However, the accuracy is not yet satisfying for triplet excited states. \cite{Bruneval_2015,Jacquemin_2017a,Jacquemin_2017b}
Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require more accuracy.
Unfortunately, defining a systematic way to go beyond $GW$, the so-called vertex corrections, is a tricky task.
Lewis and Berkelbach have shown that naive vertex corrections can even worsen the results with respect to the initial $GW$ results. \cite{Lewis_2019}
We refer the reader to the recent review by Golze and co-workers for an extensive list of current challenges in many-body perturbation theory (see Ref.~\onlinecite{Golze_2019}) and we will now focus on another flaw throughout this manuscript.}
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,Loos_2020e,Berger_2021,DiSabatino_2021}
In addition, systems \ant{whose quasi-particle equation admits two solutions with} a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Forster_2021}
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 the recent successes of regularization schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} 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.
This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a}
The SRG has also been successful in the context of nuclear structure theory, where it was first developed as a mature computational tool thanks to the work of several research groups.
The SRG transformation aims at decoupling an internal (or reference) space from an external space while incorporating information about their coupling in the reference space.
This process can often result in the appearance of intruder states. \cite{Evangelista_2014b,ChenyangLi_2019a}
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.
By definition, intruder states have energies that are close to the reference energy, and therefore are the last to be decoupled.
By stopping the SRG transformation once all external configurations except the intruder states have been decoupled,
correlation effects between the internal and external spaces can be incorporated (or folded) without the presence of intruder states.
The goal of this manuscript is to determine if the SRG formalism can effectively address the issue of intruder states in many-body perturbation theory, as it has in other areas of electronic and nuclear structure theory.
\ant{This open question will lead us to an \textit{intruder-state-free first-principle static approximation of the self-energy} that can be used for qs$GW$ calculations.}
\ant{The self-energy consider in this work will always be the $GW$ one [Eq.~\eqref{eq:gw_selfenergy}] but the subsequent derivations can be straightforwardly transposed to other approximations such as GF(2) or $GT$.
In addition, we assume a Hartree-Fock (HF) starting point throughout the manuscript.}
The self-energy can be physically understood as a \ant{correction to the HF problem (represented by $\bF$) accounting for dynamical screening effects}.
Note that $\bSig(\omega)$ is dynamical, \titou{\ie} it depends on the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.
where $\bX$ is the matrix of eigenvectors of the particle-hole direct RPA (dRPA) problem in the Tamm-Dancoff approximation (TDA). This problem is defined as
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~\eqref{eq:GW_selfenergy}.
The non-TDA case is discussed in Appendix~\ref{sec:nonTDA}.
Throughout the manuscript $p,q,r,s$ indices are used for general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
The indices $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
Because of the frequency dependence, fully solving the quasi-particle equation 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.
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).
Therefore, one can \titou{optimize} 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}
\PFL{Maybe it is worth mentioning here that is is a fairly heuristic approach that is obviously system dependent?}
Alternatively, one could solve this set of quasi-particle equations self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007,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.
This procedure is iterated until convergence for $\epsilon_p$ is reached.
\PFL{This is not quite right. It is probably going to be easier to explain when you're going to introduce the explicit expressions of these quantities.}
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.
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 $\bSig^{\qs}$.
Then the qs$GW$ problem is solved using the usual HF algorithm with $\bF$ replaced by $\bF+\bSig^{\qs}$.
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}
The $\ii\eta$ term that is usually added in the denominators of the self-energy [see Eq.~(\ref{eq:GW_selfenergy})] is the usual imaginary-shift regularizer used in various other theories flawed by intruder states. \cite{Battaglia_2022}\ant{more ref...}
Various other regularizers are possible and in particular one of us has shown that a regularizer inspired by the SRG had some advantages over the imaginary shift. \cite{Monino_2022}
But it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
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 usual $GW$ non-linear equation can be obtained by applying L\"odwin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} which gives the following expression for the self-energy \cite{Bintrim_2021}
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} 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 is $\order{K^3}$ while it is $\order{K}$ in the latter.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Ref.~\cite{Tolle_2022}).
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.
The similarity renormalization group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
depends on a flow parameter $s$, such that $\bH(s=0)$ is the initial untransformed Hamiltonian and $\bH(s=\infty)$ is the (block)-diagonal Hamiltonian.
To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce an approximation for $\boldsymbol{\eta}(s)$.
which implies that the matrix elements of the off-diagonal part will decrease in a monotonic way.
Even more, the coupling coefficients associated with the highest energy determinants are removed first as will be evidenced by the perturbative analysis after.
The main flaw of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \ant{ref}
However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions so we will not be affected by this problem. \cite{Evangelista_2014, Hergert_2016}
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.
The last equation can be solved by introducing $\bU$ the matrix that diagonalizes $\bC^{(0)}=\bU\bD^{(0)}\bU^{-1}$ such that the differential equation for $\bV^{(0)}$ becomes
which gives the same system of equations as in the previous subsection except that $\bV^{(0)}$ and $\bV^{(0),\dagger}$ should be replaced by $\bV^{(1)}$ and $\bV^{(1),\dagger}$.
Once again the two first equations are easily solved
At $s=0$ the elements $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~\eqref{eq:GW_sERI} while for $s\to\infty$ they go to zero.
Therefore, $W_{p,(q,v)}^{(1)}(s)$ are renormalized two-electrons screened integrals.
Note the close similarity of the first-order element expressions with the ones of Evangelista in Ref.~\onlinecite{Evangelista_2014b} obtained in a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
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.
Collecting every second-order terms in the flow equation and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
Interestingly, the static limit, \ie$s\to\infty$ limit, of Eq.~\eqref{eq:GW_renorm} defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} which matrix elements read as
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.
which depends on one regularising parameter $s$ analogously to $\eta$ in the usual case.
The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
Indeed, previously we mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
So is it possible to use the SRG machinery developed above to remove discontinuities?
In fact, not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
However, doing a change of variable such that
\begin{align}
e^{-s}&= 1-e^{-t}& 1 - e^{-s}&= e^{-t}
\end{align}
hence choosing a finite value of $t$ is well-designed to avoid discontinuities in the dynamic.
In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
This section starts by considering the Neon atom and the water molecule in the aug-cc-pVTZ cartesian basis set in Fig.~\ref{fig:fig1}.
The HF values (orange lines) lie below the reference CCSD(T) ones, a result which is now well-understood.
Indeed, this is due to an over(under? \ant{I will check this...}) screening of the interactions in the mean-field treatment. \cite{Lewis_2019}
The usual Sym-qs$GW$ scheme (blue lines) brings a quantitative improvement as both IP energies are now within \SI{0.5}{\electronvolt} of the reference.
The Neon atom is a well-behaved system and could be converged without regularization parameter while for water it was set to 0.01 to help convergence.
Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IP energies as a function of the flow parameter.
At $s=0$, the IPs are equal to their HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
For $s\to\infty$ both IPs reach a plateau that are significantly better than their $s=0$ starting point.
Even more, the values associated with these plateau are more accurate than their Sym-qs$GW$ counterparts.
The SRG-qs$GW$ IPs do not increase smoothly between the HF values and their limits as for small $s$ values they are actually worst than the HF IPs.
The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
Add sentence about $GW$ better than GF2 when the results will be here.
The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
The GF(2) IP admits the following perturbation expansion...
Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
But the mechanism causing the increase/decrease of the $GW$ IPs as a function of $s$ should be closely related to the GF(2) one exposed above.
\begin{itemize}
\item Li2 interesting because HF underestimate IP
\item LiH interesting because Sym-qs$GW$ worst than HF
\item BeO interesting because usually difficult to converge in qs$GW$ (cf Forster 2021)
\acknowledgements{This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
$\boldsymbol{\Omega}$ 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 that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~\eqref{eq:full_dRPA}.
Using the SRG on this matrix instead of Eq.~\eqref{eq:GWlin} gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}.