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Pierre-Francois Loos 2019-04-06 23:14:11 +02:00
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73
Manuscript/G2-srDFT.tex
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@ -13,10 +13,10 @@
\newcommand{\juju}[1]{\textcolor{purple}{#1}}
\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashJL}[1]{\textcolor{purple}{\sout{#1}}}
\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}}
\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}}
\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
\newcommand{\JL}[1]{\juju{(\underline{\bf JL}: #1)}}
\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\usepackage{hyperref}
@ -27,41 +27,16 @@
urlcolor=blue,
citecolor=blue
}
\newcommand{\cdash}{\multicolumn{1}{c}{---}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\mr}{\multirow}
\newcommand{\SI}{\textcolor{blue}{supporting information}}
\newcommand{\QP}{\textsc{quantum package}}
% second quantized operators
%\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
%\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
\newcommand{\ai}[1]{\hat{a}_{#1}}
\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
\newcommand{\vpqrs}[0]{V_{pq}^{rs}}
\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})}
\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')}
%operators
\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}}
\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}}
%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}}
%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}}
%
% numbers
\newcommand{\bfr}[1]{{\bf x}_{#1}}
\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
% densities
\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}}
% units
\newcommand{\IneV}[1]{#1 eV}
@ -69,23 +44,24 @@
\newcommand{\InAA}[1]{#1 \AA}
\newcommand{\kcal}{kcal.mol$^{-1}$}
% methods
\newcommand{\D}{\text{D}}
\newcommand{\T}{\text{T}}
\newcommand{\Q}{\text{Q}}
\newcommand{\X}{\text{X}}
\newcommand{\UEG}{\text{UEG}}
\newcommand{\HF}{\text{HF}}
\newcommand{\LDA}{\text{LDA}}
\newcommand{\PBE}{\text{PBE}}
\newcommand{\FCI}{\text{FCI}}
\newcommand{\CBS}{\text{CBS}}
\newcommand{\exFCI}{\text{exFCI}}
\newcommand{\CCSDT}{\text{CCSD(T)}}
\newcommand{\lr}{\text{lr}}
\newcommand{\sr}{\text{sr}}
\newcommand{\Nel}{N}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\E}[2]{E_{#1}^{#2}}
@ -99,6 +75,7 @@
\newcommand{\w}[2]{w_{#1}^{#2}}
\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
\newcommand{\V}[2]{V_{#1}^{#2}}
\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
\newcommand{\modX}{\text{X}}
@ -356,9 +333,9 @@ is the two-body density associated with $\wf{}{\Bas}$, $\Gam{pq}{rs}[\wf{}{\Bas}
which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$
\begin{equation}
\label{eq:WeeB}
\hWee{\Bas} = \frac{1}{2} \sum_{pqrs \in \Bas}\vpqrs \aic{r} \aic{s} \ai{q} \ai{p}
\hWee{\Bas} = \frac{1}{2} \sum_{pqrs \in \Bas}\V{pq}{rs} \aic{r} \aic{s} \ai{q} \ai{p}
\end{equation}
over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals in $\Bas$ and $\vpqrs$ are the usual two-electron Coulomb integrals.
over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals in $\Bas$ and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
\begin{subequations}
\begin{align}
@ -374,7 +351,7 @@ where
\label{eq:fbasis}
\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
\\
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \vpqrs \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
\end{multline}
it comes naturally that
\begin{equation}
@ -491,7 +468,7 @@ Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version
\label{eq:def_lda_tot}
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] \n{}{}(\br{}) \dbr{},
\end{equation}
where $\be{\UEG}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
where $\be{\UEG}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
%--------------------------------------------
@ -552,13 +529,15 @@ Following the spirit of Eq.~\eqref{eq:fbasis}, we have
\label{eq:fbasisval}
\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
\\
= \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \vpqrs \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
= \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
\end{multline}
and the valence part of the effective interaction is
\begin{subequations}
\begin{gather}
\label{eq:Wval}
\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})/\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2}),
\\
\label{eq:muval}
\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\bx{},\Bar{\bx{}}),
\end{gather}
\end{subequations}
@ -689,14 +668,28 @@ Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valenc
\end{figure*}
%\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$ and F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ in the case of N$_2$ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). N$_2$, O$_2$ and F$_2$ belong to the G2 set and can be considered as weakly correlated, whereas C$_2$ contains already a non negligible non dynamic correlation component.
All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core.
In order to estimate the CBS limit of each model we use the two-point extrapolation of Ref. \onlinecite{HalHelJorKloKocOls-CPL-98} for the correlation energies and report the corresponding atomization energy which are referred as $D_e^{Q5Z}$ and $D_e^{C(Q5)Z}$ for the cc-pVXZ and cc-pCVXZ basis sets, respectively. All through this work, the valence interaction and density was used when the frozen core approximation was done on the WFT model.
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.
In a second time, we compute the entire atomization energies of the G2 test sets composed by 55 molecules.
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.
We refer the interested reader to Refs.~\onlinecite{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-2009} and have been performed 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.
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 $\CBS$.
%\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.
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.
%\subsection{The effect of the basis set correction within the LDA and PBE approximation}

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Manuscript/SI/G2_srDFT-SI.tex
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@ -19,7 +19,6 @@
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}