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added some on top

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Emmanuel Giner 2020-03-25 15:41:34 +01:00
parent a63c719d6f
commit 6765fd3dbf
3 changed files with 664 additions and 27 deletions
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86
Revised_Manuscript/srDFT_SC.tex
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@ -143,6 +143,7 @@
% effective interaction
\newcommand{\twodm}[4]{\mel{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\murcas}[0]{\mu_{\text{CASSCF}}({\bf r})}
\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\ntwo}[0]{n_{2}}
\newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
@ -267,6 +268,11 @@
\newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\PBEspin}{PBEspin}
\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}}
\newcommand{\ontopcas}{\langle n_2^{\text{CAS}}(\br{},\br{}) \rangle}
\newcommand{\ontopextrap}{\langle \mathring{n}_{2}^{\text{CAS}}(\br{},\br{}) \rangle}
\newcommand{\ontopcipsi}{\langle n_2^{\text{CIPSI}}(\br{},\br{}) \rangle}
\newcommand{\muaverage}{\langle \murcas \rangle}
\newcommand{\largemu}{E_{c,md}^{\mu \rightarrow \infty}}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
@ -681,6 +687,55 @@ The performance of each of these functionals is tested in the following. Note th
\end{table*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*}
\caption{Integral of the on-top pair density in real space at various levels of theory (see text for details) for N$_2$, N, O$_2$ an O in the aug-cc-pVXZ basis sets (X=D,T,Q).}
\begin{ruledtabular}
\begin{tabular}{lrcccccc}
%\begin{tabular}{lrcccccc}
System & \tabc{Basis set} &\tabc{$\ontopcas$}& \tabc{$\ontopextrap$}& \tabc{$\ontopcipsi$}& \tabc{$\muaverage$} \\
\hline
\ce{N2} & aug-cc-pVDZ & 0.587712 & 0.329839 & 0.513967 & 0.946 \\
& aug-cc-pVTZ & 0.591622 & 0.385069 & 0.461386 & 1.328 \\
& aug-cc-pVQZ & 0.592422 & 0.461393 & 0.419336 & 1.706 \\[0.1cm]
\ce{N} & aug-cc-pVDZ & 0.172320 & 0.098115 & 0.127420 & 0.910 \\
& aug-cc-pVTZ & 0.173022 & 0.113155 & 0.111726 & 1.263 \\
& aug-cc-pVQZ & 0.173071 & 0.123332 & 0.106121 & 1.601 \\[0.1cm]
\hline
\ce{O2} & aug-cc-pVDZ & 1.164428 & 0.707757 & 0.971217 & 1.107 \\
& aug-cc-pVTZ & 1.166676 & 0.797154 & 0.884406 & 1.545 \\
& aug-cc-pVQZ & 1.167688 & 0.861134 & 0.841366 & 1.979 \\[0.1cm]
\ce{O} & aug-cc-pVDZ & 0.513919 & 0.314972 & 0.416040 & 1.080 \\
& aug-cc-pVTZ & 0.516070 & 0.369056 & 0.376896 & 1.499 \\
& aug-cc-pVQZ & 0.516288 & 0.383353 & 0.362491 & 1.924 \\[0.1cm]
% & &\multicolumn{4}{c}{Estimated exact:\fnm[1] 665.4} \\[0.2cm]
\hline
\end{tabular}
\end{ruledtabular}
\label{tab:d1}
\end{table*}
%& \tabc{$\largemu$}
%
%& -0.0520
%& -0.0210
%& -0.0104
%
%& -0.0107
%& -0.0064
%& -0.0032
%
%& -0.0726
%& -0.0302
%& -0.0152
%
%& -0.0318
%& -0.0138
%& -0.0069
\subsection{Computational details}
We present potential energy curves of small molecules up to the dissociation limit
@ -775,6 +830,37 @@ We report in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential en
Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have be anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94}
It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the accuracy of the atomization energy slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI atomization energy and very close to chemical accuracy.
\manu{
The overestimation of the atomization energy appearing for \ce{N2} in large basis sets reveals a kind of unbalanced treatment between the molecule and atoms in favour of the molecular system.
As the integral of the exact on-top pair density is proportional to the correlation energy in the large $\mu$ limit\cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18} (see Eq. \eqref{eq:lim_mularge}), the accuracy of a given approximation to the exact on-top pair density will have a direct influence on the accuracy of the related correlation energy.
To quantify the quality of various flavour of on-top pair densities for a given system and a given basis set $\basis$, we define the following quantities
}
\begin{equation}
\ontopcas = \int \text{d}\br{}\, n_2^{\text{CASSCF}}(\br{},\br{}),
\end{equation}
\begin{equation}
\ontopextrap = \int \text{d}\br{}\, \ntwoextrap(n_2^{\text{CASSCF}}(\br{},\br{}),\murcas),
\end{equation}
\begin{equation}
\ontopcipsi = \int \text{d}\br{}\,n_2^{\text{CIPSI}}(\br{},\br{}),
\end{equation}
%\begin{equation}
% \largemu = \int \text{d}\br{}\, \frac{(-2+\sqrt{2})\sqrt{2\pi}}{3\left(\murcas\right)^3} \ntwoextrap(n_2^{\text{CASSCF}}(\br{},\br{}),\murcas),
%\end{equation}
\begin{equation}
\muaverage = \frac{1}{N_{e}}\int \text{d}\br{}\,n^{\text{CASSCF}}(\br{}) \,\, \murcas
\end{equation}
\manu{
The quantity $n_2^{\text{CIPSI}}(\br{},\br{})$ is the on-top pair density of the largest CIPSI wave function obtained for each system in a given basis, which contains here at leas $10^7$ Slater determinants. All of these three quantities were computed excluding all contributions from the $1s$ orbitals.
We report in Table \ref{tab:d1} these quantities for N, \ce{N2}, O and \ce{O2} in different basis sets.
From this Table \ref{tab:d1} one can notice that integral of the on-top pair density at the CIPSI level is systematically lower than that at the CASSCF level, which is expected as the short-range correlation, digging the coulomb hone in a given basis set $\basis$ at near FCI level, is missing from the valence CASSCF wave function.
Also, the on-top pair density at the CIPSI level decreases roughly by $20\%$ between the aug-cc-pVDZ and aug-cc-pVQZ, whereas the on-top pair density at the CASSCF level is almost constant with respect to the basis set.
In order to estimate the integral of exact on-top pair density, we take as reference the value of $\ontopcipsi$ in the aug-cc-pVQZ basis set, although it is certainly an upper bound to the exact values.
Regarding $\ontopextrap$, such a quantity is directly linked to the basis set correction in the large $\mu$ limit.
The lowering of $\ontopextrap$ with respect to $\ontopcas$ is noticeable in a given basis set, but such $\ontopextrap$ globally increases when enlarging the basis set. This can be understood easily by remembering Eq. \eqref{eq:def_n2extrap} and realizing that the CASSCF on-top pair density is globally constant with the basis set whereas the value of $\murcas$ globally increases (as evidenced by $\muaverage$).
By comparing $\ontopextrap$ to the $\ontopcipsi$ in the aug-cc-pVQZ basis set, it is quite clear that the error
}
Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary functional can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is based on the value of the effective electron-electron interaction at coalescence and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected.
We hope to report further on this in the near future.

54
new/data_on_top/data
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@ -1,29 +1,29 @@
=========================
N2
CAS CAS-extrap CIPSI
avdz
N2 0.587712 0.337529 0.513967
N 0.172320 0.100855 0.127420
-------------------------------------
avtz
N2 0.591622 0.388739 0.461386
N 0.173022 0.114365 0.111726
-------------------------------------
avqz
N2 0.592422 0.423673 0.419336
0.173071 0.124562 0.106121
======================================
======================================
O2
CAS CAS-extrap CIPSI
avdz
O2 1.164428 0.70775731 0.97121
O 0.513919 0.31497205 0.41604
--------------------------------------
avtz
O2 1.166676 0.79715452 0.88440
O 0.516070 0.36905624 0.37689
--------------------------------------
avqz
O2 1.167688 0.861134291 0.84136
O 0.516288 0.398760190 0.36249
CAS CAS-extrap CIPSI CIPSI-extrap
avdz
N2 0.58771 0.32983 0.51397
N 0.17232 0.09811 0.12742 0.07343
------------- ---------- -------------------------
avtz
N2 0.59162 0.38506 0.46139
N 0.17302 0.11315 0.11173 0.07414
------------- ---------- -------------------------
avqz
N2 0.59242 0.42006 0.41934
N 0.17307 0.12333 0.10612 0.07582
============= ====================================
============= ====================================
O2
CAS CAS-extrap CIPSI
avdz
O2 1.16443 0.69100 0.97121
O 0.51391 0.30623 0.41604 0.24934
--------------------------------------------------
avtz
O2 1.16668 0.78996 0.88440
O 0.51607 0.35068 0.37689 0.25768
--------------------------------------------------
avqz
O2 1.16769 0.85386 0.84136
O 0.51628 0.37922 0.36249 0.26283

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new/data_on_top/data_on_top_he.eps Normal file
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