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Manuscript/sfBSE.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-01-17 10:06:24 +0100
%% Created for Pierre-Francois Loos at 2021-01-17 14:51:24 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Cohen_2008c,
abstract = {Density functional theory of electronic structure is widely and successfully applied in simulations throughout engineering and sciences. However, for many predicted properties, there are spectacular failures that can be traced to the delocalization error and static correlation error of commonly used approximations. These errors can be characterized and understood through the perspective of fractional charges and fractional spins introduced recently. Reducing these errors will open new frontiers for applications of density functional theory.},
author = {Cohen, Aron J. and Mori-S{\'a}nchez, Paula and Yang, Weitao},
date-added = {2021-01-17 14:51:05 +0100},
date-modified = {2021-01-17 14:51:15 +0100},
doi = {10.1126/science.1158722},
eprint = {https://science.sciencemag.org/content/321/5890/792.full.pdf},
issn = {0036-8075},
journal = {Science},
number = {5890},
pages = {792--794},
publisher = {American Association for the Advancement of Science},
title = {Insights into Current Limitations of Density Functional Theory},
url = {https://science.sciencemag.org/content/321/5890/792},
volume = {321},
year = {2008},
Bdsk-Url-1 = {https://science.sciencemag.org/content/321/5890/792},
Bdsk-Url-2 = {https://doi.org/10.1126/science.1158722}}
@article{Barca_2014,
author = {G. M. J. Barca and A. T. B. Gilbert and P. M. W. Gill},
date-added = {2021-01-17 10:05:12 +0100},
@ -6607,10 +6626,10 @@
year = {2007},
Bdsk-Url-1 = {https://doi.org/10.1063/1.2741248}}
@article{Cohen_2008,
@article{Cohen_2008b,
author = {Cohen, Aron J. and {Mori-S\'anchez}, Paula and Yang, Weitao},
date-added = {2020-01-01 21:36:51 +0100},
date-modified = {2020-01-01 21:36:51 +0100},
date-modified = {2021-01-17 14:51:23 +0100},
doi = {10.1103/PhysRevB.77.115123},
file = {/Users/loos/Zotero/storage/QYTAN7UQ/Cohen et al. - 2008 - Fractional charge perspective on the band gap in d.pdf},
issn = {1098-0121, 1550-235X},

54
Manuscript/sfBSE.tex
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@ -670,7 +670,7 @@ Generally we can observe that all the scheme with SF-BSE used do not increase si
\begin{figure}
\includegraphics[width=\linewidth]{Be}
\caption{
Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained with the 6-31G basis for various levels of theory:
Excitation energies (in eV) with respect to the $^1S(1s^2 2s^2)$ singlet ground state of \ce{Be} obtained with the 6-31G basis for various levels of theory:
SF-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black).
All the spin-flip calculations have been performed with a UHF reference.
\label{fig:Be}}
@ -682,31 +682,43 @@ 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 (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, especially, 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.
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.
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.
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.
Note that this avoided crossing is not visible with non-spin-flip methods, such as CIS, TD-DFT, and BSE, as these are ``blind'' to double excitations.
Three methods, in their standard and spin-flip versions, are studied here (CIS, TD-BH\&HLYP and BSE) and are compared to the reference EOM-CCSD excitation energies (that is equivalent to FCI in the case of \ce{H2}).
All these calculations are performed in the cc-pVQZ basis.
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.
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 right panel gives results of the TD-BH\&HLYP calculation with and without spin-flip.
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 $\text{EF}\,{}^1\Sigma_g^+$ and $\text{F}\,{}^1\Sigma_g^+$ 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.
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.
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.
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.
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.
\titou{RKS or UKS?}
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 E${}^1 \Sigma_g^+$ state when spin-flip is not used.
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?}
%%% FIG 2 %%%
@ -715,8 +727,10 @@ This is because for these methods we are in the space of single excitation and d
\includegraphics[width=1\linewidth]{H2_BHHLYP.pdf}
\includegraphics[width=1\linewidth]{H2_BSE.pdf}
\caption{
Excitation energies of the three states of interest [with respect to the singlet ground state] of \ce{H2} obtained with the cc-pVQZ basis. Three sets of curves are drawn, the solid curves are the references (EOM-CCSD), the dashed curves are obtained with spin-flip method and the dotted curves are obtained without using spin-flip. The top panel shows CIS results, the center panel shows TD-BH\&HLYP results and the bottom panel shows the BSE results.
Excitation energies of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{E}\,{}^1\Sigma_g^+$ (blue) states (with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state) of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)CIS (top), (SF-)TD-BH\&HLYP (middle), and (SF-)BSE (bottom) levels of theory.
The reference EOM-CCSD excitation energies are represented as solid lines, while the results obtained with and without spin-flip results are represented as dashed and dotted lines, respectively.
All the spin-flip calculations have been performed with a UHF reference.
The raw data are reported in the {\SI}.
\label{fig:H2}}
\end{figure}
%%% %%% %%%
@ -728,7 +742,7 @@ This is because for these methods we are in the space of single excitation and d
Cyclobutadiene (CBD) is an interesting example as its electronic character of its ground state can be tune via geometrical deformation. \cite{Balkova_1994,Manohar_2008,Lefrancois_2015,Casanova_2020}
%with potential large spin contamination.
In its $D_{2h}$ rectangular $^1 A_g$ singlet ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals
In its $D_{2h}$ rectangular $^1 A_g$ singlet ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals.
However, in its $D_{4h}$ square-planar $^3 A_{2g}$ triplet round-state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals.
In this case, single-reference methods notoriously fail.
Nonetheless, the lowest triplet state of symmetry $^3 A_{2g}$ remains of single-reference character and is then a perfect starting point for spin-flip calculations.