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Manuscript/CASPT3.bib

@@ -1,13 +1,70 @@
 %% This BibTeX bibliography file was created using BibDesk.
-%% https://bibdesk.sourceforge.io/
+%% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2022-04-13 16:47:01 +0200 
+%% Created for Pierre-Francois Loos at 2022-06-07 09:28:45 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Liang_2022,
+	author = {Liang, Jiashu and Feng, Xintian and Hait, Diptarka and Head-Gordon, Martin},
+	date-added = {2022-06-07 09:26:50 +0200},
+	date-modified = {2022-06-07 09:27:51 +0200},
+	doi = {10.1021/acs.jctc.2c00160},
+	journal = {J. Chem. Theory Comput.},
+	title = {Revisiting the Performance of Time-Dependent Density Functional Theory for Electronic Excitations: Assessment of 43 Popular and Recently Developed Functionals from Rungs One to Four},
+	year = {in press},
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.2c00160}}
+
+@article{Mok_1996,
+	author = {D. K. W. Mok and R. Neumann and N. C. Handy},
+	date-added = {2022-06-06 21:58:06 +0200},
+	date-modified = {2022-06-06 21:59:28 +0200},
+	doi = {10.1021/jp9528020},
+	journal = {J. Phys. Chem.},
+	pages = {6225--6230},
+	title = {Dynamical and Nondynamical Correlation},
+	volume = {100},
+	year = {1996},
+	bdsk-url-1 = {https://doi.org/10.1021/jp9528020}}
+
+@article{Loos_2019b,
+	author = {Loos, Pierre-Francois and Jacquemin, Denis},
+	date-added = {2022-06-03 10:20:05 +0200},
+	date-modified = {2022-06-03 10:20:36 +0200},
+	journal = {ChemPhotoChem},
+	pages = {684--696},
+	title = {Evaluating 0-0 Energies with Theoretical Tools: a Short Review},
+	volume = {3},
+	year = {2019}}
+
+@article{Loos_2019a,
+	author = {Loos, Pierre-Francois and Jacquemin, Denis},
+	date-added = {2022-06-03 10:20:05 +0200},
+	date-modified = {2022-06-03 10:20:48 +0200},
+	doi = {10.1021/acs.jctc.8b01103},
+	journal = {J. Chem. Theory Comput.},
+	pages = {2481--2491},
+	title = {Chemically Accurate 0-0 Energies with not-so-Accurate Excited State Geometries},
+	volume = {15},
+	year = {2019},
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.8b01103}}
+
+@article{Loos_2018b,
+	author = {Loos, Pierre-Fran{\c c}ois and Galland, Nicolas and Jacquemin, Denis},
+	date-added = {2022-06-03 10:20:05 +0200},
+	date-modified = {2022-06-07 09:26:14 +0200},
+	doi = {10.1021/acs.jpclett.8b02058},
+	journal = {J. Phys. Chem. Lett.},
+	number = {16},
+	pages = {4646--4651},
+	title = {Theoretical 0--0 Energies with Chemical Accuracy},
+	volume = {9},
+	year = {2018},
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.8b02058}}
+
 @article{Olsen_1988,
 	author = {J. Olsen and B. O. Roos and P. Jorgensen and H. J. A. Jensen},
 	date-added = {2022-04-13 16:31:23 +0200},
@@ -17,7 +74,8 @@
 	pages = {2185},
 	title = {Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces},
 	volume = {89},
-	year = {1988}}
+	year = {1988},
+	bdsk-url-1 = {https://doi.org/10.1063/1.455063}}
 
 @article{Fulscher_1994,
 	author = {M. P. Fulscher and B. O. Roos},
@@ -28,7 +86,8 @@
 	pages = {403},
 	title = {The excited states of pyrazine: A basis set study},
 	volume = {87},
-	year = {1994}}
+	year = {1994},
+	bdsk-url-1 = {https://doi.org/10.1007/BF01113393}}
 
 @article{Tran_2019,
 	author = {T. Tran and J. Segarra-Marti and M. J. Bearpark and M. A. Robb},
@@ -39,7 +98,8 @@
 	pages = {5223},
 	title = {Molecular Vertical Excitation Energies Studied with First-Order RASSCF (RAS[1,1]): Balancing Covalent and Ionic Excited States},
 	volume = {123},
-	year = {2019}}
+	year = {2019},
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jpca.9b03715}}
 
 @article{Boggio-Pasqua_2004,
 	author = {M. Boggio-Pasqua and M. J. Bearpark and M. Klene and M. A. Robb},
@@ -50,7 +110,8 @@
 	pages = {7849},
 	title = {A computational strategy for geometry optimization of ionic and covalent excited states, applied to butadiene and hexatriene},
 	volume = {120},
-	year = {2004}}
+	year = {2004},
+	bdsk-url-1 = {https://doi.org/10.1063/1.1690756}}
 
 @article{Borden_1996,
 	author = {W. T. Borden and E. R. Davidson},
@@ -61,7 +122,8 @@
 	pages = {67},
 	title = {The Importance of Including Dynamic Electron Correlation in ab Initio Calculations},
 	volume = {29},
-	year = {1996}}
+	year = {1996},
+	bdsk-url-1 = {https://doi.org/10.1021/ar950134v}}
 
 @article{Boggio-Pasqua_2007,
 	author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Robb, Michael A.},
@@ -103,20 +165,6 @@
 	year = {1995},
 	bdsk-url-1 = {https://doi.org/10.1016/0009-2614(95)01111-L}}
 
-@misc{Liang_2022,
-	author = {Liang, Jiashu and Feng, Xintian and Hait, Diptarka and Head-Gordon, Martin},
-	copyright = {arXiv.org perpetual, non-exclusive license},
-	date-added = {2022-04-04 22:47:24 +0200},
-	date-modified = {2022-04-04 22:47:30 +0200},
-	doi = {10.48550/ARXIV.2202.13208},
-	keywords = {Chemical Physics (physics.chem-ph), Other Condensed Matter (cond-mat.other), Computational Physics (physics.comp-ph), Quantum Physics (quant-ph), FOS: Physical sciences, FOS: Physical sciences},
-	publisher = {arXiv},
-	title = {Revisiting the performance of time-dependent density functional theory for electronic excitations: Assessment of 43 popular and recently developed functionals from rungs one to four},
-	url = {https://arxiv.org/abs/2202.13208},
-	year = {2022},
-	bdsk-url-1 = {https://arxiv.org/abs/2202.13208},
-	bdsk-url-2 = {https://doi.org/10.48550/ARXIV.2202.13208}}
-
 @article{Davidson_1996,
 	author = {Davidson, Ernest R.},
 	date-added = {2022-04-04 22:37:02 +0200},

Manuscript/CASPT3.tex

@@ -144,6 +144,10 @@ Although CASPT3 calculations have been reported in the literature,
 \cite{Angeli_2006,Yanai_2007,Grabarek_2016,Li_2017,Li_2018,Li_2021,Bittererova_2001,Bokarev_2009,Frankcombe_2011,Gu_2008,Kerkines_2005,Lampart_2008,Leininger_2000,Maranzana_2020,Papakondylis_1999,Schild_2013,Sun_2018,Takatani_2009,Takatani_2010,Verma_2018,Woywod_2010,Yan_2004,Zhang_2020,Zhu_2005,Zhu_2007,Zhu_2013,Zou_2009}
 the present study provides, to the best of our knowledge, the first comprehensive benchmark of CASPT3 and allows assessing its accuracy in the framework of electronically excited states.
 We underline that, although a third-order version of NEVPT has been developed \cite{Angeli_2006} and has been used in some applications \cite{Pastore_2006a,Pastore_2006b,Pastore_2007,Angeli_2007,Camacho_2010,Angeli_2011,Angeli_2012} by Angeli and coworkers, as far as we are aware of, no NEVPT3 implementation is publicly available.
+
+\alert{Although comparing with experimental values would be interesting on its own right, this involves the computation of 0-0 energies which are much more expensive to determine as they require the equilibrium geometries of the ground and excited states as well as the zero-point vibrational energies for each state.
+Moreover, we have recently shown that the accuracy of such quantities are mainly driven by the quality of the (absorption and emission) vertical excitation energies as other geometrical and vibrational effects mostly cancel out. \cite{Loos_2018b,Loos_2019a,Loos_2019b} 
+Therefore, it is clear that vertical excitation energies are the key quantities to reproduce.}
 \\
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -159,7 +163,7 @@ We underline that, although a third-order version of NEVPT has been developed \c
 \end{figure}
 %%% %%% %%% %%%
 
-For each compound represented in Fig.~\ref{fig:mol}, we have computed the \alert{single-state} CASPT2 and CASPT3 vertical excitation energies.
+For each compound represented in Fig.~\ref{fig:mol}, we have computed \alert{vertical excitation energies based on single-state CASPT2 and CASPT3 calculations}.
 \alert{All calculations reported in the present manuscript have been performed with Dunning's aug-cc-pVTZ basis set.} \cite{Kendall_1992} 
 Geometries and reference theoretical best estimates (TBEs) for the vertical excitation energies have been extracted from the QUEST database \cite{Veril_2021} and can be downloaded at \url{https://lcpq.github.io/QUESTDB_website}.
 
@@ -170,6 +174,7 @@ The MOLPRO implementation of CASPT3 is based on a modification of the multi-refe
 For the sake of computational efficiency, the doubly-excited external configurations are internally contracted while the singly-excited internal and semi-internal configurations are left uncontracted. \cite{Werner_1996}
 These perturbative calculations have been performed by considering a state-averaged (SA) CASSCF wave function where we have included the ground state and (at least) the excited states of interest.
 In several occasions, we have added additional excited states to avoid convergence and/or root-flipping issues.
+\alert{Note that the implementation of the IPEA shift is not exactly identical in MOLPRO (used here) and in MOLCAS (used, for example, in Ref.~\onlinecite{Zobel_2017}), since in MOLPRO the singly external configurations are not contracted in the RS2 scheme.}
 
 For each system and transition, we report in the {\SupMat} the exhaustive description of the active spaces for each symmetry representation.
 Additionally, for the challenging transitions, we have steadily increased the size of the active space to carefully assess the convergence of the vertical excitation energies of interest.
@@ -186,7 +191,7 @@ This value has been slightly increased in particularly difficult cases, and such
 
 A detailed discussion of each individual molecule can be found in Ref.~\onlinecite{Sarkar_2022} and in earlier works, \cite{Loos_2018a,Loos_2020b} where theoretical and experimental literature values are discussed.
 We therefore decided to focus on global trends here.
-The exhaustive list of CASPT2 and CASPT3 transitions can be found in Table \ref{tab:BigTab} and the distribution of the errors are represented in Fig.~\ref{fig:PT2_vs_PT3}.
+The exhaustive list of \alert{CASSCF}, CASPT2, and CASPT3 transitions can be found in Table \ref{tab:BigTab} and the distribution of the errors are represented in Fig.~\ref{fig:PT2_vs_PT3}.
 The usual statistical indicators are used in the following, namely, the mean signed error (MSE), the mean absolute error (MAE), the root-mean-square error (RMSE), the standard deviation of the errors (SDE), as well as the largest positive and negative deviations [Max($+$) and Max($-$), respectively].
 These are given in Table \ref{tab:stat} considering the 265 ``safe'' TBEs (out of 280) for which chemical accuracy is assumed (absolute error below \SI{0.043}{\eV}). 
 The MAEs determined for subsets of transitions (singlet, triplet, valence, Rydberg, $n\to\pis$, $\pi\to\pis$, and double excitations) and system sizes (3 non-H atoms, 4 non-H atoms, and 5-6 non-H atoms) can be found in Table \ref{tab:stat_subset}. 
@@ -563,7 +568,8 @@ TBEs listed as ``safe'' are assumed to be chemically accurate (\ie, absolute err
 %%% %%% %%% %%%
 
 From the different statistical quantities reported in Table \ref{tab:stat}, one can highlight the following trends.
-First, as previously reported, \cite{Werner_1996,Grabarek_2016} CASPT3 vertical excitation energies are much less sensitive to the IPEA shift, which drastically alters the accuracy of CASPT2: the mean absolute deviation between the CASPT2(NOIPEA) and CASPT2(IPEA) data is \SI{0.329}{\eV} while it is only \SI{0.051}{\eV} between CASPT3(NOIPEA) and CASPT3(IPEA).
+\alert{First, as expected, CASSCF returns a large MAE of \SI{0.47}{\eV} and a relative small MSE of \SI{0.12}{\eV}, with a better accuracy for triplet states than for singlet states.}
+\alert{Second}, as previously reported, \cite{Werner_1996,Grabarek_2016} CASPT3 vertical excitation energies are much less sensitive to the IPEA shift, which drastically alters the accuracy of CASPT2: the mean absolute deviation between the CASPT2(NOIPEA) and CASPT2(IPEA) data is \SI{0.329}{\eV} while it is only \SI{0.051}{\eV} between CASPT3(NOIPEA) and CASPT3(IPEA).
 Consequently, the MAEs of CASPT3(IPEA) and CASPT3(NOIPEA) are amazingly close (\SI{0.11}{} and \SI{0.09}{\eV}), while the MAEs of CASPT2(IPEA) and CASPT2(NOIPEA) are remarkably different (\SI{0.11}{} and \SI{0.27}{\eV}).
 Likewise, the MSEs of CASPT2(IPEA) and CASPT2(NOIPEA), \SI{0.06}{} and \SI{-0.26}{\eV}, clearly evidence the well-known global underestimation of the CASPT2(NOIPEA) excitation energies in molecular systems when large basis sets are used.  
 For CASPT3, the MSE with IPEA shift is only slightly larger without IPEA (\SI{0.10}{} and \SI{0.05}{\eV}, respectively).
@@ -577,7 +583,7 @@ Note that combining CASPT2 and CASPT3 via an hybrid protocol such as CASPT2.5, a
 It is worth mentioning that CASPT3(NOIPEA) yields MAEs for each subset that is almost systematically below \SI{0.1}{\eV}, except for the singlet subset which contains some states showing large (positive) deviations at both the CASPT2 and CASPT3 levels.
 This is most notably the case for the $^1 B_u(\pi,\pis)$ state of butadiene, the $^1B_2(\pi,\pis)$ state of cyclopentadiene, the $^1A_1(\pi,\pis)$ state of cyclopropenone, the second $^1B_{1u}(\pi,\pis)$ state of pyrazine, the $^1B_2(\pi,\pis)$ state of pyridazine, and the $^1E'(\pi,\pis)$ state of triazine, for which both CASPT2(IPEA) and CASPT3(NOIPEA) overestimate the corresponding vertical transition energies by at least \SI{0.4}{\eV} with respect to the TBEs. 
 This can be tracked down to the relatively small active spaces that we have considered here and, more precisely, to the lack of direct $\sig$-$\pi$ coupling in the active space that is known to be important in ionic states, for example. \cite{Davidson_1996,Borden_1996,Boggio-Pasqua_2004,Angeli_2009,Garniron_2018,Tran_2019,BenAmor_2020} 
-For this family of states, it is particularly important to describe the dynamic response of the $\sig$-electron framework to the field of the $\pi$-electron system, a phenomenon known as dynamic $\sig$ polarization. 
+For this family of states, it is particularly important to describe the dynamic response of the $\sig$-electron framework to the field of the $\pi$-electron system, a phenomenon known as dynamic $\sig$ polarization \alert{(that should not be confused with so-called left-right polarization \cite{Mok_1996})}. 
 Because the dynamic $\sig$ polarization is generally more important for the ionic excited state than for the ground state, its contribution is expected to lower the vertical transition energy. 
 Furthermore, this part of the dynamic $\sig$-$\pi$ correlation needs to be included at the orbital optimization stage, otherwise the orbitals become too diffuse, resulting in artificial valence-Rydberg mixing which cannot be disentangled using non-degenerate perturbation theory such as the version of CASPT2 and CASPT3 considered here. \cite{Angeli_2009}
 

Response_Letter/Response_Letter.tex

@@ -62,7 +62,8 @@ We have modified the conclusion to stress these two points.
 {Make it VERY clear that the excitation energies are Single-State CASPTX excitation energies.}
 \\
 \alert{
-The reviewer is right. We have added this additional information in the "Computational Details" section of the manuscript.
+The reviewer is right. 
+We have added this additional information in the "Computational Details" section of the manuscript.
 }
 
 \item 
@@ -111,6 +112,12 @@ The question is, does one want to be as good as CASPT2 or as good as the experim
 I would say the latter, so while the TBEs are useful, a comparison with the experimental values should be provided and the results also discussed in this light.}
 \\
 \alert{
+Concerning the first point, we have added, in the "Computational Details" section, a comment stressing the fact that the implementation of the IPEA shift in MOLPRO and MOLCAS are not exact identical (yet very similar): \textit{``Note that the implementation of the IPEA shift is not exactly identical in MOLPRO (used here) and in MOLCAS (used, for example, in Ref. 56), since in MOLPRO the singly external configurations are not  contracted in the RS2 scheme.''}\\
+Concerning the second point, comparison with experimental values are much trickier as it involves computing 0-0 energies which are the average of the vertical excitation energies from the ground- and excited-state equilibrium geometries corrected for zero-point vibrational energies (ZPVEs).
+As shown in our earlier works [see ChemPhotoChem 3, 684 (2019) for a recent review], these are much more computationally expensive and depends strongly on the quality of the (absorption and emission) vertical energies.
+The two other effects (ground- and excited-state geometries and ZPVEs) are less important.
+Therefore, it is clear that vertical excitation energies are the key quantities to reproduce.
+We have added a comment on this point at the end of the "Introdcution" section.
 }
 
 \item 
@@ -125,7 +132,7 @@ All calculations reported in the present manuscript have been performed with the
 We have modified the corresponding sentence to stress this further.
 Having an augmented triple-$\zeta$ basis allows us to describe faithfully all excited states (including the Rydberg states).
 Therefore, there is no basis set effect between the CASPT2/CASPT3 and TBE excitation energies.
-Moreover, because the basis set effects are extremely transferable within wave function methods, we can safely assume that the present conclusions would be identical in a small or larger basis set.
+Moreover, because the basis set effects are extremely transferable within wave function methods, we can safely assume that the present conclusions would be identical in a smaller or larger basis set.
 }
 
 \end{enumerate}