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Pierre-Francois Loos 2020-10-20 14:31:39 +02:00
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Response_Letter/Response_Letter.tex
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@ -41,15 +41,15 @@ We look forward to hearing from you.
{The systems chosen are said to represent examples of valence, charge-transfer, and Rydberg excited states. {The systems chosen are said to represent examples of valence, charge-transfer, and Rydberg excited states.
Some explanation is required for why they represent charge-transfer and Rydberg excited states.} Some explanation is required for why they represent charge-transfer and Rydberg excited states.}
\\ \\
\alert{} \alert{A new paragraph describing the electronic states of each system has been added to the manuscript (see page 3).}
\item \item
{For HeH+ example, the internuclear separation is taken to be near equilibrium, if I understand it correctly. {For HeH+ example, the internuclear separation is taken to be near equilibrium, if I understand it correctly.
What then is the charge-transfer character of the excited states, i.e. how much of a significant change in the charge distribution do the excited states have compared to the ground state?} What then is the charge-transfer character of the excited states, i.e. how much of a significant change in the charge distribution do the excited states have compared to the ground state?}
\\ \\
\alert{As now mentioned in the manuscript, a Mulliken or Lowdin population analysis reveal that 1.53 electrons are located on He and 0.47 electron on H in the ground state. \alert{As now mentioned in the manuscript (see page 3), a Mulliken or L\"owdin population analysis reveal that 1.53 electrons are located on He and 0.47 electrons on H in the ground state.
Thus, electronic excitations correspond to a charge transfer from He to H. Thus, electronic excitations in \ce{HeH+} correspond to a charge transfer from He to H.
A paragraph has been added to discuss this point.} A paragraph has been added to discuss this point alongside additional references.}
\item \item
{For the He example, if I recall correctly, the lowest double-excitation of He lies in the continuum, i.e. is not a bound state but rather a resonance. {For the He example, if I recall correctly, the lowest double-excitation of He lies in the continuum, i.e. is not a bound state but rather a resonance.
@ -58,7 +58,7 @@ We look forward to hearing from you.
Perhaps the finite basis set makes this resonance appear bound.} Perhaps the finite basis set makes this resonance appear bound.}
\\ \\
\alert{The reviewer is right. \alert{The reviewer is right.
In He, the lowest doubly-excited state is an auto-ionising resonance state, extremely high in energy and lies in the continuum. In He, the lowest doubly-excited state has an auto-ionising resonance state, extremely high in energy and lies in the continuum.
Highly-accurate calculations estimate an excitation energy of $57.85$ hartree for this $1s^2 \rightarrow 2s^2$ transition. Highly-accurate calculations estimate an excitation energy of $57.85$ hartree for this $1s^2 \rightarrow 2s^2$ transition.
In a minimal basis set though, it is of Rydberg nature as it corresponds to a transition from a relatively compact $s$-type function to a more diffuse one. In a minimal basis set though, it is of Rydberg nature as it corresponds to a transition from a relatively compact $s$-type function to a more diffuse one.
All of these information have been added to the revised version of the manuscript with additional references.} All of these information have been added to the revised version of the manuscript with additional references.}
@ -68,7 +68,8 @@ We look forward to hearing from you.
However I believe this is only true for two-electron systems. However I believe this is only true for two-electron systems.
For the $N$-electron case the exact-exchange kernel defined within TDDFT is frequency-dependent (see e.g. Hellgren and Gross, PRA 88, 052507 (2013) and references therein, Hesselmann and Goerling JCP 134, 034120 (2011)) } For the $N$-electron case the exact-exchange kernel defined within TDDFT is frequency-dependent (see e.g. Hellgren and Gross, PRA 88, 052507 (2013) and references therein, Hesselmann and Goerling JCP 134, 034120 (2011)) }
\\ \\
\alert{The reviewer is right. We have mentioned that, in a density-functional context, the exchange kernel can be frequency dependent if exact exchange is considered, and we have cited the two references provided by the reviewer.} \alert{The reviewer is right.
We have mentioned that, in a density-functional context, the exchange kernel can be frequency dependent if exact exchange is considered, and we have cited the two references provided by the reviewer.}
\item \item
{The authors comment that the D-TDDFT kernel (Eq 14) ``is known to work best in the weak correlation regime where the true excitations have a clear single and double excitation character'' (third last para end of sec IIIB). {The authors comment that the D-TDDFT kernel (Eq 14) ``is known to work best in the weak correlation regime where the true excitations have a clear single and double excitation character'' (third last para end of sec IIIB).
@ -120,13 +121,16 @@ We look forward to hearing from you.
For example the dressed TDDFT method is suppose to work best in the case of doubles strongly coupled to single excitations which from the tables seems not to be the case in these systems (w1updown and w2updown are not very close in energy) so it's rather remarkable how well dressed TDDFT predicts the energy of w2updown.} For example the dressed TDDFT method is suppose to work best in the case of doubles strongly coupled to single excitations which from the tables seems not to be the case in these systems (w1updown and w2updown are not very close in energy) so it's rather remarkable how well dressed TDDFT predicts the energy of w2updown.}
\\ \\
\alert{An entire paragraph explaining the pecularities of each system has been added. \alert{An entire paragraph explaining the pecularities of each system has been added.
As mentioned in the revised manuscript, the dressed TDDFT method works indeed best when the single and double excitations are close in energy. As mentioned in the revised manuscript, the dressed TDDFT method works best when the single and double excitations are close in energy.
However I believe that more importantly, it works best when these two excitations are well separated from the others which is the case here.} However, we believe that, more importantly, it works best when these two excitations are well separated from the others which is the case here.
(This has been mentioned in the revised manuscript.)}
\item \item
{2. No explanation or reference is provided accompanying the statement that the model systems that are chosen are prototypical of valence, charge-transfer and Rydberg excitations and weather these different excitations can be well represented in the reduced (two-level) space used.} {2. No explanation or reference is provided accompanying the statement that the model systems that are chosen are prototypical of valence, charge-transfer and Rydberg excitations and weather these different excitations can be well represented in the reduced (two-level) space used.}
\\ \\
\alert{We have added a reference to the work of Senjean \textit{et al.}~where \ce{H2} and \ce{HeH+} are used as prototypical systems. The He atom is considered in the work of Romaniello \textit{et al.}~which is also cited. } \alert{We have added a reference to the work of Senjean \textit{et al.}~where \ce{H2} and \ce{HeH+} are used as prototypical systems.
The He atom is considered in the work of Romaniello \textit{et al.}~which is also cited.
A new paragraph describing the electronic states of each system has been added to the manuscript (see page 3 and the answer to reviewer \#2's comments).}
\item \item
{3. The molecules H2 and HeH+ are studied at the equilibrium distance, right? {3. The molecules H2 and HeH+ are studied at the equilibrium distance, right?
@ -134,7 +138,10 @@ We look forward to hearing from you.
If you claim the problem is an example of charge-transfer excitations then it would be good to study the stretched molecules... If you claim the problem is an example of charge-transfer excitations then it would be good to study the stretched molecules...
Would a larger basis set be needed in such case?} Would a larger basis set be needed in such case?}
\\ \\
\alert{We have modified the manuscript to add stretched H2 and HeH+ (taken at five times the equilibrium distance).} \alert{We agree with the referee that it would be very interesting to study these molecules at stretched geometries.
However, singlet and triplet instabilities (which makes the excitation energies complex in some cases) prevent us to perform such study.
We believe that studying frequency-dependent kernels is this type of situations is extremely valuable but it is probably outside of scope of the present study.
The appearance of such instabilities has been mentioned in our revised manuscript (see Conclusion).}
\item \item
{4. The statement about the vanishing of the matrix element $\langle S|H|D \rangle=0$ in H2 could be discussed better or a reference added.} {4. The statement about the vanishing of the matrix element $\langle S|H|D \rangle=0$ in H2 could be discussed better or a reference added.}
@ -149,7 +156,7 @@ We look forward to hearing from you.
Can you really study Rydberg excitations using a minimal basis?} Can you really study Rydberg excitations using a minimal basis?}
\\ \\
\alert{For He, 6-31G is not a minimal basis. \alert{For He, 6-31G is not a minimal basis.
Thus, there's some orbital relaxation in the excited states, and the matrix element $\langle S|H|D \rangle$ does not vanish.} Thus, orbital relaxation effects are at play in the excited states, and the matrix element $\langle S|H|D \rangle$ does not vanish.}
\item \item
{6. It is also not discussed how the exact solution is computed. {6. It is also not discussed how the exact solution is computed.
@ -157,7 +164,7 @@ We look forward to hearing from you.
Is the eigensystem computed within the same minimal basis set? Is the eigensystem computed within the same minimal basis set?
How are the excitation energies that appear as ``exact'' in the tables computed?} How are the excitation energies that appear as ``exact'' in the tables computed?}
\\ \\
\alert{The exact Hamiltonian (within this minimal basis) is provided by Eq.~(12).jwjehwyuiuuyylk \alert{The exact Hamiltonian (within this minimal basis) is provided by Eq.~(12).
These are exact results within the one-electron space spanned by these basis functions. These are exact results within the one-electron space spanned by these basis functions.
The exact excitation energies are calculated as differences of these total energies. The exact excitation energies are calculated as differences of these total energies.
This is now explicitly stated.} This is now explicitly stated.}
@ -177,14 +184,14 @@ We look forward to hearing from you.
\\ \\
\alert{Following the reviewer's advice, we have added a short description of the STO-3G and 6-31G basis for these three systems. \alert{Following the reviewer's advice, we have added a short description of the STO-3G and 6-31G basis for these three systems.
For a one-center system like He, a double-zeta basis must be considered in order to have two levels. For a one-center system like He, a double-zeta basis must be considered in order to have two levels.
For two-center systems like H2 and HeH+, a minimal basis on each center must be considered to produce two-level systems.} For two-center systems like \ce{H2} and \ce{HeH+}, a minimal basis on each center must be considered to produce two-level systems.}
\item \item
{9. How does the dimension of the basis set affect the performance of the different dynamical kernel {9. How does the dimension of the basis set affect the performance of the different dynamical kernel
under study?} under study?}
\\ \\
\alert{This is an open question that we hope to answer in the near future. \alert{This is an open question that we hope to answer in the near future.
To do so, one must be able to solve the non-linear, frequency dependent eigensystem in a large basis, which is not straightforward from a tehcnical point of view. To do so, one must be able to solve the non-linear, frequency dependent eigensystem in a large basis, which is not straightforward from a technical point of view.
Moreover, one must be able to characterized unambiguously the different excited states.} Moreover, one must be able to characterized unambiguously the different excited states.}
\item \item
@ -220,16 +227,15 @@ omega2 up down= $\langle D|H|D \rangle - \langle 0|H|0 \rangle$ ?}
\\ \\
\alert{The reviewer is right, this is inconsistent. \alert{The reviewer is right, this is inconsistent.
The quantities gathered in Table I are now provided in eV, like the rest of the results. The quantities gathered in Table I are now provided in eV, like the rest of the results.
The figures are now also plotted in eV.
Yes, the ``exact'' values in Tables II, III and IV are the same as in Table I. Yes, the ``exact'' values in Tables II, III and IV are the same as in Table I.
These exact transition energies are computed as the difference of total energies (ie eigenvalues) provided by the Hamiltonian defined in Eq.~(12). These exact transition energies are computed as the difference of total energies (i.e. eigenvalues) provided by the Hamiltonian defined in Eq.~(12).
It does not correspond to the expressions provided by the reviewer as there is, in general, coupling between these terms.} It does not correspond to the expressions provided by the reviewer as there is, in general, coupling between these terms.}
\item \item
{15. page 5, second para: Do you mean by ``static excitations'' w1updown and w1upup?} {15. page 5, second para: Do you mean by ``static excitations'' w1updown and w1upup?}
\\ \\
\alert{Yes. This has been more clearly specified in the revised manuscript. \alert{Yes. This has been more clearly specified in the revised manuscript.
Thank you for mentioning that it wasn't clear enough.} Thank you for mentioning that it was not clear enough.}
\end{itemize} \end{itemize}

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dynker.tex
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@ -244,9 +244,10 @@ These three systems provide prototypical examples of valence, charge-transfer, a
In the case of \ce{H2}, the HOMO and LUMO orbitals have $\sigma_g$ and $\sigma_u$ symmetries, respectively. In the case of \ce{H2}, the HOMO and LUMO orbitals have $\sigma_g$ and $\sigma_u$ symmetries, respectively.
The electronic configuration of the ground state is $\sigma_g^2$, and the doubly-excited state of configuration $\sigma_u^2$ has an auto-ionising resonance nature. \cite{Bottcher_1974,Barca_2018a,Marut_2020} The electronic configuration of the ground state is $\sigma_g^2$, and the doubly-excited state of configuration $\sigma_u^2$ has an auto-ionising resonance nature. \cite{Bottcher_1974,Barca_2018a,Marut_2020}
The singly-excited states correspond to $\sigma_g \sigma_u$ configurations. The singly-excited states correspond to $\sigma_g \sigma_u$ configurations.
In He, highly-accurate calculations reveal that the lowest doubly-excited state is an auto-ionising resonance state, extremely high in energy and lies in the continuum. \cite{Madden_1963,Burges_1995,Marut_2020} In He, highly-accurate calculations reveal that the lowest doubly-excited state of configuration $1s^2$ is an auto-ionising resonance state, extremely high in energy and lies in the continuum. \cite{Madden_1963,Burges_1995,Marut_2020}
However, in a minimal basis set such as STO-3G, it is of Rydberg nature as it corresponds to a transition from a relatively compact $s$-type function to a more diffuse one. However, in a minimal basis set such as STO-3G, it is of Rydberg nature as it corresponds to a transition from a relatively compact $s$-type function to a more diffuse orbital of the same symmetry.
In the heteronuclear diatomic \ce{HeH+}, a Mulliken or Lowdin population analysis associates $1.53$ electrons on the \ce{He} center and $0.47$ electrons on \ce{H} for the ground state, \cite{SzaboBook} with an opposite trend for the excited states. Electronic excitation corresponds then to a charge transfer from the \ce{He} nucleus to the proton.} In the heteronuclear diatomic \ce{HeH+}, a Mulliken or L\"owdin population analysis associates $1.53$ electrons on the \ce{He} center and $0.47$ electrons on the \ce{H} nucleus for the ground state, \cite{SzaboBook} with an opposite trend for the excited states.
Thus, electronic excitations in \ce{HeH+} correspond to a charge transfer from the \ce{He} nucleus to the proton.}
The numerical values of the various quantities defined above are gathered in Table \ref{tab:params} for each system. The numerical values of the various quantities defined above are gathered in Table \ref{tab:params} for each system.
%%% TABLE I %%% %%% TABLE I %%%
@ -359,7 +360,7 @@ It would be, of course, a different story in a larger basis set where the coupli
%%% FIGURE 1 %%% %%% FIGURE 1 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{Maitra} \includegraphics[width=\linewidth]{fig1}
\caption{ \caption{
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in \alert{eV}) for both the singlet (gray and black) and triplet (orange) manifolds of \ce{HeH+}. $\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in \alert{eV}) for both the singlet (gray and black) and triplet (orange) manifolds of \ce{HeH+}.
The static TDHF Hamiltonian (dashed) and dynamic D-TDHF Hamiltonian (solid) are considered. The static TDHF Hamiltonian (dashed) and dynamic D-TDHF Hamiltonian (solid) are considered.
@ -493,7 +494,7 @@ Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dyn
%%% FIGURE 2 %%% %%% FIGURE 2 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{dBSE} \includegraphics[width=\linewidth]{fig2}
\caption{ \caption{
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ \alert{(in eV)} for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}. $\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ \alert{(in eV)} for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}.
The static BSE Hamiltonian (dashed) and dynamic dBSE Hamiltonian (solid) are considered. The static BSE Hamiltonian (dashed) and dynamic dBSE Hamiltonian (solid) are considered.
@ -684,7 +685,7 @@ In the case of BSE2, the perturbative partitioning (pBSE2) is simply
%%% FIGURE 4 %%% %%% FIGURE 4 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{dBSE2} \includegraphics[width=\linewidth]{fig3}
\caption{ \caption{
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ \alert{(in eV)} for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}. $\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ \alert{(in eV)} for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}.
The static BSE2 Hamiltonian (dashed) and dynamic dBSE2 Hamiltonian (solid) are considered. The static BSE2 Hamiltonian (dashed) and dynamic dBSE2 Hamiltonian (solid) are considered.
@ -717,6 +718,7 @@ Using a simple two-model system, we have explored the physics of three dynamical
Prototypical examples of valence, charge-transfer, and Rydberg excited states have been considered. Prototypical examples of valence, charge-transfer, and Rydberg excited states have been considered.
From these, we have observed that, overall, the dynamical correction usually improves the static excitation energies, and that, although one can access double excitations, the accuracy of the BSE and BSE2 kernels for double excitations is rather average. From these, we have observed that, overall, the dynamical correction usually improves the static excitation energies, and that, although one can access double excitations, the accuracy of the BSE and BSE2 kernels for double excitations is rather average.
If one has no interest in double excitations, a perturbative treatment is an excellent alternative to a non-linear resolution of the dynamical equations. If one has no interest in double excitations, a perturbative treatment is an excellent alternative to a non-linear resolution of the dynamical equations.
\alert{Although it would be interesting to study the performance of such kernels in the case of stretched bonds, the appearance of singlet and triplet instabilities makes such type of invesgations particularly difficult.}
We hope that the present contribution will foster new developments around dynamical kernels for optical excitations, in particular to access double excitations in molecular systems. We hope that the present contribution will foster new developments around dynamical kernels for optical excitations, in particular to access double excitations in molecular systems.

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