modif in intro + first draft of the reworked section II
This commit is contained in:
parent
881b8d6c98
commit
2f59de64a0
@ -14961,7 +14961,7 @@
|
||||
year = {2012},
|
||||
bdsk-url-1 = {https://doi.org/10.1016/j.cpc.2011.12.006}}
|
||||
|
||||
@article{Lewis_2019a,
|
||||
@article{Lewis_2019,
|
||||
author = {Alan M. Lewis and Timothy C. Berkelbach},
|
||||
date-added = {2019-10-12 14:31:33 +0200},
|
||||
date-modified = {2019-10-12 14:32:30 +0200},
|
||||
|
@ -51,7 +51,8 @@
|
||||
\newcommand{\titou}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}}
|
||||
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
|
||||
\newcommand{\ant}[1]{\textcolor{green}{#1}}
|
||||
\newcommand{\ant}[1]{\textcolor{teal}{#1}}
|
||||
\newcommand{\ANT}[1]{\ant{(\underline{\bf ANT}: #1)}}
|
||||
|
||||
% addresses
|
||||
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
@ -99,7 +100,8 @@ The self-energy encapsulates all the Hartree-exchange-correlation effects which
|
||||
%Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation.
|
||||
Approximating $\Sigma$ as the first-order term of its perturbative expansion with respect to the screened Coulomb potential $W$ yields the so-called $GW$ approximation \cite{Hedin_1965,Martin_2016}
|
||||
\begin{equation}
|
||||
\Sigma^{\GW}(1,2) = \ii G(1,2) W(1,2).
|
||||
\label{eq:gw_selfenergy}
|
||||
\Sigma^{\GW}(1,2) = \ii G(1,2) W(1,2).
|
||||
\end{equation}
|
||||
Diagrammatically, $GW$ corresponds to a resummation of the direct ring diagrams and is thus particularly well suited for weak correlation.
|
||||
|
||||
@ -107,12 +109,18 @@ Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in
|
||||
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}
|
||||
|
||||
Despite a wide range of successes, many-body perturbation theory is not flawless. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017,Duchemin_2020}
|
||||
\PFL{to be expanded as discussed.}
|
||||
In particular, 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 exhibit some discontinuities. \cite{Veril_2018,Loos_2018b,Loos_2020e,Berger_2021,DiSabatino_2021}
|
||||
\ant{ For example, modelling core electron spectroscopy requires core ionisation energies which have been proved to be challenging for routine $GW$ calculations. \cite{Golze_2018}
|
||||
Many-body perturbation theory can also be used to access optical excitation energies through the Bethe-Salpeter. However, the accuracy is not yet satisfying for triplet excited states. \cite{Bruneval_2015,Jacquemin_2017a,Jacquemin_2017b}
|
||||
Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require more accuracy.
|
||||
Unfortunately, defining a systematic way to go beyond $GW$, the so-called vertex corrections, is a tricky task.
|
||||
Lewis and Berkelbach have shown that naive vertex corrections can even worsen the results with respect to the initial $GW$ results. \cite{Lewis_2019}
|
||||
We refer the reader to the recent review by Golze and co-workers for an extensive list of current challenges in many-body perturbation theory (see Ref.~\onlinecite{Golze_2019}) and we will now focus on another flaw throughout this manuscript.}
|
||||
|
||||
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 exhibit some discontinuities. \cite{Veril_2018,Loos_2018b,Loos_2020e,Berger_2021,DiSabatino_2021}
|
||||
Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is supposed to be valid.
|
||||
These discontinuities are due to a transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022}
|
||||
This is another occurrence of the infamous intruder-state problem. \cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001}
|
||||
In addition, systems for which \titou{two quasi-particle solutions} have a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Forster_2021}
|
||||
In addition, systems \ant{whose quasi-particle equation admits two solutions with} a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Forster_2021}
|
||||
|
||||
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 quasi-particle equation. \cite{Monino_2022}
|
||||
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} this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants.
|
||||
@ -130,7 +138,7 @@ By stopping the SRG transformation once all external configurations except the i
|
||||
correlation effects between the internal and external spaces can be incorporated (or folded) without the presence of intruder states.
|
||||
|
||||
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.
|
||||
\PFL{I think we should also mention that it may provide static approximations of the self-energy from first principles via this downfolding. What do you think?}
|
||||
\ant{This open question will lead us to an \textit{intruder-state-free first-principle static approximation of the self-energy} that can be used for qs$GW$ calculations.}
|
||||
|
||||
The manuscript is organized as follows.
|
||||
We begin by reviewing the $GW$ approximation in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
|
||||
@ -148,79 +156,104 @@ This section starts by
|
||||
\label{sec:gw}
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\PFL{Antoine, please move the various expressions related to the $GW$ quantities in this section.}
|
||||
\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 $GT$.
|
||||
In addition, we assume a Hartree-Fock (HF) starting point throughout the manuscript.}
|
||||
|
||||
\titou{Here and in the following, we assume a Hartree-Fock (HF) starting point.}
|
||||
The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation
|
||||
\begin{equation}
|
||||
\label{eq:quasipart_eq}
|
||||
\qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
|
||||
\end{equation}
|
||||
where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is \titou{(the correlation part of)} the self-energy.
|
||||
where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the self-energy.
|
||||
Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
|
||||
The self-energy can be physically understood as a dynamical \titou{screening} correction to the HF problem represented by $\bF$.
|
||||
The self-energy can be physically understood as a \ant{correction to the HF problem (represented by $\bF$) accounting for dynamical screening effects}.
|
||||
Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently.
|
||||
Note that $\bSig$ 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.
|
||||
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.
|
||||
|
||||
Because of this frequency dependence, fully solving this equation is a rather complicated task.
|
||||
The matrix elements of $\bSig(\omega)$ have the following analytic expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
|
||||
\begin{equation}
|
||||
\label{eq:GW_selfenergy}
|
||||
\Sigma_{pq}(\omega)
|
||||
= \sum_{iv} \frac{W_{p,(i,v)} W_{q,(i,v)}}{\omega - \epsilon_i + \Omega_{v} - \ii \eta}
|
||||
+ \sum_{av} \frac{W_{p,(a,v)}W_{q,(a,v)}}{\omega - \epsilon_a - \Omega_{v} + \ii \eta},
|
||||
\end{equation}
|
||||
with the screened two-electron integrals defined as
|
||||
\begin{equation}
|
||||
\label{eq:GW_sERI}
|
||||
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia},
|
||||
\end{equation}
|
||||
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
|
||||
\begin{equation}
|
||||
\bA \bX = \boldsymbol{\Omega} \bX,
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
|
||||
\end{equation}
|
||||
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~\eqref{eq:GW_selfenergy}.
|
||||
The non-TDA case is discussed in Appendix~\ref{sec:nonTDA}.
|
||||
Throughout the manuscript $p,q,r,s$ indices are used for general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
|
||||
The indices $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
|
||||
|
||||
Because of the frequency dependence, fully solving the quasi-particle equation is a rather complicated task.
|
||||
Hence, several approximate schemes have been developed to bypass self-consistency.
|
||||
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.
|
||||
In this case, one gets $K$ quasi-particle equations that read
|
||||
tr\begin{equation}
|
||||
This results in $K$ quasi-particle equations that read
|
||||
\begin{equation}
|
||||
\label{eq:G0W0}
|
||||
\epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
|
||||
\end{equation}
|
||||
where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
|
||||
The previous equation is non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$ (where the index $s$ is numbering solutions).
|
||||
The previous equations are non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$ (where the index $s$ is numbering solutions).
|
||||
These solutions can be characterized by their spectral weight given by the renormalization factor $Z_{p,s}$
|
||||
\begin{equation}
|
||||
\label{eq:renorm_factor}
|
||||
0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
|
||||
\end{equation}
|
||||
The solution with the largest weight is referred to as the quasi-particle while the others are known as satellites (or shake-up transitions).
|
||||
However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights and the quasi-particle is not well-defined.
|
||||
In fact, these cases are related to the discontinuities and convergence problems discussed earlier (see Sec.~\ref{sec:intro}) because the additional solutions with large weights are the previously mentioned intruder states.
|
||||
However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasi-particle is not well-defined.
|
||||
These additional solutions with large weights are the previously mentioned intruder states.
|
||||
|
||||
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
|
||||
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.
|
||||
Therefore, one can \titou{optimize} 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}
|
||||
\PFL{Maybe it is worth mentioning here that is is a fairly heuristic approach that is obviously system dependent?}
|
||||
Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
|
||||
To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~\eqref{eq:G0W0} and then this equation is solved for $\omega$ again until convergence is reached.
|
||||
Alternatively, one could solve this set of quasi-particle equations self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
|
||||
The solutions $\epsilon_p$ are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equation are solved for $\omega$ again.
|
||||
This procedure is iterated until convergence for $\epsilon_p$ is reached.
|
||||
\PFL{This is not quite right. It is probably going to be easier to explain when you're going to introduce the explicit expressions of these quantities.}
|
||||
However, if the quasi-particle solution is not well-defined, reaching self-consistency can be quite difficult, if not impossible.
|
||||
Even at convergence, the starting point dependence is not totally removed as the results still depend on the starting orbitals. \cite{Marom_2012}
|
||||
\ANT{Is it better now ?}
|
||||
However, if one of the quasi-particle equations does not have a well-defined quasi-particle solution, reaching self-consistency can be quite difficult, if not impossible.
|
||||
Even at convergence, the starting point dependence is not totally removed as the results still depend on the initial molecular orbitals. \cite{Marom_2012}
|
||||
|
||||
To update both energies and orbitals, one must take into account the off-diagonal elements in Eq.~\eqref{eq:quasipart_eq}.
|
||||
To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
|
||||
The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$.
|
||||
In order to update both the orbital energies and coefficients, one must consider the off-diagonal elements in $\bSig(\omega)$.
|
||||
To take into account the off-diagonal elements without solving the dynamic quasi-particle equation [Eq.~\eqref{eq:quasipart_eq}], one can resort to the quasi-particle self-consistent (qs) $GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
|
||||
Then the qs$GW$ problem is solved using the usual HF algorithm with $\bF$ replaced by $\bF + \bSig^{\qs}$.
|
||||
Various choices for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
|
||||
\begin{equation}
|
||||
\label{eq:sym_qsgw}
|
||||
\Sigma_{pq}^\qs = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
|
||||
\end{equation}
|
||||
This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the \titou{form of the} effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
|
||||
This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
|
||||
One of the main results of this manuscript is the derivation from first principles of an alternative static Hermitian form.
|
||||
This will be done in the next section.
|
||||
This will be done in the next sections.
|
||||
|
||||
In this case, as well self-consistency can be difficult to reach in cases where multiple solutions have large spectral weights.
|
||||
Multiple solutions arise due to the $\omega$ dependence of the self-energy.
|
||||
Therefore, by suppressing this dependence the static qs approximation relies on the fact that there is one well-defined quasi-particle solution.
|
||||
If it is not the case, the qs scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
|
||||
Once again, in cases where multiple solutions have large spectral weights, qs$GW$ self-consistency can be difficult to reach.
|
||||
Multiple solutions of Eq.~(\ref{eq:G0W0}) arise due to the $\omega$ dependence of the self-energy.
|
||||
Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasi-particle solution.
|
||||
If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
|
||||
|
||||
Convergence problems arise when a shake-up configuration has an energy similar to the associated quasi-particle solution, this satellite is the so-called intruder state.
|
||||
The satellites causing convergence problems are the so-called intruder states.
|
||||
The intruder state problem can be dealt with by introducing \textit{ad hoc} regularizers.
|
||||
The $\ii \eta$ term that is usually added in the denominator of the self-energy (see below) is the usual imaginary-shift regularizer used in various other theories flawed by intruder states. \cite{Battaglia_2022} \ant{more ref...}
|
||||
Various other regularizers are possible and in particular one of us has shown that a regularizer inspired by the SRG had some advantages, in the $GW$ case, over the imaginary shift one. \cite{Monino_2022}
|
||||
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 regularizer used in various other theories flawed by intruder states. \cite{Battaglia_2022} \ant{more ref...}
|
||||
Various other regularizers are possible and in particular one of us has shown that a regularizer inspired by the SRG had some advantages over the imaginary shift. \cite{Monino_2022}
|
||||
But it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
|
||||
This is the aim of this work.
|
||||
This is the aim of the rest of this work.
|
||||
|
||||
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.
|
||||
Applying the SRG to $GW$ could gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
|
||||
However, to do so one needs to identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
|
||||
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.
|
||||
From now on, we will restrict ourselves to the $GW$ in the Tamm-Dancoff approximation (TDA) case but the same derivation could be done for the non-TDA $GW$ and GF(2) self-energies.
|
||||
The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively.
|
||||
The upfolded $GW$ quasi-particle equation is the following
|
||||
The upfolded $GW$ quasi-particle equation is
|
||||
\begin{equation}
|
||||
\label{eq:GWlin}
|
||||
\begin{pmatrix}
|
||||
@ -244,9 +277,9 @@ The upfolded $GW$ quasi-particle equation is the following
|
||||
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_{i}^{\GW} + \epsilon_{j}^{\GW} - \epsilon_{a}^{\GW}) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} ,
|
||||
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_{i} + \epsilon_{j} - \epsilon_{a} ) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} ,
|
||||
\\
|
||||
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_{a}^{\GW} + \epsilon_{b}^{\GW} - \epsilon_{i}^{\GW}) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd},
|
||||
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_{a} + \epsilon_{b} - \epsilon_{i} ) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd},
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
and the corresponding coupling blocks read
|
||||
@ -255,46 +288,19 @@ and the corresponding coupling blocks read
|
||||
&
|
||||
V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
|
||||
\end{align}
|
||||
Throughout the manuscript $p,q,r,s$ indices are used for general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
|
||||
The indices $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
|
||||
The usual $GW$ non-linear equation
|
||||
\begin{equation}
|
||||
\label{eq:GWnonlin}
|
||||
\left( \bF + \bSig(\omega) \right) \bX = \omega \bX,
|
||||
\end{equation}
|
||||
can be obtained by applying L\"odwin partitioning technique to Eq.~\eqref{eq:GWlin} \cite{Lowdin_1963,Bintrim_2021} which gives the following the expression for the self-energy
|
||||
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}
|
||||
\begin{align}
|
||||
\bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1} (\bV^{\hhp})^{\mathsf{T}} \\
|
||||
&+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1} (\bV^{\pph}\right)^{\mathsf{T}}, \notag
|
||||
\end{align}
|
||||
which can be further developed as
|
||||
\begin{equation}
|
||||
\label{eq:GW_selfenergy}
|
||||
\Sigma_{pq}(\omega)
|
||||
= \sum_{iv} \frac{W_{p,(i,v)} W_{q,(i,v)}}{\omega - \epsilon_i + \Omega_{v} - \ii \eta}
|
||||
+ \sum_{av} \frac{W_{p,(a,v)}W_{q,(a,v)}}{\omega - \epsilon_a - \Omega_{v} + \ii \eta},
|
||||
\end{equation}
|
||||
with the screened integrals defined as
|
||||
\begin{equation}
|
||||
\label{eq:GW_sERI}
|
||||
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia},
|
||||
\end{equation}
|
||||
where $\bX$ is the matrix of eigenvectors of the particle-hole direct RPA (dRPA) problem in the TDA defined as
|
||||
\begin{equation}
|
||||
\bA \bX = \boldsymbol{\Omega} \bX,
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
|
||||
\end{equation}
|
||||
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~\eqref{eq:GW_selfenergy}.
|
||||
which can be further developed to give exactly Eq.~(\ref{eq:GW_selfenergy}).
|
||||
|
||||
Equations \eqref{eq:GWlin} and \eqref{eq:GWnonlin} have exactly the same solutions but one is linear and the other not.
|
||||
The price to pay for this linearity is that the size of the matrix in the former equation is $\order{K^3}$ while it is $\order{K}$ in the latter one.
|
||||
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case).
|
||||
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other not.
|
||||
The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is $\order{K}$ in the latter.
|
||||
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Ref.~\cite{Tolle_2022}).
|
||||
|
||||
As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $ V^\text{2p1h}$ are coupling the 1h and 1p configuration to the dressed 2h1p and 2p1h configurations.
|
||||
Therefore, these blocks will be the target of our SRG transformation but before going into more detail we will review the SRG formalism.
|
||||
Therefore, these blocks will be the target of the SRG transformation but before going into more detail we will review the SRG formalism.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{The similarity renormalization group}
|
||||
@ -606,7 +612,7 @@ But the mechanism causing the increase/decrease of the $GW$ IPs as a function of
|
||||
\item BeO interesting because usually difficult to converge in qs$GW$ (cf Forster 2021)
|
||||
\end{itemize}
|
||||
|
||||
%%% FIG 2 %%%
|
||||
%%% FIG 2 %%%
|
||||
\begin{figure*}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{fig2.pdf}
|
||||
@ -636,6 +642,7 @@ 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).}
|
||||
The authors thank Francesco Evangelista for inspiring discussions.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -650,11 +657,11 @@ The data that supports the findings of this study are available within the artic
|
||||
\appendix
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Non-TDA $GW$ equations}
|
||||
\section{Non-TDA $GW$ and $GW_{\text{TDHF}}$ equations}
|
||||
\label{sec:nonTDA}
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The $GW$ self-energy without TDA is the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
|
||||
The matrix elements of the $GW$ self-energy without TDA are the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
|
||||
\begin{equation}
|
||||
\label{eq:GWnonTDA_sERI}
|
||||
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
|
||||
@ -726,7 +733,7 @@ and the corresponding coupling blocks read
|
||||
Using the SRG on this matrix instead of Eq.~\eqref{eq:GWlin} gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{GF(2) equations}
|
||||
\section{GF(2) equations \ant{NOT SURE THAT WE KEEP IT}}
|
||||
\label{sec:GF2}
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
Loading…
Reference in New Issue
Block a user