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Pierre-Francois Loos 2020-02-07 15:55:01 +01:00
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BSE-PES.tex
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@ -515,24 +515,33 @@ Additional graphs and tables for other basis sets can be found in the {\SI}.
\begin{squeezetable}
\begin{table*}
\caption{
Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained with the cc-pVQZ basis set at various levels of theory.
Equilibrium bond length $\Req$ (in bohr) and correlation energy $\Ec$ (in millihartree) for the ground state of diatomic molecules obtained with the cc-pVQZ basis set at various levels of theory.
When irregularities appear in the PES, the values are reported in parenthesis and they have been obtained by fitting a Morse potential to the PES.
The error (in \%) compared to the reference CC3 values are reported in square brackets.
}
\label{tab:Req}
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
& \mc{8}{c}{Molecules} \\
\cline{2-9}
Method & \ce{H2} & \ce{LiH} & \ce{LiF} & \ce{HCl} & \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} \\
\begin{tabular}{llrrrrrrrr}
& & \mc{8}{c}{Molecules} \\
\cline{3-10}
Method & & \ce{H2} & \ce{LiH} & \ce{LiF} & \ce{HCl} & \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} \\
\hline
CC3 & 1.402 & 3.019 & 2.963 & 2.403 & 2.075 & 2.136 & 2.390 & 2.663 \\
CCSD & 1.402[$+0.00\%$]& 3.020[$+0.03\%$] & 2.953[$-0.34\%$] & 2.398[$-0.21\%$] & 2.059[$-0.77\%$] & 2.118[$-0.84\%$] & 2.380[$-0.42\%$] & 2.621[$-1.58\%$] \\
CC2 & 1.391[$-0.78\%$]& 2.989[$-0.99\%$] & 2.982[$+0.64\%$] & 2.396[$-0.29\%$] & 2.106[$+1.49\%$] & 2.156[$+0.94\%$] & 2.393[$+0.13\%$] & 2.665[$+0.08\%$] \\
MP2 & 1.391[$-0.78\%$]& 3.008[$-0.36\%$] & 2.970[$+0.24\%$] & 2.395[$-0.33\%$] & 2.091[$+0.77\%$] & 2.137[$+0.05\%$] & 2.382[$-0.33\%$] & 2.634[$-1.09\%$] \\
BSE@{\GOWO}@HF & 1.399[$-0.21\%$]& 3.017[$-0.07\%$] & (2.974)[$+0.37\%$] & \gb{(2.408)} & \gb{(2.070)} & \gb{(2.130)} & \gb{(2.383)} & (2.640)[$-0.86\%$] \\
RPA@{\GOWO}@HF & 1.382[$-1.43\%$]& 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & \gb{(2.389)} & \gb{(2.045)} & \gb{(2.110)} & \gb{(2.367)} & (2.571)[$-3.45\%$] \\
RPAx@HF & 1.394[$-0.57\%$]& 3.011[$-0.26\%$] & 2.944[$-0.64\%$] & 2.391[$-0.50\%$] & \gb{(2.041)} & \gb{(2.105)} & \gb{(2.367)} & \gb{(2.563)}\\
RPA@HF & 1.386[$-1.14\%$]& 2.994[$-0.83\%$] & 2.946[$-0.57\%$] & 2.382[$-0.87\%$] & \gb{(2.042)} & \gb{(2.104)} & \gb{(2.365)} & \gb{(2.571)}\\
CC3 & $\Req$ & 1.402 & 3.019 & 2.963 & 2.403 & 2.075 & 2.136 & 2.390 & 2.663 \\
& $\Ec$ & 40.382 & 69.974 & 383.686 & 382.188 & 494.393 & 477.580 & 447.472 & 668.875 \\
CCSD & $\Req$ & 1.402[$+0.00\%$] & 3.020[$+0.03\%$] & 2.953[$-0.34\%$] & 2.398[$-0.21\%$] & 2.059[$-0.77\%$] & 2.118[$-0.84\%$] & 2.380[$-0.42\%$] & 2.621[$-1.58\%$] \\
& $\Ec$ & 40.382[$+0.00\%$] & 69.845[$-0.18\%$] & 372.580[$-2.89\%$] & 370.764[$-2.99\%$] & 470.627[$-4.81\%$] & 455.214[$-4.68\%$] & 432.856[$-3.27\%$] & 644.001[$-3.72\%$] \\
CC2 & $\Req$ & 1.391[$-0.78\%$] & 2.989[$-0.99\%$] & 2.982[$+0.64\%$] & 2.396[$-0.29\%$] & 2.106[$+1.49\%$] & 2.156[$+0.94\%$] & 2.393[$+0.13\%$] & 2.665[$+0.08\%$] \\
& $\Ec$ & 33.259[$-17.64\%$] & 57.289[$-18.13\%$] & 376.712[$-1.82\%$] & 356.909[$-6.61\%$] & 488.017[$-1.29\%$] & 465.492[$-2.53\%$] & 427.285[$-4.51\%$] & 654.878[$-2.09\%$] \\
MP2 & $\Req$ & 1.391[$-0.78\%$] & 3.008[$-0.36\%$] & 2.970[$+0.24\%$] & 2.395[$-0.33\%$] & 2.091[$+0.77\%$] & 2.137[$+0.05\%$] & 2.382[$-0.33\%$] & 2.634[$-1.09\%$] \\
& $\Ec$ & 33.164[$-17.87\%$] & 57.930[$-17.21\%$] & 372.997[$-2.79\%$] & 355.652[$-6.94\%$] & 477.951[$-3.33\%$] & 455.020[$-4.72\%$] & 421.600[$-5.78\%$] & 644.349[$-3.67\%$] \\
BSE@{\GOWO}@HF & $\Req$ & 1.399[$-0.21\%$] & 3.017[$-0.07\%$] & (2.974)[$+0.37\%$] & \gb{(2.408)} & \gb{(2.070)} & \gb{(2.130)} & \gb{(2.383)} & (2.640)[$-0.86\%$] \\
& $\Ec$ & 46.498[$+15.15\%$] & 78.075[$+11.58\%$] & 388.907[$+1.36\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] \\
RPA@{\GOWO}@HF & $\Req$ & 1.382[$-1.43\%$] & 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & \gb{(2.389)} & \gb{(2.045)} & \gb{(2.110)} & \gb{(2.367)} & (2.571)[$-3.45\%$] \\
& $\Ec$ & 57.567[$+42.56\%$] & 101.092[$+44.47\%$] & 473.053[$+23.29\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] \\
RPAx@HF & $\Req$ & 1.394[$-0.57\%$] & 3.011[$-0.26\%$] & 2.944[$-0.64\%$] & 2.391[$-0.50\%$] & \gb{(2.041)} & \gb{(2.105)} & \gb{(2.367)} & \gb{(2.563)} \\
& $\Ec$ & 37.886[$-6.18\%$] & 65.203[$-6.82\%$] & 343.604[$-10.45\%$] & 344.249[$-9.93\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] \\
RPA@HF & $\Req$ & 1.386[$-1.14\%$] & 2.994[$-0.83\%$] & 2.946[$-0.57\%$] & 2.382[$-0.87\%$] & \gb{(2.042)} & \gb{(2.104)} & \gb{(2.365)} & \gb{(2.571)} \\
& $\Ec$ & 57.332[$+41.98\%$] & 100.164[$+43.15\%$] & 465.905[$+21.43\%$] & 442.675[$+15.83\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] \\
\end{tabular}
\end{ruledtabular}
\end{table*}
@ -563,7 +572,7 @@ However, in the case of \ce{LiF}, the attentive reader would have observed a sma
As observed in Refs.~\onlinecite{vanSetten_2015,Maggio_2017,Loos_2018} and explained in details in Refs.~\onlinecite{Veril_2018,Duchemin_2020}, these irregularities, which makes particularly tricky the location of the minima, are due to ``jumps'' between distinct solutions of the $GW$ quasiparticle equation.
Including a broadening via the increasing the value of $\eta$ in the $GW$ self-energy and the screened Coulomb operator soften the problem, but does not remove it completely.
Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
When irregularities are present in the PES, we have fitted a Morse potential of the form $M(r) = D_e\{1-\exp[-\alpha(r-r_0)]\}^2$ to the PES in order to provide an estimate of the equilibrium bond length.
When irregularities are present in the PES, we have fitted a Morse potential of the form $M(R) = D_0\qty{1-\exp[-\alpha\qty(R-\Req)]}^2$ to the PES in order to provide an estimate of the equilibrium bond length.
These values are reported in parenthesis in Table \ref{tab:Req}.
For the smooth PES where one can obtain both the genuine minimum and the fitted minimum (\ie, based on the Morse curve), this procedure has been shown to be very accurate with an error of the order of $10^{-3}$ bohr in most cases.