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110
Manuscript/G2-srDFT.bib
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@ -1,13 +1,101 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-04-07 14:59:34 +0200
%% Created for Pierre-Francois Loos at 2019-04-07 21:16:12 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{FelPet-JCP-09,
Author = {D. Feller and K. A. Peterson},
Date-Added = {2019-04-07 20:41:03 +0200},
Date-Modified = {2019-04-07 20:41:44 +0200},
Doi = {10.1063/1.478747},
Journal = {J. Chem. Phys.},
Pages = {8384},
Title = {Re-examination of atomization energies for the Gaussian-2 set of molecules},
Volume = {110},
Year = {1999},
Bdsk-Url-1 = {https://doi.org/10.1063/1.478747}}
@article{FelPetDix-JCP-08,
Author = {D. Feller and K. A. Peterson and D. A. Dixon},
Date-Added = {2019-04-07 20:39:13 +0200},
Date-Modified = {2019-04-07 20:40:12 +0200},
Doi = {10.1063/1.3008061},
Journal = {J. Chem. Phys.},
Pages = {204105},
Title = {A survey of factors contributing to accurate theoretical predictions of atomization energies and molecular structures},
Volume = {129},
Year = {2008},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3008061}}
@article{FelPetHil-JCP-11,
Author = {D. Feller and K. A. Peterson and J. G Hill},
Date-Added = {2019-04-07 20:37:05 +0200},
Date-Modified = {2019-04-07 20:38:11 +0200},
Journal = {J. Chem. Phys.},
Keywords = {10.1063/1.3613639},
Pages = {044102},
Title = {On the effectiveness of CCSD(T) complete basis set extrapolations for atomization energies},
Volume = {135},
Year = {2011}}
@article{FelPet-JCP-13,
Author = {D. Feller and K. A. Peterson},
Date-Added = {2019-04-07 20:35:12 +0200},
Date-Modified = {2019-04-07 20:36:01 +0200},
Doi = {10.1063/1.4819125},
Journal = {J. Chem. Phys.},
Pages = {084110},
Title = {An expanded calibration study of the explicitly correlated CCSD(T)-F12b method using large basis set standard CCSD(T) atomization energies},
Volume = {139},
Year = {2013},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4819125}}
@article{Gro-JCP-09,
Author = {J. C. Grossman},
Date-Added = {2019-04-07 20:32:32 +0200},
Date-Modified = {2019-04-07 20:33:29 +0200},
Doi = {10.1063/1.1487829},
Journal = {J. Chem. Phys.},
Pages = {1434},
Title = {Benchmark quantum Monte Carlo calculations},
Volume = {117},
Year = {2002},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1487829}}
@article{KesSylKohTewMar-JCP-18,
Author = {M. K. Kesharwani and N. Sylvetsky and A. Kohn and D. P. Tew and Jan M. L. Martin},
Date-Added = {2019-04-07 20:30:12 +0200},
Date-Modified = {2019-04-07 20:36:23 +0200},
Doi = {10.1063/1.5048665},
Journal = {J. Chem. Phys.},
Pages = {154109},
Title = {Do CCSD and approximate CCSD-F12 variants converge to the same basis set limits? The case of atomization energies},
Volume = {149},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5048665}}
@article{NemTowNee-JCP-10,
Author = {Nemec, Norbert and Towler, Michael D. and Needs, R. J.},
Date-Added = {2019-04-07 20:28:31 +0200},
Date-Modified = {2019-04-07 20:28:40 +0200},
Doi = {10.1063/1.3288054},
Issn = {1089-7690},
Journal = {J. Chem. Phys.},
Month = {Jan},
Number = {3},
Pages = {034111},
Publisher = {AIP Publishing},
Title = {Benchmark all-electron ab initio quantum Monte Carlo calculations for small molecules},
Url = {http://dx.doi.org/10.1063/1.3288054},
Volume = {132},
Year = {2010},
Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.3288054}}
@article{Hylleraas30,
Author = {E. A. Hylleraas},
Date-Added = {2019-04-07 14:28:20 +0200},
@ -369,11 +457,13 @@
Volume = {58},
Year = {1973}}
@article{malrieu,
@article{HurMalRan-JCP-73,
Author = {B. Huron and J.P. Malrieu and P. Rancurel},
Date-Modified = {2019-04-07 21:03:11 +0200},
Doi = {10.1063/1.1679199},
Journal = {J. Chem. Phys.},
Pages = {5745},
Title = {Iterative perturbation calculations of ground and excited state energies from multiconfigura- tional zeroth-order wavefunctions},
Volume = {58},
Year = {1973},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1679199}}
@ -454,8 +544,9 @@
Volume = {141},
Year = {2014}}
@article{f2_dmc,
@article{GinSceCaf-JCP-15,
Author = {Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel},
Date-Modified = {2019-04-07 20:49:51 +0200},
Eid = 044115,
Journal = {J. Chem. Phys.},
Number = {4},
@ -2688,10 +2779,14 @@
@article{CurRagTruPop-JCP-91,
Author = {L. A. Curtiss and K. Raghavachari and G. W. Trucks and J. A. Pople},
Date-Modified = {2019-04-07 20:42:48 +0200},
Doi = {10.1063/1.460205},
Journal = {J. Chem. Phys.},
Pages = {7221},
Title = {Gaussian2 theory for molecular energies of first and secondrow compounds},
Volume = {94},
Year = {1991}}
Year = {1991},
Bdsk-Url-1 = {https://doi.org/10.1063/1.460205}}
@article{DahLeeBar-IJQC-05,
Author = {Nils Erik Dahlen and Robert Van Leeuwen and Ulf Von Barth},
@ -7815,13 +7910,6 @@
Volume = {5},
Year = {2009}}
@article{NemTowNee-JCP-10,
Author = {N. Nemec and M. D. Towler and R. J. Needs},
Journal = {J. Chem. Phys.},
Pages = {034111},
Volume = {132},
Year = {2010}}
@article{NesPey-JPB-90,
Author = {B. M. Nestmann and S. D. Peyerimhoff},
Journal = {J. Phys. B},

41
Manuscript/G2-srDFT.tex
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@ -80,8 +80,8 @@
\newcommand{\V}[2]{V_{#1}^{#2}}
\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
\newcommand{\modX}{\text{X}}
\newcommand{\modY}{\text{Y}}
\newcommand{\modX}{\mathcal{X}}
\newcommand{\modY}{\mathcal{Y}}
% basis sets
\newcommand{\Bas}{\mathcal{B}}
@ -679,36 +679,45 @@ Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valenc
We begin our investigation of the performance of the basis set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis sets (X $=$ D, T, Q and 5).
In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 test set, whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
In a second time, we compute the entire atomization energies of the G2 test sets composed by 55 molecules.
In a second time, we compute the entire atomization energies of the G2 test set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family.
This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08,Gro-JCP-09,FelPet-JCP-09,NemTowNee-JCP-10,FelPetHil-JCP-11,PetTouUmr-JCP-12,FelPet-JCP-13,KesSylKohTewMar-JCP-18}).
%The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
As a method $\modX$ we employ either $\CCSDT$ or $\exFCI$.
Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm.
As a method $\modX$ we employ either CCSD(T) or exFCI.
Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
We refer the interested reader to Refs.~\onlinecite{SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
Throughout this study, we have $\modY = \HF$ as we use the Hartree-Fock (HF) one-electron density to compute the complementary energy.
The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
\titou{For the quadrature grid, we employ ... radial and angular points.}
Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been performed at the B3LYP/6-31G(2df,p) level of theory.
Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
Frozen core calculations are defined as such: an \ce{He} core is frozen from \ce{B} to \ce{Mg}, while a \ce{Ne} core is frozen from \ce{Al} to \ce{Xe}.
In the context of the basis set correction, the set of valence spinorbitals $\Val$ involved in the definition of the valence part of the effective interaction [see Eq.~\eqref{eq:Wval}] refers to the non-frozen spinorbitals.
In the context of the basis set correction, the set of valence spinorbitals $\Val$ involved in the definition of the effective interaction [see Eq.~\eqref{eq:Wval}] refers to the non-frozen spinorbitals.
The ``valence'' correction was used consistently when the FC approximation was applied.
In order to estimate the complete basis set (CBS) limit of each model we employ the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOls-CPL-98} for the correlation energies.
The corresponding atomization energy are referred as $\CBS$.
In order to estimate the complete basis set (CBS) limit for each model, we employed the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98} for the correlation energies.
We refer to these atomization energies as $\CBS$.
%\subsection{Convergence of the atomization energies with the WFT models }
As the exFCI calculations were converged with a precision of about 0.2 mH, we can consider the atomization energies computed at this level as near FCI values which we will consider as the reference for a given system in a given basis set. The results for these molecules are shown in Table \ref{tab:diatomics}.
As one can notice from the data, the convergence of the exFCI atomization energies is slow with respect to the basis set, and the chemical accuracy is barely reached for C$_2$, O$_2$ and F$_2$ even at the cc-pV5Z basis set. Also, the atomization energies are always too small, reflecting the fact that, in a given basis set, a molecule is always more poorly described than the atoms due to the larger number of interacting pairs of electrons in the molecule.
The same behaviours hold for the CCSD(T) model, and one can notice that the atomization energies of the CCSD(T) are always slightly underestimated with respect to the CIPSI ones, showing that the CCSD(T) ansatz is better suited for the atoms than for the molecule.
As the exFCI calculations were converged with a precision of about 0.1 {\kcal}, we can consider these atomization energies as near-FCI values.
They will be our references for a given system in a given basis.
The results for four diatomics mentioned above are reported in Table \ref{tab:diatomics}.
As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with cc-pV5Z.
Also, the atomization energies are consistently underestimated, reflecting that, in a given basis, the atom is always better described than the molecule due to the larger number of interacting electron pairs in the molecular system.
A similar trend holds for CCSD(T).
%, and one can notice that the atomization energies of the CCSD(T) are always slightly underestimated with respect to the CIPSI ones, showing that the CCSD(T) ansatz is better suited for the atoms than for the molecule.
%\subsection{The effect of the basis set correction within the LDA and PBE approximation}
Regarding the effect of the basis set correction, both for the CIPSI and CCSD(T) models, several observations can be done.
First, in a given basis set, the addition of the basis set correction, both at the LDA and PBE level, improves the result even if it can overestimates the estimated CBS atomization energies by a few tens of kcal/mol (the largest deviation being 0.6 kcal/mol for N$_2$ at the (FC)CCSD(T)+PBE-val level in the cc-pV5Z basis). Nevertheless, the deviations observed in the largest basis sets are typically in the range of the accuracy of the atomization energies computed with the CBS extrapolation technique.
Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values. Also, one can observe that the sensitivity to the functional is quite large for the double- and triple-zeta basis sets, where clearly the PBE functional performs better. Nevertheless, from the quadruple-zeta basis set, the LDA and PBE functional agrees within a few tens of kcal/mol.
Regarding the effect of the basis set correction, several general observations can be made for both exFCI and CCSD(T).
First, in a given basis set, the basis set correction systematically improves the result (both at the LDA and PBE level).
A small overestimation can occur compared to the CBS values by a few tenths of a {\kcal} (the largest deviation being 0.6 {\kcal} for \ce{N2} at the CCSD(T)+PBE/cc-pV5Z level).
Nevertheless, the deviation observed for the largest basis set is typically within the extrapolation error of the CBS atomization energies, which is highly satisfactory knowing the marginal computation cost of the present correction.
%Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values.
Importantly, the sensitivity with respect to the SR-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting information}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for raw data of the G2 atomization energies.
See {\SI} for raw data associated with the G2 atomization energies.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}