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5
Response_Letter/Response_Letter.tex
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@ -95,7 +95,7 @@ We look forward to hearing from you.
These numbers vary by $1.4$ mE$_h$. These numbers vary by $1.4$ mE$_h$.
The four-point extrapolated estimate that we have chosen to report as our best estimate corresponds to the smallest fitting error. The four-point extrapolated estimate that we have chosen to report as our best estimate corresponds to the smallest fitting error.
Quadratic fits yield much larger variations and we never use them in practice. Quadratic fits yield much larger variations and we never use them in practice.
All these additional information are now provided in the revised version of the manuscript (see footnote 66).} All these additional information are now provided in the revised version of the manuscript (see footnote 67).}
\item \item
{As an aside, it would seem like the fifth last point differs ever so slightly from the general trend (regardless of the choice of (r)MP2)? {As an aside, it would seem like the fifth last point differs ever so slightly from the general trend (regardless of the choice of (r)MP2)?
@ -111,7 +111,8 @@ We look forward to hearing from you.
\item \item
{It would be interesting if the authors could comment (even speculatively) on why results in the localized FB basis are significantly lower (and hence, in the authors' own words, more trustworthy) than the corresponding results in the NO basis.} {It would be interesting if the authors could comment (even speculatively) on why results in the localized FB basis are significantly lower (and hence, in the authors' own words, more trustworthy) than the corresponding results in the NO basis.}
\\ \\
\alert{It is well known that the localized orbitals significantly speed up the convergence of SCI calculations [see Refs.~???]} \alert{Localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of the electron correlation.
We have mentioned this in the revised manuscript and added two references discussing the use of localized orbitals in SCI calculations (Refs.~65 and 66).}
\item \item
{Why are the rMP2-based corrections considered superior to the corresponding corrections based on MP2? {Why are the rMP2-based corrections considered superior to the corresponding corrections based on MP2?

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benzene.bib
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@ -1,13 +1,25 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-10-09 13:24:13 +0200 %% Created for Pierre-Francois Loos at 2020-10-09 21:38:45 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{BenAmor_2011,
Author = {Ben Amor,Nadia and Bessac,Fabienne and Hoyau,Sophie and Maynau,Daniel},
Date-Added = {2020-10-09 21:28:40 +0200},
Date-Modified = {2020-10-09 21:29:52 +0200},
Doi = {10.1063/1.3600351},
Journal = {J. Chem. Phys.},
Pages = {014101},
Title = {Direct selected multireference configuration interaction calculations for large systems using localized orbitals},
Volume = {135},
Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3600351}}
@article{Deustua_2017, @article{Deustua_2017,
Author = {Deustua, J. Emiliano and Shen, Jun and Piecuch, Piotr}, Author = {Deustua, J. Emiliano and Shen, Jun and Piecuch, Piotr},
Date-Added = {2020-10-09 12:21:32 +0200}, Date-Added = {2020-10-09 12:21:32 +0200},
@ -63,15 +75,11 @@
Abstract = { The density matrix renormalization group is a method that is useful for describing molecules that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the density matrix renormalization group works, a survey of its existing applications to molecular problems, and some thoughts on the future of the method. }, Abstract = { The density matrix renormalization group is a method that is useful for describing molecules that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the density matrix renormalization group works, a survey of its existing applications to molecular problems, and some thoughts on the future of the method. },
Author = {Chan, Garnet Kin-Lic and Sharma, Sandeep}, Author = {Chan, Garnet Kin-Lic and Sharma, Sandeep},
Date-Added = {2020-10-09 11:59:58 +0200}, Date-Added = {2020-10-09 11:59:58 +0200},
Date-Modified = {2020-10-09 12:00:15 +0200}, Date-Modified = {2020-10-09 21:37:30 +0200},
Doi = {10.1146/annurev-physchem-032210-103338}, Doi = {10.1146/annurev-physchem-032210-103338},
Eprint = {https://doi.org/10.1146/annurev-physchem-032210-103338}, Journal = {Annu. Rev. Phys. Chem.},
Journal = {Annual Review of Physical Chemistry},
Note = {PMID: 21219144},
Number = {1},
Pages = {465-481}, Pages = {465-481},
Title = {The Density Matrix Renormalization Group in Quantum Chemistry}, Title = {The Density Matrix Renormalization Group in Quantum Chemistry},
Url = {https://doi.org/10.1146/annurev-physchem-032210-103338},
Volume = {62}, Volume = {62},
Year = {2011}, Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1146/annurev-physchem-032210-103338}} Bdsk-Url-1 = {https://doi.org/10.1146/annurev-physchem-032210-103338}}

5
benzene.tex
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@ -132,7 +132,7 @@ The corresponding energies are reported in Table \ref{tab:NOvsLO} as functions o
A second run has been performed with localized orbitals. A second run has been performed with localized orbitals.
Starting from the same natural orbitals, a Boys-Foster localization procedure \cite{Boys_1960} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals. Starting from the same natural orbitals, a Boys-Foster localization procedure \cite{Boys_1960} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals.
\footnote{MO indices for Boys-Foster localization procedure: \footnote{Indices of molecular orbitals for Boys-Foster localization procedure:
core [1--6]; core [1--6];
$\sigma$ [7--18]; $\sigma$ [7--18];
$\pi$ [19--21]; $\pi$ [19--21];
@ -141,7 +141,8 @@ $\sigma^*$ [25--36];
higher-lying $\pi$ [39,41--43,46,49,50,53--57,71--74,82--85,87,92,93,98]; higher-lying $\pi$ [39,41--43,46,49,50,53--57,71--74,82--85,87,92,93,98];
higher-lying $\sigma$ [37,38,40,44,45,47,48,51,52,58--70,75--81,86,88--91,94--97,99--114].} higher-lying $\sigma$ [37,38,40,44,45,47,48,51,52,58--70,75--81,86,88--91,94--97,99--114].}
Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene. Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene.
As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of $N_\text{det}$, the variational energy as well as the PT2-corrected energies are much lower with localized orbitals than with natural orbitals. \cite{Chien_2018,Eriksen_2020} As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of $N_\text{det}$, the variational energy as well as the PT2-corrected energies are much lower with localized orbitals than with natural orbitals.
\alert{Indeed, localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of electron correlation.\cite{BenAmor_2011,Chien_2018,Eriksen_2020}}
We, therefore, consider these energies more trustworthy, and we will base our best estimate of the correlation energy of benzene on these calculations. We, therefore, consider these energies more trustworthy, and we will base our best estimate of the correlation energy of benzene on these calculations.
The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of the correlation energy, $\Delta E_\text{var.}$ and $\Delta E_\text{var.} + E_\text{(r)PT2}$, as a function of $N_\text{det}$ (left panel). The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of the correlation energy, $\Delta E_\text{var.}$ and $\Delta E_\text{var.} + E_\text{(r)PT2}$, as a function of $N_\text{det}$ (left panel).
The right panel of Fig.~\ref{fig:CIPSI} is more instructive as it shows $\Delta E_\text{var.}$ as a function of $E_\text{(r)PT2}$, and their corresponding four-point linear extrapolation curves that we have used to get our final estimate of the correlation energy. The right panel of Fig.~\ref{fig:CIPSI} is more instructive as it shows $\Delta E_\text{var.}$ as a function of $E_\text{(r)PT2}$, and their corresponding four-point linear extrapolation curves that we have used to get our final estimate of the correlation energy.