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Pierre-Francois Loos 2021-01-17 15:25:29 +01:00
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Manuscript/sfBSE.tex
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@ -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 \\