figures and Manuscript
This commit is contained in:
parent
f58e21bfca
commit
962267e5df
Binary file not shown.
Binary file not shown.
|
Before Width: | Height: | Size: 513 KiB |
Binary file not shown.
Binary file not shown.
|
Before Width: | Height: | Size: 489 KiB |
Binary file not shown.
Binary file not shown.
|
Before Width: | Height: | Size: 511 KiB |
|
|
@ -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}
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user