From 29d803aa4974211739b66eed6536f544a34ca38b Mon Sep 17 00:00:00 2001 From: bsenjean Date: Wed, 6 May 2020 10:27:18 +0200 Subject: [PATCH] Added some precision about LIM --- Manuscript/FarDFT.tex | 50 +++++++++++++++++++++++++++---------------- 1 file changed, 32 insertions(+), 18 deletions(-) diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index f571e47..bbdde92 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -770,19 +770,28 @@ which require three independent calculations, as well as the MOM excitation ener \end{subequations} 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 -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 in energy compared to the doubly-excited one (with weight $w_2$). If the ordering changes, then one should read $\E{}{\bw{}=(0,1/2)}$ 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. -The same hold for the MOM excitation energies in -Eqs.~\ref{eq:MOM1} and \ref{MOM2}. +The same holds for the MOM excitation energies in +Eqs.~\ref{eq:MOM1} and \ref{eq:MOM2}. For a general expression with multiple (and possibly degenerate) states, we -refer the reader to Eq.~106 of Ref.~\cite{Senjean_2015} (note -however that another convention were used to define the ensemble). -\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.} +refer the reader to Eq.~106 of Ref.~\onlinecite{Senjean_2015}, where +LIM is shown to interpolate linearly the +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 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} \mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\ \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 HF & & aug-cc-pVDZ & 35.59 & 33.33 & & 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 & & aug-mcc-pV8Z & & & & 28.65 \\ \hline - \mc{6}{l}{Accurate\fnm[1]} & 28.75 \\ + \mc{6}{l}{Accurate\fnm[2]} & 28.75 \\ \end{tabular} \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} %%% %%% %%% %%% @@ -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). 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}. +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. 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). @@ -894,11 +906,11 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{ \begin{tabular}{llcccc} \mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\ \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 HF & & 19.09 & 8.82 & 12.92 & 6.52 \\ 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 & VWN5 & 5.34 & 5.64 & 5.46 & 5.52 \\ 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 \\ HF & LYP & & & & 6.68 \\ \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 - \mc{5}{l}{Accurate\fnm[3]} & 8.69 \\ + \mc{5}{l}{Accurate\fnm[4]} & 8.69 \\ \end{tabular} \end{ruledtabular} -\fnt[1]{KS calculation does not converge.} -\fnt[2]{Short-range multiconfigurational DFT/aug-cc-pVQZ calculations from Ref.~\onlinecite{Senjean_2015}.} -\fnt[3]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}} +\fnt[1]{Eqs.~(\ref{eq:LIM1}) and (\ref{eq:MOM1}) are used where the first weight corresponds to the doubly-excited state.} +\fnt[2]{KS calculation does not converge.} +\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} %%% %%% %%% %%% @@ -958,7 +971,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of \begin{tabular}{llcccc} \mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\ \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 HF & & 1.874 & 2.212 & 2.123 & 2.142 \\ 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 \\ HF & LYP & & & & 2.171 \\ \hline - \mc{2}{l}{Accurate\fnm[1]} & & & & 2.126 \\ + \mc{2}{l}{Accurate\fnm[2]} & & & & 2.126 \\ \end{tabular} \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}.} \end{table}