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Manuscript/Ec.tex
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@ -52,6 +52,7 @@
\newcommand{\EHF}{E_\text{HF}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\Evar}{E_\text{var}}
\newcommand{\Eextrap}{E_\text{extrap}}
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\ECIPSI}{E_\text{CIPSI}}
@ -139,7 +140,7 @@ Again, at least in theory, one can obtain the exact energy of the system by ramp
The second-order M{\o}ller-Plesset (MP2) method \cite{Moller_1934} [which scales as $\order*{\Norb^{5}}$] has been broadly adopted in quantum chemistry for several decades, and is now included in the increasingly popular double-hybrid functionals \cite{Grimme_2006} alongside exact HF exchange.
Its higher-order variants [MP3, \cite{Pople_1976}
MP4, \cite{Krishnan_1980} MP5, \cite{Kucharski_1989} and MP6 \cite{He_1996a,He_1996b} which scales as $\order*{\Norb^{6}}$, $\order*{\Norb^{7}}$, $\order*{\Norb^{8}}$, and $\order*{\Norb^{9}}$ respectively] have been investigated much more scarcely.
However, it is now widely recognised that the series of MP approximations might show erratic, slow, or divergent behavior that limit its applicability and systematic improvability. \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003}
However, it is now widely recognised that the series of MP approximations might show erratic, slow, or divergent behavior that limit its applicability and systematic improvability. \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003,Marie_2021}
Again, MP perturbation theory and CC methods can be coupled.
The CCSD(T) method, \cite{Raghavachari_1989} known as the gold-standard of quantum chemistry for weakly correlated systems, is probably the most iconic example of such coupling.
@ -194,6 +195,8 @@ Note both localized and optimized orbitals do break the spatial symmetry.
Unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Chilkuri_2021} the present wave functions do not fulfill this property as we aim for the lowest possible energy of a closed-shell singlet state.
We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
All the data (geometries, energies, etc) and supplementary material associated with the present manuscript are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.
%%%%%%%%%%%%%%%%%%%%%%%%%
\section{CIPSI with optimized orbitals}
\label{sec:OO-CIPSI}
@ -392,7 +395,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\begin{squeezetable}
\begin{table*}
\caption{Total energy $E$ (in \SI{}{\hartree}) and correlation energy $\Delta E$ (in \SI{}{\milli\hartree}) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set.
For the CIPSI estimates of the correlation energy, the fitting error associated with the four-point linear fit is reported in parenthesis.
For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted four-point linear fit is reported in parenthesis.
\label{tab:Tab5-VDZ}}
\begin{ruledtabular}
\begin{tabular}{lcccccccccc}
@ -418,7 +421,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
CCSD(T) & $-193.5439$ & $-735.6$ & $-229.4073$ & $-764.0$ & $-225.6099$ & $-774.5$ & $-209.5836$ & $-754.9$ & $-552.0458$ & $-724.8$
\\
\hline
CIPSI & & $-739.3(1)$ & & $-768.1(2)$ & & $-778.3(1)$ & & $-758.4(2)$ & & $-729.1(3)$\\
FCI & & $-739.3(1)$ & & $-768.1(2)$ & & $-778.3(1)$ & & $-758.4(2)$ & & $-729.1(3)$\\
\end{tabular}
\end{ruledtabular}
\end{table*}
@ -429,7 +432,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\begin{squeezetable}
\begin{table*}
\caption{Total energy $E$ (in \SI{}{\hartree}) and correlation energy $\Delta E$ (in \SI{}{\milli\hartree}) for the frozen-core ground state of six-membered rings in the cc-pVDZ basis set.
For the CIPSI estimates of the correlation energy, the fitting error associated with the four-point linear fit is reported in parenthesis.
For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted four-point linear fit is reported in parenthesis.
\label{tab:Tab6-VDZ}}
\begin{ruledtabular}
\begin{tabular}{lcccccccccccccc}
@ -455,7 +458,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\hline
CCSD(T) & $-231.5798$ & $-857.5$ & $-263.6024$ & $-899.4$ & $-263.5740$ & $-904.1$ & $-247.5929$ & $-877.7$ & $-263.6099$ & $-896.2$ & $-295.5680$ & $-952.2$ & $-279.6305$ & $-913.1$ \\
\hline
CIPSI & & $-863.0(4)$ & & $-904.6(6)$ & & $-908.8(2)$ & & $-883.4(0)$ & & $-900.4(4)$ & & $-957.3(2)$ & & $-918.5(5)$\\
FCI & & $-863.0(4)$ & & $-904.6(6)$ & & $-908.8(2)$ & & $-883.4(0)$ & & $-900.4(4)$ & & $-957.3(2)$ & & $-918.5(5)$\\
\end{tabular}
\end{ruledtabular}
\end{table*}
@ -467,15 +470,15 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\begin{squeezetable}
\begin{table}
\caption{
Extrapolated correlation energies $\Delta \Evar$ (in \SI{}{\milli\hartree}) for the twelve cyclic molecules represented in Fig.~\ref{fig:mol} and their associated fitting errors (in \SI{}{\milli\hartree}) obtained via weighted linear fits with a varying number of points.
Extrapolated correlation energies $\Delta \Eextrap$ (in \SI{}{\milli\hartree}) for the twelve cyclic molecules represented in Fig.~\ref{fig:mol} and their associated fitting errors (in \SI{}{\milli\hartree}) obtained via weighted linear fits with a varying number of points.
The weights are taken as the inverse square of the perturbative corrections.
For a $n$-point fit, the $n$ largest the largest variational wave functions are used.
For a $m$-point fit, the $m$ largest variational wave functions are used.
\label{tab:fit}}
\begin{ruledtabular}
\begin{tabular}{lccc}
Molecule & Number of & \mc{2}{c}{Fitting parameters} \\
\cline{3-4}
& fitting points & $\Delta \Evar$ & Fitting error \\
& fitting points & $\Delta \Eextrap$ & Fitting error \\
\hline
Cyclopentadiene & 3 & $-739.295$ & $0.199$ \\
& 4 & $-739.309$ & $0.088$ \\
@ -571,14 +574,15 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\label{sec:cipsi_res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We first study the convergence of the CIPSI correlation energy $\Delta \Evar = \Evar - \EHF$ (where $\EHF$ is the HF energy) as a function of the number of determinants.
We first study the convergence of the CIPSI energy as a function of the number of determinants.
Our motivation here is to generate FCI-quality reference correlation energies for the twelve cyclic molecules represented in Fig.~\ref{fig:mol} in order to benchmark, in a second time, the performance and convergence properties of various mainstream MP and CC methods (see Sec.~\ref{sec:mpcc_res}).
For the natural and optimized orbital sets, we report, in Fig.~\ref{fig:vsNdet}, the evolution of the variational correlation energy $\Delta \Evar$ and its perturbatively corrected value $\Delta \Evar + \EPT$ with respect to the number of determinants for each cyclic molecule.
For the natural and optimized orbital sets, we report, in Fig.~\ref{fig:vsNdet}, the evolution of the variational correlation energy $\Delta \Evar = \Evar - \EHF$ (where $\EHF$ is the HF energy) and its perturbatively corrected value $\Delta \Evar + \EPT$ with respect to the number of determinants $\Ndet$ for each cyclic molecule.
As compared to natural orbitals (solid red lines), one can see that, for a given number of determinants, the use of optimized orbitals greatly lowers $\Delta \Evar$ (solid blue lines).
Adding the perturbative correction $\EPT$ yields similar curves for both sets of orbitals (dashed lines).
This indicates that, for a given number of determinants, $\EPT$ (which provides a qualitative idea to the distance to the FCI limit) is much smaller for optimized orbitals than for natural orbitals.
This indicates that, for a given number of determinants, $\EPT$ (which, we recall, provides a qualitative idea to the distance to the FCI limit) is much smaller for optimized orbitals than for natural orbitals.
This is further evidenced in Fig.~\ref{fig:vsEPT2} where we show the behavior of $\Delta \Evar$ as a function of $\EPT$ for both sets of orbitals.
The four-point weighted linear fit using the four largest variational wave functions are also represented (dashed black lines), while the CCSDTQ correlation energy (solid black line) is also reported for comparison purposes.
The four-point weighted linear fit using the four largest variational wave functions are also represented (dashed black lines).
The CCSDTQ correlation energy (solid black line) is also reported for comparison purposes in Figs.~\ref{fig:vsNdet} and \ref{fig:vsEPT2}
From Fig.~\ref{fig:vsEPT2}, it is clear that the behavior of $\Delta \Evar$ is much more linear and produces smaller $\EPT$ values, hence facilitating the extrapolation procedure to the FCI limit (see below).
%%% FIG 4 %%%
@ -591,15 +595,20 @@ From Fig.~\ref{fig:vsEPT2}, it is clear that the behavior of $\Delta \Evar$ is m
\end{figure}
%%% %%% %%%
Figure~\ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for the natural, localized, and optimized sets of orbitals in the case of benzene.
Figure~\ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for the natural, localized, and optimized sets of orbitals in the particular case of benzene.
As mentioned in Sec.~\ref{sec:compdet}, although both sets break the spatial symmetry to take advantage of the local nature of electron correlation, optimized orbitals further improve on the use of localized orbitals.
More quantitatively, optimized orbitals produce the same variational energy as localized orbitals with, roughly, a ten-fold reduction in the number of determinants.
A similar improvement is observed going from natural to localized orbitals.
\titou{Comment on PT2 for localized orbitals.}
Accordingly to these observations, all our FCI correlation energy estimates have been produced with the set of optimized orbitals.
To do so, we have then extrapolated to the orbital-optimized CIPSI calculations to $\EPT = 0$ via a weighted linear extrapolation using the four largest variational wave functions.
These estimates are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively.
To do so, we have then extrapolated the orbital-optimized variational CIPSI correlation energies to $\EPT = 0$ via a weighted four-point linear fit using the four largest variational wave functions (see Fig.~\ref{fig:vsEPT2}).
The fitting weights have been taken as the inverse square of the perturbative correction at each point.
Our final FCI correlation energy estimates are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively, alongside their corresponding fitting error.
The stability of these estimates are illustrated by the results gathered in Table \ref{tab:fit} where we report, for each system, the extrapolated correlation energies $\Delta \Eextrap$ and their associated fitting errors obtained via weighted linear fits varying the number of fitting points from 3 to 8.
Although we cannot provide a mathematically rigorous error bar, the data of Table \ref{tab:fit} show that the extrapolation procedure is robust and that our FCI estimates are very likely accurate to a few tenths of a millihartree.
Logically, the FCI estimates for the five-membered rings seem slightly more accurate than for the larger six-membered rings.
Note that it is pleasing to see that, although different geometries are considered, our present estimate of the frozen-core correlation energy of the benzene molecule in the cc-pVDZ basis is very close to the one reported in Refs.~\onlinecite{Eriksen_2020,Loos_2020e}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Benchmark of CC and MP methods}
@ -630,6 +639,21 @@ These estimates are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ}
\end{figure*}
%%% %%% %%%
Using the CIPSI estimates of the FCI correlation energy produced in Sec.~\ref{sec:cipsi_res}, we now study the performance and convergence properties of three series of methods: i) MP2, MP3, MP4, and MP5, ii) CC2, CC3, and CC4, and iii) CCSD, CCSDT, and CCSDTQ.
Additionally, we also report CCSD(T) correlation energies.
All these data are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively.
In Fig.~\ref{fig:MPCC}, we show, for each molecule, the convergence of the correlation energy for each series of methods as a function of the computational cost of the corresponding method.
The FCI correlation energy estimate is represented as a black line for reference.
First, let us investigate the series of MP approximations which is known, as mentioned in Sec.~\ref{sec:intro}, to potentially exhibit ``surprising'' behavior depending on the type of correlation at play.\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003}
(See Ref.~\onlinecite{Marie_2021} for a detailed discussion).
For each system, the MP series decreases monotonically up to MP4 but raises quite significantly when one takes into account the fifth-order correction.
We note that the MP4 correlation energy is always quite accurate and is only a few millihartree higher than the FCI value (except in the case of tetrazine where the MP4 number is very slightly below the reference value): MP5 is thus systematically worse than MP4 for these systems.
Importantly here, one notices that MP4 (which scales as $\order*{N^7}$) is systematically on par with the more expensive $\order*{N^10}$ CCSDTQ method.
Second, we investigate the approximate CC series of methods CC2, CC3, and CC4.
As observed in our recent study on excitation energies, \cite{Loos_2021} CC4 is an outstanding approximation to its CCSDTQ parent.
Moreover, we observe here that CC3 and CC4 provide correlation energies only one or two millihartree different, which shows that CC3 is particularly effective for ground-state energetics.
%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}