Manuscript

Manuscript/sfBSE.tex

@@ -597,6 +597,8 @@ 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.
+
 %%% TABLE I %%%
 \begin{squeezetable}
 \begin{table}
@@ -754,7 +756,8 @@ The excitation energies corresponding to the first singlet and triplet single ex
 \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. 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 and the F${}^1 \Sigma_g^+$ state after the avoided crossing. But the E${}^1 \Sigma_g^+$ state is well describe after this avoided crossing and the F${}^1 \Sigma_g^+$ state is well describe before the avoided crossing. SF-CIS does not give a good description of the double excitation. 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. 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. 
+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. 
+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*}