From d6dad8419b87f1cfee1af9db588d86393699913b Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 18 Jan 2021 15:40:29 +0100 Subject: [PATCH] done with H2 for now --- Manuscript/sfBSE.tex | 30 +- Notebooks/sf-BSE.nb | 2255 +++++++++++++++++------------------------- 2 files changed, 933 insertions(+), 1352 deletions(-) diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index 66e84c6..642ae94 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -688,30 +688,24 @@ All these calculations are performed in the cc-pVQZ basis, and both the spin-con The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dashed lines) excitation energies as a function of $R(\ce{H-H})$. 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 $\text{E}$ and $\text{F}$ states. +We observe that both CIS and SF-CIS poorly describe the $\text{B}\,{}^1\Sigma_u^+$ state in the dissociation limit with an error greater than $1$ eV, while CIS, unlike SF-CIS, is much more accurate around the equilibrium geometry \titou{(spin-contamination of the SF-CIS wave function?)}. +Similar observations can be made for the $\text{E}\,{}^1\Sigma_g^+$ state with a good description at the CIS level for all bond lengths. 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 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 double excitations. -However, CIS is quite accurate for the $\text{E}\,{}^1\Sigma_g^+$. -\titou{Spin-contamination of the E state?} +Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than after \titou{(spin-contamination of the SF-CIS wave function?)}, while this state is completely absent at the CIS level. +Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations. +At the SF-CIS level, the avoided crossing between the $\text{E}$ and $\text{F}$ states is qualitatively reproduced and placed at a slightly larger bond length than at the EOM-CCSD level. 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. Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI}. +SF-TD-BH\&HLYP shows, at best, qualitative agreement with EOM-CCSD, while the TD-BH\&HLYP excitation energies of the $\text{B}$ and $\text{E}$ states are only trustworthy around equilibrium but inaccurate at dissociation. +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} -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. -However SF-BSE does not describe well the $\text{E}\,{}^1\Sigma_g^+$ 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 $\text{F}\,{}^1\Sigma_g^+$ state, indeed we have an error of 0.008-0.6 eV. BSE results for the $\text{B}\,{}^1\Sigma_u^+$ state are close to the reference until 2.0 \AA~ and the give bad agreement for the dissociation limit. -For the $\text{E}\,{}^1\Sigma_g^+$ 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 $\text{E}\,{}^1\Sigma_g^+$ state when spin-flip is not used. -This is because for these methods we are in the space of single excitation and de-excitation. +In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented. A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI}. +SF-BSE provides surprisingly accurate excitation energies for the $\text{B}\,{}^1\Sigma_u^+$ state with errors between $0.05$ and $0.3$ eV, outperforming in the process the standard BSE formalism. +However SF-BSE does not describe well the $\text{E}\,{}^1\Sigma_g^+$ state with error ranging from half an eV to $1.6$ eV \titou{(spin-contamination of the SF-BSE wave function?)}. +Similar performances are observed at the BSE level around equilibrium with a clear improvement in the dissociation limit. +Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\Sigma_g^+$ doubly-excited state, resulting in an avoided crossing around $R(\ce{H-H}) = 1.6$ \AA. %%% FIG 2 %%% \begin{figure} diff --git a/Notebooks/sf-BSE.nb b/Notebooks/sf-BSE.nb index ea110bc..e355fd1 100644 --- a/Notebooks/sf-BSE.nb +++ b/Notebooks/sf-BSE.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest 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