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updated manuscript

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Emmanuel Giner 2020-03-26 15:19:43 +01:00
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Revised_Manuscript/srDFT_SC.tex
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@ -144,6 +144,7 @@
\newcommand{\twodm}[4]{\mel{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} \newcommand{\twodm}[4]{\mel{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})} \newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\murcas}[0]{\mu_{\text{CASSCF}}({\bf r})} \newcommand{\murcas}[0]{\mu_{\text{CASSCF}}({\bf r})}
\newcommand{\murcipsi}[0]{\mu_{\text{CIPSI}}({\bf r})}
\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})} \newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\ntwo}[0]{n_{2}} \newcommand{\ntwo}[0]{n_{2}}
\newcommand{\ntwohf}[0]{n_2^{\text{HF}}} \newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
@ -270,8 +271,10 @@
\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}} \newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}}
\newcommand{\ontopcas}{\langle n_2^{\text{CAS}}(\br{},\br{}) \rangle} \newcommand{\ontopcas}{\langle n_2^{\text{CAS}}(\br{},\br{}) \rangle}
\newcommand{\ontopextrap}{\langle \mathring{n}_{2}^{\text{CAS}}(\br{},\br{}) \rangle} \newcommand{\ontopextrap}{\langle \mathring{n}_{2}^{\text{CAS}}(\br{},\br{}) \rangle}
\newcommand{\ontopextrapcipsi}{\langle \mathring{n}_{2}^{\text{CIPSI}}(\br{},\br{}) \rangle}
\newcommand{\ontopcipsi}{\langle n_2^{\text{CIPSI}}(\br{},\br{}) \rangle} \newcommand{\ontopcipsi}{\langle n_2^{\text{CIPSI}}(\br{},\br{}) \rangle}
\newcommand{\muaverage}{\langle \murcas \rangle} \newcommand{\muaverage}{\langle \murcas \rangle}
\newcommand{\muaveragecipsi}{\langle \murcipsi \rangle}
\newcommand{\largemu}{E_{c,md}^{\mu \rightarrow \infty}} \newcommand{\largemu}{E_{c,md}^{\mu \rightarrow \infty}}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
@ -689,29 +692,28 @@ The performance of each of these functionals is tested in the following. Note th
\begin{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).} \caption{Integral of the on-top pair density in real space at various levels of theory (see text for details) for N$_2$ and N in the aug-cc-pVXZ basis sets (X=D,T,Q).}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lrcccccc} \begin{tabular}{lrccccccc}
%\begin{tabular}{lrcccccc} %\begin{tabular}{lrccccccc}
System & \tabc{Basis set} &\tabc{$\ontopcas$}& \tabc{$\ontopextrap$}& \tabc{$\ontopcipsi$}& \tabc{$\muaverage$} \\ System & \tabc{Basis set} &\tabc{$\ontopcas$}& \tabc{$\ontopextrap$}& \tabc{$\ontopcipsi$} & \tabc{$\ontopextrapcipsi$}& \tabc{$\muaverage$} & \tabc{$\muaveragecipsi$} \\
\hline \hline
\ce{N2} & aug-cc-pVDZ & 0.587712 & 0.329839 & 0.513967 & 0.946 \\ \ce{N2} & aug-cc-pVDZ & 0.58771 & 0.32983 & 0.51396 & 0.29114 & 0.946 & 0.962 \\
& aug-cc-pVTZ & 0.591622 & 0.385069 & 0.461386 & 1.328 \\ & aug-cc-pVTZ & 0.59162 & 0.38506 & 0.46138 & 0.30537 & 1.328 & \\
& aug-cc-pVQZ & 0.592422 & 0.461393 & 0.419336 & 1.706 \\[0.1cm] & aug-cc-pVQZ & 0.59242 & 0.42006 & 0.41933 & 0.29991 & 1.706 & \\[0.1cm]
\ce{N} & aug-cc-pVDZ & 0.172320 & 0.098115 & 0.127420 & 0.910 \\ \ce{N} & aug-cc-pVDZ & 0.17232 & 0.09811 & 0.12742 & 0.07343 & 0.910 & 0.922 \\
& aug-cc-pVTZ & 0.173022 & 0.113155 & 0.111726 & 1.263 \\ & aug-cc-pVTZ & 0.17302 & 0.11315 & 0.11172 & 0.07414 & 1.263 & 1.299 \\
& aug-cc-pVQZ & 0.173071 & 0.123332 & 0.106121 & 1.601 \\[0.1cm] & aug-cc-pVQZ & 0.17307 & 0.12333 & 0.10612 & 0.07582 & 1.601 & 1.653 \\[0.1cm]
\hline %\hline
\ce{O2} & aug-cc-pVDZ & 1.164428 & 0.707757 & 0.971217 & 1.107 \\ %\ce{O2} & aug-cc-pVDZ & 1.16442 & 0.70775 & 0.97121 & & 1.107 & 1.107 \\
& aug-cc-pVTZ & 1.166676 & 0.797154 & 0.884406 & 1.545 \\ % & aug-cc-pVTZ & 1.16667 & 0.79715 & 0.88440 & & 1.545 & 1.545 \\
& aug-cc-pVQZ & 1.167688 & 0.861134 & 0.841366 & 1.979 \\[0.1cm] % & aug-cc-pVQZ & 1.16768 & 0.86113 & 0.84136 & & 1.979 & 1.979 \\[0.1cm]
%
\ce{O} & aug-cc-pVDZ & 0.513919 & 0.314972 & 0.416040 & 1.080 \\ %\ce{O} & aug-cc-pVDZ & 0.51391 & 0.31497 & 0.41604 & & 1.080 & 1.080 \\
& aug-cc-pVTZ & 0.516070 & 0.369056 & 0.376896 & 1.499 \\ % & aug-cc-pVTZ & 0.51607 & 0.36905 & 0.37689 & & 1.499 & 1.499 \\
& aug-cc-pVQZ & 0.516288 & 0.383353 & 0.362491 & 1.924 \\[0.1cm] % & aug-cc-pVQZ & 0.51628 & 0.38335 & 0.36249 & & 1.924 & 1.924 \\[0.1cm]
% & &\multicolumn{4}{c}{Estimated exact:\fnm[1] 665.4} \\[0.2cm]
\hline \hline
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
@ -835,31 +837,57 @@ It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N
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. 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. 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 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} \begin{equation}
\label{eq:ontopcas}
\ontopcas = \int \text{d}\br{}\, n_2^{\text{CASSCF}}(\br{},\br{}), \ontopcas = \int \text{d}\br{}\, n_2^{\text{CASSCF}}(\br{},\br{}),
\end{equation} \end{equation}
\begin{equation} \begin{equation}
\label{eq:ontopextrap}
\ontopextrap = \int \text{d}\br{}\, \ntwoextrap(n_2^{\text{CASSCF}}(\br{},\br{}),\murcas), \ontopextrap = \int \text{d}\br{}\, \ntwoextrap(n_2^{\text{CASSCF}}(\br{},\br{}),\murcas),
\end{equation} \end{equation}
\begin{equation} \begin{equation}
\label{eq:ontopcipsi}
\ontopcipsi = \int \text{d}\br{}\,n_2^{\text{CIPSI}}(\br{},\br{}), \ontopcipsi = \int \text{d}\br{}\,n_2^{\text{CIPSI}}(\br{},\br{}),
\end{equation} \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} \begin{equation}
\label{eq:ontopextrapcipsi}
\ontopextrapcipsi = \int \text{d}\br{}\, \ntwoextrap(n_2^{\text{CIPSI}}(\br{},\br{}),\murcipsi),
\end{equation}
\begin{equation}
\label{eq:muaverage}
\muaverage = \frac{1}{N_{e}}\int \text{d}\br{}\,n^{\text{CASSCF}}(\br{}) \,\, \murcas \muaverage = \frac{1}{N_{e}}\int \text{d}\br{}\,n^{\text{CASSCF}}(\br{}) \,\, \murcas
\end{equation} \end{equation}
\begin{equation}
\label{eq:muaveragecipsi}
\muaveragecipsi = \frac{1}{N_{e}}\int \text{d}\br{}\,n^{\text{CIPSI}}(\br{}) \,\, \murcipsi
\end{equation}
}
\manu{ \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. The quantity $n_2^{\text{CIPSI}}(\br{},\br{})$ is the on-top pair density of the largest variational wave function for a given CIPSI calculation in a given basis, which contains here at leas $10^7$ Slater determinants. The quantity $\murcipsi$ is the $\murpsi$ obtained with the definition of Eq. \eqref{eq:def_mur} with the two-body tensor $\Gam{pq}{rs}$ and on-top pair density $\twodmrdiagpsi$ associated with the largest variational CIPSI wave function for a given basis.
We report in Table \ref{tab:d1} these quantities for N, \ce{N2}, O and \ce{O2} in different basis sets. All quantities from Eqs. \eqref{eq:ontopcas} to \eqref{eq:muaverage} were computed excluding all contributions from the $1s$ orbitals. }
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. \manu{
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. We report in Table \ref{tab:d1} these quantities for N and \ce{N2} in different basis sets.
Regarding $\ontopextrap$, such a quantity is directly linked to the basis set correction in the large $\mu$ limit. From this Table \ref{tab:d1} one can notice that the 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.
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 $\muaverage$). Also, the on-top pair density at the CIPSI level decreases in a monotonous way, 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.
By comparing $\ontopextrap$ to the $\ontopcipsi$ in the aug-cc-pVQZ basis set, it is quite clear that the error Regarding the extrapolated on-top pair densities, $\ontopextrap$ and $\ontopextrapcipsi$, it is interesting to notice that they are substantially lower with respect to their original on-top pair density, which are $\ontopcas$ and $\ontopcipsi$.
Nevertheless, the behaviour of $\ontopextrap$ and $\ontopextrapcipsi$ are qualitatively different : $\ontopextrap$ globally increases when enlarging the basis set whereas $\ontopextrapcipsi$ remains qualitatively constant. More precisely, in the case of \ce{N2} the value $\ontopextrap$ increases by about 50$\%$ between the aug-cc-pVDZ and aug-cc-pVQZ basis sets, whereas $\ontopextrapcipsi$ fluctuates by about 5$\%$ within the same basis sets.
The behaviour of $\ontopextrap$ can be understood easily by noticing that (see Eq. \eqref{eq:def_n2extrap})
\begin{equation}
\lim_{\mu \rightarrow \infty} \ntwoextrap(n_2,\mu) = n_2,
\end{equation}
that the CASSCF on-top pair density is globally constant with the basis set and that the value of $\murcas$ globally increases (as evidenced by $\muaverage$).
Eventually, at the CBS limit, $\murcas \rightarrow \infty$ and therefore one obtains
\begin{equation}
\lim_{\basis \rightarrow \text{CBS}} \ontopextrap = \ontopcas.
\end{equation}
On the other hand, the stability of $\ontopextrapcipsi$ is quite remarkable and must come from i) the fact that the on-top pair density at the CIPSI level already captures the coulomb hole within the basis set $\basis$, and ii) the $\murcipsi$ together with the large-$\mu$ limit extrapolation of the on-top pair density (see Eq. \eqref{eq:def_n2extrap}) are quantitatively correct.
Therefore, in order to estimate the integral of exact on-top pair density, we take as reference the value of $\ontopextrapcipsi$ in the aug-cc-pVQZ basis set.
}
\manu{
In the case of the present work, it is important to keep in mind that $\ontopextrap$ is directly linked to the basis set correction in the large $\mu$ limit, and more precisely the correlation energy (in absolute value) is a growing function of $\ontopextrap$. Therefore, the error on $\ontopextrap$ with respect to the estimated exact (here taken as $\ontopextrapcipsi$ in the aug- cc-pVQZ basis set) provides an indication on the magnitude of the error on the basis set correction for a given system and a given basis set.
In the aug-cc-pVQZ, for \ce{N2} $\ontopextrap - \ontopextrapcipsi = 0.120$ whereas $2\times(\ontopextrap - \ontopextrapcipsi) = 0.095$. We can then conclude that the overestimation of the on-top pair density and therefore of the basis set correction is more important on the \ce{N2} molecule at equilibrium distance than on the dissociated molecule, explaining probably the overestimation of the atomization energy.
To confirm such statement, we computed the basis set correction at the equilibrium geometry of \ce{N2} and the isolated N atoms using $\murcipsi$ and $\ntwoextrap(n_2^{\text{CIPSI}}(\br{},\br{}),\murcipsi)$ in the aug-cc-pVTZ and aug-cc-PVQZ basis sets, and obtained the following values for the atomization energies: 362.12 mH in aug-cc-pVTZ and ????? in the aug-cc-pVQZ, which are more accurate values than those obtained using $\murcas$ and $\ntwoextrap(n_2^{\text{CASSCF}}(\br{},\br{}),\murcas)$.
} }
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. 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.