modifs in Sec IV
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@ -119,8 +119,8 @@ We refer the reader to the recent review by Golze and co-workers (see Ref.~\onli
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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.
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In some cases, this transfer of spectral weight makes it difficult to distinguish between a quasiparticle and a satellite.
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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}
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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}
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These multiple solutions hinder the convergence of partially self-consistent schemes, \cite{Veril_2018,Forster_2021,Monino_2022} such as quasiparticle self-consistent $GW$ \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} (qs$GW$) and eigenvalue-only self-consistent $GW$ \cite{Shishkin_2007a,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016} (ev$GW$).
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The simpler one-shot $G_0W_0$ scheme \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007a} 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}
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These convergence problems and discontinuities can even happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
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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}
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@ -210,7 +210,7 @@ The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to
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Because of the frequency dependence of the self-energy, fully solving the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task.
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Hence, several approximate schemes have been developed to bypass self-consistency.
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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.
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The most popular strategy is the one-shot (perturbative) $G_0W_0$ scheme, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
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Assuming a HF starting point, this results in $K$ quasiparticle equations that read
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\begin{equation}
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\label{eq:G0W0}
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@ -228,37 +228,36 @@ However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions wit
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These 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.~\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.
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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}
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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}
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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.
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Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbital energies instead.
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As commonly done, one can even ``tune'' the starting point to obtain the best possible one-shot GW quasiparticle energies \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
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Alternatively, one may solve this set of quasiparticle equations self-consistently leading to the ev$GW$ scheme.
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\titou{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.}
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This procedure is iterated until convergence for $\epsilon_p$ is reached.
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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.
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Even at convergence, the starting point dependence is not totally removed as the results still depend on the initial molecular orbitals. \cite{Marom_2012}
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However, if one of the quasiparticle equations does not have a well-defined quasiparticle solution, reaching self-consistency can be challenging, if not impossible.
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Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012}
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In order to update both the orbital energies and coefficients, one must consider the off-diagonal elements in $\bSig(\omega)$.
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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 $\bSig^{\qs}$.
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Then the qs$GW$ problem is solved using the usual HF algorithm with $\bF$ replaced by $\bF + \bSig^{\qs}$.
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Various choices for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
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In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig(\omega)$.
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To avoid solving the non-Hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
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Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qs}$.
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Various choices for $\bSig^\qs$ are possible but the most popular 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[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
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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.
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This will be done in the next sections.
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which was first 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 functions by Ismail-Beigi. \cite{Ismail-Beigi_2017}
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One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy.
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Once again, in cases where multiple solutions have large spectral weights, qs$GW$ self-consistency can be difficult to reach.
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Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
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Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
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Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasiparticle solution.
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If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
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If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
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The satellites causing convergence problems are the so-called intruder states.
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The intruder state problem can be dealt with by introducing \textit{ad hoc} regularisers.
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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 regulariser used in various other theories flawed by intruder states. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}
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Various other regularisers are possible and in particular one of us has shown that a regulariser inspired by the SRG had some advantages over the imaginary shift. \cite{Monino_2022}
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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.
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This is the aim of the rest of this work.
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The satellites causing convergence problems are the above-mentioned intruder states.
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One can deal with them by introducing \textit{ad hoc} regularizers.
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The $\ii \eta$ term \titou{that is usually added in the denominators of the self-energy} [see Eq.~\eqref{eq:GW_selfenergy}] is the usual imaginary-shift regularizer used in various other theories \titou{...} by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}
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Various other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
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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 one of the aims of the present work.
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Applying the SRG to $GW$ could gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle.
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However, to do so one needs to identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
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@ -298,7 +297,7 @@ and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
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&
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W^\text{2p1h}_{p,a\nu} & = W_{p,a\nu}.
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\end{align}
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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}
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The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021}
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\begin{equation}
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\begin{split}
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\bSig(\omega)
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@ -307,14 +306,14 @@ The usual $GW$ non-linear equation can be obtained by applying L\"odwin partitio
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& + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1} \T{(\bW^{\pph})},
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\end{split}
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\end{equation}
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which can be further developed to give exactly Eq.~(\ref{eq:GW_selfenergy}).
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which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.
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Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other not.
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Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other one is not.
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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.
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We refer to Ref.~\cite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Ref.~\cite{Tolle_2022}).
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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 Refs.~\onlinecite{Tolle_2022,Scott_2023}).
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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 the SRG transformation but before going into more detail we will review the SRG formalism.
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As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $\bW^\text{2h1p}$ and $\bW^\text{2p1h}$ are coupling the 1h and 1p configuration to the 2h1p and 2p1h configurations.
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Therefore, these blocks will be the target of the SRG transformation but before going into more detail we review the SRG formalism.
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%%%%%%%%%%%%%%%%%%%%%%
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\section{The similarity renormalization group}
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@ -380,9 +379,10 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
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\label{sec:srggw}
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%%%%%%%%%%%%%%%%%%%%%%
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Finally, the SRG formalism exposed above will be applied to $GW$.
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Finally, the SRG formalism exposed above is applied to $GW$.
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The first step is to define the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
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As hinted at the end of Sec.~\ref{sec:gw}, the diagonal and off-diagonal parts are defined as
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As hinted at the end of Sec.~\ref{sec:gw}, it is natural to define the diagonal and off-diagonal parts as
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\begin{subequations}
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\begin{align}
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\label{eq:diag_and_offdiag}
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\bH^\text{d}(s) &=
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@ -390,32 +390,33 @@ As hinted at the end of Sec.~\ref{sec:gw}, the diagonal and off-diagonal parts a
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\bF & \bO & \bO \\
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\bO & \bC^{\text{2h1p}} & \bO \\
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\bO & \bO & \bC^{\text{2p1h}} \\
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\end{pmatrix}
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& \\
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\end{pmatrix},
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\\
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\bH^\text{od}(s) &=
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\begin{pmatrix}
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\bO & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
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(\bW^{\text{2h1p}})^{\mathrm{T}} & \bO & \bO \\
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(\bW^{\text{2p1h}})^{\mathrm{T}} & \bO & \bO \\
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\end{pmatrix}
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\end{pmatrix},
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\end{align}
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where we have omitted the $s$ dependence of the matrix elements for the sake of brevity.
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\end{subequations}
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where we omit the $s$ dependence of the matrices for the sake of brevity.
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Then, the aim is to solve order by order the flow equation [see Eq.~\eqref{eq:flowEquation}] knowing that the initial conditions are
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\begin{align}
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\bHd{0}(0) &= \begin{pmatrix} \bF & \bO \\ \bO & \bC \end{pmatrix},
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&
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\bHod{1}(0) &= \begin{pmatrix} \bO & \bW \\ \bW^{\dagger} & \bO \end{pmatrix},
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\end{align}
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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.
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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.
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In particular, in this manuscript, the focus will be on the second-order renormalized quasiparticle equation.
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and $\bHod{0}(0) = \bHd{1}(0) = \bO$, where the matrices $\bC$ and $\bV$ collect the 2h1p and 2p1h channels.
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Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:GWlin} before applying the downfolding process to obtain a renormalized version of the quasiparticle equation.
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In particular, we focus here on the second-order renormalized quasiparticle equation.
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%///////////////////////////%
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\subsection{Zeroth-order matrix elements}
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% ///////////////////////////%
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The choice of the Wegner generator associated with the form of the flow equation [see Eq.~(\ref{eq:flowEquation})] implies that the off-diagonal corrections are of order $\order{\lambda}$ while the correction to the diagonal blocks are at least $\order{\lambda^2}$.
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Therefore, the zeroth order Hamiltonian is independent of $s$ and we have
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The choice of Wegner's generator in the flow equation [see Eq.~\eqref{eq:flowEquation}] implies that the off-diagonal correction is of order $\order*{\lambda}$ while the correction to the diagonal block is at least $\order*{\lambda^2}$.
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Therefore, the zeroth-order Hamiltonian is independent of $s$ and we have
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\begin{equation}
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\bH^{(0)}(s) = \bH^{(0)}(0).
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\end{equation}
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@ -424,34 +425,53 @@ Therefore, the zeroth order Hamiltonian is independent of $s$ and we have
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\subsection{First-order matrix elements}
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%///////////////////////////%
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Knowing that $\bHod{0}(s)=\bO$, the first order flow equation is
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Knowing that $\bHod{0}(s)=\bO$, the first-order flow equation is
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\begin{equation}
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\dv{\bH^{(1)}}{s} = \comm{\comm{\bHd{0}}{\bHod{1}}}{\bHd{0}}
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\dv{\bH^{(1)}}{s} = \comm{\comm{\bHd{0}}{\bHod{1}}}{\bHd{0}},
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\end{equation}
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which gives the following system of equations
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\ANT{Do you know a cleaner way to write this system? The vertical spaces are too large...}
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\begin{subequations}
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\begin{align}
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\dv{\bF^{(0)}}{s}&=\bO & \dv{\bC^{(0)}}{s}&=\bO
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\label{eq:F0_C0}
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\dv{\bF^{(0)}}{s}&=\bO, & \dv{\bC^{(0)}}{s}&=\bO,
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\end{align}
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and
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\begin{multline}
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\dv{\bW^{(1),\dagger}}{s}{(s)} = 2 \bC^{(0)}\bW^{(1),\dagger}(s)\bF^{(0)} - \bW^{(1),\dagger}(s)(\bF^{(0)})^2 \\ - (\bC^{(0)})^2\bW^{(1),\dagger}(s)
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\label{eq:W1}
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\dv{\bW^{(1)}}{s}
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= 2 \bF^{(0)}\bW^{(1)}\bC^{(0)}
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\\
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- (\bF^{(0)})^2\bW^{(1)} - \bW^{(1)}(\bC^{(0)})^2.
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\end{multline}
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\begin{multline}
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\dv{\bW^{(1)}}{s}{(s)} = 2 \bF^{(0)}\bW^{(1)}(s)\bC^{(0)} - (\bF^{(0)})^2\bW^{(1)}(s) \\ - \bW^{(1)}(s)(\bC^{(0)})^2.
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\end{multline}
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\end{subequations}
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The two first equations imply
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%\begin{subequations}
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% \begin{align}
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% \label{eq:W1}
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% \begin{split}
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% \dv{\bW^{(1),\dagger}}{s}
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% & = 2 \bC^{(0)}\bW^{(1),\dagger}\bF^{(0)}
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% \\
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% & - \bW^{(1),\dagger}(\bF^{(0)})^2 - (\bC^{(0)})^2\bW^{(1),\dagger},
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% \end{split}
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% \\
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% \begin{split}
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% \label{eq:W1dag}
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% \dv{\bW^{(1)}}{s}
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% & = 2 \bF^{(0)}\bW^{(1)}\bC^{(0)}
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% \\
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% & - (\bF^{(0)})^2\bW^{(1)} - \bW^{(1)}(s)(\bC^{(0)})^2.
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% \end{split}
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% \end{align}
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%\end{subequations}
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Equation \eqref{eq:F0_C0} implies
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\begin{align}
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\bF^{(1)}(s) &= \bF^{(1)}(0) = \bO & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO.
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\bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
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\end{align}
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and thanks to the diagonal structure of $\bF^{(0)}$ and $\bC^{(0)}$ the differential equations for the coupling elements are easily solved and give
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and, thanks to the diagonal structure of $\bF^{(0)}$ and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
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\begin{equation}
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W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
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\end{equation}
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At $s=0$ the elements $W_{p,q\nu}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~\eqref{eq:GW_sERI} while for $s\to\infty$ they go to zero.
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Therefore, $W_{p,q\nu}^{(1)}(s)$ are renormalized two-electrons screened integrals.
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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}).
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At $s=0$ the elements $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero.
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Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
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It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} following a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
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%///////////////////////////%
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\subsection{Second-order matrix elements}
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@ -460,40 +480,47 @@ Note the close similarity of the first-order element expressions with the ones o
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The second-order renormalized quasiparticle equation is given by
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\begin{equation}
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\label{eq:GW_renorm}
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\left( \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) \right) \bX = \omega \bX,
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% \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) ] \bX = \omega \bX,
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\qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
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\end{equation}
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with
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\begin{subequations}
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\begin{align}
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\widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s),\\
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\label{eq:srg_sigma}
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\widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger}.
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
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 $\bF^{(2)}$
|
||||
Collecting every second-order terms in the flow equation and performing the block matrix products results in the following differential equation
|
||||
\begin{multline}
|
||||
\label{eq:diffeqF2}
|
||||
\dv{\bF^{(2)}}{s} = \bF^{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF^{(0)} \\ - 2 \bW^{(1)}\bC^{(0)}\bW^{(1),\dagger} .
|
||||
\dv{\bF^{(2)}}{s} = \bF^{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF^{(0)} \\ - 2 \bW^{(1)}\bC^{(0)}\bW^{(1),\dagger},
|
||||
\end{multline}
|
||||
This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
|
||||
which can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
|
||||
\begin{multline}
|
||||
F_{pq}^{(2)}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W^{\dagger}_{r\nu,q} \times \\
|
||||
\left(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}\right),
|
||||
F_{pq}^{(2)}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W^{\dagger}_{r\nu,q} \\
|
||||
\times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}],
|
||||
\end{multline}
|
||||
with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$.
|
||||
with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ \titou{(where $\epsilon_F$ is the ...)}.
|
||||
|
||||
At $s=0$, this second-order correction is null while for $s\to\infty$ it tends towards the following static limit
|
||||
At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends towards the following static limit
|
||||
\begin{equation}
|
||||
\label{eq:static_F2}
|
||||
F_{pq}^{(2)}(\infty) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W^{\dagger}_{r\nu,q}.
|
||||
F_{pq}^{(2)}(\infty) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} \titou{W_{q,r\nu}}.
|
||||
\end{equation}
|
||||
Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
|
||||
Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
|
||||
Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}(s)$.
|
||||
This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
|
||||
|
||||
%///////////////////////////%
|
||||
\subsection{Alternative form of the static self-energy}
|
||||
% ///////////////////////////%
|
||||
|
||||
Interestingly, the static limit, \ie the $s\to\infty$ 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
|
||||
\begin{equation}
|
||||
\label{eq:sym_qsGW}
|
||||
\Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu} \left( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2} \right) W_{p,r\nu} W^{\dagger}_{r\nu,q}.
|
||||
\Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}} ) W_{p,r\nu} \titou{W_{q,r\nu}}.
|
||||
\end{equation}
|
||||
This alternative static form will be refered to as SRG-qs$GW$ in the following.
|
||||
Both approximations are closely related as they share the same diagonal terms when $\eta=0$.
|
||||
@ -505,7 +532,7 @@ Indeed, in traditional qs$GW$ calculation increasing $\eta$ to ensure convergenc
|
||||
Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
|
||||
\begin{multline}
|
||||
\label{eq:SRG_qsGW}
|
||||
\Sigma_{pq}^{\text{SRG}}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu} \times \\ \left(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}\right)
|
||||
\Sigma_{pq}^{\text{SRG}}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu} \\ \times \qty[(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
|
||||
\end{multline}
|
||||
which depends on one regularising parameter $s$ analogously to $\eta$ in the usual qs$GW$.
|
||||
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 $\tilde{\bF}$.
|
||||
@ -643,8 +670,9 @@ In addition, the MgO and O3 molecules (which are part of GW100 as well) has been
|
||||
Here comes the conclusion.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\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).}
|
||||
\acknowledgements{
|
||||
The authors thank Francesco Evangelista for inspiring discussions.
|
||||
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).}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -652,10 +680,6 @@ The authors thank Francesco Evangelista for inspiring discussions.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
The data that supports the findings of this study are available within the article.% and its supplementary material.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\bibliography{SRGGW}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\appendix
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -756,4 +780,8 @@ Within the TDA the renormalized matrix elements have the same $s$ dependence as
|
||||
% \end{align}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\bibliography{SRGGW}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\end{document}
|
||||
|
||||
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