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\newcommand { \LCPQ } { Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\' e de Toulouse, CNRS, UPS, France}
\begin { document}
\title { \large A similarity renormalization group approach to Green's function methods}
%\title{Excited states, symmetry breaking, and unphysical solutions in CASSCF theory}
%\title{Characterising state-specific CASSCF theory: Excited states, symmetry breaking, and unphysical solutions}
%\title{Exploring the CASSCF energy landscape: Excited states, symmetry breaking, and unphysical solutions}
\author { Antoine \surname { Marie} }
\email { amarie@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
\author { Pierre-Fran\c { c} ois \surname { Loos} }
\email { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
\begin { abstract}
\end { abstract}
\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { SRG-$ G _ 0 W _ 0 $ , ev$ GW $ and SRG-ev$ GW $ statistics}
\label { app:appendixA}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In this section, the values obtained with the two alternative SRG-based methods derived in the main manuscript, SRG-$ G _ 0 W _ 0 $ and SRG-ev$ GW $ , are reported along with their corresponding histogram plot of the errors.
For the sake of completeness, the SRG-regularized self-energy and quasiparticle equation used for the SRG-$ G _ 0 W _ 0 $ and SRG-ev$ GW $ calculations are reported below:
\begin { equation}
\epsilon _ p^ { \text { HF} } + \Sigma ^ \text { SRG-$ GW $ } _ { pp} (\omega ) - \omega = 0,
\end { equation}
with
\begin { equation}
\Sigma ^ \text { SRG-$ GW $ } _ { pp} (\omega ) = \sum _ { i\nu } \frac { (W_ { pi} ^ { \nu } )^ 2 } { \omega - \epsilon _ i + \Omega _ { \nu } } e^ { -2(\epsilon _ p - \epsilon _ i + \Omega _ { \nu } )^ 2 s} + \sum _ { a\nu } \frac { (W_ { pa} ^ { \nu } )^ 2} { \omega - \epsilon _ a - \Omega _ { \nu } } e^ { -2(\epsilon _ p - \epsilon _ a - \Omega _ { \nu } )^ 2 s} ,
\end { equation}
Therefore, the SRG-$ G _ 0 W _ 0 $ values are obtained by solving once these equations (one-shot procedure) without linearization, while the SRG-ev$ GW $ results correspond to solutions of these equations where self-consistency on the $ \epsilon _ p $ 's has been reached.
One observe in Table \ref { tab:tab1} that the $ G _ 0 W _ 0 $ and SRG-$ G _ 0 W _ 0 $ values are the same for all systems (up to $ \num { e - 2 } $ \si { \electronvolt } ).
Figure \ref { fig:supporting} shows that ev$ GW $ provides a slight improvement over $ G _ 0 W _ 0 $ , while ev$ GW $ and SRG-ev$ GW $ perform similarly.
One interesting fact is that the convergence of SRG-ev$ GW $ deteriorates faster than for SRG-qs$ GW $ with respect to $ s $ .
We suspect that it is due to the absence of the off-diagonal terms.
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%%% FIG 1 %%%
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\begin { figure*} [h]
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\includegraphics [width=\linewidth] { supporting}
\caption {
Histogram of the errors [with respect to $ \Delta $ CCSD(T)] for the principal IP of the $ GW $ 50 test set calculated using $ G _ 0 W _ 0 $ @HF, SRG-$ G _ 0 W _ 0 $ @HF, ev$ GW $ , and SRG-ev$ GW $ .
All calculations are performed with the aug-cc-pVTZ basis.
\label { fig:supporting} }
\end { figure*}
%%% %%% %%% %%%
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%%% FIG 2 %%%
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\begin { figure*} [h]
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\includegraphics [width=\linewidth] { supporting2}
\caption {
Histogram of the errors [with respect to $ \Delta $ CCSD(T)] for the principal EA of the $ GW $ 50 test set calculated using $ G _ 0 W _ 0 $ @HF, SRG-$ G _ 0 W _ 0 $ @HF, ev$ GW $ , and SRG-ev$ GW $ .
All calculations are performed with the aug-cc-pVTZ basis.
\label { fig:supporting} }
\end { figure*}
%%% %%% %%% %%%
%%% TABLE I %%%
\begin { table*}
\caption { Principal IP and EA (in eV) of the $ GW $ 50 test set calculated using $ \Delta $ CCSD(T) (reference), $ G _ 0 W _ 0 $ @HF, SRG-$ G _ 0 W _ 0 $ @HF, ev$ GW $ , and SRG-ev$ GW $ .
The statistical descriptors associated with the errors with respect to the reference values are also reported.
All calculations are performed with the aug-cc-pVTZ basis.}
\label { tab:tab1}
\begin { ruledtabular}
\begin { tabular} { ldddddddddd}
& \mc { 5} { c} { Principal IP} & \mc { 5} { c} { Principal EA} \\
\cline { 2-6} \cline { 7-11}
& \mcc { $ \Delta \text { CCSD ( T ) } $ } & \mcc { $ G _ 0 W _ 0 $ @HF} & \mcc { SRG-$ G _ 0 W _ 0 $ @HF} & \mcc { ev$ GW $ } & \mcc { SRG-ev$ GW $ } & \mcc { $ \Delta \text { CCSD ( T ) } $ } & \mcc { $ G _ 0 W _ 0 $ @HF} & \mcc { SRG-$ G _ 0 W _ 0 $ @HF} & \mcc { ev$ GW $ } & \mcc { SRG-ev$ GW $ } \\
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Mol. & \mcc { (Ref.)} & \mcc { ($ \eta = \num { e - 3 } $ )} & \mcc { ($ s = \num { e 3 } $ )} & \mcc { ($ \eta = \num { e - 1 } $ )} & \mcc { ($ s = \num { 50 } $ )} & \mcc { (Ref.)} & \mcc { ($ \eta = \num { e - 3 } $ )} & \mcc { ($ s = \num { e 3 } $ )} & \mcc { ($ \eta = \num { e - 1 } $ )} & \mcc { ($ s = \num { 50 } $ )} \\
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\hline
\ce { He} & 24.54 & 24.59 & 24.59 & 24.58 & 24.57 & -2.66 & -2.66 & -2.66 & -2.66 & -2.66 \\
\ce { Ne} & 21.47 & 21.46 & 21.46 & 21.30 & 21.29 & -5.09 & -5.25 & -5.25 & -5.24 & -5.24\\
\ce { H2} & 16.40 & 16.49 & 16.49 & 16.52 & 16.51 & -1.35 & -1.28 & -1.28 & -1.28 & -1.28 \\
\ce { Li2} & 5.25 & 5.38 & 5.38 & 5.44 & 5.42 & 0.34 & 0.17 & 0.17 & 0.16 & 0.17 \\
\ce { LiH} & 8.02 & 8.22 & 8.22 & 8.26 & 8.23 & -0.29 & 0.27 & 0.27 & 0.27 & 0.27 \\
\ce { HF} & 16.15 & 16.25 & 16.25 & 16.10 & 16.09 & -0.66 & -0.71 & -0.71 & -0.71 & -0.71 \\
\ce { Ar} & 15.60 & 15.72 & 15.72 & 15.67 & 15.66 & -2.55 & -2.68 & -2.68 & -2.67 & -2.67 \\
\ce { H2O} & 12.69 & 12.90 & 12.90 & 12.80 & 12.79 & -0.61 & -0.68 & -0.68 & -0.68 & -0.68 \\
\ce { LiF} & 11.47 & 11.40 & 11.40 & 11.20 & 11.18 & 0.35 & 0.33 & 0.33 & 0.33 & 0.33 \\
\ce { HCl} & 12.67 & 12.78 & 12.78 & 12.76 & 12.75 & -0.57 & -0.64 & -0.64 & -0.64 & -0.64 \\
\ce { BeO} & 9.95 & 9.74 & 9.74 & 9.64 & 9.61 & 2.17 & 2.28 & 2.28 & 2.30 & 2.31 \\
\ce { CO} & 13.99 & 14.80 & 14.80 & 14.77 & 14.76 & -1.57 & -1.66 & -1.66 & -1.65 & -1.65 \\
\ce { N2} & 15.54 & 17.10 & 17.10 & 17.10 & 17.09 & -2.37 & -2.10 & -2.10 & -2.10 & -2.10 \\
\ce { CH4} & 14.39 & 14.76 & 14.76 & 14.76 & 14.75 & -0.65 & -0.70 & -0.70 & -0.69 & -0.69 \\
\ce { BH3} & 13.31 & 13.68 & 13.68 & 13.70 & 13.69 & -0.09 & -0.46 & -0.46 & -0.46 & -0.45 \\
\ce { NH3} & 10.91 & 11.22 & 11.22 & 11.19 & 11.17 & -0.61 & -0.68 & -0.68 & -0.68 & -0.68 \\
\ce { BF} & 11.15 & 11.34 & 11.34 & 11.37 & 11.36 & -0.80 & -0.90 & -0.90 & -0.90 & -0.90 \\
\ce { BN} & 12.05 & 11.76 & 11.76 & 11.78 & 11.76 & 3.02 & 3.90 & 3.90 & 3.95 & 3.95 \\
\ce { SH2} & 10.39 & 10.51 & 10.51 & 10.51 & 10.50 & -0.52 & -0.60 & -0.60 & -0.60 & -0.60 \\
\ce { F2} & 15.81 & 16.35 & 16.35 & 16.15 & 16.14 & 0.32 & -0.53 & -0.53 & -0.47 & -0.47 \\
\ce { MgO} & 7.97 & 8.40 & 8.40 & 8.34 & 8.28 & 1.54 & 1.64 & 1.64 & 1.65 & 1.66 \\
\ce { O3} & 12.85 & 13.56 & 13.56 & 13.53 & 13.51 & 1.82 & 2.19 & 2.19 & 2.25 & 2.25 \\
\ce { C2H2} & 11.45 & 11.57 & 11.57 & 11.60 & 11.59 & -0.80 & -0.71 & -0.71 & -0.71 & -0.71 \\
\ce { HCN} & 13.76 & 13.86 & 13.86 & 13.87 & 13.86 & -0.53 & -0.52 & -0.52 & -0.52 & -0.52 \\
\ce { B2H6} & 12.27 & 12.81 & 12.81 & 12.81 & 12.80 & -0.52 & -0.56 & -0.56 & -0.56 & -0.56 \\
\ce { CH2O} & 10.93 & 11.39 & 11.39 & 11.34 & 11.32 & -0.60 & -0.61 & -0.61 & -0.60 & -0.60 \\
\ce { C2H4} & 10.69 & 10.74 & 10.74 & 10.78 & 10.77 & -1.90 & -0.75 & -0.75 & -0.74 & -0.74 \\
\ce { SiH4} & 12.79 & 13.22 & 13.22 & 13.23 & 13.22 & -0.53 & -0.59 & -0.59 & -0.59 & -0.59 \\
\ce { PH3} & 10.60 & 10.79 & 10.79 & 10.82 & 10.81 & -0.51 & -0.58 & -0.58 & -0.58 & -0.58 \\
\ce { CH4O} & 11.09 & 11.55 & 11.55 & 11.48 & 11.47 & -0.59 & -0.64 & -0.64 & -0.64 & -0.64 \\
\ce { H2NNH2} & 9.49 & 9.84 & 9.84 & 9.80 & 9.79 & -0.60 & -0.69 & -0.69 & -0.68 & -0.68 \\
\ce { HOOH} & 11.51 & 11.96 & 11.96 & 11.85 & 11.83 & -0.96 & -0.75 & -0.75 & -0.75 & -0.75 \\
\ce { KH} & 6.32 & 6.44 & 6.44 & 6.48 & 6.42 & 0.30 & 0.28 & 0.28 & 0.28 & 0.28 \\
\ce { Na2} & 4.93 & 4.98 & 4.98 & 5.03 & 5.02 & 0.36 & 0.26 & 0.26 & 0.24 & 0.26 \\
\ce { HN3} & 10.77 & 11.12 & 11.12 & 11.11 & 11.10 & -0.51 & -0.6 & -0.6 & -0.59 & -0.59 \\
\ce { CO2} & 13.80 & 14.24 & 14.24 & 14.16 & 14.15 & -0.88 & -0.98 & -0.98 & -0.97 & -0.97 \\
\ce { PN} & 11.90 & 12.33 & 12.33 & 12.34 & 12.33 & -0.02 & -0.03 & -0.03 & 0.01 & 0.01 \\
\ce { CH2O2} & 11.54 & 12.00 & 12.00 & 11.90 & 11.88 & -0.63 & -0.69 & -0.69 & -0.68 & -0.68 \\
\ce { C4} & 11.43 & 11.77 & 11.77 & 11.77 & 11.76 & 2.38 & 2.24 & 2.24 & 2.34 & 2.35 \\
\ce { C3H6} & 10.83 & 11.20 & 11.20 & 11.20 & 11.19 & -0.94 & -0.75 & -0.75 & -0.75 & -0.75 \\
\ce { C2H3F} & 10.63 & 10.84 & 10.84 & 10.85 & 10.84 & -0.65 & -0.69 & -0.69 & -0.68 & -0.68 \\
\ce { C2H4O} & 10.29 & 10.84 & 10.84 & 10.76 & 10.75 & -0.54 & -0.56 & -0.56 & -0.56 & -0.56 \\
\ce { C2H6O} & 10.82 & 11.37 & 11.37 & 11.31 & 11.30 & -0.58 & -0.65 & -0.65 & -0.64 & -0.64 \\
\ce { C3H8} & 12.13 & 12.61 & 12.61 & 12.60 & 12.59 & -0.63 & -0.70 & -0.70 & -0.70 & -0.70 \\
\ce { NaCl} & 9.10 & 9.20 & 9.20 & 9.16 & 9.13 & 0.67 & 0.64 & 0.64 & 0.64 & 0.64 \\
\ce { P2} & 10.72 & 10.49 & 10.49 & 10.52 & 10.51 & 0.43 & 0.47 & 0.47 & 0.53 & 0.54 \\
\ce { MgF2} & 13.93 & 13.94 & 13.94 & 13.74 & 13.72 & 0.29 & 0.15 & 0.15 & 0.16 & 0.16 \\
\ce { OCS} & 11.23 & 11.52 & 11.52 & 11.50 & 11.49 & -1.43 & -1.03 & -1.03 & -1.02 & -1.01 \\
\ce { SO2} & 10.48 & 11.38 & 11.38 & 11.34 & 11.33 & 2.24 & 2.82 & 2.82 & 2.87 & 2.88 \\
\ce { C2H3Cl} & 10.17 & 10.39 & 10.39 & 10.39 & 10.38 & -0.61 & -0.66 & -0.66 & -0.65 & -0.65 \\
\hline
MSE & & 0.29 & 0.29 & 0.26 & 0.25 & & 0.02 & 0.02 & 0.03 & 0.00 \\
MAE & & 0.33 & 0.33 & 0.32 & 0.31 & & 0.16 & 0.16 & 0.16 & 0.17 \\
RMSE & & 0.43 & 0.43 & 0.41 & 0.40 & & 0.28 & 0.28 & 0.29 & 0.28 \\
SDE & & 0.31 & 0.31 & 0.31 & 0.32 & & 0.29 & 0.29 & 0.29 & 0.29 \\
Min & & -0.29 & -0.29 & -0.31 & -0.34 & & -0.85 & -0.85 & -0.79 & -0.82 \\
Max & & 1.56 & 1.56 & 1.56 & 1.55 & & 1.15 & 1.15 & 1.16 & 1.14 \\
\end { tabular}
\end { ruledtabular}
\end { table*}
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\end { document}
