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%% This BibTeX bibliography file was created using BibDesk.
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%% Created for Pierre-Francois Loos at 2021-01-16 14:12:40 +0100
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%% Created for Pierre-Francois Loos at 2021-01-17 09:52:25 +0100
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@article{Vuckovic_2017,
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author = {Vuckovic, Stefan and Gori-Giorgi, Paola},
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date-added = {2021-01-17 09:51:44 +0100},
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date-modified = {2021-01-17 09:52:00 +0100},
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doi = {10.1021/acs.jpclett.7b01113},
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eprint = {https://doi.org/10.1021/acs.jpclett.7b01113},
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journal = {The Journal of Physical Chemistry Letters},
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note = {PMID: 28581751},
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number = {13},
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pages = {2799-2805},
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title = {Simple Fully Nonlocal Density Functionals for Electronic Repulsion Energy},
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url = {https://doi.org/10.1021/acs.jpclett.7b01113},
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volume = {8},
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year = {2017},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.7b01113}}
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@article{Becke_1993b,
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author = {Becke,Axel D.},
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date-added = {2021-01-13 09:37:07 +0100},
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@ -12169,24 +12185,6 @@
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year = {2016},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00177}}
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@article{Vuckovic_2017,
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abstract = {From a simplified version of the mathematical structure of the strong coupling limit of the exact exchange-correlation functional, we construct an approximation for the electronic repulsion energy at physical coupling strength, which is fully nonlocal. This functional is self-interaction free and yields energy densities within the definition of the electrostatic potential of the exchange-correlation hole that are locally accurate and have the correct asymptotic behavior. The model is able to capture strong correlation effects that arise from chemical bond dissociation, without relying on error cancellation. These features, which are usually missed by standard density functional theory (DFT) functionals, are captured by the highly nonlocal structure, which goes beyond the ``Jacob's ladder'' framework for functional construction, by using integrals of the density as the key ingredient. Possible routes for obtaining the full exchange-correlation functional by recovering the missing kinetic component of the correlation energy are also implemented and discussed.},
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author = {Vuckovic, Stefan and {Gori-Giorgi}, Paola},
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date-added = {2020-01-01 21:36:51 +0100},
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date-modified = {2020-01-01 21:36:52 +0100},
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doi = {10.1021/acs.jpclett.7b01113},
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file = {/Users/loos/Zotero/storage/YJUN4JS9/Vuckovic and Gori-Giorgi - 2017 - Simple Fully Nonlocal Density Functionals for Elec.pdf},
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issn = {1948-7185},
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journal = {J. Phys. Chem. Lett.},
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language = {en},
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month = jul,
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number = {13},
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pages = {2799-2805},
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title = {Simple {{Fully Nonlocal Density Functionals}} for {{Electronic Repulsion Energy}}},
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volume = {8},
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year = {2017},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.7b01113}}
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@article{Vuckovic_2017a,
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author = {Vuckovic, Stefan and Levy, Mel and {Gori-Giorgi}, Paola},
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date-added = {2020-01-01 21:36:51 +0100},
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@ -611,9 +611,11 @@ Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this deg
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However, the SF-TD-BH\&HLYP excitation energy of the $^1P(1s^2 2s^1 2p^1)$ state is still off by $1.6$ eV as compared to the FCI reference.
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For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significantly improved.
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The center part of Fig.~\ref{fig:Be} (blue lines) shows the SF-BSE results alongside the SF-CIS excitation energies (purple line).
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The center part of Fig.~\ref{fig:Be} (blue lines) shows the SF-BSE results alongside the SF-CIS excitation energies (purple lines).
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All of these are computed with 100\% of exact exchange with the inclusion of correlation in the case of SF-BSE thanks to the introduction of the screening.
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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 $^1D(1s^2 2p^2)$ 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.
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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.
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For the last excited state $^1D(1s^2 2p^2)$ the largest error is $0.85$ eV with SF-dBSE@{\qsGW}, so we have a bad description of this state due to spin contamination.
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Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies.
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%%% TABLE I %%%
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\begin{squeezetable}
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@ -675,119 +677,36 @@ They are close the FCI results, because of the fact that there is one hundred pe
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\end{figure}
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%%% %%% %%% %%%
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%%% TABLE II %%%
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%\begin{squeezetable}
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%\begin{table*}
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% \caption{
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% Spin-flip excitations (in eV) of \ce{H2} obtained for various methods with the cc-pVQZ basis.
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% The $GW$ calculations are performed with a HF starting point.
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% \label{tab:H2}}
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%\begin{ruledtabular}
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%\begin{tabular}{lccccccccccccc}% seven columns now, not six...
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%
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%Distance (\AA) & \multicolumn{3}{c}{SF-CIS} & \multicolumn{3}{c}{SF-TDDFT} & \multicolumn{3}{c}{SF-BSE@{\GOWO}} & \multicolumn{3}{c}{EOM-CCSD}\\ \hline
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%
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% & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ \\
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%
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% \hline
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%
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%0.50 & \\
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%0.55 & \\
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%0.60 & \\
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%0.65 & \\
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%0.70 & \\
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%0.75 & \\
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%0.80 & \\
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%0.85 & \\
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%0.90 & \\
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%0.95 & \\
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%1.00 & \\
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%1.05 & \\
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%1.10 & \\
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%1.15 & \\
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%1.20 & \\
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%1.25 & \\
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%1.30 & \\
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%1.35 & \\
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%1.40 & \\
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%1.45 & \\
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%1.50 & \\
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%1.55 & \\
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%1.60 & \\
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%1.65 & \\
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%1.70 & \\
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%1.75 & \\
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%1.80 & \\
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%1.85 & \\
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%1.90 & \\
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%1.95 & \\
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%2.00 & \\
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%2.05 & \\
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%2.10 & \\
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%2.15 & \\
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%2.20 & \\
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%2.25 & \\
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%2.30 & \\
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%2.35 & \\
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%2.40 & \\
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%2.45 & \\
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%2.50 & \\
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%2.55 & \\
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%2.60 & \\
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%2.65 & \\
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%2.70 & \\
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%2.75 & \\
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%2.80 & \\
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%2.85 & \\
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%2.90 & \\
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%2.95 & \\
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%3.00 & \\
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%3.30 & \\
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%3.40 & \\
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%3.50 & \\
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%3.55 & \\
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%3.60 & \\
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%3.65 & \\
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%3.70 & \\
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%3.80 & \\
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%
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%
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%\end{tabular}
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%\end{ruledtabular}
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%\end{table*}
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%\end{squeezetable}
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%%%% %%% %%% %%%
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%%% FIG 1 %%%
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%\begin{figure*}
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%\includegraphics[width=0.7\linewidth]{Be}
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%\caption{
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%Spin-flip excitation energies (in eV) of \ce{Be} obtained for various methods with the 6-31G basis.
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%\label{fig:Be}}
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%\end{figure*}
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%===============================
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\subsection{Hydrogen molecule}
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\label{sec:H2}
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%===============================
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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.
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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.
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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 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 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.
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Our second example deals with the dissociation of the \ce{H2} molecule, which is a prototypical system for testing new electronic structure methods (see, for example, Refs.~\onlinecite{Caruso_2013,Vuckovic_2017}, and references therein).
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The $\text{X}\,{}^1 \Sigma_g^+$ ground state of \ce{H2} has an electronic configuration $(1\sigma_g)^2$ configuration.
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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.
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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.
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Because these latter two excited states interact strongly and form an avoided crossing around $R_{\ce{H-H}} = 1.4$ \AA, they are usually labeled as the $\text{EF}\,{}^1 \Sigma_g^+$ state.
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Note that this avoided crossing is not visible with non-spin-flip methods such as CIS, TD-DFT, and BSE, has these are ``blind'' to double excitations.
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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.
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Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and without spin-flip.
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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.
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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.
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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.
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SF-CIS describes better the F${}^1 \Sigma_g^+$ state before the avoided crossing than at the dissociation limit.
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In general, SF-CIS does not give a good description of the double excitation.
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As expected CIS does not find the double excitation to the F${}^1 \Sigma_g^+$ state.
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The right panel gives results of the TD-BH\&HLYP calculation with and without spin-flip.
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TD-BH\&HLYP shows bad results for all the states of interest with and without spin-flip.
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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.
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In the last panel we have results for BSE calculation with and without spin-flip.
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SF-BSE gives a good representation of the B${}^1 \Sigma_u^+$ state with error of 0.05-0.3 eV.
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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.
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For the E${}^1 \Sigma_g^+$ state BSE gives closer results to the reference than SF-BSE.
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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.
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There is no avoided crossing or perturbation in the curve for the E${}^1 \Sigma_g^+$ state when spin-flip is not used.
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This is because for these methods we are in the space of single excitation and de-excitation.
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%%% FIG 2 %%%
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@ -818,8 +737,18 @@ The $D_{2h}$ and $D_{4h}$ optimized geometries of the $^1 A_g$ and $^3 A_{2g}$ s
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EOM-CCSD and SF-ADC calculations have been taken from Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}.
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All of them have been obtained with a UHF reference like the SF-BSE calculations performed here.
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Table~\ref{tab:CBD_D2h} shows results obtained for the $D_{2h}$ rectangular equilibrium geometry and Table~\ref{tab:CBD_D4h} shows results obtained for $D_{4h}$ square equilibrium geometry. These results are given with respect to the singlet ground state. For each geometry three states are under investigation, for the $D_{2h}$ CBD we look at the $1 ~^3 B_{1g}$, $1~^1 B_{1g}$ and $2 ~^1 A_{1g}$ states. For the $D_{4h}$ CBD we look at the $1 ~^3 A_{2g}$, $2~^1 A_{1g}$ and $1 ~^1 B_{2g}$ states. Several methods using spin-flip are compared to the spin-flip version of BSE with and without dynamical corrections. In Table~\ref{tab:CBD_D2h}, comparing the results of our work and the most accurate ADC level, i.e., SF-ADC(3) with SF-BSE@{\GOWO} we have a difference in the excitation energy of 0.017 eV for the $1 ~^3 B_{1g}$ state. This difference grows to 0.572 eV for the $1 ~^1 B_{1g}$ state and then it is 0.212 eV for the $2 ~^1 A_{1g}$ state. Adding dynamical corrections in SF-dBSE@{\GOWO} do not improve the accuracy of the excitation energies comparing to SF-ADC(3). Indeed, we have a difference of 0.052 eV for the $1 ~^3 B_{1g}$ state, 0.393 eV for the $1 ~^1 B_{1g}$ state and 0.293 eV for the $2 ~^1 A_{1g}$ state.
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Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-ADC(3) we have an interesting result. Indeed, we have a wrong ordering in our first excited state, we find that the triplet state $1 ~^3 A_{2g}$ is lower in energy that the singlet state $B_{1g}$ in contrary to all of the results extracted in Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}. Then adding dynamical corrections in SF-dBSE@{\GOWO} not only improve the difference of excitation energies with SF-ADC(3) it gives the right ordering for the first excited state, meaning that we retrieve the triplet state $1 ~^3 A_{2g}$ above the singlet state $B_{1g}$. So here we have an example where the dynamical corrections are necessary to get the right chemistry.
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Table~\ref{tab:CBD_D2h} shows results obtained for the $D_{2h}$ rectangular equilibrium geometry and Table~\ref{tab:CBD_D4h} shows results obtained for $D_{4h}$ square equilibrium geometry. These results are given with respect to the singlet ground state.
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For each geometry three states are under investigation, for the $D_{2h}$ CBD we look at the $1 ~^3 B_{1g}$, $1~^1 B_{1g}$ and $2 ~^1 A_{1g}$ states.
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For the $D_{4h}$ CBD we look at the $1 ~^3 A_{2g}$, $2~^1 A_{1g}$ and $1 ~^1 B_{2g}$ states.
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Several methods using spin-flip are compared to the spin-flip version of BSE with and without dynamical corrections.
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In Table~\ref{tab:CBD_D2h}, comparing the results of our work and the most accurate ADC level, \ie, SF-ADC(3) with SF-BSE@{\GOWO} we have a difference in the excitation energy of 0.017 eV for the $1 ~^3 B_{1g}$ state.
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This difference grows to 0.572 eV for the $1 ~^1 B_{1g}$ state and then it is 0.212 eV for the $2 ~^1 A_{1g}$ state.
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Adding dynamical corrections in SF-dBSE@{\GOWO} do not improve the accuracy of the excitation energies comparing to SF-ADC(3).
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Indeed, we have a difference of 0.052 eV for the $1 ~^3 B_{1g}$ state, 0.393 eV for the $1 ~^1 B_{1g}$ state and 0.293 eV for the $2 ~^1 A_{1g}$ state.
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Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-ADC(3) we have an interesting result.
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Indeed, we have a wrong ordering in our first excited state, we find that the triplet state $1 ~^3 A_{2g}$ is lower in energy that the singlet state $B_{1g}$ in contrary to all of the results extracted in Refs.~\onlinecite{Manohar_2008,Lefrancois_2015}.
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Then adding dynamical corrections in SF-dBSE@{\GOWO} not only improve the difference of excitation energies with SF-ADC(3) it gives the right ordering for the first excited state, meaning that we retrieve the triplet state $1\,{}^3 A_{2g}$ above the singlet state $B_{1g}$.
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So here we have an example where the dynamical corrections are necessary to get the right chemistry.
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%%% FIG 3 %%%
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\begin{figure*}
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