Merge branch 'master' of https://git.irsamc.ups-tlse.fr/loos/CBD

Response_Letter/Response_Letter.tex

@@ -61,7 +61,25 @@ The latter actually could potentially provide a faster convergence to the A1g st
 However, this same property leads to a distinct inability to properly access the B1g ground state via a single excitation in EOM-CC. 
 Some illuminating comments on this issue would be welcome.}
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+\alert{We thank the reviewer for this interesting comment. 
+Indeed, at the $D_{4h}$ T1 optimized geometry, we have used the conventional standard orientation where two $C_2$ axes run through the carbon atoms. 
+In this conventional orientation, the singlet ground state $1 ^1B_{1g}$ remains $1 ^1B_{1g}$ in the $D_{2h}$ point group and the singlet excited state $1 ^1A_{1g}$ becomes $1 ^1Ag$ in the $D_{2h}$ point group. 
+As pointed out by the reviewer, upon rotating the molecular framework by 45 degrees in the ($xy$) plane, the two $C_2$ axes then bisect the carbon-carbon bonds. 
+This induces a different orbital picture. The correlation between the orbitals and states in the new molecular framework are illustrated in the figure below at the CASSCF(4,4) level. 
+In this new orientation, the two singlet states $1 ^1B_{1g}$ and $1 ^1A_{1g}$ become both $1 ^1A_{g}$ in the $D_{2h}$ point group. 
+Because of the different orbital picture (the frontier orbitals are localized on two carbon atoms in the standard orientation and on four carbon atoms in the other orientation), the new CI coefficients resulting from this rotation bring also a different wavefunction representation. 
+Whereas the $1 ^1B_{1g}$ ground state is described in a one-electron-excitation picture in the standard orientation (the $1 ^1A_{1g}$ excited state involves a double excitation), the corresponding $1 ^1B_{1g}$ ground state in the new orientation involves a two-electron-excitation picture (the $1 ^1A_{1g}$ excited state also involves a double excitation). 
+Of course, these two representations are perfectly equivalent at the CASSCF level which describes single and double excitations on the same footing. 
+This is obviously not the case in linear response theory, as pointed out by the reviewer.
+As mentioned in our manuscript in section IIb, for the $D_{4h}$ arrangement, we have considered the lowest closed-shell singlet state $1 ^1A_{1g}$ as reference. 
+Because this state has a substantial double-excitation character, we expect a significant error at the CCSD level. 
+The $1 ^1B_{1g}$ ground state is obtained as a singly excited state from that reference, while the $1 ^1B_{2g}$ excited state should also be a mixture involving a double excitation. 
+In the other (non-standard) orientation, the lowest $^1A_g$ state correlates with the $1 ^1B_{1g}$ ground state, which in this orientation has a strong double-excitation character. 
+Then, the $1 ^1 A_{1g}$ excited state has also a strong double-excitation character, while the $1 ^1B_{2g}$ excited state is obtained by one-electron excitation. 
+Thus, whatever the orientation of the molecule, we will face the same problem for the reference state. 
+Note that in the case of the SF formalism, these three singlet states should all be described correctly if one takes the $1 ^3A_{2g}$ state as a reference high spin state, whatever the orientation.}
+
+\includegraphics[width=\textwidth]{MOs}
 
 \item{The authors note a significant improvement in the MRPT results as the active space is enlarged. 
 However, it seems to me that the most appropriate active space (for the D4h geometry at least) is in fact (2e,2o) [i.e. $Eg^2$ at D4h]. 
@@ -69,7 +87,14 @@ Within this space, the CI coefficients become fixed at D4h, leading to an “exa
 Perhaps the major problem with the MRPT results is not active space insufficiency, then, but intruder states? 
 Can the authors perform MRCI+Q  or MRAQCC calculations for comparison?}
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+\alert{We agree with the reviewer that at the $D_{4h}$ geometry the (2e,2o) active space would be enough to describe the pure static correlation. 
+However, to calculate the automerization barrier, we need to make the energy difference between the energy obtained for the ground state at the $D_{4h}$ geometry and that at the $D_{2h}$ geometry. 
+At this last geometry, the correct description of the static correlation requires including (4e,4o) in the active space (i.e., all valence $\pi$ orbitals).
+In addition, there are states with ionic character which required including the dynamic electron correlation (in particular the $\sigma$-$\pi$ polarization). 
+Thus, the improvement of our results by including all $\sigma_{CC}$ is rather expected. 
+Note that we have minimized the intruder state problem by using an appropriate level shift and that this potential problem is not present at the NEVPT2 level.
+As suggested by the reviewer, we have now added some results at the MRCI+Q level.
+}
 
 \item 
 {It seems that extrapolated CCSDTQ/aQZ values are available for the automerization barrier.