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@ -49,29 +49,30 @@ Here comes the abstract.
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\label{sec:intro}
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%=================================================================%
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One-body Green's functions provide a natural and elegant way to access charged excitations energies of a physical system. \cite{Martin_2016,Golze_2019}
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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}
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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}
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Unfortunately, fully solving the Hedin's equations is out of reach and one must resort to approximations.
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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}
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Unfortunately, fully solving Hedin's equations is out of reach and one must resort to approximations.
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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}
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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
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\begin{equation}
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\label{eq:dyson}
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G = G_0 + G_0\Sigma G.
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\end{equation}
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The self-energy encapsulates all the exchange-correlation effects which are not taken in account in the reference system.
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Approximating $\Sigma$ as the first order truncation of its perturbation expansion in terms of the screened interaction $W$ gives the so-called $GW$ approximation. \cite{Hedin_1965, Martin_2016}
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The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken in account in the reference system.
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%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.
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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}
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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}
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The GF(2) approximation is also known has the second Born approximation. \ant{ref ?}
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The GF(2) approximation is also known as the second Born approximation. \ant{ref ?}
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Despite a wide range of successes, many-body perturbation theory is not flawless.
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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}
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Even more worrying these discontinuities can happen in the weakly correlated regime where GW is thought to be valid.
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These discontinuities are due to the transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022}
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Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is thought to be valid.
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These discontinuities are due to a transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022}
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This is another occurrence of the infamous intruder-state problem. \cite{Roos_1995,Olsen_2000,Choe_2001} \ant{more ref}
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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}
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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}
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In a recent study, Monino and Loos showed that these discontinuities could be removed by introducing a regularizer inspired by the similarity renormalisation group (SRG) in the quasi-particle equation. \cite{Monino_2022}
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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}
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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.
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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.
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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}
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@ -79,10 +80,10 @@ The SRG has also been successful in the context of nuclear theory, \cite{Bogner_
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The SRG transformation aims at decoupling a reference space from an external space while folding information about the coupling in the reference space.
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This is often during such decoupling that intruder states appear. \ant{ref}
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Yet, SRG is particularly well-suited to avoid them because the speed to which each external configurations is decoupled is proportional to the energy difference between each external configurations and the reference space.
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Because by definition intruder states have energies really close to the reference energies therefore they will be the last decoupled.
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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.
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Because intruder states have energies really close to the reference energies they will be the last ones decoupled.
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Therefore the SRG continuous transformation can be stopped once every external configurations except the intruder ones have been decoupled.
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Doing so, it gives a way to fold in information in the reference space while avoiding intruder states.
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Doing so, it gives a way to fold in information about the coupling in the reference space while avoiding intruder states.
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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.
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We begin by reviewing the $GW$ formalism in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
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@ -100,20 +101,20 @@ This section starts by
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\label{sec:gw}
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%%%%%%%%%%%%%%%%%%%%%%
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Within approximate many-body perturbation theory based on Hedin's equations the central equation is the so-called quasi-particle equation
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The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation
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\begin{equation}
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\label{eq:quasipart_eq}
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\left[ \bF + \bSig(\omega = \epsilon_p) \right] \psi_p = \epsilon_p \psi_p,
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\end{equation}
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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.
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The self-energy can be physically understood as a dynamical screening correction to the Hartree-Fock (HF) problem.
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Because $\bSig$ is dynamical it depends on both the eigenvalues $\epsilon_p$ and eigenvectors $\psi_p$ while $\bF$ depends only on the eigenvectors.
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Therefore, similarly to the HF case, this equation needs to be solved self-consistently.
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The self-energy can be physically understood as a dynamical screening correction to the Hartree-Fock (HF) problem represented by $\bF$.
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Similarly to the HF case, this equation needs to be solved self-consistently.
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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.
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However, because of the $\omega$ dependence, fully solving this equation is a rather complicated task, hence several approximate solving schemes has been developed.
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Because of this $\omega$ dependence, fully solving this equation is a rather complicated task, hence several approximate solving schemes has been developed.
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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.
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In this case, there are $K$ quasi-particle equations which read as
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\begin{equation}
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tr\begin{equation}
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\label{eq:G0W0}
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\epsilon_p^{\HF} + \Sigma_{p}(\omega) - \omega = 0,
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\end{equation}
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@ -126,43 +127,44 @@ These solutions can be characterised by their spectral weight defined as the ren
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\end{equation}
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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.
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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.
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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 state.
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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.
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One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
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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.
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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}
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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}
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To do so one use the energy of the quasi-particle solution of the previous iteration to build Eq.~(\ref{eq:G0W0}) and then solves for $\omega$ again until convergence is reached.
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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.
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However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach.
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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}
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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}).
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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 $\bSig^{\qs}$.
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The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$.
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Various choice for $\bSig^\qs$ are possible but the most used one is the following hermitian one
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Various choice for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
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\begin{equation}
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\label{eq:sym_qsgw}
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\Sigma_{pq}^\qs = \frac{1}{2}\Re\left(\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) \right).
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\end{equation}
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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}
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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 next section.
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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.
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In this case as well self-consistency is hard to reach in cases where multiple solutions have large spectral weights.
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In this case as well self-consistency can be difficult to reach in cases where multiple solutions have large spectral weights.
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Multiple solutions arise due to the $\omega$ dependence of the self-energy.
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Therefore, by suppressing this dependence the static qs approximation relies on the fact that there is one well-defined quasi-particle solution.
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If it is not the case, the qs scheme will oscillates between the solutions with large weights. \cite{Forster_2021}
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Therefore 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.
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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.
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The intruder state problem can be dealt with by introducing \textit{ad hoc} regularizers.
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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...}
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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 one in the $GW$ case. \cite{Monino_2022}
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But it would be more rigorous to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
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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}
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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.
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This is the aim of this work.
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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.
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However, to do so one needs to identify the coupling terms in Eq.~(\ref{eq:quasipart_eq}) which is not straightforward.
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However, to do so one needs to identify the coupling terms in Eq.~(\ref{eq:quasipart_eq}), which is not straightforward.
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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.
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From now on, we will restrict ourselves to the $GW$ case but the same derivation could be done for the GF(2) and $GT$ self-energy and the corresponding formula are given in Appendix~\ref{sec:GF2}. \ant{do we really give GT equations?}
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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.
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The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively.
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The upfolded $GW$ quasi-particle equation is the following
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\begin{equation}
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\label{eq:GWlin}
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@ -200,7 +202,7 @@ and the corresponding coupling blocks read
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&
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V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
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\end{align}
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The $GW$ non-linear equation can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021}
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The usual $GW$ non-linear equation can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021}
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\begin{equation}
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\label{eq:GWnonlin}
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\left( \bF + \bSig(\omega) \right) \bX = \omega \bX,
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@ -210,7 +212,7 @@ with
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\bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1} (\bV^{\hhp})^{\mathsf{T}} \\
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&+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1} (\bV^{\pph}\right)^{\mathsf{T}}, \notag
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\end{align}
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which can be further developed to give the usual
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which can be further developed to give
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\begin{equation}
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\label{eq:GW_selfenergy}
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\Sigma_{pq}(\omega)
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@ -222,7 +224,7 @@ with the screened integrals defined as
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\label{eq:GW_sERI}
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W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia},
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\end{equation}
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where $\bX$ is the matrix of eigenvectors of the particle-hole direct RPA (dRPA) problem in the Tamm-Dancoff approximation (TDA) defined as
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where $\bX$ is the matrix of eigenvectors of the particle-hole direct RPA (dRPA) problem in the TDA defined as
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\begin{equation}
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\bA \bX = \boldsymbol{\Omega} \bX,
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\end{equation}
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@ -231,11 +233,10 @@ with
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A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
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\end{equation}
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$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~(\ref{eq:GW_selfenergy}).
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The case of the non-TDA approximation is discussed in Appendix~\ref{sec:nonTDA}.
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Equations~(\ref{eq:GWlin}) and~(\ref{eq:GWnonlin}) have exactly the same solutions but one is linear and the other not.
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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.
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We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding process of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case).
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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).
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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.
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Therefore, these blocks will be the target of our SRG transformation but before going in more details we will review the SRG formalism.
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@ -252,7 +253,8 @@ Therefore, the transformed Hamiltonian
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\bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
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\end{equation}
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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.
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An evolution equation for $\bH(s)$ can be easily obtained by deriving Eq~(\ref{eq:SRG_Ham}) and this gives the flow equation
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An evolution equation for $\bH(s)$ can be easily obtained by deriving Eq~(\ref{eq:SRG_Ham}) with respect to $s$.
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This gives the flow equation
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\begin{equation}
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\label{eq:flowEquation}
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\dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)},
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@ -262,12 +264,12 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
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\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
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\end{equation}
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To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce approximation for $\boldsymbol{\eta}(s)$.
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Before defining such an approximation, we need to define what are the blocks to suppress in order to obtain a block-diagonal Hamiltonian.
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Therefore, the Hamiltonian is separated in two parts as
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Before defining such an approximation, we need to define what are the blocks to suppress to obtain a block-diagonal Hamiltonian.
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The Hamiltonian is separated in two parts as
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\begin{equation}
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\bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}}.
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\bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}},
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\end{equation}
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By definition, we have the following condition on $\bH^\text{od}$
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and by definition we have the following condition on $\bH^\text{od}$
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\begin{equation}
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\bH^\text{od}(s=\infty) = \boldsymbol{0}.
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\end{equation}
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@ -467,7 +469,13 @@ In fact, the dynamic part after the change of variable is closely related to the
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%=================================================================%
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\section{Computational details}
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\label{sec:comp_det}
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%=================================================================%
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% =================================================================%
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The two qs$GW$ variants considered in this work has been implemented in a in-house program.
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The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
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The geometry have been optimized at the CC3 level in the aug-cc-pvtz basis set without frozen core using the CFOUR program.
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The reference CCSD(T) IP energies have obtained using default parameters of Gaussian 16.
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This means that the cations used an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted formalism.
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%=================================================================%
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\section{Results}
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@ -576,4 +584,54 @@ Using the SRG on this matrix instead of Eq.~(\ref{eq:GWlin}) gives the same expr
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\label{sec:GF2}
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%%%%%%%%%%%%%%%%%%%%%%
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The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of M\"oller-Plesset perturbation theory.
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The matrix elements of its second-order version read as
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\begin{align}
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\label{eq:GF2_selfenergy}
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\Sigma_{pq}^{\text{GF(2)}}(\omega)
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&= \frac{1}{\sqrt{2}} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \\
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&+ \frac{1}{\sqrt{2}} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta}
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\end{align}
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This self-energy can be upfolded similarly to the $GW$ case and one obtain the following ``super-matrix''
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\begin{equation}
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\label{eq:unfolded_matrice}
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\bH =
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\begin{pmatrix}
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\bF & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
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(\bV^{\text{2h1p}})^{\mathsf{T}} & \bC^{\text{2h1p}} & \bO \\
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(\bV^{\text{2p1h}})^{\mathsf{T}} & \bO & \bC^{\text{2p1h}} \\
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\end{pmatrix}
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\end{equation}
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The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}$ is given below.
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\begin{align}
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\label{eq:GF2_unfolded}
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V^\text{2h1p}_{p,ija} & = \frac{1}{\sqrt{2}}\aeri{pa}{ij}
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\\
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V^\text{2p1h}_{p,iab} & = \frac{1}{\sqrt{2}}\aeri{pi}{ab}
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\\
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||||
C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik}
|
||||
\\
|
||||
C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd}
|
||||
\end{align}
|
||||
Note that this matrix is exactly the ADC(2) matrix for charged excitations.
|
||||
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) 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}
|
||||
\end{equation}
|
||||
\begin{equation}
|
||||
(V^\text{2h1p}_{p,iab})^{(1)}(s) = \frac{1}{\sqrt{2}}\aeri{pi}{ab} e^{- (\epsilon_p + \epsilon_i - \epsilon_a - \epsilon_b)^2 s}
|
||||
\end{equation}
|
||||
|
||||
We define $ \Delta_{pq,rs} = \epsilon_p + \epsilon_q - \epsilon_r - \epsilon_s $
|
||||
|
||||
\begin{align}
|
||||
F_{pq}^{(2)}(s) &= \sum_{ria} \frac{\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W_{p,(r,v)} \notag \\
|
||||
&\times W^{\dagger}_{(r,v),q}\left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right).
|
||||
\end{align}
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user