loos/srDFT_G2
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

final LCT's commit !

Browse Source
This commit is contained in:
eginer 2019-04-19 17:43:28 +02:00
parent 25db61848d
commit c504b24840
1 changed files with 10 additions and 10 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

20
Manuscript/G2-srDFT.tex
View File

@ -244,7 +244,7 @@ where
\n{2,\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2})
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation} \end{equation}
and $\Gam{pq}{rs} =\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO), and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
\begin{equation} \begin{equation}
\label{eq:fbasis} \label{eq:fbasis}
\f{\Bas}{}(\br{1},\br{2}) \f{\Bas}{}(\br{1},\br{2})
@ -254,14 +254,14 @@ and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb inte
With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}
\label{eq:int_eq_wee} \label{eq:int_eq_wee}
\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, \mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{equation} \end{equation}
where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$. where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
\begin{eqnarray} \begin{eqnarray}
\iint r_{12}^{-1} \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \phantom{xxxxxxxxx} \frac{1}{2}\iint \frac{1}{r_{12}} \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \phantom{xxxxxxxxx}
\nonumber\\ \nonumber\\
\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, \frac{1}{2}\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{eqnarray} \end{eqnarray}
it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originating from the e-e cusp. Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originating from the e-e cusp.
@ -270,7 +270,7 @@ As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}
Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}
\label{eq:lim_W} \label{eq:lim_W}
\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\ \lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = \frac{1}{r_{12}}
\end{equation} \end{equation}
for any $(\br{1},\br{2})$ such that $\n{2,\Bas}{}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit. for any $(\br{1},\br{2})$ such that $\n{2,\Bas}{}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit.
@ -378,7 +378,7 @@ inspired by the recent functional proposed by some of the authors \cite{FerGinTo
\beta(\{n_\sigma\},\{\nabla n_\sigma\},\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{\n{2}{\UEG}(0,\{\n{\sigma}{}\})}. \beta(\{n_\sigma\},\{\nabla n_\sigma\},\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{\n{2}{\UEG}(0,\{\n{\sigma}{}\})}.
\end{gather} \end{gather}
\end{subequations} \end{subequations}
The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its UEG version, i.e.~$\n{2,\Bas}{}(\br{},\br{}) \approx \n{2}{\UEG}(0,\{\n{\sigma}{}(\br{})\})$, with $\n{2}{\UEG}(0,\{n_\sigma\}) = 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ and the UEG on-top pair-distribution function $g(0,n)$ whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its UEG version, i.e.~$\n{2,\Bas}{}(\br{},\br{}) \approx \n{2}{\UEG}(0,\{\n{\sigma}{}(\br{})\})$, with $\n{2}{\UEG}(0,\{n_\sigma\}) = 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ and the UEG on-top pair-distribution function $g(0,n)$, the $0$ standing for $r_{12}=0$, whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2,\Bas}{}(\br{},\br{})$. This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2,\Bas}{}(\br{},\br{})$.
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}. Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
@ -439,13 +439,13 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
%%% FIGURE 1 %%% %%% FIGURE 1 %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=0.33\linewidth]{C2_VXZ} \includegraphics[width=0.30\linewidth]{C2_VXZ}
\hspace{1cm} \hspace{1cm}
\includegraphics[width=0.33\linewidth]{O2_VXZ} \includegraphics[width=0.30\linewidth]{O2_VXZ}
\\ \\
\includegraphics[width=0.33\linewidth]{N2_VXZ} \includegraphics[width=0.30\linewidth]{N2_VXZ}
\hspace{1cm} \hspace{1cm}
\includegraphics[width=0.33\linewidth]{F2_VXZ} \includegraphics[width=0.30\linewidth]{F2_VXZ}
\caption{ \caption{
Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets. Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets.
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}). The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).