diff --git a/Manuscript/Be.pdf b/Manuscript/Be.pdf index 04e1e76..af619c9 100644 Binary files a/Manuscript/Be.pdf and b/Manuscript/Be.pdf differ diff --git a/Manuscript/CBD_D2h.pdf b/Manuscript/CBD_D2h.pdf new file mode 100644 index 0000000..de602cc Binary files /dev/null and b/Manuscript/CBD_D2h.pdf differ diff --git a/Manuscript/CBD_D4h.pdf b/Manuscript/CBD_D4h.pdf new file mode 100644 index 0000000..e5e28dc Binary files /dev/null and b/Manuscript/CBD_D4h.pdf differ diff --git a/Manuscript/sfBSE.bib b/Manuscript/sfBSE.bib index 0aa2945..a0b5983 100644 --- a/Manuscript/sfBSE.bib +++ b/Manuscript/sfBSE.bib @@ -1,7 +1,7 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2021-01-13 09:41:10 +0100 +%% Created for Pierre-Francois Loos at 2021-01-14 16:29:49 +0100 %% Saved with string encoding Unicode (UTF-8) @@ -1017,10 +1017,12 @@ @article{Loos_2020a, author = {P. F. Loos and B. Pradines and A. Scemama and E. Giner and J. Toulouse}, date-added = {2020-12-09 09:59:26 +0100}, - date-modified = {2020-12-09 09:59:26 +0100}, + date-modified = {2021-01-14 16:29:23 +0100}, + doi = {10.1021/acs.jctc.9b01067}, journal = {J. Chem. Theory Comput.}, - pages = {in press}, + pages = {1018--1028}, title = {A Density-Based Basis-Set Incompleteness Correction for GW Methods}, + volume = {16}, year = {2020}, Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01103}} diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index 79b340e..6a70dfb 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -610,19 +610,19 @@ In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where f \begin{tabular}{lcccc} & \mc{4}{c}{Excitation energies (eV)} \\ \cline{2-5} - Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$ - & $^3P(1s^22p^2)$ & $^1D(1s^22p^2)$ \\ + Method & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$ + & $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ \\ \hline SF-TD-BLYP\fnm[1] & 3.210 & 3.210 & 6.691 & 7.598 \\ SF-TD-B3LYP\fnm[1] & 3.332 & 4.275 & 6.864 & 7.762 \\ SF-TD-BH\&HLYP\fnm[1] & 2.874 & 4.922 & 7.112 & 8.188 \\ - SF-BSE@{\GOWO}\fnm[2] & 2.399 & 6.191 & 7.792 & 9.373 \\ - SF-BSE@{\evGW}\fnm[2] & 2.407 & 6.199 & 7.788 & 9.388 \\ - SF-BSE@{\qsGW}\fnm[2] & 2.376 & 6.241 & 7.668 & 9.417 \\ - SF-dBSE@{\GOWO}\fnm[2] & 2.363 & 6.263 & 7.824 & 9.424 \\ - SF-dBSE@{\evGW}\fnm[2] & 2.369 & 6.273 & 7.820 & 9.441 \\ - SF-dBSE@{\qsGW}\fnm[2] & 2.335 & 6.317 & 7.689 & 9.470 \\ - SF-CIS\fnm[3] & 2.111 & 6.036 & 7.480 & 8.945 \\ + SF-CIS\fnm[2] & 2.111 & 6.036 & 7.480 & 8.945 \\ + SF-BSE@{\GOWO}\fnm[3] & 2.399 & 6.191 & 7.792 & 9.373 \\ + SF-BSE@{\evGW}\fnm[3] & 2.407 & 6.199 & 7.788 & 9.388 \\ + SF-BSE@{\qsGW}\fnm[3] & 2.376 & 6.241 & 7.668 & 9.417 \\ + SF-dBSE@{\GOWO}\fnm[3] & 2.363 & 6.263 & 7.824 & 9.424 \\ + SF-dBSE@{\evGW}\fnm[3] & 2.369 & 6.273 & 7.820 & 9.441 \\ + SF-dBSE@{\qsGW}\fnm[3] & 2.335 & 6.317 & 7.689 & 9.470 \\ SF-ADC(2)-s\fnm[2] & 2.433 & 6.255 & 7.745 & 9.047 \\ SF-ADC(2)-x\fnm[2] & 2.866 & 6.581 & 7.664 & 8.612 \\ SF-ADC(3)\fnm[2] & 2.863 & 6.579 & 7.658 & 8.618 \\ @@ -630,13 +630,13 @@ In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where f \end{tabular} \end{ruledtabular} \fnt[1]{Value from Ref.~\onlinecite{Casanova_2020}.} - \fnt[2]{This work.} - \fnt[3]{Value from Ref.~\onlinecite{Krylov_2001a}.} + \fnt[2]{Value from Ref.~\onlinecite{Krylov_2001a}.} + \fnt[3]{This work.} \end{table} \end{squeezetable} %%% %%% %%% %%% -%%% FIG. 1 %%% +%%% FIG 1 %%% \begin{figure} \includegraphics[width=\linewidth]{Be} \caption{ @@ -645,6 +645,7 @@ In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where f All the spin-flip calculations have been performed with a UHF reference. \label{fig:Be}} \end{figure} +%%% %%% %%% %%% %%% TABLE II %%% %\begin{squeezetable} @@ -761,18 +762,17 @@ Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and wi In the last panel we have results for BSE calculation with and without spin-flip. SF-BSE gives a good representation of the B${}^1 \Sigma_u^+$ state with error of 0.05-0.3 eV. However SF-BSE does not describe well the E${}^1 \Sigma_g^+$ state with error of 0.5-1.6 eV. SF-BSE shows a good agreement with the EOM-CCSD reference for the double excitation to the F${}^1 \Sigma_g^+$ state, indeed we have an error of 0.008-0.6 eV. BSE results for the B${}^1 \Sigma_u^+$ state are close to the reference until 2.0 \AA and the give bad agreement for the dissociation limit. For the E${}^1 \Sigma_g^+$ state BSE gives closer results to the reference than SF-BSE. However we can observe that for all the methods that we compared, when the spin-flip is not used standard methods can not retrieve double excitation. There is no avoided crossing or perturbation in the curve for the E${}^1 \Sigma_g^+$ state when spin-flip is not used. This is because for these methods we are in the space of single excitation and de-excitation. +%%% FIG 2 %%% \begin{figure} -\includegraphics[width=1\linewidth]{H2_CIS.pdf} -\includegraphics[width=1\linewidth]{H2_BHHLYP.pdf} -\includegraphics[width=1\linewidth]{H2_BSE.pdf} + \includegraphics[width=1\linewidth]{H2_CIS.pdf} + \includegraphics[width=1\linewidth]{H2_BHHLYP.pdf} + \includegraphics[width=1\linewidth]{H2_BSE.pdf} \caption{ Excitation energies of the three states of interest [with respect to the singlet ground state] of \ce{H2} obtained with the cc-pVQZ basis. Three sets of curves are drawn, the solid curves are the references (EOM-CCSD), the dashed curves are obtained with spin-flip method and the dotted curves are obtained without using spin-flip. The top panel shows CIS results, the center panel shows TD-BH\&HLYP results and the bottom panel shows the BSE results. All the spin-flip calculations have been performed with a UHF reference. \label{fig:H2}} \end{figure} - - - +%%% %%% %%% %=============================== \subsection{Cyclobutadiene} @@ -793,11 +793,24 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations Table~\ref{tab:CBD_D2h} shows results obtained for the $D_{2h}$ rectangular equilibrium geometry and Table~\ref{tab:CBD_D4h} shows results obtained for $D_{4h}$ square equilibrium geometry. These results are given with respect to the singlet ground state. For each geometry three states are under investigation, for the $D_{2h}$ CBD we look at the $1 ~^3 B_{1g}$, $1~^1 B_{1g}$ and $2 ~^1 A_{1g}$ states. For the $D_{4h}$ CBD we look at the $1 ~^3 A_{2g}$, $2~^1 A_{1g}$ and $1 ~^1 B_{2g}$ states. Several methods using spin-flip are compared to the spin-flip version of BSE with and without dynamical corrections. In Table~\ref{tab:CBD_D2h}, comparing the results of our work and the most accurate ADC level, i.e., SF-ADC(3) with SF-BSE@{\GOWO} we have a difference in the excitation energy of 0.017 eV for the $1 ~^3 B_{1g}$ state. This difference grows to 0.572 eV for the $1 ~^1 B_{1g}$ state and then it is 0.212 eV for the $2 ~^1 A_{1g}$ state. Adding dynamical corrections in SF-dBSE@{\GOWO} do not improve the accuracy of the excitation energies comparing to SF-ADC(3). Indeed, we have a difference of 0.052 eV for the $1 ~^3 B_{1g}$ state, 0.393 eV for the $1 ~^1 B_{1g}$ state and 0.293 eV for the $2 ~^1 A_{1g}$ state. Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-ADC(3) we have an interesting result. Indeed, we have a wrong ordering in our first excited state, we find that the triplet state $1 ~^3 A_{2g}$ is lower in energy that the singlet state $B_{1g}$ in contrary to all of the results extracted in Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}. Then adding dynamical corrections in SF-dBSE@{\GOWO} not only improve the difference of excitation energies with SF-ADC(3) it gives the right ordering for the first excited state, meaning that we retrieve the triplet state $1 ~^3 A_{2g}$ above the singlet state $B_{1g}$. So here we have an example where the dynamical corrections are necessary to get the right chemistry. +%%% FIG 3 %%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{CBD_D2h} + \includegraphics[width=0.45\linewidth]{CBD_D4h} +\caption{ + Vertical excitation energies of CBD. + Left: $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state (see Table \ref{tab:CBD_D2h} for the raw data). + Right: $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3 A_{2g}$ state (see Table \ref{tab:CBD_D4h} for the raw data). + All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set. + \label{fig:CBD}} +\end{figure*} +%%% %%% %%% + %%% TABLE ?? %%% \begin{table} \caption{ - Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ singlet ground state. - All the spin-flip calculations have been performed with a UHF reference. + Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state. + All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set. \label{tab:CBD_D2h}} \begin{ruledtabular} \begin{tabular}{lccc} @@ -828,8 +841,8 @@ Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-A %%% TABLE ?? %%% \begin{table} \caption{ - Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $\text{X}\,{}^1B_{1g}$ singlet ground state. - All the spin-flip calculations have been performed with a UHF reference. + Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states of CBD at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state. + All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set. \label{tab:CBD_D4h}} \begin{ruledtabular} \begin{tabular}{lccc} diff --git a/Notebooks/sf-BSE.nb b/Notebooks/sf-BSE.nb index 51e6c61..21e2dce 100644 --- a/Notebooks/sf-BSE.nb +++ b/Notebooks/sf-BSE.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 261405, 4767] -NotebookOptionsPosition[ 258809, 4710] -NotebookOutlinePosition[ 259202, 4726] -CellTagsIndexPosition[ 259159, 4723] +NotebookDataLength[ 706091, 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