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Manuscript/sfBSE.tex
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@ -684,10 +684,10 @@ All these calculations are performed with the cc-pVQZ basis.
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 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. 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 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?)}. 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.
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. 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. 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 \titou{(spin-contamination of the SF-CIS wave function?)}, while this state is completely absent at the CIS level. Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than after, 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. 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. 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.
@ -698,11 +698,16 @@ Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general
In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented. In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented.
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. 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?)}. 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.
Similar performances are observed at the BSE level around equilibrium with a clear improvement in the dissociation limit. 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. 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.
A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI} where it is shown that dynamical effects do not affect the present conclusions. A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI} where it is shown that dynamical effects do not affect the present conclusions.
The right side of Fig.~\ref{fig:H2} shows the amount of spin contamination as a function of the bond length for SF-CIS (top), SF-TD-BH\&HLYP (center), and SF-BSE (bottom).
Overall, one can see that $\expval*{\hS^2}$ behaves similarly for SF-CIS and SF-BSE with a small spin contamination of the $\text{B}\,{}^1\Sigma_u^+$ at short bond length. In contrast, the $\text{B}$ state is much more spin contaminated at the SF-TD-BH\&HLYP level.
For all spin-flip methods, the $\text{E}$ is strongly spin contaminated as expected, while the $\expval*{\hS^2}$ values associated with the $\text{F}$ state
only deviate significantly from zero for short bond length and around the avoided crossing where it strongly couples with the spin contaminated $\text{E}$ state.
%%% FIG 2 %%% %%% FIG 2 %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=0.4\linewidth]{H2_CIS} \includegraphics[width=0.4\linewidth]{H2_CIS}
@ -719,7 +724,7 @@ A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be foun
\hspace{0.05\linewidth} \hspace{0.05\linewidth}
\includegraphics[width=0.4\linewidth]{H2_BSE_S2} \includegraphics[width=0.4\linewidth]{H2_BSE_S2}
\caption{ \caption{
Excitation energies with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state (left) and $\expval*{\hS^2}$ (right) of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{E}\,{}^1\Sigma_g^+$ (blue) states 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. Excitation energies with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state (left) and expectation value of the spin operator $\expval*{\hS^2}$ (right) of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{E}\,{}^1\Sigma_g^+$ (blue) states 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 are represented as dashed and dotted lines, respectively. The reference EOM-CCSD excitation energies are represented as solid lines, while the results obtained with and without spin-flip are represented as dashed and dotted lines, respectively.
All the spin-conserved and spin-flip calculations have been performed with an unrestricted reference. All the spin-conserved and spin-flip calculations have been performed with an unrestricted reference.
The raw data are reported in the {\SI}. The raw data are reported in the {\SI}.