lots of small changes in intro
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@ -8,9 +8,9 @@
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%%% Latin %%%
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%%% Latin %%%
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\newcommand{\ie}{\textit{i.e.}~}
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\newcommand{\ie}{\textit{i.e.}\xspace}
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\newcommand{\eg}{\textit{e.g.}~}
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\newcommand{\eg}{\textit{e.g.}\xspace}
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\newcommand{\etal}{\textit{et al.}~}
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\newcommand{\etal}{\textit{et al.}\xspace}
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%%% Operators %%%
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%%% Operators %%%
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@ -19,7 +19,7 @@
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\newcommand{\Hsim}{\hat{\bar{H}}} % Similarity transformed Hamiltonian
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\newcommand{\Hsim}{\hat{\bar{H}}} % Similarity transformed Hamiltonian
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\newcommand{\hC}{\Hat{C}} % CI operator
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\newcommand{\hC}{\Hat{C}} % CI operator
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\newcommand{\hT}{\Hat{T}} % Cluster operator
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\newcommand{\hT}{\Hat{T}} % Cluster operator
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\newcommand{\T}[1]{\Hat{\mathnormal{T}}_{#1}} % Cluster operator of a given excitation number
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\newcommand{\T}[1]{#1^{\intercal}}
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\newcommand{\hsig}{\Hat{\sigma}} % Unitary cluster operator
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\newcommand{\hsig}{\Hat{\sigma}} % Unitary cluster operator
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\newcommand{\hK}{\Hat{K}} % Anti-hermitian orbital rotation operator
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\newcommand{\hK}{\Hat{K}} % Anti-hermitian orbital rotation operator
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\newcommand{\hS}{\Hat{S}} % Anti-hermitian CI coefficients rotation operator
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\newcommand{\hS}{\Hat{S}} % Anti-hermitian CI coefficients rotation operator
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@ -135,4 +135,4 @@
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\newcommand{\Ne}{N}
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\newcommand{\Ne}{N}
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\newcommand{\Norb}{K}
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\newcommand{\Norb}{K}
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\newcommand{\Nocc}{O}
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\newcommand{\Nocc}{O}
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\newcommand{\Nvir}{V}
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\newcommand{\Nvir}{V}
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@ -1,5 +1,5 @@
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace}
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\usepackage[version=4]{mhchem}
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\usepackage[version=4]{mhchem}
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\usepackage[utf8]{inputenc}
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\usepackage[utf8]{inputenc}
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@ -105,8 +105,8 @@ Approximating $\Sigma$ as the first-order term of its perturbative expansion wit
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\end{equation}
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\end{equation}
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Diagrammatically, $GW$ corresponds to a resummation of the (time-dependent) direct ring diagrams via the computation of the random-phase approximation (RPA) polarizability \cite{Ren_2012,Chen_2017} and is thus particularly well suited for weak correlation.
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Diagrammatically, $GW$ corresponds to a resummation of the (time-dependent) direct ring diagrams via the computation of the random-phase approximation (RPA) polarizability \cite{Ren_2012,Chen_2017} and is thus particularly well suited for weak correlation.
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Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017}
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\titou{Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017}
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The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013}
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The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013}}
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Despite a wide range of successes, many-body perturbation theory has well-documented limitations. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017a,Duchemin_2020}
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Despite a wide range of successes, many-body perturbation theory has well-documented limitations. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017a,Duchemin_2020}
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For example, modeling core electron spectroscopy requires core ionization energies which have been proven to be challenging for routine $GW$ calculations. \cite{Golze_2018,Golze_2020,Li_2022}
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For example, modeling core electron spectroscopy requires core ionization energies which have been proven to be challenging for routine $GW$ calculations. \cite{Golze_2018,Golze_2020,Li_2022}
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@ -122,7 +122,7 @@ These discontinuities have been traced back to a transfer of spectral weight bet
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In addition, systems, where the quasiparticle equation admits two solutions with similar spectral weights, are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Veril_2018,Forster_2021,Monino_2022}
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In addition, systems, where the quasiparticle equation admits two solutions with similar spectral weights, are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Veril_2018,Forster_2021,Monino_2022}
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In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasiparticle equation. \cite{Monino_2022}
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In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasiparticle equation. \cite{Monino_2022}
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Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} the present work investigates the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants.
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Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} the present work investigates the application of the SRG formalism to many-body perturbation theory in its $GW$ \titou{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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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 introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and coworkers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a,Zhang_2019,ChenyangLi_2021,Wang_2021,Wang_2023}
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This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and coworkers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a,Zhang_2019,ChenyangLi_2021,Wang_2021,Wang_2023}
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The SRG has also been successful in the context of nuclear structure theory, where it was first developed as a mature computational tool thanks to the work of several research groups.
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The SRG has also been successful in the context of nuclear structure theory, where it was first developed as a mature computational tool thanks to the work of several research groups.
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@ -138,6 +138,7 @@ correlation effects between the internal and external spaces can be incorporated
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The goal of this manuscript is to determine if the SRG formalism can effectively address the issue of intruder states in many-body perturbation theory, as it has in other areas of electronic and nuclear structure theory.
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The goal of this manuscript is to determine if the SRG formalism can effectively address the issue of intruder states in many-body perturbation theory, as it has in other areas of electronic and nuclear structure theory.
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This open question will lead us to an intruder-state-free static approximation of the self-energy derived from first-principles that can be employed in \titou{qs$GW$} calculations.
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This open question will lead us to an intruder-state-free static approximation of the self-energy derived from first-principles that can be employed in \titou{qs$GW$} calculations.
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\titou{Here, we focus on the $GW$ approximation but the subsequent derivations can be straightforwardly applied to other approximations such as GF(2) or $T$-matrix.}
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The manuscript is organized as follows.
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The manuscript is organized as follows.
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We begin by reviewing the $GW$ approximation in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
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We begin by reviewing the $GW$ approximation in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
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@ -155,49 +156,50 @@ This section starts by
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\label{sec:gw}
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\label{sec:gw}
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%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%
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\ant{The self-energy consider in this work will always be the $GW$ one [Eq.~\eqref{eq:gw_selfenergy}] but the subsequent derivations can be straightforwardly transposed to other approximations such as GF(2) or $GT$.
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The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation which, within the $GW$ approximation, reads
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In addition, we assume a Hartree-Fock (HF) starting point throughout the manuscript.}
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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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\begin{equation}
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\label{eq:quasipart_eq}
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\label{eq:quasipart_eq}
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\qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
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\qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
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\end{equation}
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\end{equation}
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where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the self-energy.
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where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the $GW$ self-energy.
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Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
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Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
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The self-energy can be physically understood as a \ant{correction to the HF problem (represented by $\bF$) accounting for dynamical screening effects}.
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The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects.
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Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently.
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Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently.
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Note that $\bSig(\omega)$ is dynamical, \titou{\ie} it depends on the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.
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\titou{Note that $\bSig(\omega)$ is dynamical, \ie it depends on the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.}
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The matrix elements of $\bSig(\omega)$ have the following analytic expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
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The matrix elements of $\bSig(\omega)$ have the following analytic expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
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\begin{equation}
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\begin{equation}
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\label{eq:GW_selfenergy}
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\label{eq:GW_selfenergy}
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\Sigma_{pq}(\omega)
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\Sigma_{pq}(\omega)
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= \sum_{iv} \frac{W_{p,(i,v)} W_{q,(i,v)}}{\omega - \epsilon_i + \Omega_{v} - \ii \eta}
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= \sum_{i\nu} \frac{W_{pi\nu} W_{qi\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta}
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+ \sum_{av} \frac{W_{p,(a,v)}W_{q,(a,v)}}{\omega - \epsilon_a - \Omega_{v} + \ii \eta},
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+ \sum_{a\nu} \frac{W_{pa\nu}W_{qa\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
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\end{equation}
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\end{equation}
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with the screened two-electron integrals defined as
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where $\eta$ is a positive infinitesimal and the screened two-electron integrals are
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\begin{equation}
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\begin{equation}
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\label{eq:GW_sERI}
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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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W_{pq\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu}+\bY_{\nu})_{ia},
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\end{equation}
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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). This problem is defined as
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where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct RPA problem defined as
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\begin{equation}
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\begin{equation}
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\bA \bX = \boldsymbol{\Omega} \bX,
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\mqty( \bA & \bB \\ -\bA & -\bB ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \boldsymbol{\Omega},
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\end{equation}
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\end{equation}
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with
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with
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\begin{equation}
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\begin{subequations}
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A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
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\begin{align}
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\end{equation}
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A_{ia,jb} & = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
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$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~\eqref{eq:GW_selfenergy}.
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\\
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The non-TDA case is discussed in Appendix~\ref{sec:nonTDA}.
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B_{ia,jb} & = \eri{ij}{ab}.
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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.
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\end{align}
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The indices $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
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\end{subequations}
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The diagonal matrix $\boldsymbol{\Omega}$ contains the 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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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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Because of the frequency dependence, fully solving the quasi-particle equation is a rather complicated task.
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Because of the frequency dependence, fully solving the quasi-particle equation is a rather complicated task.
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Hence, several approximate schemes have been developed to bypass self-consistency.
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Hence, several approximate schemes have been developed to bypass self-consistency.
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The most popular one is the one-shot (perturbative) scheme, known as $G_0W_0$, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
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The most popular one is the one-shot (perturbative) scheme, known as $G_0W_0$, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
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This results in $K$ quasi-particle equations that read
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\titou{Assuming a HF starting point,} this results in $K$ quasi-particle equations that read
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\begin{equation}
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\begin{equation}
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\label{eq:G0W0}
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\label{eq:G0W0}
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\epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
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\epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
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@ -257,8 +259,8 @@ The upfolded $GW$ quasi-particle equation is
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\label{eq:GWlin}
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\label{eq:GWlin}
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\begin{pmatrix}
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\begin{pmatrix}
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\bF & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
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\bF & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
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(\bV^{\text{2h1p}})^{\mathrm{T}} & \bC^{\text{2h1p}} & \bO \\
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\T{(\bV^{\text{2h1p}})} & \bC^{\text{2h1p}} & \bO \\
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(\bV^{\text{2p1h}})^{\mathrm{T}} & \bO & \bC^{\text{2p1h}} \\
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\T{(\bV^{\text{2p1h}})} & \bO & \bC^{\text{2p1h}} \\
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\end{pmatrix}
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\end{pmatrix}
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\begin{pmatrix}
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\begin{pmatrix}
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\bX \\
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\bX \\
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@ -288,10 +290,14 @@ and the corresponding coupling blocks read
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V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
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V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
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\end{align}
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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\"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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\begin{align}
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\begin{equation}
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\bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1} (\bV^{\hhp})^{\mathsf{T}} \\
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\begin{split}
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&+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1} (\bV^{\pph}\right)^{\mathsf{T}}, \notag
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\bSig(\omega)
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\end{align}
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& = \bV^{\hhp} \qty(\omega \mathbb{1} - \bC^{\hhp})^{-1} \T{(\bV^{\hhp})}
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\\
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& + \bV^{\pph} \qty(\omega \mathbb{1} - \bC^{\pph})^{-1} \T{(\bV^{\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 give exactly Eq.~(\ref{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 not.
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