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}
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}
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}
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 self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken in 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.
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}
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}
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}
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.
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}
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.
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.
Because intruder states have energies really close to the reference energies they will be the last ones decoupled.
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}.
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.
where $\bF$ is the Fock matrix, \cite{SzaboBook} and $\bSig(\omega)$ is the self-energy, both are $K \times K$ matrices with $K$ the number of one-body basis functions.
The self-energy can be physically understood as a dynamical screening correction to the Hartree-Fock (HF) problem represented by $\bF$.
Similarly to the HF case, this equation needs to be solved self-consistently.
Note that $\bSig$ is dynamical, \ie it depends on both the eigenvalues $\epsilon_p$ and eigenvectors $\psi_p$ while $\bF$ depends only on the eigenvectors.
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.
In this case, there are $K$ quasi-particle equations which read as
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.
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}
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.
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}
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 $\bSig^{\qs}$.
The algorithm to solve the qs problem is totally analog to the HF case 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}
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.
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 $\ii eta$ term that is usually added in the denominator of the self-energy (see below) 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, in the $GW$ case, over the imaginary shift one. \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.
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.
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.
The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively.
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 $v=(ia)$.
$\boldsymbol{\Omega}$ 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.
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.
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.
Therefore, these blocks will be the target of our SRG transformation but before going in more details we will review the SRG formalism.
The similarity renormalization group method aims at continuously transforming an 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 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}
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
Note that at $s=0$ the elements $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~(\ref{eq:GW_sERI}) and that 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 and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
Therefore, the SRG flow gradually transforms the dynamic degrees of freedom of $\bSig(\omega)$ in static ones, starting from the ones that have the largest denominators in Eq.~(\ref{eq:static_F2}).
Interestingly, the static limit, \ie$s\to\infty$ limit, of Eq.~(\ref{eq:GW_renorm}) defines an alternative qs$GW$ approximation to the one defined by Eq.~(\ref{eq:sym_qsgw}).
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.
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
The fact that the $s\to\infty$ static limit does not well converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.
To conclude this section, we will discuss the case of discontinuities.
Indeed, we have previously said 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 remove discontinuities by using the SRG machinery developed above?
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}.
\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 which 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}
which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~(\ref{eq:full_dRPA}).
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
Using the SRG on this matrix instead of Eq.~(\ref{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}.