small changes
This commit is contained in:
parent
2f59de64a0
commit
4683af29b7
@ -122,8 +122,8 @@ These discontinuities are due to a transfer of spectral weight between two solut
|
|||||||
This is another occurrence of the infamous intruder-state problem. \cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001}
|
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 \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 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}
|
In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regulariser inspired by the similarity renormalisation 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.
|
Encouraged by the recent successes of regularisation 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.
|
||||||
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.
|
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.
|
||||||
This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a}
|
This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a}
|
||||||
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.
|
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.
|
||||||
@ -205,7 +205,7 @@ This results in $K$ quasi-particle equations that read
|
|||||||
\end{equation}
|
\end{equation}
|
||||||
where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
|
where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
|
||||||
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).
|
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 $Z_{p,s}$
|
These solutions can be characterized by their spectral weight given by the renormalisation factor $Z_{p,s}$
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\label{eq:renorm_factor}
|
\label{eq:renorm_factor}
|
||||||
0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
|
0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
|
||||||
@ -244,10 +244,10 @@ Therefore, by suppressing this dependence the static approximation relies on the
|
|||||||
If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
|
If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
|
||||||
|
|
||||||
The satellites causing convergence problems are the so-called intruder states.
|
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 intruder state problem can be dealt with by introducing \textit{ad hoc} regularisers.
|
||||||
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...}
|
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...}
|
||||||
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}
|
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}
|
||||||
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.
|
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.
|
||||||
This is the aim of the rest of this work.
|
This is the aim of the rest of this work.
|
||||||
|
|
||||||
Applying the SRG to $GW$ could 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.
|
||||||
@ -303,11 +303,11 @@ As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $
|
|||||||
Therefore, these blocks will be the target of the 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}
|
\section{The similarity renormalisation group}
|
||||||
\label{sec:srg}
|
\label{sec:srg}
|
||||||
%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
|
||||||
The similarity renormalization group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
|
The similarity renormalisation group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
|
||||||
Therefore, the transformed Hamiltonian
|
Therefore, the transformed Hamiltonian
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\label{eq:SRG_Ham}
|
\label{eq:SRG_Ham}
|
||||||
@ -364,7 +364,7 @@ Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as w
|
|||||||
Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations.
|
Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations.
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%
|
||||||
\section{Regularized $GW$ approximation}
|
\section{Regularised $GW$ approximation}
|
||||||
\label{sec:srggw}
|
\label{sec:srggw}
|
||||||
%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
|
||||||
@ -547,7 +547,7 @@ However, doing a change of variable such that
|
|||||||
e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t}
|
e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t}
|
||||||
\end{align}
|
\end{align}
|
||||||
hence choosing a finite value of $t$ is well-designed to avoid discontinuities in the dynamic.
|
hence choosing a finite value of $t$ is well-designed to avoid discontinuities in the dynamic.
|
||||||
In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
|
In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regulariser introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
|
||||||
|
|
||||||
%=================================================================%
|
%=================================================================%
|
||||||
\section{Computational details}
|
\section{Computational details}
|
||||||
@ -579,30 +579,36 @@ Then the accuracy of the IP yielded by the Sym and SRG schemes will be statistic
|
|||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\linewidth]{fig1.pdf}
|
\includegraphics[width=\linewidth]{fig1.pdf}
|
||||||
\caption{
|
\caption{
|
||||||
Add caption
|
Add caption \ANT{Should we add $G_0W_0$?}
|
||||||
\label{fig:fig1}}
|
\label{fig:fig1}}
|
||||||
\end{figure*}
|
\end{figure*}
|
||||||
%%% %%% %%% %%%
|
%%% %%% %%% %%%
|
||||||
|
|
||||||
This section starts by considering the Neon atom and the water molecule in the aug-cc-pVTZ cartesian basis set in Fig.~\ref{fig:fig1}.
|
This section starts by considering the Neon atom and the water molecule in the aug-cc-pVTZ cartesian basis set in Fig.~\ref{fig:fig1}.
|
||||||
The HF values (orange lines) lie below the reference CCSD(T) ones, a result which is now well-understood.
|
Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to CCSD(T).
|
||||||
Indeed, this is due to an over(under? \ant{I will check this...}) screening of the interactions in the mean-field treatment. \cite{Lewis_2019}
|
The HF IPs (cyan lines) are overestimated, this is a consequence of the missing correlation, a result which is now well-understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
|
||||||
The usual Sym-qs$GW$ scheme (blue lines) brings a quantitative improvement as both IP energies are now within \SI{0.5}{\electronvolt} of the reference.
|
The usual Sym-qs$GW$ scheme (green lines) brings a quantitative improvement as both IP energies are now within \SI{0.5}{\electronvolt} of the reference.
|
||||||
The Neon atom is a well-behaved system and could be converged without regularization parameter while for water it was set to 0.01 to help convergence.
|
The Neon atom is a well-behaved system and could be converged without regularisation parameter while for water $\eta$ was set to 0.01 to help convergence.
|
||||||
|
|
||||||
Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IP energies as a function of the flow parameter.
|
Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IPs as a function of the flow parameter (blue curves).
|
||||||
At $s=0$, the IPs are equal to their HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
|
At $s=0$, the IPs are equal to their HF counterparts as expected from the discussion of Sec.~\ref{sec:srggw}.
|
||||||
For $s\to\infty$ both IPs reach a plateau that are significantly better than their $s=0$ starting point.
|
For $s\to\infty$ both IPs reach a plateau at an error that is significantly smaller than their $s=0$ starting point.
|
||||||
Even more, the values associated with these plateau are more accurate than their Sym-qs$GW$ counterparts.
|
Even more, the values associated with these plateau are more accurate than their Sym-qs$GW$ counterparts.
|
||||||
The SRG-qs$GW$ IPs do not increase smoothly between the HF values and their limits as for small $s$ values they are actually worst than the HF IPs.
|
However, the SRG-qs$GW$ error do not decrease smoothly between the HF values and their limits as for small $s$ values they are actually worst than the HF IPs.
|
||||||
|
|
||||||
|
In addition, we also considered the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening.
|
||||||
|
The TDA IPs are now understimated unlike their RPA counterparts.
|
||||||
|
The SRG-qs$GW$ absolute error for the IPs clearly deteriorates when going from RPA to TDA.
|
||||||
|
On the other hand, for qs$GW$ the TDA-based IP is better than the RPA one for Neon while it is the other way around for water.
|
||||||
|
This trend will be investigated in more details in the next subsection.
|
||||||
|
|
||||||
|
\ANT{Maybe we should add GF(2) because it allows us to explain the behavior of the SRG curve using perturbation theory.}
|
||||||
The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
|
The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
|
||||||
Add sentence about $GW$ better than GF2 when the results will be here.
|
Add sentence about $GW$ better than GF2 when the results will be here.
|
||||||
The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
|
The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
|
||||||
We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
|
We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
|
||||||
The GF(2) IP admits the following perturbation expansion...
|
The GF(2) IP admits the following perturbation expansion...
|
||||||
|
|
||||||
|
|
||||||
Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
|
Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
|
||||||
But the mechanism causing the increase/decrease of the $GW$ IPs as a function of $s$ should be closely related to the GF(2) one exposed above.
|
But the mechanism causing the increase/decrease of the $GW$ IPs as a function of $s$ should be closely related to the GF(2) one exposed above.
|
||||||
|
|
||||||
|
Binary file not shown.
Loading…
Reference in New Issue
Block a user