several changes here and there + fix compilation problem (the new .rty wasn't staged...)
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@ -14908,6 +14908,30 @@
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year = {2013},
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bdsk-url-1 = {https://doi.org/10.1103/PhysRevLett.111.073003}}
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@article{Surjan_1996,
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title = {Damping of Perturbation Corrections in Quasidegenerate Situations},
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author = {Surj{\'a}n, P. R. and Szabados, {\'A}.},
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year = {1996},
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journal = {The Journal of Chemical Physics},
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volume = {104},
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number = {9},
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pages = {3320--3324},
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issn = {0021-9606},
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doi = {10.1063/1.471814}
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}
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@article{Forsberg_1997,
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title = {Multiconfiguration Perturbation Theory with Imaginary Level Shift},
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author = {Forsberg, Niclas and Malmqvist, Per-{\AA}ke},
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year = {1997},
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journal = {Chemical Physics Letters},
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volume = {274},
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number = {1},
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pages = {196--204},
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issn = {0009-2614},
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doi = {10.1016/S0009-2614(97)00669-6}
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}
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@article{Tew_2007,
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author = {D. P. Tew and W. Klopper and C. Neiss and C. Hattig},
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date-added = {2019-10-24 20:19:01 +0200},
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@ -57,7 +57,7 @@
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\newcommand{\bW}{\boldsymbol{W}}
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\newcommand{\bX}{\boldsymbol{X}}
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\newcommand{\bY}{\boldsymbol{Y}}
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\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
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\newcommand{\bZ}{\boldsymbol{Z}}
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bI}{\boldsymbol{1}}
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@ -198,7 +198,7 @@ with
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\end{align}
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\end{subequations}
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The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
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\titou{The TDA case is discussed in Appendix \ref{sec:nonTDA}.}
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The TDA case is discussed in Appendix \ref{sec:nonTDA}.
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Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
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The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral excitations.
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@ -249,12 +249,11 @@ If it is not the case, the qs$GW$ self-consistent scheme will oscillate between
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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{Battaglia_2022} \ant{more ref...}
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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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\ANT{The matrix element expressions should be changed}
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Applying the SRG to $GW$ could 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.~\eqref{eq:quasipart_eq}, which is not straightforward.
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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.
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@ -397,24 +396,11 @@ As hinted at the end of Sec.~\ref{sec:gw}, the diagonal and off-diagonal parts a
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where we have omitted the $s$ dependence of the matrix elements 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}
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\bF{}{} & \bO \\
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\bO & \bC{}{}
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\end{pmatrix} &
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\bHod{0}(0) &=\begin{pmatrix}
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\bO & \bO \\
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\bO & \bO
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\end{pmatrix} \notag \\
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\bHd{1}(0) &=\begin{pmatrix}
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\bO & \bO \\
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\bO & \bO
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\end{pmatrix} &
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\bHod{1}(0) &= \begin{pmatrix}
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\bO & \bW{}{} \\
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\bW^{\dagger} & \bO \notag
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\end{pmatrix} \notag
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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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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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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 quasi-particle equation.
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In particular, in this manuscript, the focus will be on the second-order renormalized quasi-particle equation.
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@ -558,7 +544,7 @@ The first step will be to analyse in depth the behavior of the two static self-e
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Then the accuracy of the IP yielded by the traditional and SRG schemes will be statistically gauged over a test set of molecules.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Flow parameter dependence of the qs$GW$ and SRG-qs$GW$ schemes}
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\subsection{Flow parameter dependence of the SRG-qs$GW$ scheme}
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\label{sec:flow_param_dep}
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%%%%%%%%%%%%%%%%%%%%%%
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@ -585,8 +571,8 @@ For $s\to\infty$, the IP reaches a plateau at an error that is significantly sma
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Even more, the value associated with this plateau is slightly more accurate than its qs$GW$ counterpart.
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However, the SRG-qs$GW$ error do not decrease smoothly between the initial HF value and the $s\to\infty$ limit as for small $s$ values it is actually worst than the HF starting point.
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This behavior as a function of $s$ can be \ant{approximately} streamlined by applying matrix perturbation theory on Eq.~(\ref{eq:GWlin}).
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Through second order, the principal IP is
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This behavior as a function of $s$ can be approximately rationalized by applying matrix perturbation theory on Eq.~(\ref{eq:GWlin}).
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Through second order in the coupling block, the principal IP is
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\begin{equation}
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\label{eq:2nd_order_IP}
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I_k = - \epsilon_k - \sum_{i\nu} \frac{W_{k,i\nu}^2}{\epsilon_k - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{W_{k,a\nu}^2}{\epsilon_k - \epsilon_a - \Omega_\nu} + \order{3}
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@ -609,6 +595,9 @@ The Lithium dimer \ce{Li2} will be considered as a case where HF actually undere
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The Lithium hydrid will also be investigated because in this case the usual qs$GW$ IP is worst than the HF one.
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Finally, the Beryllium oxyde will be studied as a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
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Note that for \ce{LiH} and \ce{BeO} the TDA actually improves the accuracy compared to RPA-based qs$GW$ schemes.
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However, as we will see in the next subsection these are just particular molecular systems and in average the RPA polarizability performs better than the TDA one.
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% \ANT{Maybe we should add GF(2) because it allows us to explain the behavior of the SRG curve using perturbation theory.}
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% The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
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% Add sentence about $GW$ better than GF2 when the results will be here.
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@ -629,13 +618,13 @@ Finally, the Beryllium oxyde will be studied as a prototypical example of a mole
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Comparison of the Sym-qs and SRG-qs approximations}
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\subsection{Statistical performance \ANT{Need a better subsection title...}}
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\label{sec:SRG_vs_Sym}
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%%%%%%%%%%%%%%%%%%%%%%
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The test set considered in this study is composed of the GW20 set of molecules introduced by Lewis and Berkelbach. \cite{Lewis_2019}
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This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set.
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We also added the MgO and O3 molecules which are part of GW100 and are known to be difficult to converged for qs$GW$. \cite{vanSetten_2015,Forster_2021}
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This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set. \cite{vanSetten_2015}
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In addition, the MgO and O3 molecules (which are part of GW100 as well) has been added to the test set because they are known to be difficult to be plagued by intruder states. \cite{vanSetten_2015,Forster_2021}
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%=================================================================%
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\section{Conclusion}
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