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modifications in Sec IV

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Pierre-Francois Loos 2023-02-05 10:23:46 +01:00
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@ -326,7 +326,7 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation. By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.
However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward. However, to do so, one must 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 which elegantly highlights the coupling terms. A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version which elegantly highlights the coupling terms in the process.
The upfolded $GW$ quasiparticle equation is \cite{Bintrim_2021,Tolle_2022} The upfolded $GW$ quasiparticle equation is \cite{Bintrim_2021,Tolle_2022}
\begin{equation} \begin{equation}
@ -374,8 +374,8 @@ The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitio
\end{equation} \end{equation}
which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}. which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other is not. Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} \titou{yield exactly the same energies} but one is linear and the other is 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. The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is only $\order{K}$ 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 Refs.~\onlinecite{Tolle_2022,Scott_2023}). 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}).
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. 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.
@ -399,13 +399,27 @@ Therefore, it is natural to define, within the SRG formalism, the diagonal and o
\end{align} \end{align}
\end{subequations} \end{subequations}
where we omit the $s$ dependence of the matrices for the sake of brevity. where we omit the $s$ dependence of the matrices for the sake of brevity.
Then, the aim is to solve order by order the flow equation [see Eq.~\eqref{eq:flowEquation}] knowing that the initial conditions are Then, the aim is to solve, order by order, the flow equation \eqref{eq:flowEquation} knowing that the initial conditions are
\begin{subequations}
\begin{align} \begin{align}
\bHd{0}(0) &= \begin{pmatrix} \bF & \bO \\ \bO & \bC \end{pmatrix}, \bHd{0}(0) & = \mqty( \bF & \bO \\ \bO & \bC ),
& &
\bHod{1}(0) &= \begin{pmatrix} \bO & \bW \\ \bW^{\dagger} & \bO \end{pmatrix}, \bHod{0}(0) & = \bO,
\\
\bHd{1}(0) & = \bO,
&
\bHod{1}(0) & = \mqty( \bO & \bW \\ \bW^{\dagger} & \bO ),
\end{align} \end{align}
and $\bHod{0}(0) = \bHd{1}(0) = \bO$, where the matrices $\bC$ and $\bV$ collect the 2h1p and 2p1h channels. \end{subequations}
where the supermatrices
\begin{subequations}
\begin{align}
\bC & = \mqty( \bC^{\text{2h1p}} & \bO \\ \bO & \bC^{\text{2p1h}} )
\\
\bW & = \mqty( \bW^{\text{2h1p}} & \bW^{\text{2p1h}} )
\end{align}
\end{subequations}
collect the 2h1p and 2p1h channels.
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. 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.
In particular, we focus here on the second-order renormalized quasiparticle equation. In particular, we focus here on the second-order renormalized quasiparticle equation.
@ -463,13 +477,13 @@ Equation \eqref{eq:F0_C0} implies
\begin{align} \begin{align}
\bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO, \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
\end{align} \end{align}
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 and, thanks to the \titou{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
\begin{equation} \begin{equation}
W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s} W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
\end{equation} \end{equation}
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. 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.
Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals. Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
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}). It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} \titou{in a different context} following a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
%///////////////////////////% %///////////////////////////%
\subsection{Second-order matrix elements} \subsection{Second-order matrix elements}
@ -490,7 +504,7 @@ with
\end{align} \end{align}
\end{subequations} \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. 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 Collecting every second-order term in the flow equation and performing the block matrix products results in the following differential equation
\begin{multline} \begin{multline}
\label{eq:diffeqF2} \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},
@ -500,7 +514,7 @@ which can be solved by simple integration along with the initial condition $\bF^
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} \\ 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}], \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}],
\end{multline} \end{multline}
with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ \titou{(where $\epsilon_F$ is the ...)}. with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
At $s=0$, the second-order correction vanishes 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} \begin{equation}
@ -508,23 +522,24 @@ At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends
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}}. 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} \end{equation}
Note that, in the limit $s\to\infty$, 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 $\widetilde{\bF}(s)$. \titou{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}. 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} \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 \PFL{This part has to be rewritten because it is too confusing.}
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} with matrix elements
\begin{equation} \begin{equation}
\label{eq:sym_qsGW} \label{eq:sym_qsGW}
\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}}. \Sigma_{pq}^{\titou{\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} \end{equation}
This alternative static form will be refered to as SRG-qs$GW$ in the following. This alternative static form will be referred to as SRG-qs$GW$ in the following.
Both approximations are closely related as they share the same diagonal terms when $\eta=0$. Both approximations are closely related as they share the same diagonal terms when $\eta=0$.
Also, note that the SRG static approximation is naturally Hermitian as opposed to the usual case where it is enforced by symmetrization. Also, note that the SRG static approximation is naturally Hermitian as opposed to the usual case where it is enforced by symmetrization.
However, as will be discussed in more detail in the results section, the convergence of the qs$GW$ scheme using $\widetilde{\bF}(\infty)$ is very poor. However, as we shall discuss further in Sec.~\ref{sec:results}, the convergence of the qs$GW$ scheme using \titou{$\widetilde{\bF}(\infty)$} is very poor.
This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero. This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
Indeed, in traditional qs$GW$ calculation increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable. Indeed, in traditional qs$GW$ calculation increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable.
Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
@ -532,9 +547,9 @@ Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
\label{eq:SRG_qsGW} \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 \qty[(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ], \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} \end{multline}
which depends on one regularising parameter $s$ analogously to $\eta$ in the usual qs$GW$. which depends on one regularizing 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 $\tilde{\bF}$. 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 $\tilde{\bF}$.
Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states. Therefore, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.
To conclude this section, we will discuss the case of discontinuities. To conclude this section, we will discuss the case of discontinuities.
Indeed, previously we mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level. Indeed, previously we mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
@ -545,7 +560,7 @@ 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 regulariser 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 regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
%=================================================================% %=================================================================%
\section{Computational details} \section{Computational details}
@ -570,8 +585,8 @@ However, in order to perform a black-box comparison of the methods these paramet
\label{sec:results} \label{sec:results}
%=================================================================% %=================================================================%
The results section is divided in two parts. The results section is divided into two parts.
The first step will be to analyse in depth the behavior of the two static self-energy approximations in a few test cases. The first step will be to analyze in depth the behavior of the two static self-energy approximations in a few test cases.
Then the accuracy of the IP yielded by the traditional and SRG schemes will be statistically gauged over a test set of molecules. Then the accuracy of the IP yielded by the traditional and SRG schemes will be statistically gauged over a test set of molecules.
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
@ -592,7 +607,7 @@ Then the accuracy of the IP yielded by the traditional and SRG schemes will be s
This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set. This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set.
Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to (w.r.t.) the CCSD(T) reference value. Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to (w.r.t.) the CCSD(T) reference value.
The HF IP (dashed black line) is 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 HF IP (dashed black line) is 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.}
\PFL{Check Szabo\&Ostlund, section on Koopman's theorem.} \PFL{Check Szabo\&Ostlund, section on Koopman's theorem.}
The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference. The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference.
%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. %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.
@ -618,9 +633,9 @@ Therefore, for small $s$ only the last term of Eq.~\eqref{eq:2nd_order_IP} will
As soon as $s$ is large enough to decouple the 2h1p block as well the IP will start to decrease and eventually go below the $s=0$ initial value as observed in Fig.~\ref{fig:fig1}. As soon as $s$ is large enough to decouple the 2h1p block as well the IP will start to decrease and eventually go below the $s=0$ initial value as observed in Fig.~\ref{fig:fig1}.
In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening are also considered in Fig.~\ref{fig:fig1}. In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening are also considered in Fig.~\ref{fig:fig1}.
The TDA IPs are now underestimated unlike their RPA counterparts. The TDA IPs are now underestimated, unlike their RPA counterparts.
For both static self-energies, the TDA leads to a slight increase of the absolute error. For both static self-energies, the TDA leads to a slight increase in the absolute error.
This trend will be investigated in more details in the next subsection. This trend will be investigated in more detail in the next subsection.
Now the flow parameter dependence of the SRG-qs$GW$ method will be investigated in three less well-behaved molecular systems. Now the flow parameter dependence of the SRG-qs$GW$ method will be investigated in three less well-behaved molecular systems.
The left panel of Fig.~\ref{fig:fig2} shows the results for the Lithium dimer, \ce{Li2} is an interesting case because the HP IP is actually underestimated. The left panel of Fig.~\ref{fig:fig2} shows the results for the Lithium dimer, \ce{Li2} is an interesting case because the HP IP is actually underestimated.
@ -629,16 +644,16 @@ Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportion
In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig1}. In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig1}.
Both TDA results are worse than their RPA counterparts but in this case the SRG-qs$GW_\TDA$ is more accurate than the qs$GW_\TDA$. Both TDA results are worse than their RPA counterparts but in this case the SRG-qs$GW_\TDA$ is more accurate than the qs$GW_\TDA$.
Now turning to the Lithium hydrid heterodimer, see middle panel of Fig.~\ref{fig:fig2}. Now turning to the lithium hydride heterodimer, see the middle panel of Fig.~\ref{fig:fig2}.
In this case the qs$GW$ IP is actually worse than the HF one which is already pretty accurate. In this case, the qs$GW$ IP is actually worse than the HF one which is already pretty accurate.
However, the SRG-qs$GW$ can improve slightly the accuracy with respect to HF. However, the SRG-qs$GW$ can improve slightly the accuracy with respect to HF.
Finally, the Beryllium oxyde is considered as a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021} Finally, the beryllium oxide is considered a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
The SRG-qs$GW$ could be converged without any problem even for large values of $s$. The SRG-qs$GW$ could be converged without any problem even for large values of $s$.
Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart. Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.
Note that for \ce{LiH} and \ce{BeO} the TDA actually improves the accuracy compared to RPA-based qs$GW$ schemes. Note that for \ce{LiH} and \ce{BeO} the TDA actually improves the accuracy compared to RPA-based qs$GW$ schemes.
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. However, as we will see in the next subsection these are just particular molecular systems and on average the RPA polarizability performs better than the TDA one.
Also the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig2} but this is the other way around. Also, the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig2} but this is the other way around.
Therefore, it seems that the effect of the TDA can not be systematically predicted. Therefore, it seems that the effect of the TDA can not be systematically predicted.
%%% FIG 2 %%% %%% FIG 2 %%%