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Added some precision about LIM

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Bruno Senjean 2020-05-06 10:27:18 +02:00
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@ -770,19 +770,28 @@ which require three independent calculations, as well as the MOM excitation ener
\end{subequations} \end{subequations}
which also require three separate calculations at a different set of ensemble weights. which also require three separate calculations at a different set of ensemble weights.
As readily seen in Eqs.~(\ref{eq:LIM1}) and (\ref{eq:LIM2}), LIM is a recursive strategy where the first excitation energy has to be determined As readily seen in Eqs.~(\ref{eq:LIM1}) and (\ref{eq:LIM2}), LIM is a recursive strategy where the first excitation energy has to be determined
in order to compute the second one. In the above equations, we in order to compute the second one.
In the above equations, we
assumed that the singly-excited state (with weight $w_1$) was lower assumed that the singly-excited state (with weight $w_1$) was lower
in energy compared to the doubly-excited one (with weight $w_2$). in energy compared to the doubly-excited one (with weight $w_2$).
If the ordering changes, then one should read $\E{}{\bw{}=(0,1/2)}$ If the ordering changes, then one should read $\E{}{\bw{}=(0,1/2)}$
instead of $\E{}{\bw{}=(1/2,0)}$ in Eqs.~(\ref{eq:LIM1}) and (\ref{eq:LIM2}) which then correspond to the excitation energies of the instead of $\E{}{\bw{}=(1/2,0)}$ in Eqs.~(\ref{eq:LIM1}) and (\ref{eq:LIM2}) which then correspond to the excitation energies of the
doubly-excited state and the singly-excited one, respectively. doubly-excited state and the singly-excited one, respectively.
The same hold for the MOM excitation energies in The same holds for the MOM excitation energies in
Eqs.~\ref{eq:MOM1} and \ref{MOM2}. Eqs.~\ref{eq:MOM1} and \ref{eq:MOM2}.
For a general expression with multiple (and possibly degenerate) For a general expression with multiple (and possibly degenerate)
states, we states, we
refer the reader to Eq.~106 of Ref.~\cite{Senjean_2015} (note refer the reader to Eq.~106 of Ref.~\onlinecite{Senjean_2015}, where
however that another convention were used to define the ensemble). LIM is shown to interpolate linearly the
\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.} ensemble energy between equi-ensembles.
Note that two calculations are needed for the first excitation energy
within LIM, but only one is required for each higher excitation
energies. By construction, for ensemble energies that are quadratic with
respect to the weight (which is almost always the case in this paper), the
first excitation energy within LIM
and MOM can actually be obtained in a single calculation at
$\ew{} = 1/4$ and
$\ew{} = 1/2$, respectively.
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. The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI.
@ -799,7 +808,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\begin{tabular}{llccccc} \begin{tabular}{llccccc}
\mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\ \mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\
\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\fnm[1] & MOM\fnm[1] \\
\hline \hline
HF & & aug-cc-pVDZ & 35.59 & 33.33 & & 28.65 \\ HF & & aug-cc-pVDZ & 35.59 & 33.33 & & 28.65 \\
& & aug-cc-pVTZ & 35.01 & 33.51 & & 28.65 \\ & & aug-cc-pVTZ & 35.01 & 33.51 & & 28.65 \\
@ -842,10 +851,11 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
HF & LYP & aug-mcc-pV8Z & & & & 29.18 \\ HF & LYP & aug-mcc-pV8Z & & & & 29.18 \\
HF & & aug-mcc-pV8Z & & & & 28.65 \\ HF & & aug-mcc-pV8Z & & & & 28.65 \\
\hline \hline
\mc{6}{l}{Accurate\fnm[1]} & 28.75 \\ \mc{6}{l}{Accurate\fnm[2]} & 28.75 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{FCI/aug-mcc-pV8Z calculation from Ref.~\onlinecite{Barca_2018a}.} \fnt[1]{Eqs.~(\ref{eq:LIM2}) and (\ref{eq:MOM2}) are used where the first weight corresponds to the singly-excited state.}
\fnt[2]{FCI/aug-mcc-pV8Z calculation from Ref.~\onlinecite{Barca_2018a}.}
\end{table} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -857,6 +867,8 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
To investigate the weight dependence of the xc functional in the strong correlation regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr). To investigate the weight dependence of the xc functional in the strong correlation regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr).
Note that, for this particular geometry, the doubly-excited state becomes the lowest excited state with the same symmetry as the ground state. Note that, for this particular geometry, the doubly-excited state becomes the lowest excited state with the same symmetry as the ground state.
Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble defined in Sec.~\ref{sec:H2}. Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble defined in Sec.~\ref{sec:H2}.
One should just be careful when reading the equations, as they correspond to the case where the
singly-excited state is lower in energy than the doubly-excited one.
We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH = 3.7$ bohr. We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH = 3.7$ bohr.
It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}]. It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve). The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve).
@ -894,11 +906,11 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\begin{tabular}{llcccc} \begin{tabular}{llcccc}
\mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\ \mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
\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\fnm[1] & MOM\fnm[1] \\
\hline \hline
HF & & 19.09 & 8.82 & 12.92 & 6.52 \\ HF & & 19.09 & 8.82 & 12.92 & 6.52 \\
HF & VWN5 & 19.40 & 8.81 & 13.02 & 6.49 \\ HF & VWN5 & 19.40 & 8.81 & 13.02 & 6.49 \\
HF & eVWN5 & 19.59 & 8.95 & 13.11 & \fnm[1] \\ HF & eVWN5 & 19.59 & 8.95 & 13.11 & \fnm[2] \\
S & & 5.31 & 5.67 & 5.46 & 5.56 \\ S & & 5.31 & 5.67 & 5.46 & 5.56 \\
S & VWN5 & 5.34 & 5.64 & 5.46 & 5.52 \\ S & VWN5 & 5.34 & 5.64 & 5.46 & 5.52 \\
S & eVWN5 & 5.53 & 5.79 & 5.56 & 5.72 \\ S & eVWN5 & 5.53 & 5.79 & 5.56 & 5.72 \\
@ -910,14 +922,15 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
B3 & LYP & & & & 5.55 \\ B3 & LYP & & & & 5.55 \\
HF & LYP & & & & 6.68 \\ HF & LYP & & & & 6.68 \\
\hline \hline
\mc{2}{l}{srLDA ($\mu = 0.4$) \fnm[2]} & 6.39 & & 6.47 & \\ \mc{2}{l}{srLDA ($\mu = 0.4$) \fnm[3]} & 6.39 & & 6.47 & \\
\hline \hline
\mc{5}{l}{Accurate\fnm[3]} & 8.69 \\ \mc{5}{l}{Accurate\fnm[4]} & 8.69 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{KS calculation does not converge.} \fnt[1]{Eqs.~(\ref{eq:LIM1}) and (\ref{eq:MOM1}) are used where the first weight corresponds to the doubly-excited state.}
\fnt[2]{Short-range multiconfigurational DFT/aug-cc-pVQZ calculations from Ref.~\onlinecite{Senjean_2015}.} \fnt[2]{KS calculation does not converge.}
\fnt[3]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}} \fnt[3]{Short-range multiconfigurational DFT/aug-cc-pVQZ calculations from Ref.~\onlinecite{Senjean_2015}.}
\fnt[4]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}}
\end{table} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -958,7 +971,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
\begin{tabular}{llcccc} \begin{tabular}{llcccc}
\mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\ \mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
\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\fnm[1] & MOM\fnm[1] \\
\hline \hline
HF & & 1.874 & 2.212 & 2.123 & 2.142 \\ HF & & 1.874 & 2.212 & 2.123 & 2.142 \\
HF & VWN5 & 1.988 & 2.260 & 2.190 & 2.193 \\ HF & VWN5 & 1.988 & 2.260 & 2.190 & 2.193 \\
@ -974,9 +987,10 @@ Excitation energies (in hartree) associated with the lowest double excitation of
B3 & LYP & & & & 2.150 \\ B3 & LYP & & & & 2.150 \\
HF & LYP & & & & 2.171 \\ HF & LYP & & & & 2.171 \\
\hline \hline
\mc{2}{l}{Accurate\fnm[1]} & & & & 2.126 \\ \mc{2}{l}{Accurate\fnm[2]} & & & & 2.126 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Eqs.~(\ref{eq:LIM2}) and (\ref{eq:MOM2}) are used where the first weight corresponds to the singly-excited state.}
\fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.} \fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.}
\end{table} \end{table}