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Pierre-Francois Loos 2020-05-04 23:56:23 +02:00
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Manuscript/FarDFT.tex
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@ -711,7 +711,7 @@ We shall come back to this point later on.
Third, we add up correlation effects via the VWN5 local correlation functional. \cite{Vosko_1980} Third, we add up correlation effects via the VWN5 local correlation functional. \cite{Vosko_1980}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}. For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies, especially at small weights, where the CC-SVWN5 excitation energy is almost spot on. The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (green curve in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies (red curve in Figs.~\ref{fig:Ew_H2} and \ref{fig:Om_H2}), especially at small weights, where the CC-SVWN5 excitation energy is almost spot on.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Weight-dependent correlation functional} \subsubsection{Weight-dependent correlation functional}
@ -847,7 +847,7 @@ We note also that, by construction, we have
\end{equation} \end{equation}
showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model. showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model.
As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is slightly less concave than its CC-SVWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}). As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is very slightly less concave than its CC-SVWN5 counterpart and it also improves (not by much) the excitation energy (see purple curve in Fig.~\ref{fig:Om_H2}).
For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets. For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
In particular, we report the excitation energies obtained with GOK-DFT In particular, we report the excitation energies obtained with GOK-DFT
in the zero-weight limit (\ie, $\ew{} = 0$) and for equi-weights (\ie, $\ew{} = 1/3$). in the zero-weight limit (\ie, $\ew{} = 0$) and for equi-weights (\ie, $\ew{} = 1/3$).
@ -875,7 +875,7 @@ which also require three separate calculations at a different set of ensemble we
%They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie, %They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional. The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI, with again a small improvement as compared to CC-SVWN5. The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI.
It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}). It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
\bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\ew{} = 1/4$ and $\ew{} = 1/2$, respectively, instead of performing an interpolation between two different calculations.} \bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\ew{} = 1/4$ and $\ew{} = 1/2$, respectively, instead of performing an interpolation between two different calculations.}
Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between the ghost-interaction-free pure-state limits. Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between the ghost-interaction-free pure-state limits.
@ -892,41 +892,41 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\cline{1-2} \cline{4-5} \cline{1-2} \cline{4-5}
\tabc{x} & \tabc{c} & Basis & $\ew{} = 0$ & $\ew{} = 1/3$ & LIM & MOM \\ \tabc{x} & \tabc{c} & Basis & $\ew{} = 0$ & $\ew{} = 1/3$ & LIM & MOM \\
\hline \hline
HF & & aug-cc-pVDZ & 35.59 & 33.33 & 34.00 & 28.65 \\ HF & & aug-cc-pVDZ & 35.59 & 33.33 & & 28.65 \\
& & aug-cc-pVTZ & 35.01 & 33.51 & 35.80 & 28.65 \\ & & aug-cc-pVTZ & 35.01 & 33.51 & & 28.65 \\
& & aug-cc-pVQZ & 34.66 & 33.54 & 35.82 & 28.65 \\ & & aug-cc-pVQZ & 34.66 & 33.54 & & 28.65 \\
\\ \\
HF & VWN5 & aug-cc-pVDZ & 37.83 & 33.86 & 34.91 & 29.17 \\ HF & VWN5 & aug-cc-pVDZ & 37.83 & 33.86 & & 29.17 \\
& & aug-cc-pVTZ & 37.61 & 33.99 & 37.23 & 29.17 \\ & & aug-cc-pVTZ & 37.61 & 33.99 & & 29.17 \\
& & aug-cc-pVQZ & 37.07 & 34.01 & 37.21 & 29.17 \\ & & aug-cc-pVQZ & 37.07 & 34.01 & & 29.17 \\
\\ \\
HF & eVWN5 & aug-cc-pVDZ & 38.09 & 34.00 & 35.00 & 29.34 \\ HF & eVWN5 & aug-cc-pVDZ & 38.09 & 34.00 & & 29.34 \\
& & aug-cc-pVTZ & 37.61 & 34.13 & 37.28 & 29.34 \\ & & aug-cc-pVTZ & 37.61 & 34.13 & & 29.34 \\
& & aug-cc-pVQZ & 37.32 & 34.14 & 37.27 & 29.34 \\ & & aug-cc-pVQZ & 37.32 & 34.14 & & 29.34 \\
\\ \\
S & & aug-cc-pVDZ & 19.44 & 28.00 & 23.54 & 26.60 \\ S & & aug-cc-pVDZ & 19.44 & 28.00 & 25.09 & 26.60 \\
& & aug-cc-pVTZ & 19.47 & 28.11 & 23.62 & 26.67 \\ & & aug-cc-pVTZ & 19.47 & 28.11 & 25.20 & 26.67 \\
& & aug-cc-pVQZ & 19.41 & 28.13 & 23.62 & 26.67 \\ & & aug-cc-pVQZ & 19.41 & 28.13 & 25.22 & 26.67 \\
\\ \\
S & VWN5 & aug-cc-pVDZ & 21.04 & 28.49 & 24.40 & 27.10 \\ S & VWN5 & aug-cc-pVDZ & 21.04 & 28.49 & 25.90 & 27.10 \\
& & aug-cc-pVTZ & 21.14 & 28.58 & 24.46 & 27.17 \\ & & aug-cc-pVTZ & 21.14 & 28.58 & 25.99 & 27.17 \\
& & aug-cc-pVQZ & 21.13 & 28.59 & 24.46 & 27.17 \\ & & aug-cc-pVQZ & 21.13 & 28.59 & 26.00 & 27.17 \\
\\ \\
S & eVWN5 & aug-cc-pVDZ & 21.28 & 28.64 & 24.49 & 27.27 \\ S & eVWN5 & aug-cc-pVDZ & 21.28 & 28.64 & 25.99 & 27.27 \\
& & aug-cc-pVTZ & 21.39 & 28.74 & 24.55 & 27.34 \\ & & aug-cc-pVTZ & 21.39 & 28.74 & 26.08 & 27.34 \\
& & aug-cc-pVQZ & 21.38 & 28.75 & 24.55 & 27.34 \\ & & aug-cc-pVQZ & 21.38 & 28.75 & 26.09 & 27.34 \\
\\ \\
CC-S & & aug-cc-pVDZ & 26.83 & 29.29 & 26.53 & 26.60 \\ CC-S & & aug-cc-pVDZ & 26.83 & 29.29 & 28.83 & 26.60 \\
& & aug-cc-pVTZ & 26.88 & 29.41 & 26.61 & 26.67 \\ & & aug-cc-pVTZ & 26.88 & 29.41 & 28.96 & 26.67 \\
& & aug-cc-pVQZ & 26.82 & 29.43 & 26.62 & 26.67 \\ & & aug-cc-pVQZ & 26.82 & 29.43 & 28.97 & 26.67 \\
\\ \\
CC-S & VWN5 & aug-cc-pVDZ & 28.54 & 29.85 & 27.48 & 27.10 \\ CC-S & VWN5 & aug-cc-pVDZ & 28.54 & 29.85 & 29.73 & 27.10 \\
& & aug-cc-pVTZ & 28.66 & 29.96 & 27.56 & 27.17 \\ & & aug-cc-pVTZ & 28.66 & 29.96 & 29.83 & 27.17 \\
& & aug-cc-pVQZ & 28.64 & 29.97 & 27.56 & 27.17 \\ & & aug-cc-pVQZ & 28.64 & 29.97 & 29.84 & 27.17 \\
\\ \\
CC-S & eVWN5 & aug-cc-pVDZ & 28.78 & 29.99 & 27.56 & 27.27 \\ CC-S & eVWN5 & aug-cc-pVDZ & 28.78 & 29.99 & 29.82 & 27.27 \\
& & aug-cc-pVTZ & 28.90 & 30.10 & 27.64 & 27.34 \\ & & aug-cc-pVTZ & 28.90 & 30.10 & 29.92 & 27.34 \\
& & aug-cc-pVQZ & 28.89 & 30.11 & 27.65 & 27.34 \\ & & aug-cc-pVQZ & 28.89 & 30.11 & 29.93 & 27.34 \\
\hline \hline
B & LYP & aug-mcc-pV8Z & & & & 28.42 \\ B & LYP & aug-mcc-pV8Z & & & & 28.42 \\
B3 & LYP & aug-mcc-pV8Z & & & & 27.77 \\ B3 & LYP & aug-mcc-pV8Z & & & & 27.77 \\
@ -1037,7 +1037,7 @@ We then restrict ourselves to a triensemble keeping in mind the possible theoret
The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.912\,574$, $\beta = +2.715\,267$, and $\gamma = +2.163\,422$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}). The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.912\,574$, $\beta = +2.715\,267$, and $\gamma = +2.163\,422$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange. The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value. The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy for $\ew{} = 0$ is only $18$ millihartree off the reference value.
As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}. As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}.
This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy. This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy.
As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation. As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
@ -1054,21 +1054,21 @@ Excitation energies (in hartree) associated with the lowest double excitation of
\cline{1-2} \cline{3-4} \cline{1-2} \cline{3-4}
\tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/3$ & LIM & MOM \\ \tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/3$ & LIM & MOM \\
\hline \hline
HF & & 1.874 & 2.212 & 2.080 & 2.142 \\ HF & & 1.874 & 2.212 & 2.123 & 2.142 \\
HF & VWN5 & 1.988 & 2.260 & 2.153 & 2.193 \\ HF & VWN5 & 1.988 & 2.260 & 2.190 & 2.193 \\
HF & eVWN5 & 2.000 & 2.265 & 2.156 & 2.196 \\ HF & eVWN5 & 2.000 & 2.265 & 2.193 & 2.196 \\
S & & 1.062 & 2.056 & 1.547 & 2.030 \\ S & & 1.062 & 2.056 & 1.675 & 2.030 \\
S & VWN5 & 1.163 & 2.104 & 1.612 & 2.079 \\ S & VWN5 & 1.163 & 2.104 & 1.735 & 2.079 \\
S & eVWN5 & 1.174 & 2.109 & 1.615 & 2.083 \\ S & eVWN5 & 1.174 & 2.109 & 1.738 & 2.083 \\
CC-S & & 1.996 & 2.264 & 1.988 & 2.030 \\ CC-S & & 1.996 & 2.264 & 2.148 & 2.030 \\
CC-S & VWN5 & 2.107 & 2.318 & 2.060 & 2.079 \\ CC-S & VWN5 & 2.107 & 2.318 & 2.215 & 2.079 \\
CC-S & eVWN5 & 2.108 & 2.323 & 2.063 & 2.083 \\ CC-S & eVWN5 & 2.108 & 2.323 & 2.218 & 2.083 \\
\hline \hline
B & LYP & & & & 2.147 \\ B & LYP & & & & 2.147 \\
B3 & LYP & & & & 2.150 \\ B3 & LYP & & & & 2.150 \\
HF & LYP & & & & 2.171 \\ HF & LYP & & & & 2.171 \\
\hline \hline
\mc{5}{l}{Accurate\fnm[1]} & 2.126 \\ \mc{2}{l}{Accurate\fnm[1]} & & & & 2.126 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.} \fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.}