Manuscript updates

Manuscript/CBD.bib

@@ -6,7 +6,27 @@
 
 %% Saved with string encoding Unicode (UTF-8) 
 
+@article{Head-Gordon_2003,
+	author = {Martin, Head-Gordon},
+	doi = {10.1016/S0009-2614(03)00422-6},
+	journal = {Chem. Phys. Lett.},
+	title = {Characterizing unpaired electrons from the one-particle density matrix},
+	volume = {372},
+    number = {3},
+    pages = {508-511},
+    year = {2003},
+	bdsk-url-1 = {https://doi.org/10.1016/S0009-2614(03)00422-6}}
 
+@article{Orms_2018,
+	author = {Natalie, Orms and Dirk.R, Rehn and Andreas, Dreuw and Anna I. Krylov},
+	doi = {10.1021/acs.jctc.7b01012},
+	journal = {J. Chem. Theory Comput.},
+	title = {Characterizing Bonding Patterns in Diradicals and Triradicals by Density-Based Wave Function Analysis: A Uniform Approach},
+	volume = {14},
+     number = {2},
+     pages = {638-648},
+     year = {2018},
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.7b01012}}
 
 @article{Gulania_2021,
 	author = {Sahil, Gulania and Eirik, F. Kj{\o}nstad and John, F. Stanton and Henrik, Koch and Anna I. Krylov},
@@ -2798,7 +2818,7 @@
 	date-modified = {2022-03-23 11:33:09 +0100},
 	doi = {10.1021/acs.jctc.8b00406},
 	journal = {J. Chem. Theory Comput.},
-	pages = {4360},
+	pages = {4360-4379},
 	title = {A Mountaineering Strategy to Excited States: {{Highly-accurate}} Reference Energies and Benchmarks},
 	volume = {14},
 	year = {2018},

Manuscript/CBD.tex

@@ -140,6 +140,9 @@ Finally, another option to deal with these chemical scenarios is to rely on the
 One can address part of this issue by increasing the excitation order or by complementing the spin-incomplete configuration set with the missing configurations. \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018}
 %both solutions being associated with an increased computational cost. 
 
+\alert{It is also important to note that we can obtain insights into the electronic structure with the number of unpaired electrons using Head-Gordon \cite{Head-Gordon_2003} index which provide a quantification of the polyradical character associated to a given electronic state. \cite{Orms_2018}}
+
+
 In the present work, we define highly-accurate reference values and investigate the accuracy of each family of computational methods mentioned above on the automerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries. 
 Computational details are reported in Sec.~\ref{sec:compmet}.
 Section \ref{sec:res} is devoted to the discussion of our results. 

Response_Letter/Response_Letter.tex

@@ -70,7 +70,7 @@ In this new orientation, the two singlet states $1 ^1B_{1g}$ and $1 ^1A_{1g}$ be
 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.
+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. 
@@ -89,11 +89,11 @@ Can the authors perform MRCI+Q  or MRAQCC calculations for comparison?}
 \\
 \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).
+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.
+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