large changes, every data for GW50
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@ -17104,6 +17104,17 @@
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volume = {73},
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year = {2006}}
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@misc{Coveney_2023,
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title = {A Regularized Second-Order Correlation Method from {{Green}}'s Function Theory},
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author = {Coveney, Christopher J. N. and Tew, David P.},
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year = {2023},
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number = {arXiv:2302.13296},
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eprint = {2302.13296},
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eprinttype = {arxiv},
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doi = {10.48550/arXiv.2302.13296},
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archiveprefix = {arXiv}
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}
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@article{Ou_2016,
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author = {Ou, Qi and Subotnik, Joseph E.},
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doi = {10.1021/acs.jpca.6b03294},
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@ -130,7 +130,7 @@ The simpler one-shot $G_0W_0$ scheme \cite{Strinati_1980,Hybertsen_1985a,Hyberts
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These convergence problems and discontinuities can even happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
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In a recent study, Monino and Loos showed that the 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, such 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 in $GW$-based methods.
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Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, such as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022,Coveney_2023} the present work investigates the application of the SRG formalism in $GW$-based methods.
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In particular, we focus here on the possibility of curing the qs$GW$ convergence issues using the SRG.
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The SRG formalism has been developed independently by Wegner \cite{Wegner_1994} in the context of condensed matter systems and Glazek \& Wilson \cite{Glazek_1993,Glazek_1994} in light-front quantum field theory.
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@ -268,7 +268,7 @@ The satellites causing convergence problems are the above-mentioned intruder sta
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One can deal with them by introducing \textit{ad hoc} regularizers.
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For example, the $\ii\eta$ term in the denominators of Eq.~\eqref{eq:GW_selfenergy}, sometimes referred to as a broadening parameter linked to the width of the quasiparticle peak, is similar to the usual imaginary-shift regularizer employed in various other theories plagued by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}.
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However, this $\eta$ parameter stems from a regularization of the convolution that yields the self-energy and should theoretically be set to zero. \cite{Martin_2016}
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Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and, in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
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Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021,Coveney_2023} and, in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
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Nonetheless, 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.
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This is one of the aims of the present work.
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@ -302,7 +302,7 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
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\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
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\end{equation}
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\titou{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 $\boldsymbol{\eta}(s)$.}
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\ant{The flow equation can be approximately solved by introduction of an approximate form of $\boldsymbol{\eta}(s)$.}
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In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
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\begin{equation}
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\boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)},
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@ -337,7 +337,7 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
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\label{sec:srggw}
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%%%%%%%%%%%%%%%%%%%%%%
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\titou{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.}
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\ant{Finally, the two previous subsections will be combined by applying the SRG method to the $GW$ formalism.}
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However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
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A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
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Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022}
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@ -581,7 +581,7 @@ For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \
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%%% FIG 2 %%%
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\begin{figure*}
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\includegraphics[width=0.8\linewidth]{fig1.pdf}
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\includegraphics[width=0.8\linewidth]{fig2.pdf}
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\caption{
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Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$.}
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\label{fig:plot}
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@ -626,10 +626,24 @@ Indeed, it has been previously mentioned that intruder states are responsible fo
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Is it then possible to rely on the SRG machinery to remove discontinuities?
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Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite value of $s$ is suitable to avoid intruder states in its static part.
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However, performing the following bijective transformation
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\titou{\begin{align}
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\ant{\begin{align}
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e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t},
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% s = t/2 - \ln 2 - \ln[\sinh(t/2)]
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\end{align}}
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\end{align}
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on the renormalized quasiparticle equation,
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\begin{multline}
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F_{pq}^{(2)}(t)
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= \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu}
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\\
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\times e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] t},
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\end{multline}
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\begin{equation}
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\begin{split}
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\widetilde{\bSig}_{pq}(\omega; t)
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&= \sum_{i\nu} \frac{W_{pi}^{\nu} W_{qi}^{\nu}}{\omega - \epsilon_i + \Omega_{\nu}}\qty[1- e^{-\qty[(\Delta_{pi}^{\nu})^2 + (\Delta_{qi}^{\nu})^2 ] t}] \\
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&+ \sum_{a\nu} \frac{W_{pa}^{\nu} W_{qa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu}}\qty[1- e^{-\qty[(\Delta_{pa}^{\nu})^2 + (\Delta_{qa}^{\nu})^2 ] t}],
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\end{split}
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\end{equation}}
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reverses the situation and makes finite values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation.
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Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
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@ -639,8 +653,8 @@ Note that, after this transformation, the form of the regularizer is actually cl
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%=================================================================%
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% Reference comp det
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Our set of molecules is composed by closed-shell organic compounds that correspond to the 50 smallest atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015}
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Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR}
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Our set of molecules is composed by closed-shell compounds that correspond to the 50 smallest (wrt the number of electrons) atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015}
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Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level (without frozen core) \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR}
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The reference CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using Gaussian 16 \cite{g16} with default parameters, that is, within the restricted and unrestriced HF formalism for the neutral and charged species, respectively.
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% GW comp det
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@ -651,7 +665,7 @@ We use (restricted) HF guess orbitals and energies for all self-consistent $GW$
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The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively.
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In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme.
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However, in order to perform a black-box comparison, these parameters have been fixed to these default values.
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\titou{The $\eta$ value used in the convetional $G_0W_0$ and qs$GW$ calculations corresponds to the largest value where one successfully converges all systems.}
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\ant{The $\eta$ value has been set to \num{e-3} for the conventional $G_0W_0$ calculation while for (SRG-)qs$GW$ calculations the $\eta$ value has been chosen as the largest value where one successfully converges the 50 systems of the test set.}
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%=================================================================%
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\section{Results}
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@ -662,25 +676,21 @@ The results section is divided into two parts.
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The first step will be to analyze in depth the behavior of the two static self-energy approximations in a few test cases.
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Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-qs$GW$ schemes are statistically gauged over the test set of molecules described in Sec.~\ref{sec:comp_det}.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Flow parameter dependence of the SRG-qs$GW$ scheme}
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\label{sec:flow_param_dep}
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%%%%%%%%%%%%%%%%%%%%%%
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%%% FIG 2 %%%
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%%% FIG 3 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{fig2.pdf}
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\includegraphics[width=\linewidth]{fig3.pdf}
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\caption{
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Principal IP of the water molecule in the aug-cc-pVTZ basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method with and without TDA.
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Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
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\PFL{Should we have a similar figure for EAs? (maybe not water though)}
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\ANT{I did the plot, let's discuss it at the next meeting}
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\label{fig:fig2}}
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\end{figure}
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%%% %%% %%% %%%
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%%% FIG 3 %%%
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%%% FIG 4 %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{fig3.pdf}
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\includegraphics[width=\linewidth]{fig4.pdf}
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\caption{
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Principal IP of the \ce{Li2}, \ce{LiH} and \ce{BeO} in the aug-cc-pVTZ basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method with and without TDA.
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Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
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@ -688,6 +698,11 @@ Then, the accuracy of the principal IPs and EAs produced by the qs$GW$ and SRG-q
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\end{figure*}
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Flow parameter dependence of the SRG-qs$GW$ scheme}
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\label{sec:flow_param_dep}
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%%%%%%%%%%%%%%%%%%%%%%
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This section starts by considering a prototypical molecular system, the water molecule, in the aug-cc-pVTZ basis set.
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Figure \ref{fig:fig2} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
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The IP at the HF level (dashed black line) is too large; this is a consequence of the missing correlation and the lack of orbital relaxation for the cation, a result that is well understood. \cite{SzaboBook,Lewis_2019}
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@ -714,14 +729,15 @@ As $s$ increases, the first states that decouple from the HOMO are the 2p1h conf
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Therefore, for small $s$, only the last term of Eq.~\eqref{eq:2nd_order_IP} is partially included, resulting in a positive correction to the IP.
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As soon as $s$ is large enough to decouple the 2h1p block, the IP starts decreasing and eventually goes below the $s=0$ initial value as observed in Fig.~\ref{fig:fig2}.
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\ANT{I don't know if we should remove this paragraph and the TDA curves in Fig 3 and 4 or not...}
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In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening (dubbed qs$GW^\TDA$ and SRG-qs$GW^\TDA$) are also considered in Fig.~\ref{fig:fig2}.
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The TDA values are now underestimated the IP, unlike their RPA counterparts.
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The TDA values are now underestimating the IP, unlike their RPA counterparts.
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For both static self-energies, the TDA leads to a slight increase in the absolute error.
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This trend is investigated in more detail in the next subsection.
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\trashant{This trend is investigated in more detail in the next subsection.}
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Next, we investigate the flow parameter dependence of SRG-qs$GW$ for three more challenging molecular systems.
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The left panel of Fig.~\ref{fig:fig3} shows the results for the lithium dimer, \ce{Li2}, which is an interesting case because, unlike in water, the HF approximation underestimates the reference IP.
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On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are too large.
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On the other hand, the qs$GW$ and SRG-qs$GW$ IPs are too large.
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Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
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In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig2}.
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Both TDA results are worse than their RPA versions but, in this case, SRG-qs$GW^\TDA$ is more accurate than qs$GW^\TDA$.
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@ -730,7 +746,7 @@ We now turn to lithium hydride, \ce{LiH} (see middle panel of Fig.~\ref{fig:fig2
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In this case, the qs$GW$ IP is actually worse than the fairly accurate HF value.
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However, SRG-qs$GW$ improves slightly the accuracy as compared to HF.
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Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example where it is particularly difficult to converge self-consistent $GW$ calculations because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
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Finally, beryllium oxide, \ce{BeO}, is considered as it is a well-known example where it is particularly difficult to converge self-consistent $GW$ calculations due to the presence of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
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The SRG-qs$GW$ calculations could be converged without any issue even for large $s$ values.
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Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.
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@ -746,216 +762,139 @@ Therefore, it seems that the effect of the TDA cannot be systematically predicte
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\label{sec:SRG_vs_Sym}
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%%%%%%%%%%%%%%%%%%%%%%
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%%% FIG 4 %%%
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%%% FIG 5 %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{fig4.pdf}
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\includegraphics[width=\linewidth]{fig5.pdf}
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\caption{
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Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first ionization potential calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
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Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first ionization potential of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
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\label{fig:fig4}}
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\end{figure*}
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%%% %%% %%% %%%
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\begin{table*}
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\caption{First ionization potential (left) and first electron attachment (right) 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. \ANT{Maybe change the values of SRG with the one for s=1000}}
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\label{tab:tab1}
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\begin{ruledtabular}
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\begin{tabular}{l|ddddd|ddddd}
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Mol. & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} \\
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& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
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\hline
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\ce{He} & 24.54 & 24.98 & 24.59 & 24.58 & 24.54 & 2.66 & 2.70 & 2.66 & 2.66 & 2.66 \\
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\ce{Ne} & 21.47 & 23.15 & 21.46 & 21.83 & 21.59 & 5.09 & 5.47 & 5.25 & 5.19 & 5.19 \\
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\ce{H2} & 16.40 & 16.16 & 16.49 & 16.45 & 16.45 & 1.35 & 1.33 & 1.28 & 1.28 & 1.28 \\
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\ce{Li2} & 5.25 & 4.96 & 5.38 & 5.40 & 5.37 & -0.34 & 0.08 & -0.17 & -0.18 & -0.21 \\
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\ce{LiH} & 8.02 & 8.21 & 8.22 & 8.25 & 8.15 & 0.29 & -0.20 & -0.27 & -0.27 & -0.27 \\
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\ce{HF} & 16.15 & 17.69 & 16.25 & 16.45 & 16.34 & 0.66 & 0.81 & 0.71 & 0.70 & 0.70 \\
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\ce{Ar} & 15.60 & 16.08 & 15.71 & 15.61 & 15.63 & 2.55 & 2.97 & 2.68 & 2.64 & 2.65 \\
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\ce{H2O} & 12.69 & 13.88 & 12.90 & 12.98 & 12.88 & 0.61 & 0.80 & 0.68 & 0.65 & 0.66 \\
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\ce{LiF} & 11.47 & 12.91 & 11.40 & 11.75 & 11.58 & -0.35 & -0.29 & -0.33 & -0.32 & -0.33 \\
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\ce{HCl} & 12.67 & 12.98 & 12.78 & 12.77 & 12.72 & 0.57 & 0.79 & 0.64 & 0.63 & 0.63 \\
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\ce{BeO} & 9.95 & 10.45 & 9.74 & 10.32 & 10.18 & -2.17 & -1.80 & -2.28 & -2.10 & -2.13 \\
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\ce{CO} & 13.99 & 15.11 & 14.80 & 14.34 & 14.33 & 1.57 & 1.80 & 1.66 & 1.61 & 1.62 \\
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\ce{N2} & 15.54 & 16.68 & 17.10 & 15.93 & 15.91 & 2.37 & 2.20 & 2.10 & 2.10 & 2.10 \\
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\ce{CH4} & 14.39 & 14.83 & 14.76 & 14.67 & 14.63 & 0.65 & 0.79 & 0.70 & 0.68 & 0.68 \\
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\ce{BH3} & 13.31 & 13.59 & 13.68 & 13.62 & 13.59 & 0.09 & 0.81 & 0.46 & 0.29 & 0.30 \\
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\ce{NH3} & 10.91 & 11.69 & 11.21 & 11.18 & 11.10 & 0.61 & 0.80 & 0.68 & 0.66 & 0.66 \\
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\ce{BF} & 11.14 & 11.04 & 11.34 & 11.19 & 11.17 & 0.80 & 1.06 & 0.90 & 0.87 & 0.87 \\
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\ce{BN} & 12.05 & 11.55 & 11.76 & 11.89 & 11.90 & -3.02 & -2.97 & -3.90 & -3.41 & -3.44 \\
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\ce{SH2} & 10.39 & 10.49 & 10.51 & 10.50 & 10.45 & 0.52 & 0.76 & 0.60 & 0.58 & 0.59 \\
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\ce{F2} & 15.81 & 18.15 & 16.35 & 16.27 & 16.22 & -0.32 & 1.71 & 0.53 & -0.10 & -0.07 \\
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\ce{MgO} & 7.97 & 8.75 & 8.40 & 8.54 & 8.36 & -1.54 & -1.40 & -1.64 & -1.72 & -1.71 \\
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\ce{O3} & 12.85 & 13.29 & 13.56 & 13.34 & 13.27 & -1.82 & -1.32 & -2.19 & -2.22 & -2.17 \\
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\ce{C2H2} & & & & & & & & & & \\
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\ce{NCH} & & & & & & & & & & \\
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\ce{B2H6} & & & & & & & & & & \\
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\ce{H2CO} & & & & & & & & & & \\
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\ce{C2H4} & & & & & & & & & & \\
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\ce{SiH4} & & & & & & & & & & \\
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\ce{PH3} & & & & & & & & & & \\
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\ce{CH4O} & & & & & & & & & & \\
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\ce{H2NNH2} & & & & & & & & & & \\
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\ce{HOOH} & & & & & & & & & & \\
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\ce{KH} & & & & & & & & & & \\
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\ce{Na2} & & & & & & & & & & \\
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\ce{HN3} & & & & & & & & & & \\
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\ce{CO2} & & & & & & & & & & \\
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\ce{PN} & & & & & & & & & & \\
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\ce{CH2O2} & & & & & & & & & & \\
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\ce{C4} & & & & & & & & & & \\
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\ce{C3H6} & & & & & & & & & & \\
|
||||
\ce{C2H3F} & & & & & & & & & & \\
|
||||
\ce{C2H4O} & & & & & & & & & & \\
|
||||
\ce{C2H6O} & & & & & & & & & & \\
|
||||
\ce{C3H8} & & & & & & & & & & \\
|
||||
\ce{NaCl} & & & & & & & & & & \\
|
||||
\ce{P2} & & & & & & & & & & \\
|
||||
\ce{F2Mg} & & & & & & & & & & \\
|
||||
\ce{OCS} & & & & & & & & & & \\
|
||||
\ce{SO2} & & & & & & & & & & \\
|
||||
\ce{C2H3Cl} & & & & & & & & & & \\
|
||||
\hline
|
||||
& \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} \\
|
||||
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
|
||||
\hline
|
||||
MSE & & 0.64 & 0.26 & 0.24 & 0.17 & & -0.30 & -0.02 & 0.00 & 0.00 \\
|
||||
MAE & & 0.74 & 0.32 & 0.25 & 0.19 & & 0.32 & 0.19 & 0.11 & 0.12 \\
|
||||
SDE & & 0.71 & 0.39 & 0.18 & 0.15 & & 0.43 & 0.31 & 0.17 & 0.17 \\
|
||||
Min & & -0.50 & -0.29 & -0.16 & -0.15 & & -2.03 & -0.85 & -0.22 & -0.25 \\
|
||||
Max & & 2.35 & 1.56 & 0.56 & 0.42 & & 0.17 & 0.88 & 0.41 & 0.42 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
|
||||
Table \ref{tab:tab1} shows the principal IP of the 50 molecules considered in this work computed at various levels of theory.
|
||||
As mentioned previously the HF approximation overestimates the IPs with a mean signed error (MSE) of \SI{0.64}{\eV} and a mean absolute error (MAE) of \SI{0.74}{\eV}.
|
||||
Performing a one-shot $G_0W_0$ calculation on top of this mean-field starting point, $G_0W_0$@HF, reduces by more than a factor two the MSE and MAE, \SI{0.26}{\eV} and \SI{0.32}{\eV}, respectively.
|
||||
As mentioned previously the HF approximation overestimates the IPs with a mean signed error (MSE) of \SI{0.56}{\eV} and a mean absolute error (MAE) of \SI{0.69}{\eV}.
|
||||
Performing a one-shot $G_0W_0$ calculation on top of this mean-field starting point, $G_0W_0$@HF, reduces by more than a factor two the MSE and MAE, \SI{0.29}{\eV} and \SI{0.33}{\eV}, respectively.
|
||||
However, there are still outliers with large errors.
|
||||
For example, the IP of \ce{N2} is overestimated by \SI{1.56}{\eV}.
|
||||
Self-consistency can mitigate the error of the outliers as the maximum error at the qs$GW$ level is now \SI{0.56}{\eV} and the standard deviation of the error (SDE) is decreased from \SI{0.39}{\eV} for $G_0W_0$ to \SI{0.18}{\eV} for qs$GW$.
|
||||
In addition, the MSE and MAE (\SI{0.24}{\eV} and \SI{0.25}{\eV}, respectively) are also slightly improved with respect to $G_0W_0$@HF.
|
||||
Self-consistency can mitigate the error of the outliers as the maximum error at the qs$GW$ level is now \SI{0.57}{\eV} and the standard deviation of the error (SDE) is decreased from \SI{0.31}{\eV} for $G_0W_0$ to \SI{0.18}{\eV} for qs$GW$.
|
||||
In addition, the MSE and MAE (\SI{0.23}{\eV} and \SI{0.25}{\eV}, respectively) are also slightly improved with respect to $G_0W_0$@HF.
|
||||
|
||||
Now turning to the new results of this manuscript, \ie the alternative self-consistent scheme SRG-qs$GW$.
|
||||
Table \ref{tab:tab1} shows the SRG-qs$GW$ values for $s=100$.
|
||||
Now turning to the new method of this manuscript, \ie the SRG-qs$GW$ alternative self-consistent scheme.
|
||||
Table \ref{tab:tab1} shows the SRG-qs$GW$ values for $s=\num{e3}$.
|
||||
The statistical descriptors corresponding to this alternative static self-energy are all improved with respect to qs$GW$.
|
||||
In particular, the MSE and MAE are decreased by \SI{0.06}{\eV}
|
||||
Of course, these are slight improvements but this is done with no additional computational cost and can be implemented really easily by changing the form of the static approximation.
|
||||
The evolution of the statistical descriptors with respect to the various methods considered in Table~\ref{tab:tab1} is graphically illustrated by Fig.~\ref{fig:fig4}.
|
||||
The decrease of the MSE and SDE correspond to a shift of the maximum toward zero and a shrink of the distribution width, respectively.
|
||||
|
||||
%%% FIG 5 %%%
|
||||
\begin{table*}
|
||||
\caption{First ionization potential (left) and first electron attachment (right) 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{ruledtabular}
|
||||
\begin{tabular}{l|ddddd|ddddd}
|
||||
Mol. & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} \\
|
||||
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} \\
|
||||
\hline
|
||||
\ce{He} & 24.54 & 24.98 & 24.59 & 24.58 & 24.55 & -2.66 & -2.70 & -2.66 & -2.66 & -2.66 \\
|
||||
\ce{Ne} & 21.47 & 23.15 & 21.46 & 21.83 & 21.59 & -5.09 & -5.47 & -5.25 & -5.19 & -5.19 \\
|
||||
\ce{H2} & 16.40 & 16.16 & 16.49 & 16.45 & 16.45 & -1.35 & -1.33 & -1.28 & -1.28 & -1.28 \\
|
||||
\ce{Li2} & 5.25 & 4.96 & 5.38 & 5.40 & 5.37 & 0.34 & -0.08 & 0.17 & 0.18 & 0.21 \\
|
||||
\ce{LiH} & 8.02 & 8.21 & 8.22 & 8.25 & 8.15 & -0.29 & 0.20 & 0.27 & 0.27 & 0.27 \\
|
||||
\ce{HF} & 16.15 & 17.69 & 16.25 & 16.45 & 16.34 & -0.66 & -0.81 & -0.71 & -0.70 & -0.70 \\
|
||||
\ce{Ar} & 15.60 & 16.08 & 15.72 & 15.61 & 15.63 & -2.55 & -2.97 & -2.68 & -2.64 & -2.65 \\
|
||||
\ce{H2O} & 12.69 & 13.88 & 12.90 & 12.98 & 12.88 & -0.61 & -0.80 & -0.68 & -0.65 & -0.66 \\
|
||||
\ce{LiF} & 11.47 & 12.91 & 11.40 & 11.75 & 11.58 & 0.35 & 0.29 & 0.33 & 0.33 & 0.33 \\
|
||||
\ce{HCl} & 12.67 & 12.98 & 12.78 & 12.77 & 12.72 & -0.57 & -0.79 & -0.64 & -0.63 & -0.63 \\
|
||||
\ce{BeO} & 9.95 & 10.45 & 9.74 & 10.32 & 10.18 & 2.17 & 1.80 & 2.28 & 2.10 & 2.13 \\
|
||||
\ce{CO} & 13.99 & 15.11 & 14.80 & 14.34 & 14.33 & -1.57 & -1.80 & -1.66 & -1.61 & -1.62 \\
|
||||
\ce{N2} & 15.54 & 16.68 & 17.10 & 15.93 & 15.91 & -2.37 & -2.20 & -2.10 & -2.10 & -2.10 \\
|
||||
\ce{CH4} & 14.39 & 14.83 & 14.76 & 14.67 & 14.63 & -0.65 & -0.79 & -0.70 & -0.68 & -0.68 \\
|
||||
\ce{BH3} & 13.31 & 13.59 & 13.68 & 13.62 & 13.59 & -0.09 & -0.81 & -0.46 & -0.29 & -0.30 \\
|
||||
\ce{NH3} & 10.91 & 11.69 & 11.22 & 11.18 & 11.10 & -0.61 & -0.80 & -0.68 & -0.66 & -0.66 \\
|
||||
\ce{BF} & 11.15 & 11.04 & 11.34 & 11.19 & 11.18 & -0.80 & -1.06 & -0.90 & -0.87 & -0.86 \\
|
||||
\ce{BN} & 12.05 & 11.55 & 11.76 & 11.89 & 11.90 & 3.02 & 2.97 & 3.90 & 3.41 & 3.44 \\
|
||||
\ce{SH2} & 10.39 & 10.49 & 10.51 & 10.50 & 10.45 & -0.52 & -0.76 & -0.60 & -0.58 & -0.59 \\
|
||||
\ce{F2} & 15.81 & 18.15 & 16.35 & 16.27 & 16.22 & 0.32 & -1.71 & -0.53 & 0.10 & 0.07 \\
|
||||
\ce{MgO} & 7.97 & 8.75 & 8.40 & 8.54 & 8.36 & 1.54 & 1.40 & 1.64 & 1.72 & 1.71 \\
|
||||
\ce{O3} & 12.85 & 13.29 & 13.56 & 13.34 & 13.27 & 1.82 & 1.32 & 2.19 & 2.23 & 2.17 \\
|
||||
\ce{C2H2} & 11.45 & 11.16 & 11.57 & 11.46 & 11.43 & -0.80 & -0.80 & -0.71 & -0.71 & -0.71 \\
|
||||
\ce{NCH} & 13.76 & 13.50 & 13.86 & 13.75 & 13.73 & -0.53 & -0.61 & -0.52 & -0.55 & -0.54 \\
|
||||
\ce{B2H6} & 12.27 & 12.84 & 12.81 & 12.67 & 12.64 & -0.52 & -0.64 & -0.56 & -0.55 & -0.55 \\
|
||||
\ce{H2CO} & 10.93 & 12.09 & 11.39 & 11.33 & 11.25 & -0.60 & -0.70 & -0.61 & -0.62 & -0.62 \\
|
||||
\ce{C2H4} & 10.69 & 10.26 & 10.74 & 10.70 & 10.67 & -1.90 & -0.86 & -0.75 & -0.73 & -0.74 \\
|
||||
\ce{SiH4} & 12.79 & 13.23 & 13.22 & 13.15 & 13.11 & -0.53 & -0.69 & -0.59 & -0.57 & -0.58 \\
|
||||
\ce{PH3} & 10.60 & 10.60 & 10.79 & 10.76 & 10.73 & -0.51 & -0.71 & -0.58 & -0.56 & -0.57 \\
|
||||
\ce{CH4O} & 11.09 & 12.30 & 11.55 & 11.49 & 11.39 & -0.59 & -0.76 & -0.64 & -0.62 & -0.63 \\
|
||||
\ce{H2NNH2} & 9.49 & 10.38 & 9.84 & 9.81 & 9.73 & -0.60 & -0.82 & -0.69 & -0.65 & -0.65 \\
|
||||
\ce{HOOH} & 11.51 & 13.17 & 11.96 & 11.95 & 11.86 & -0.96 & -0.89 & -0.75 & -0.72 & -0.72 \\
|
||||
\ce{KH} & 6.32 & 6.61 & 6.44 & 6.50 & 6.38 & 0.30 & 0.21 & 0.28 & 0.28 & 0.28 \\
|
||||
\ce{Na2} & 4.93 & 4.53 & 4.98 & 5.03 & 5.01 & 0.36 & -0.01 & 0.26 & 0.27 & 0.30 \\
|
||||
\ce{HN3} & 10.77 & 11.00 & 11.12 & 10.92 & 10.89 & -0.51 & -0.75 & -0.6 & -0.56 & -0.56 \\
|
||||
\ce{CO2} & 13.80 & 14.82 & 14.24 & 14.12 & 14.06 & -0.88 & -1.22 & -0.98 & -0.95 & -0.95 \\
|
||||
\ce{PN} & 11.90 & 12.00 & 12.33 & 12.12 & 12.09 & -0.02 & -0.72 & -0.03 & 0.02 & 0.00 \\
|
||||
\ce{CH2O2} & 11.54 & 12.94 & 12.00 & 11.97 & 11.88 & -0.63 & -0.79 & -0.69 & -0.66 & -0.67 \\
|
||||
\ce{C4} & 11.43 & 11.61 & 11.77 & 11.57 & 11.54 & 2.38 & 0.58 & 2.24 & 2.29 & 2.30 \\
|
||||
\ce{C3H6} & 10.83 & 11.25 & 11.20 & 11.07 & 11.03 & -0.94 & -0.88 & -0.75 & -0.73 & -0.73 \\
|
||||
\ce{C2H3F} & 10.63 & 10.48 & 10.84 & 10.73 & 10.69 & -0.65 & -0.80 & -0.69 & -0.68 & -0.68 \\
|
||||
\ce{C2H4O} & 10.29 & 11.64 & 10.84 & 10.74 & 10.66 & -0.54 & -0.69 & -0.56 & -0.57 & -0.57 \\
|
||||
\ce{C2H6O} & 10.82 & 12.05 & 11.37 & 11.25 & 11.15 & -0.58 & -0.78 & -0.65 & -0.62 & -0.62 \\
|
||||
\ce{C3H8} & 12.13 & 12.73 & 12.61 & 12.51 & 12.46 & -0.63 & -0.83 & -0.70 & -0.67 & -0.67 \\
|
||||
\ce{NaCl} & 9.10 & 9.60 & 9.20 & 9.25 & 9.16 & 0.67 & 0.56 & 0.64 & 0.64 & 0.64 \\
|
||||
\ce{P2} & 10.72 & 10.05 & 10.49 & 10.43 & 10.40 & 0.43 & -0.35 & 0.47 & 0.48 & 0.47 \\
|
||||
\ce{F2Mg} & 13.93 & 15.46 & 13.94 & 14.23 & 14.07 & 0.29 & -0.03 & 0.15 & 0.21 & 0.21 \\
|
||||
\ce{OCS} & 11.23 & 11.44 & 11.52 & 11.37 & 11.32 & -1.43 & -1.27 & -1.03 & -0.97 & -0.98 \\
|
||||
\ce{SO2} & 10.48 & 11.47 & 11.38 & 10.85 & 10.82 & 2.24 & 1.84 & 2.82 & 2.74 & 2.68 \\
|
||||
\ce{C2H3Cl} & 10.17 & 10.13 & 10.39 & 10.27 & 10.24 & -0.61 & -0.79 & -0.66 & -0.65 & -0.65 \\
|
||||
\hline
|
||||
& \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRG-qs$GW$} \\
|
||||
& \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} & \mcc{(Reference)} & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e3}$} \\
|
||||
\hline
|
||||
MSE & & 0.56 & 0.29 & 0.23 & 0.17 & & -0.25 & 0.02 & 0.04 & 0.04 \\
|
||||
MAE & & 0.69 & 0.33 & 0.25 & 0.19 & & 0.31 & 0.16 & 0.13 & 0.12 \\
|
||||
SDE & & 0.68 & 0.31 & 0.18 & 0.16 & & 0.43 & 0.29 & 0.23 & 0.22 \\
|
||||
Min & & -0.67 & -0.29 & -0.29 & -0.32 & & -2.03 & -0.85 & -0.22 & -0.25 \\
|
||||
Max & & 2.34 & 1.56 & 0.57 & 0.42 & & 1.04 & 1.15 & 1.17 & 1.16 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
|
||||
%%% FIG 6 %%%
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{fig5.pdf}
|
||||
\includegraphics[width=\linewidth]{fig6.pdf}
|
||||
\caption{
|
||||
Temporary figure about convergence
|
||||
SRG-qs$GW$ and qs$GW$ MAE of the IPs for the GW50 test set. The bottom and top axes are equivalent and related by $\eta=1/2s^2$. A different marker has been used for qs$GW$ at $\eta=0.05$ because the MAE includes only 48 molecules.
|
||||
\label{fig:fig5}}
|
||||
\end{figure}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
The difference in
|
||||
In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme has been found to be much easier to converge than its qs$GW$ counterpart.
|
||||
Indeed, up to $s=\num{e3}$ self-consistency can be attained without any problems for the 50 compounds.
|
||||
For $s=\num{5e3}$, convergence could not be attained for 11 molecules out of 50, meaning that some intruder states were included in the static correction for this value of $s$.
|
||||
However, this is not a problem as the MAE of the test set is already converged for this value of $s$.
|
||||
This is illustrated by the blue curve of Fig.~\ref{fig:fig5} which shows the evolution of the MAE with respect to $s$ and $\eta=1/2s^2$.
|
||||
The plateau of the MAE is reached far before the convergence problem arrives.
|
||||
|
||||
In addition to this improvement of the accuracy, the SRG-qs$GW$ scheme is also much easier to converge than its qs$GW$ counterpart.
|
||||
Indeed, up to $s=10^3$ self-consistency can be attained without any problems (mean and max number of iterations = n for s=100).
|
||||
For $s=10^4$, convergence could not be attained for 12 molecules out of 22, meaning that some intruder states were included in the static correction for this value of $s$.
|
||||
This is illustrated in the case of \ce{H2O} in the upper panel Fig.
|
||||
However, this is not a problem as the MAE is already well converged before the intruder states are added to the SRG-qs$GW$ static self-energy (lower panel).
|
||||
On the other hand, the qs$GW$ convergence behaviour seems to be more erratic.
|
||||
At $\eta=\num{e-2}$ ($s=\num{5e3}$), convergence could not be achieved for 13 molecules while 2 molecules were already problematic at $\eta=\num{5e-2}$ ($s=200$).
|
||||
But this convergence problems are much more dramatic than for SRG-qs$GW$ because the MAE definitely do not reach a convergence plateau when these problem arise.
|
||||
For example, out of the 37 molecules that could be converged for $\eta=\num{e-2}$ the variation of the IP with respect to $\eta=\num{5e-2}$ can go up to \SI{0.1}{\eV}.
|
||||
|
||||
On the other hand, the qs$GW$ convergence with respect to $\eta$ is more difficult to evaluate.
|
||||
The whole set considered in this work could be converged for $\eta=0.1$.
|
||||
However, as soon as we decrease $\eta$ self-consistency could not be attained for the whole set of molecules using the black-box convergence parameters (see Sec.~\ref{sec:comp_det}).
|
||||
Unfortunately, the convergence of the IP is not as tight as in the SRG case.
|
||||
The values of the IP that could be converged for $\eta=0.01$ can vary between $10^{-3}$ and $10^{-1}$ with respect to $\eta=0.1$.
|
||||
This difference of behaviour is due to the energy (in)dependence of the regularizers.
|
||||
Indeed, the SRG regularizer first includes the terms that are important for the energy and finally adds the intruder states.
|
||||
On the other hand, the imaginary shift regularizer acts equivalently on intruder states and terms that contribute to the energy.
|
||||
|
||||
% \begin{table}
|
||||
% \caption{First ionization potential in eV calculated using $G_0W_0^{\text{TDA}}$@HF, qs$GW^{\text{TDA}}$ and SRG-qs$GW^{\text{TDA}}$. The statistical descriptors are computed for the errors with respect to the reference.}
|
||||
% \label{tab:tab1}
|
||||
% \begin{ruledtabular}
|
||||
% \begin{tabular}{lddd}
|
||||
% Mol. & \mcc{$G_0W_0^{\text{TDA}}$@HF} & \mcc{qs$GW^{\text{TDA}}$} & \mcc{SRG-qs$GW^{\text{TDA}}$} \\
|
||||
% & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{5e-2}$} & \mcc{$s=\num{e2}$} \\
|
||||
% \hline
|
||||
% \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 & 0.07 & -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 \\
|
||||
% \end{tabular}
|
||||
% \end{ruledtabular}
|
||||
% \end{table}
|
||||
|
||||
% \begin{table}
|
||||
% \caption{First electron attachment 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.}
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% \label{tab:tab2}
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% \begin{ruledtabular}
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% \begin{tabular}{lddddd}
|
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% Mol. & \mcc{$\Delta\text{CCSD(T)}$} & \mcc{HF} & \mcc{$G_0W_0$@HF} & \mcc{qs$GW$} & \mcc{SRg-qs$GW$} \\
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% & & & \mcc{$\eta=\num{e-3}$} & \mcc{$\eta=\num{e-1}$} & \mcc{$s=\num{e2}$} \\
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% \hline
|
||||
% \ce{He} & 2.66 & 2.70 & 2.66 & 2.66 & 2.66 \\
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% \ce{Ne} & 5.09 & 5.47 & 5.25 & 5.19 & 5.19 \\
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% \ce{H2} & 1.35 & 1.33 & 1.28 & 1.28 & 1.28 \\
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% \ce{Li2} & -0.34 & 0.08 & -0.17 & -0.18 & -0.21 \\
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% \ce{LiH} & 0.29 & -0.20 & -0.27 & -0.27 & -0.27 \\
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% \ce{HF} & 0.66 & 0.81 & 0.71 & 0.70 & 0.70 \\
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% \ce{Ar} & 2.55 & 2.97 & 2.68 & 2.64 & 2.65 \\
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% \ce{H2O} & 0.61 & 0.80 & 0.68 & 0.65 & 0.66 \\
|
||||
% \ce{LiF} & -0.35 & -0.29 & -0.33 & -0.32 & -0.33 \\
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% \ce{HCl} & 0.57 & 0.79 & 0.64 & 0.63 & 0.63 \\
|
||||
% \ce{BeO} & -2.17 & -1.80 & -2.28 & -2.10 & -2.13 \\
|
||||
% \ce{CO} & 1.57 & 1.80 & 1.66 & 1.61 & 1.62 \\
|
||||
% \ce{N2} & 2.37 & 2.20 & 2.10 & 2.10 & 2.10 \\
|
||||
% \ce{CH4} & 0.65 & 0.79 & 0.70 & 0.68 & 0.68 \\
|
||||
% \ce{BH3} & 0.09 & 0.81 & 0.46 & 0.29 & 0.30 \\
|
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% \ce{NH3} & 0.61 & 0.80 & 0.68 & 0.66 & 0.66 \\
|
||||
% \ce{BF} & 0.80 & 1.06 & 0.90 & 0.87 & 0.87 \\
|
||||
% \ce{BN} & -3.02 & -2.97 & -3.90 & -3.41 & -3.44 \\
|
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% \ce{SH2} & 0.52 & 0.76 & 0.60 & 0.58 & 0.59 \\
|
||||
% \ce{F2} & -0.32 & 1.71 & 0.53 & -0.10 & -0.07 \\
|
||||
% \ce{MgO} & -1.54 & -1.40 & -1.64 & -1.72 & -1.71 \\
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% \ce{O3} & -1.82 & -1.32 & -2.19 & -2.22 & -2.17 \\
|
||||
% \hline
|
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% MSE & & -0.30 & -0.02 & 0.00 & 0.00 \\
|
||||
% MAE & & 0.32 & 0.19 & 0.11 & 0.12 \\
|
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% SDE & & 0.43 & 0.31 & 0.17 & 0.17 \\
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% Min & & -2.03 & -0.85 & -0.22 & -0.25 \\
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||||
% Max & & 0.17 & 0.88 & 0.41 & 0.42 \\
|
||||
% \end{tabular}
|
||||
% \end{ruledtabular}
|
||||
% \end{table}
|
||||
|
||||
%%% FIG 6 %%%
|
||||
%%% FIG 7 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[width=\linewidth]{fig6.pdf}
|
||||
\includegraphics[width=\linewidth]{fig7.pdf}
|
||||
\caption{
|
||||
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first electron attachment calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
|
||||
Histogram of the errors (with respect to $\Delta$CCSD(T)) for the first electron attachment of the GW50 test set calculated using HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$.
|
||||
\label{fig:fig6}}
|
||||
\end{figure*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
Finally, we compare the performance of HF, $G_0W_0$@HF, qs$GW$ and SRG-qs$GW$ again but for the principal electron attachement (EA) energies.
|
||||
The raw results are given in Tab.~\ref{tab:tab2} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig6}.
|
||||
The raw results are given in Tab.~\ref{tab:tab1} while the corresponding histograms of the error distribution are plotted in Fig.~\ref{fig:fig6}.
|
||||
The HF EA are understimated in averaged with some large outliers while $G_0W_0$@HF mitigates the average error there are still large outliers.
|
||||
The performance of the two qs$GW$ schemes are quite similar for EA, \ie a MAE of \SI{\sim 0.1}{\electronvolt} and the error of the outliers is reduced with respect to $G_0W_0$@HF.
|
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|
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