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add G0W0TDA + minor corrections

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Antoine Marie 2023-02-08 13:45:05 +01:00
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Manuscript/SRGGW.tex
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@ -287,6 +287,7 @@ This transformation can be performed continuously via a unitary matrix $\bU(s)$,
\bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
\end{equation}
where $s$ is the so-called flow parameter that controls the extent of the decoupling.
\ant{This flow parameter is related to an energy cut-off $\Lambda=\frac{1}{\sqrt{s}}$ such that at a finite value of $s$ the coupling elements relating states with an energy difference larger than $\Lambda$ are zero.}
By definition, we have $\bH(s=0) = \bH$ [or $\bU(s=0) = \bI$] and $\bH^\text{od}(s=\infty) = \bO$.
An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.~\eqref{eq:SRG_Ham} with respect to $s$, yielding the flow equation
@ -298,6 +299,7 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
\begin{equation}
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
\end{equation}
\ANT{I just realized that $\eta$ is used for the flow generator and the imaginary shift, is this a problem?}
To solve the flow equation at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$.
In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
@ -531,10 +533,12 @@ At $s=0$, the second-order correction vanishes, hence giving
\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)}.
\end{equation}
For $s\to\infty$, it tends towards the following static limit
\ant{
\begin{equation}
\label{eq:static_F2}
\lim_{s\to\infty} 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_{q,r\nu}.
\lim_{s\to\infty} \widetilde{\bF}(s) = \epsilon_p \delta_{pq} + \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}.
\end{equation}
}
while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
\begin{equation}
\lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = 0.
@ -556,16 +560,16 @@ This transformation is done gradually starting from the states that have the lar
\subsection{Alternative form of the static self-energy}
% ///////////////////////////%
Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and hermitian, the new alternative form of the self-energy reported in Eq.~\eqref{eq:static_F2} can be naturally used in qs$GW$ calculations to replace Eq.~\eqref{eq:sym_qsgw}.
Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistently is once again quite difficult to achieve, if not impossible.
Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and hermitian, the new alternative form of the \ant{self-energy} reported in Eq.~\eqref{eq:static_F2} can be naturally used in qs$GW$ calculations to replace Eq.~\eqref{eq:sym_qsgw}.
Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible.
However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~\eqref{eq:GW_renorm}.
This yields a $s$-dependent static self-energy which matrix elements read
\begin{multline}
\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) = \ant{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_{q,r\nu} \\ \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
\end{multline}
Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by brute-force symmetrization.
Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every state.
\ant{Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.}
Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.
It is well-known that in traditional qs$GW$ calculations, increasing the imaginary shift $\ii\eta$ to ensure convergence in difficult cases is most often unavoidable.
@ -573,10 +577,10 @@ Similarly, in SRG-qs$GW$, one might need to decrease the value of $s$ to ensure
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, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.
It is instructive to plot both regularizing functions, this is done in Fig.~\ref{fig:plot}.
The surfaces correspond to a value of the regularizing parameter value of 1.
The SRG surface is much smoother than its qs counterpart.
In fact the SRG regularization has less work to do because for $\eta=0$ there is a single singularity at $x=y=0$.
It is instructive to plot the functional form of both regularizing functions, this is done in Fig.~\ref{fig:plot}.
The surfaces are plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first order Taylor expansion around $(0,0)$ of both functional form are equal.
One can observe that the SRG surface is much smoother than its qs counterpart.
This is due to the fact that the SRG functional has less irregularities for $\eta=0$, in fact there is a single singularity at $x=y=0$.
On the other hand the function $f_{\text{qs}}(x,y;0)$ is singular on two entire axis, $x=0$ and $y=0$.
The smoothness of the SRG surface might induce a smoother convergence of SRG-qs$GW$ compared to qs$GW$.
The convergence properties and the accuracy of both static approximations will be quantitatively gauged in Sec.~\ref{sec:results}.
@ -647,11 +651,11 @@ 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.
Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
Figure~\ref{fig:fig2} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
\ant{The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result which is now well understood.} \cite{SzaboBook,Lewis_2019}
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.
Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IP as a function of the flow parameter (blue curve).
Figure~\ref{fig:fig2} also displays the SRG-qs$GW$ IP as a function of the flow parameter (blue curve).
At $s=0$, the IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
For $s\to\infty$, the IP reaches a plateau at an error that is significantly smaller than their $s=0$ starting point.
Even more, the value associated with this plateau is slightly more accurate than its qs$GW$ counterpart.
@ -669,18 +673,18 @@ The denominator of the last term is always positive while the 2h1p term is negat
When $s$ is increased, the first states that will be decoupled from the HOMO will be the 2p1h ones because their energy difference with the HOMO is larger than the ones of the 2h1p block.
Therefore, for small $s$ only the last term of Eq.~\eqref{eq:2nd_order_IP} will be partially included resulting in a positive correction to the IP.
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:fig2}.
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:fig2}.
The TDA IPs are now underestimated, unlike their RPA counterparts.
For both static self-energies, the TDA leads to a slight increase in the absolute error.
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.
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:fig3} shows the results for the Lithium dimer, \ce{Li2} is an interesting case because the HP IP is actually underestimated.
On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are overestimated
Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
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:fig2}.
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 hydride heterodimer, see the middle panel of Fig.~\ref{fig:fig2}.
@ -692,7 +696,7 @@ Once again, a plateau is attained and the corresponding value is slightly more a
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 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:fig3} but this is the other way around.
Therefore, it seems that the effect of the TDA can not be systematically predicted.
@ -710,7 +714,7 @@ In addition, the MgO and O3 molecules (which are part of GW100 as well) has been
\centering
\caption{First ionization potential in eV calculated using $\Delta$CCSD(T) (reference), HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$. The statistical descriptors are computed for the errors with respect to the reference.}
\label{tab:tab1}
\begin{tabular}{lccccc}
\hline
\hline
@ -785,48 +789,47 @@ We will now gauge the effect of the TDA for the screening on the accuracy of the
\centering
\caption{First ionization potential in eV calculated using \ANT{TO COMPLETE}. The statistical descriptors are computed for the errors with respect to the reference.}
\label{tab:tab1}
\begin{tabular}{lccccc}
-24.454162, -20.845919, -16.530702, -5.453359, -8.144555, -15.641055, -15.602545, -12.417807, -10.751003, -12.696334, -9.32538, -14.603221, -17.364737, -14.667361, -13.662675, -10.910541, -11.379826, -11.850654, -10.467744, -15.55351, -8.098392, -13.68235
\begin{tabular}{lccc}
\hline
\hline
molecule & qs$GW_{\text{TDA}}$ & SRG-qs$GW_{\text{TDA}}$ & & & \\
& $\eta=0.05$ & $s=100$ & & & \\
molecule & $G_0W_{\text{TDA},0}$@HF & qs$GW_{\text{TDA}}$ & SRG-qs$GW_{\text{TDA}}$ \\
& $\eta=0.001$ & $\eta=0.05$ & $s=100$ \\
\hline
\ce{He} & 24.48 & 24.39 & & & \\
\ce{Ne} & 21.23 & 20.92 & & & \\
\ce{H2} & 16.46 & 16.50 & & & \\
\ce{Li2} & 5.50 & 5.46 & & & \\
\ce{LiH} & 8.17 & 8.05 & & & \\
\ce{HF} & 15.79 & 15.66 & & & \\
\ce{Ar} & 15.42 & 15.46 & & & \\
\ce{H2O} & 12.40 & 12.31 & & & \\
\ce{LiF} & 11.02 & 10.85 & & & \\
\ce{HCl} & 12.65 & 12.59 & & & \\
\ce{BeO} & 10.21 & 10.05 & & & \\
\ce{CO} & 13.82 & 13.84 & & & \\
\ce{N2} & 15.15 & 15.21 & & & \\
\ce{CH4} & 14.50 & 14.47 & & & \\
\ce{BH3} & 13.57 & 13.54 & & & \\
\ce{NH3} & 10.75 & 10.68 & & & \\
\ce{BF} & 11.11 & 11.12 & & & \\
\ce{BN} & 12.05 & 12.04 & & & \\
\ce{SH2} & 10.44 & 10.38 & & & \\
\ce{F2} & 15.23 & 15.22 & & & \\
\ce{MgO} & 7.76 & 7.58 & & & \\
\ce{O3} & 12.22 & 12.22 & & & \\
\ce{He} & 24.45 & 24.48 & 24.39 \\
\ce{Ne} & 20.85 & 21.23 & 20.92 \\
\ce{H2} & 16.53 & 16.46 & 16.50 \\
\ce{Li2} & 5.45 & 5.50 & 5.46 \\
\ce{LiH} & 8.14 & 8.17 & 8.05 \\
\ce{HF} & 15.64 & 15.79 & 15.66 \\
\ce{Ar} & 15.60 & 15.42 & 15.46 \\
\ce{H2O} & 12.42 & 12.40 & 12.31 \\
\ce{LiF} & 10.75 & 11.02 & 10.85 \\
\ce{HCl} & 12.70 & 12.65 & 12.59 \\
\ce{BeO} & 9.33 & 10.21 & 10.05 \\
\ce{CO} & 14.60 & 13.82 & 13.84 \\
\ce{N2} & 17.36 & 15.15 & 15.21 \\
\ce{CH4} & 14.67 & 14.50 & 14.47 \\
\ce{BH3} & 13.66 & 13.57 & 13.54 \\
\ce{NH3} & 10.91 & 10.75 & 10.68 \\
\ce{BF} & 11.38 & 11.11 & 11.12 \\
\ce{BN} & 11.85 & 12.05 & 12.04 \\
\ce{SH2} & 10.47 & 10.44 & 10.38 \\
\ce{F2} & 15.55 & 15.23 & 15.22 \\
\ce{MgO} & 8.10 & 7.76 & 7.58 \\
\ce{O3} & 13.68 & 12.22 & 12.22 \\
\hline
MSE & -0.12 & -0.18 & & & \\
MAE & 0.22 & 0.25 & & & \\
SDE & 0.26 & 0.27 & & & \\
Min & -0.63 & -0.63 & & & \\
Max & 0.26 & 0.22 & & & \\
MSE & 0.07 & -0.12 & -0.18 \\
MAE & 0.37 & 0.22 & 0.25 \\
SDE & 0.55 & 0.26 & 0.27 \\
Min & -0.72 & -0.63 & -0.63 \\
Max & 1.82 & 0.26 & 0.22 \\
\hline
\hline
\end{tabular}
\end{table}
Part on approximation and correction for W:
TDA vs RPA,
SRG TDA same accuracy as Sym RPA !,
TDHF G0W0 not that bad in GW100 but bad in GW22, qsGW TDHF even worse even with SRG,
Maybe that would be nice to add SRG G0W0 to see if it mitigates the outliers of GW20 cf Bruneval 2021,
That would be nice to understand clearly why qsGWTDHF is worse (screening, gap, etc)