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Pierre-Francois Loos 2020-08-22 09:46:07 +02:00
parent b883906b87
commit 7ddfc221d1
7 changed files with 282 additions and 266 deletions
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1
benzene.dat
View File

@ -21,3 +21,4 @@
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489
benzene.nb
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@ -8837,7 +8844,7 @@ Cell[BoxData[
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Axes->{True, True}, Axes->{True, True},
AxesLabel->{None, None}, AxesLabel->{None, None},
AxesOrigin->{1.8333333333333337`, -854.4583072531007}, AxesOrigin->{1.8333333333333337`, -855.5417835678822},
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Frame->{{False, False}, {False, False}}, Frame->{{False, False}, {False, False}},
FrameLabel->{{None, None}, {None, None}}, FrameLabel->{{None, None}, {None, None}},
@ -8857,7 +8864,7 @@ Cell[BoxData[
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PlotRange->{{1.8333333333333337`, PlotRange->{{1.8333333333333337`,
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PlotRangeClipping->True, PlotRangeClipping->True,
PlotRangePadding->{{ PlotRangePadding->{{
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56
benzene.tex
View File

@ -78,21 +78,21 @@ The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecit
\label{tab:energy} \label{tab:energy}
} }
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{ldc} \begin{tabular}{llc}
Method & \tabc{$E_c$} & Ref. \\ Method & \tabc{$E_c$} & Ref. \\
\hline \hline
ASCI & -860.0(2) & \onlinecite{Eriksen_2020} \\ ASCI & $-860.0(2)$ & \onlinecite{Eriksen_2020} \\
iCIPT2 & -861.1(5) & \onlinecite{Eriksen_2020} \\ iCIPT2 & $-861.1(5)$ & \onlinecite{Eriksen_2020} \\
CCSDTQ & -862.4 & \onlinecite{Eriksen_2020} \\ CCSDTQ & $-862.4$ & \onlinecite{Eriksen_2020} \\
DMRG & -862.8(7) & \onlinecite{Eriksen_2020} \\ DMRG & $-862.8(7)$ & \onlinecite{Eriksen_2020} \\
FCCR(2) & -863.0 & \onlinecite{Eriksen_2020} \\ FCCR(2) & $-863.0$ & \onlinecite{Eriksen_2020} \\
CAD-FCIQMC & -863.4 & \onlinecite{Eriksen_2020} \\ CAD-FCIQMC & $-863.4$ & \onlinecite{Eriksen_2020} \\
AS-FCIQMC & -863.7(3) & \onlinecite{Eriksen_2020} \\ AS-FCIQMC & $-863.7(3)$ & \onlinecite{Eriksen_2020} \\
SHCI & -864.2(2) & \onlinecite{Eriksen_2020} \\ SHCI & $-864.2(2)$ & \onlinecite{Eriksen_2020} \\
\hline \hline
ph-AFQMC & -864.3(4) & \onlinecite{Lee_2020} \\ ph-AFQMC & $-864.3(4)$ & \onlinecite{Lee_2020} \\
\hline \hline
CIPSI & -8xx.x(x) & This work \\ CIPSI & \titou{$-86x.x(x)$} & This work \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
@ -117,43 +117,48 @@ Being late to the party, we obviously cannot report blindly our CIPSI results.
However, following the philosophy of Eriksen \textit{et al.}, \cite{Eriksen_2020} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind. However, following the philosophy of Eriksen \textit{et al.}, \cite{Eriksen_2020} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind.
We then follow our usual ``protocol'' \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} by performing a preliminary SCI calculation using Hartree-Fock orbitals in order to generate a SCI wave function with at least $10^7$ determinants. We then follow our usual ``protocol'' \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} by performing a preliminary SCI calculation using Hartree-Fock orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
Natural orbitals (NOs) are then computed based on this wave function, and a new, larger SCI calculation is performed with this new set of orbitals. Natural orbitals (NOs) are then computed based on this wave function, and a new, larger SCI calculation is performed with this new set of orbitals.
This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element. The magnitude of $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit. The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element. The magnitude of $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems. As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems.
We then linearly extrapolate the total SCI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wave functions. We then linearly extrapolate the total SCI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wave functions.
Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value. Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by \titou{the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value.}
We believe that it provides a very safe estimate of the extrapolation error. We believe that it provides a very safe estimate of the extrapolation error.
%Note that all the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as described in Ref.~\onlinecite{Applencourt_2018}. %Note that all the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as described in Ref.~\onlinecite{Applencourt_2018}.
The corresponding energies are reported in Table \ref{tab:NOvsLO} as functions of the number of determinants in the variational space $N_\text{det}$.
A second run has been performed with localized orbitals. A second run has been performed with localized orbitals.
A Pipek-Mezey localization procedure \cite{Pipek_1989} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, and vi) the rest. \titou{More information needed here.} Starting from the Hartree-Fock orbitals, a Pipek-Mezey localization procedure \cite{Pipek_1989} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, and vi) the higher virtual orbitals. \titou{More information needed here.}
As one can see from the results of Table \ref{tab:NOvsLO}, the variational energy as well as the PT2 corrected energy is much lower with localized orbitals for a same number of determinants. As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of $N_\text{det}$, the variational energy as well as the PT2-corrected energies are much lower with localized orbitals than with NOs. We, therefore, consider these energies more trustworthy, and we will based our best estimate of the correlation energy of benzene on these calculations.
The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ as a function of $N_\text{det}$ (left panel).
The right panel of Fig.~\ref{fig:CIPSI} shows $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$, and their corresponding \titou{two}-point linear extrapolation curves that we have used to get our final estimate of the correlation energy.
% Results and discussion % Results
Our final numbers are gathered in Table \ref{tab:extrap_dist_table}, where, following the notations of Ref.~\onlinecite{Eriksen_2020}, we report, in addition to the final variational energies $\Delta E_{\text{var.}}$, the
extrapolation distances, $\Delta E_{\text{dist}}$, defined as the difference between the final computed energy, $\Delta E_{\text{final}}$, and the extrapolated energy, $\Delta E_{\text{extrap.}}$ associated with the ASCI, iCI, SHCI, CIPSI, and DMRG results.
The three flavours of SCI fall into an interval ranging from $-860.0$ m$E_h$ (ASCI) to $-864.2$ m$E_h$ (SHCI), while the other methods yield correlation energies ranging from $-863.7$ to $-862.8$ m$E_h$. Our final CIPSI number is \titou{$-86x.xx$} m$E_h$.
% Timings
The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer. The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
Each Irene's AMD node is a dual-socket AMD Rome (Epyc) CPU@2.60 GHz with 256GiB of RAM, with a total of 64 physical CPU cores. Each Irene's AMD node is a dual-socket AMD Rome (Epyc) CPU@2.60 GHz with 256GiB of RAM, with a total of 64 physical CPU cores.
These nodes are connected via Infiniband HDR100. These nodes are connected via Infiniband HDR100.
The three flavours of SCI fall into an interval ranging from $-863.7$ to $-862.8$ m$E_h$.
The CIPSI number is ?
%%$ FIG. 1 %%% %%$ FIG. 1 %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=0.4\linewidth]{fig1a} \includegraphics[width=0.4\linewidth]{fig1a}
\hspace{0.08\linewidth} \hspace{0.08\linewidth}
\includegraphics[width=0.4\linewidth]{fig1b} \includegraphics[width=0.4\linewidth]{fig1b}
\caption{ \caption{
Convergence of the CIPSI correlation energy for benzene using localized orbitals. Convergence of the CIPSI correlation energy using localized orbitals.
Left: $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of the number of determinants in the variational space. Left: $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of the number of determinants in the variational space.
Right: $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$. Right: $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$.
The two-point linear extrapolation curves (dashed lines) are also reported. The \titou{two}-point linear extrapolation curves (dashed lines) are also reported.
The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes. The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes.
\label{fig:CIPSI} \label{fig:CIPSI}
} }
\end{figure*} \end{figure*}
%%% TABLE II %%% %%% TABLE II %%%
\begin{squeezetable} %\begin{squeezetable}
\begin{table*} \begin{table*}
\caption{Variational energy $E_\text{var.}$, second-order perturbative correction $E_\text{PT2}$ and its renormalized version $E_\text{rPT2}$ (in $E_h$) as a function of the number of determinants $N_\text{det}$ for the ground-state of the benzene molecule computed in the cc-pVDZ basis set. \caption{Variational energy $E_\text{var.}$, second-order perturbative correction $E_\text{PT2}$ and its renormalized version $E_\text{rPT2}$ (in $E_h$) as a function of the number of determinants $N_\text{det}$ for the ground-state of the benzene molecule computed in the cc-pVDZ basis set.
The statistical error on $E_\text{PT2}$, corresponding to one standard deviation, are reported in parenthesis.} The statistical error on $E_\text{PT2}$, corresponding to one standard deviation, are reported in parenthesis.}
@ -187,10 +192,11 @@ The statistical error on $E_\text{PT2}$, corresponding to one standard deviation
2\,621\,440 & $-231.439\,324$ & $-231.553\,845(572)$ & $-231.551\,544(560)$ & $-231.473\,751$ & $-231.555\,261(403)$ & $-231.554\,159(397)$ \\ 2\,621\,440 & $-231.439\,324$ & $-231.553\,845(572)$ & $-231.551\,544(560)$ & $-231.473\,751$ & $-231.555\,261(403)$ & $-231.554\,159(397)$ \\
5\,242\,880 & $-231.450\,156$ & $-231.557\,541(534)$ & $-231.555\,558(524)$ & $-231.485\,829$ & $-231.558\,303(362)$ & $-231.557\,451(358)$ \\ 5\,242\,880 & $-231.450\,156$ & $-231.557\,541(534)$ & $-231.555\,558(524)$ & $-231.485\,829$ & $-231.558\,303(362)$ & $-231.557\,451(358)$ \\
10\,485\,760 & $-231.461\,927$ & $-231.559\,390(481)$ & $-231.557\,796(474)$ & $-231.497\,515$ & $-231.562\,568(322)$ & $-231.561\,901(319)$ \\ 10\,485\,760 & $-231.461\,927$ & $-231.559\,390(481)$ & $-231.557\,796(474)$ & $-231.497\,515$ & $-231.562\,568(322)$ & $-231.561\,901(319)$ \\
20\,971\,520 & $-231.474\,019$ & $-231.561\,315(430)$ & $-231.560\,063(424)$ & $-231.508\,714$ & $-231.564\,707(275)$ & $-231.564\,223(273)$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table*}
\end{squeezetable} %\end{squeezetable}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE II %%% %%% TABLE II %%%
@ -199,7 +205,7 @@ The statistical error on $E_\text{PT2}$, corresponding to one standard deviation
The final variational energies $\Delta E_{\text{var.}}$ are also reported. The final variational energies $\Delta E_{\text{var.}}$ are also reported.
See Ref.~\onlinecite{Eriksen_2020} for more details. See Ref.~\onlinecite{Eriksen_2020} for more details.
All the energies are given in m$E_h$. All the energies are given in m$E_h$.
\label{extrap_dist_table} \label{tab:extrap_dist_table}
} }
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lcccc} \begin{tabular}{lcccc}
@ -208,7 +214,7 @@ The statistical error on $E_\text{PT2}$, corresponding to one standard deviation
ASCI & $-737.1$ & $-835.4$ & $-860.0$ & $-24.6$ \\ ASCI & $-737.1$ & $-835.4$ & $-860.0$ & $-24.6$ \\
iCI & $-730.0$ & $-833.7$ & $-861.1$ & $-27.4$ \\ iCI & $-730.0$ & $-833.7$ & $-861.1$ & $-27.4$ \\
SHCI & $-827.2$ & $-852.8$ & $-864.2$ & $-11.4$ \\ SHCI & $-827.2$ & $-852.8$ & $-864.2$ & $-11.4$ \\
CIPSI & $-8xx.x$ & $-8xx.x$ & $-8xx.x$ & $-xx.x$ \\ CIPSI & \titou{$-8xx.x$} & \titou{$-8xx.x$} & \titou{$-86x.x$} & \titou{$-xx.x$} \\
DMRG & $-859.2$ & $-859.2$ & $-862.8$ & $-3.6$ \\ DMRG & $-859.2$ & $-859.2$ & $-862.8$ & $-3.6$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}

1
benzene_noS2.dat
View File

@ -20,3 +20,4 @@
2621440 -231.43932438 -231.55384542 0.00057169 -231.55154435 0.00056021 2621440 -231.43932438 -231.55384542 0.00057169 -231.55154435 0.00056021
5242880 -231.45015608 -231.55754133 0.00053356 -231.55555800 0.00052371 5242880 -231.45015608 -231.55754133 0.00053356 -231.55555800 0.00052371
10485760 -231.46192749 -231.55939032 0.00048147 -231.55779568 0.00047359 10485760 -231.46192749 -231.55939032 0.00048147 -231.55779568 0.00047359
20971520 -231.47401920 -231.56131493 0.00043027 -231.56006331 0.00042410

1
benzene_noS2_local.dat
View File

@ -20,3 +20,4 @@
2621440 -231.47375051 -231.55526062 0.00040253 -231.55415883 0.00039709 2621440 -231.47375051 -231.55526062 0.00040253 -231.55415883 0.00039709
5242880 -231.48582938 -231.55830266 0.00036203 -231.55745147 0.00035778 5242880 -231.48582938 -231.55830266 0.00036203 -231.55745147 0.00035778
10485760 -231.49751473 -231.56256801 0.00032228 -231.56190078 0.00031897 10485760 -231.49751473 -231.56256801 0.00032228 -231.56190078 0.00031897
20971520 -231.50871375 -231.56470650 0.00027492 -231.56422301 0.00027255

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