diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index 4029e4f..12fd2e1 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -682,7 +682,7 @@ Generally we can observe that all the scheme with SF-BSE used do not increase si \label{sec:H2} %=============================== -Our second example deals with the dissociation of the \ce{H2} molecule, which is a prototypical system for testing new electronic structure methods and, especially, their accuracy in the presence of strong correlation (see, for example, Refs.~\onlinecite{Caruso_2013,Barca_2014,Vuckovic_2017}, and references therein). +Our second example deals with the dissociation of the \ce{H2} molecule, which is a prototypical system for testing new electronic structure methods and, specifically, their accuracy in the presence of strong correlation (see, for example, Refs.~\onlinecite{Caruso_2013,Barca_2014,Vuckovic_2017}, and references therein). The $\text{X}\,{}^1 \Sigma_g^+$ ground state of \ce{H2} has an electronic configuration $(1\sigma_g)^2$ configuration. The variation of the excitation energies associated with the three lowest singlet excited states with respect to the elongation of the \ce{H-H} bond are of particular interest here. The lowest singly excited state $\text{B}\,{}^1 \Sigma_u^+$ has a $(1\sigma_g )(1\sigma_u)$ configuration, while the singly excited state $\text{E}\,{}^1 \Sigma_g^+$ and the doubly excited state $\text{F}\,{}^1 \Sigma_g^+$ have $(1\sigma_g ) (2\sigma_g)$ and $(1\sigma_u )(1\sigma_u)$ configurations, respectively. @@ -695,21 +695,22 @@ The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dash The EOM-CCSD reference energies are represented by solid lines. We observe that both CIS and SF-CIS poorly describe the $\text{B}\,{}^1\Sigma_u^+$ state, especially in the dissociation limit with an error greater than $1$ eV. The same analysis can be done for the $\text{F}\,{}^1\Sigma_g^+$ state at dissociation. -The EOM-CSSD curves clearly evidence the avoided crossing between the $\text{EF}\,{}^1\Sigma_g^+$ and $\text{F}\,{}^1\Sigma_g^+$ states. +The EOM-CSSD curves clearly evidence the avoided crossing between the $\text{E}$ and $\text{F}$ states. SF-CIS does not model accurately the $\text{E}\,{}^1\Sigma_g^+$ state before the avoided crossing, but the agreement between SF-CIS and EOM-CCSD is much satisfactory for bond length greater than $1.6$ \AA. -Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than at the dissociation limit. +Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than after. Nonetheless, this results in a rather good qualitative agreement with an avoided crossing placed at a slightly larger bond length than at the EOM-CCSD level. -As mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access the double excitation. +As mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations. However, CIS is quite accurate for the $\text{E}\,{}^1\Sigma_g^+$. \titou{Spin-contamination of the E state?} \titou{CIS or UCIS?} -The center panel of Fig.~\ref{fig:H2} gives results of the TD-BH\&HLYP calculation with and without spin-flip. +In the center panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results. TD-BH\&HLYP shows bad results for all the states of interest with and without spin-flip. Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general point of view. \cite{Cohen_2008a,Cohen_2008c,Cohen_2012} 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. \titou{RKS or UKS?} +Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI}. In the bottom panel of Fig.~\ref{fig:H2} we have results for BSE calculation with and without spin-flip. SF-BSE gives a good representation of the $\text{B}\,{}^1\Sigma_u^+$ state with error of 0.05-0.3 eV. @@ -719,7 +720,7 @@ However we can observe that for all the methods that we compared, when the spin- There is no avoided crossing or perturbation in the curve for the $\text{E}\,{}^1\Sigma_g^+$ state when spin-flip is not used. This is because for these methods we are in the space of single excitation and de-excitation. \titou{BSE@RHF?} - +A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI}. %%% FIG 2 %%% \begin{figure} @@ -789,8 +790,8 @@ So here we have an example where the dynamical corrections are necessary to get \cline{2-4} Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{1g}$ \\ \hline - SF-TD-BLYP \fnm[3] & & & \\ - SF-TD-B3LYP \fnm[3] &1.750 &2.260 &4.094 \\ + SF-TD-BLYP\fnm[3] & & & \\ + SF-TD-B3LYP\fnm[3] &1.750 &2.260 &4.094 \\ 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 \\