diff --git a/Manuscript/H2_BSE_RHF.pdf b/Manuscript/H2_BSE_RHF.pdf deleted file mode 100644 index 2ad8e6a..0000000 Binary files a/Manuscript/H2_BSE_RHF.pdf and /dev/null differ diff --git a/Manuscript/H2_BSE_RHF.png b/Manuscript/H2_BSE_RHF.png deleted file mode 100644 index e75aa60..0000000 Binary files a/Manuscript/H2_BSE_RHF.png and /dev/null differ diff --git a/Manuscript/H2_CIS.pdf b/Manuscript/H2_CIS.pdf index 8b824ef..23d8e13 100644 Binary files a/Manuscript/H2_CIS.pdf and b/Manuscript/H2_CIS.pdf differ diff --git a/Manuscript/H2_CIS.png b/Manuscript/H2_CIS.png deleted file mode 100644 index f683391..0000000 Binary files a/Manuscript/H2_CIS.png and /dev/null differ diff --git a/Manuscript/H2_TDDFT.pdf b/Manuscript/H2_TDDFT.pdf deleted file mode 100644 index b78af28..0000000 Binary files a/Manuscript/H2_TDDFT.pdf and /dev/null differ diff --git a/Manuscript/H2_TDDFT.png b/Manuscript/H2_TDDFT.png deleted file mode 100644 index ef77765..0000000 Binary files a/Manuscript/H2_TDDFT.png and /dev/null differ diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index 60c3461..fdabf5e 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -597,7 +597,7 @@ As a first example, we consider the simple case of the beryllium atom which was Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration. The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $P$ and $D$ spatial symmetries (respectively), obtained with the 6-31G basis set are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}. -In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where from SF-TD-BLYP to SF-TD-BH\&HLYP there is an increase of the exact exchange in the functional. As noticed in \cite{Casanova_2020}, for the BLYP functional we have the $^3P(1s^22s2p)$ and the $^1P(1s^22s2p)$ states that appear to be degenerated because their energies are given by the $2s/2p$ gap. The exact exchange break this degeneracy. However even with the increase of the exact exchange in the functionals, SF-TD-DFT excitation energy for the $^1P(1s^22s2p)$ state is at 1.6 eV from the FCI reference. For the other states, SF-TD-DFT excitation energies are in much better agreement with FCI. The middle part shows the SF-BSE results. They are close the FCI results with an error of 0.02-0.6 eV depending on the scheme of SF-BSE. For the last excited state $^1D(1s^22p^2)$ the largest error is 0.85 eV with SF-dBSE@{\qsGW}, so we have a bad description of this state due to spin contamination. Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies. +In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where from SF-TD-BLYP to SF-TD-BH\&HLYP there is an increase of the exact exchange in the functional. As noticed in \cite{Casanova_2020}, for the BLYP functional we have the $^3P(1s^22s2p)$ and the $^1P(1s^22s2p)$ states that appear to be degenerated because their energies are given by the $2s/2p$ gap. The exact exchange break this degeneracy. However even with the increase of the exact exchange in the functionals, SF-TD-BH\&HLYP excitation energy for the $^1P(1s^22s2p)$ state is at 1.6 eV from the FCI reference. For the other states, SF-TD-BH\&HLYP excitation energies are in much better agreement with FCI. The middle part shows the SF-BSE results. They are close the FCI results, because of the fact that there is one hundred percent exact exchange in the BSE method, with an error of 0.02-0.6 eV depending on the scheme of SF-BSE. For the last excited state $^1D(1s^22p^2)$ the largest error is 0.85 eV with SF-dBSE@{\qsGW}, so we have a bad description of this state due to spin contamination. Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies. %%% TABLE I %%% \begin{squeezetable} @@ -756,21 +756,20 @@ In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where f \label{sec:H2} %=============================== -The second system of interest is the \ce{H2} molecule where we stretch the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. Three excited states are investigated during the stretching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. Three methods with and without spin-flip are used to study these states. These methods are CIS, TD-BHHLYP and BSE and are compared to the reference, here the EOM-CCSD method. %that is equivalent to the FCI for the \ce{H2} molecule. -Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and without spin-flip. We can observe that both SF-CIS and CIS poorly describe the B${}^1 \Sigma_u^+$ state, especially at the dissociation limit with an error of more than 1 eV. The same analysis can be done for the F${}^1 \Sigma_g^+$ state at the dissociation limit. EOM-CSSD curves show us an avoided crossing between the E${}^1 \Sigma_g^+$ and F${}^1 \Sigma_g^+$ states due to their same symmetry. SF-CIS does not represent well the E${}^1 \Sigma_g^+$ state before the avoided crossing. But the E${}^1 \Sigma_g^+$ state is well describe after this avoided crossing. SF-CIS describes better the F${}^1 \Sigma_g^+$ state before the avoided crossing than at the dissociation limit. In general, SF-CIS does not give a good description of the double excitation. As expected CIS does not find the double excitation to the F${}^1 \Sigma_g^+$ state. The rigth panel gives results of the TD-BHHLYP calculation with and without spin-flip. TD-BHHLYP shows bad results for all the states of interest with and without spin-flip. Indeed, for the three states we have a difference in the excitation energy at the dissociation limit of several eV with and without spin-flip. +The second system of interest is the \ce{H2} molecule where we stretch the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. The three lowest singlets states are investigated during the stretching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. Three methods with and without spin-flip are used to study these states. These methods are CIS, TD-BH\&HLYP and BSE and are compared to the reference, here the EOM-CCSD method. %that is equivalent to the FCI for the \ce{H2} molecule. +Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and without spin-flip. We can observe that both SF-CIS and CIS poorly describe the B${}^1 \Sigma_u^+$ state, especially at the dissociation limit with an error of more than 1 eV. The same analysis can be done for the F${}^1 \Sigma_g^+$ state at the dissociation limit. EOM-CSSD curves show us an avoided crossing between the E${}^1 \Sigma_g^+$ and F${}^1 \Sigma_g^+$ states due to their same symmetry. SF-CIS does not represent well the E${}^1 \Sigma_g^+$ state before the avoided crossing. But the E${}^1 \Sigma_g^+$ state is well describe after this avoided crossing. SF-CIS describes better the F${}^1 \Sigma_g^+$ state before the avoided crossing than at the dissociation limit. In general, SF-CIS does not give a good description of the double excitation. As expected CIS does not find the double excitation to the F${}^1 \Sigma_g^+$ state. The rigth panel gives results of the TD-BH\&HLYP calculation with and without spin-flip. TD-BH\&HLYP shows bad results for all the states of interest with and without spin-flip. Indeed, for the three states we have a difference in the excitation energy at the dissociation limit of several eV with and without spin-flip. 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 BSE does not retrieve the double excitation as it was pointed out in the theoretical section. -\begin{figure*} -\includegraphics[width=0.49\linewidth]{H2_CIS.pdf} -\hspace{0.05cm} -\includegraphics[width=0.49\linewidth]{H2_TDDFT.pdf} -\includegraphics[width=0.49\linewidth]{H2_BSE_RHF.pdf} +\begin{figure} +\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 left panel shows CIS results, the right panel shows TD-BHHLYP results and the last panel shows the BSE results. + 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*} +\end{figure} @@ -806,10 +805,13 @@ Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-A \cline{2-4} Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{1g}$ \\ \hline - EOM-SF-CIS\fnm[1] & & & \\ - EOM-SF-CCSD\fnm[1] & & & \\ - EOM-SF-CCSD(fT)\fnm[1] & & & \\ - EOM-SF-CCSD(dT)\fnm[1] & & & \\ + SF-TD-BLYP \fnm[3] & & & \\ + SF-TD-B3LYP \fnm[3] & & & \\ + SF-TD-BH\&HLYP\fnm[3] &1.583 &2.813 & 4.528\\ + SF-CIS\fnm[1] & & & \\ + EOM-SF-CCSD\fnm[1] &1.654 & 3.416&4.360 \\ + EOM-SF-CCSD(fT)\fnm[1] & 1.516& 3.260&4.205 \\ + EOM-SF-CCSD(dT)\fnm[1] &1.475 &3.215 &4.176 \\ SF-ADC(2)-s\fnm[2] & 1.572& 3.201& 4.241\\ SF-ADC(2)-x\fnm[2] &1.576 &3.134 &3.792 \\ SF-ADC(3)\fnm[2] & 1.455&3.276 &4.328 \\ @@ -835,13 +837,16 @@ Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-A \cline{2-4} Method & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\ \hline - EOM-SF-CIS\fnm[1] & & & \\ - EOM-SF-CCSD\fnm[1] & & & \\ - EOM-SF-CCSD(fT)\fnm[1] & & & \\ - EOM-SF-CCSD(dT)\fnm[1] & & & \\ + SF-TD-BLYP \fnm[3] & & & \\ + SF-TD-B3LYP \fnm[3] & & & \\ + SF-TD-BH\&HLYP\fnm[3] & & & \\ + SF-CIS\fnm[1] &0.317 & 3.125&2.650 \\ + EOM-SF-CCSD\fnm[1] &0.369 & 1.824& 2.143\\ + EOM-SF-CCSD(fT)\fnm[1] & 0.163&1.530 &1.921 \\ + EOM-SF-CCSD(dT)\fnm[1] &0.098 &1.456 &1.853 \\ SF-ADC(2)-s\fnm[2] & 0.265 & 1.658& 1.904 \\ - SF-ADC(2)-x\fnm[2] & & & \\ - SF-ADC(3)\fnm[2] & & & \\ + SF-ADC(2)-x\fnm[2] & 0.217&1.123 &1.799 \\ + SF-ADC(3)\fnm[2] &0.083 &1.621 &1.930 \\ SF-BSE@{\GOWO}\fnm[3] & -0.049 & 1.189 & 1.480 \\ SF-dBSE@{\GOWO}\fnm[3] & 0.012 & 1.507 & 1.841 \\ \end{tabular}