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Pierre-Francois Loos 2020-04-08 22:00:14 +02:00
parent 0c1a1a6d3c
commit 252ba4ec5b
2 changed files with 379 additions and 233 deletions
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FarDFT.nb
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58
Manuscript/FarDFT.tex
View File

@ -314,7 +314,7 @@ where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-depen
\label{sec:compdet} \label{sec:compdet}
The self-consistent GOK-DFT calculations have been performed with the \texttt{QuAcK} software, freely available on \texttt{github}, where the present functional has been implemented. The self-consistent GOK-DFT calculations have been performed with the \texttt{QuAcK} software, freely available on \texttt{github}, where the present functional has been implemented.
For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found. For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found.
For all calculations, we use the aug-cc-pVXZ (X = D, T, and Q) Dunning's family of atomic basis sets. For all calculations, we use a restricted formalism and the aug-cc-pVXZ (X = D, T, and Q) Dunning's family of atomic basis sets.
Numerical quadratures are performed with the \texttt{numgrid} library using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988,Lindh_2001} Numerical quadratures are performed with the \texttt{numgrid} library using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988,Lindh_2001}
This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities). This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered. Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered.
@ -633,6 +633,8 @@ MOM excitation energies can then be obtained via GOK-DFT ensemble calculations b
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the GIC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to GIC-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 GIC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to GIC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remains in good agreement with FCI, with again a small improvement as compared to GIC-SVWN5. The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remains in good agreement with FCI, with again a small improvement as compared to GIC-SVWN5.
The GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$.
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
\caption{ \caption{
@ -657,6 +659,10 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
& & aug-cc-pVTZ & 38.54 & 27.81 & 24.46 & 27.17 \\ & & aug-cc-pVTZ & 38.54 & 27.81 & 24.46 & 27.17 \\
& & aug-cc-pVQZ & 38.81 & 27.81 & 24.46 & 27.17 \\ & & aug-cc-pVQZ & 38.81 & 27.81 & 24.46 & 27.17 \\
\\ \\
S & eVWN5 & aug-cc-pVDZ & 21.28 & 27.92 & 24.49 & 27.27 \\
& & aug-cc-pVTZ & 21.39 & 27.98 & 24.55 & 27.34 \\
& & aug-cc-pVQZ & 21.38 & 27.97 & 24.55 & 27.34 \\
\\
GIC-S & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\ GIC-S & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\
& & aug-cc-pVTZ & 26.88 & 26.59 & 26.61 & 26.67 \\ & & aug-cc-pVTZ & 26.88 & 26.59 & 26.61 & 26.67 \\
& & aug-cc-pVQZ & 26.82 & 26.60 & 26.62 & 26.67 \\ & & aug-cc-pVQZ & 26.82 & 26.60 & 26.62 & 26.67 \\
@ -682,12 +688,60 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\end{table*} \end{table*}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE I %%%
\begin{table*}
\caption{
Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 3.7$ bohr for various methods, combinations of xc functionals, and basis sets.
\label{tab:BigTab_H2}
}
\begin{ruledtabular}
\begin{tabular}{llccccc}
\mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\
\cline{1-2} \cline{4-5}
exchange & correlation & Basis & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
\hline
HF & & aug-cc-pVDZ & 19.08 & 6.58 & 12.92 & \\
& & aug-cc-pVTZ & 19.09 & 6.59 & 12.92 & \\
\\
S & & aug-cc-pVDZ & 5.31 & 5.60 & 5.46 & 5.56 \\
& & aug-cc-pVTZ & 5.31 & 5.60 & 5.46 & 5.56 \\
\\
S & VWN5 & aug-cc-pVDZ & 5.34 & 5.57 & 5.46 & 5.53 \\
& & aug-cc-pVTZ & 5.34 & 5.57 & 5.46 & 5.52 \\
\\
S & eVWN5 & aug-cc-pVDZ & 5.53 & 5.76 & 5.56 & 5.72 \\
& & aug-cc-pVTZ & 5.53 & 5.76 & 5.56 & 5.72 \\
& & aug-cc-pVQZ & & & & \\
\\
GIC-S & & aug-cc-pVDZ & 10.54 & 5.61 & 8.47 & 5.56 \\
& & aug-cc-pVTZ & 10.53 & 5.61 & 8.47 & 5.56 \\
\\
GIC-S & VWN5 & aug-cc-pVDZ & 10.67 & 5.59 & 8.55 & 5.53 \\
& & aug-cc-pVTZ & 10.67 & 5.59 & 8.55 & 5.52 \\
\\
GIC-S & eVWN5 & aug-cc-pVDZ & 10.86 & 5.77 & 8.64 & 5.72 \\
& & aug-cc-pVTZ & 10.86 & 5.77 & 8.64 & 5.72 \\
\\
% HF & FCI & aug-cc-pVDZ & & & & 8.78 \\
% HF & FCI & aug-cc-pVTZ & & & & 8.71 \\
% HF & FCI & aug-cc-pVQZ & & & & 8.70 \\
HF & FCI & aug-cc-pV5Z & & & & 8.69 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Reference \onlinecite{Mielke_2005}.}
\fnt[2]{Reference \onlinecite{Barca_2018a}.}
\end{table*}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%% %%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
\label{sec:ccl} \label{sec:ccl}
As concluding remarks, we would like to say that what we have done, we think, is awesome. We have studied the weight dependence of the ensemble energy in the framework of GOK-DFT.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%%% ACKNOWLEDGEMENTS %%% %%% ACKNOWLEDGEMENTS %%%