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Manuscript/Ex-srDFT.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-05-31 09:36:15 +0200 %% Created for Pierre-Francois Loos at 2019-06-12 14:59:56 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -789,11 +789,12 @@
@article{QP2, @article{QP2,
Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama}, Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
Date-Added = {2019-04-07 13:54:16 +0200}, Date-Added = {2019-04-07 13:54:16 +0200},
Date-Modified = {2019-05-28 22:19:25 +0200}, Date-Modified = {2019-06-12 14:59:52 +0200},
Doi = {10.1021/acs.jctc.9b00176}, Doi = {10.1021/acs.jctc.9b00176},
Journal = {J. Chem. Theory Comput.}, Journal = {J. Chem. Theory Comput.},
Pages = {3591},
Title = {Quantum Package 2.0: A Open-Source Determinant-Driven Suite Of Programs}, Title = {Quantum Package 2.0: A Open-Source Determinant-Driven Suite Of Programs},
Volume = {in press}, Volume = {15},
Year = {2019}, Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}} Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}

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Manuscript/Ex-srDFT.tex
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@ -130,7 +130,7 @@
\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}} \newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}}
\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}} \newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}}
\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}} \newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
\newcommand{\nFC}[2]{\widetilde{n}_{#1}^{#2}} \newcommand{\tn}[2]{\widetilde{n}_{#1}^{#2}}
% energies % energies
@ -268,7 +268,8 @@ The ECMD functionals admit, for any $\n{}{}$, the following two limits
& &
\lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}], \lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
\end{align} \end{align}
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65} which correspond to the WFT limit ($\mu = \infty$) and the DFT limit ($\mu = 0$).
In Eq.~\eqref{eq:large_mu_ecmd}, $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65}
The key ingredient --- the range-separated function $\rsmu{}{\Bas}(\br{})$ --- automatically adapts to the spatial non-homogeneity of the basis set incompleteness error. The key ingredient --- the range-separated function $\rsmu{}{\Bas}(\br{})$ --- automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$. \cite{GinPraFerAssSavTou-JCP-18} It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$. \cite{GinPraFerAssSavTou-JCP-18}
@ -307,48 +308,51 @@ We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,L
The local-density approximation (LDA) of the ECMD complementary functional is defined as The local-density approximation (LDA) of the ECMD complementary functional is defined as
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, \bE{\LDA}{\Bas}[\n{}{},\rsmu{}{}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
\end{equation} \end{equation}
where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\citenum{PazMorGorBac-PRB-06}. where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\citenum{PazMorGorBac-PRB-06}.
To go beyond the LDA and cure its over correlation at small $\mu$, some of the authors recently proposed a Perdew-Burke-Ernzerhof (PBE)-based ECMD functional\cite{LooPraSceTouGin-JPCL-19}, To go beyond the LDA and cure its over correlation at small $\mu$, some of the authors recently proposed a Perdew-Burke-Ernzerhof (PBE)-based ECMD functional\cite{LooPraSceTouGin-JPCL-19},
\begin{equation} \begin{equation}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \bE{\PBE}{\Bas}[\n{}{},\rsmu{}{}] =
\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
\end{equation} \end{equation}
where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient. where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient.
$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{} = 0$ (DFT limit) and the exact large-$\rsmu{}{}$ behavior where the on-top pair density appears. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} In order to avoid the computation of the exact on-top pair density, the authors proposed to use that of the UEG which leads to $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{} = 0$ and the exact large-$\rsmu{}{}$ behavior where the on-top pair density naturally appears. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
In order to avoid the computation of the exact on-top pair density, we proposed in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} to use an approximated alternative which yields
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\label{eq:epsilon_cmdpbe} \label{eq:epsilon_cmdpbe}
\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} }, \be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBE(\n{}{},s,\zeta) \rsmu{}{3} },
\\ \\
\label{eq:beta_cmdpbe} \label{eq:beta_cmdpbe}
\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)/\n{}{}}. \beta^\PBE(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)/\n{}{}}.
\end{gather} \end{gather}
\end{subequations} \end{subequations}
where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ is the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}.
We will refer to this functional as the PBE ECMD functional. We will refer to this functional as the PBE ECMD functional.
As the UEG on-top pair density might not be suitable to treat strongly correlated systems, we propose here to use an extrapolation of the exact on-top pair density based on the on-top pair density: As the UEG on-top pair density might not be suitable to treat strongly correlated systems, \titou{we propose here to use an extrapolation of the exact on-top pair density based on the on-top pair density}
\begin{equation} \begin{equation}
\n{2}{\text{extr}}(\br{},\n{2}{\Bas},\rsmu{}{\Bas}) = \frac{\n{2}{\Bas}(\br{},\br{})}{1+ \frac{2}{\sqrt{\pi}\rsmu{}{\Bas}(\br{})}}. \tn{2}{}(\br{},\n{2}{},\rsmu{}{}) = \n{2}{}(\br{},\br{}) \qty(1+ \frac{2}{\sqrt{\pi}\rsmu{}{}(\br{})})^{-1}.
\end{equation} \end{equation}
Therefore, we introduce the ``on top'' PBE (PBEot) ECMD functional which reads: Therefore, we introduce the ``on top'' PBE (PBEot) ECMD functional which reads
\begin{equation} \begin{multline}
\label{eq:def_pbe_tot} \label{eq:def_pbe_tot}
\bE{\PBEot}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \bE{\PBEot}{\Bas}[\n{}{},\n{2}{},\rsmu{}{}] =
\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBEot}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\n{2}{\Bas}(\br{},\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, \int \n{}{}(\br{})
\end{equation} \\
\times \be{\text{c,md}}{\sr,\PBEot}\qty(\n{}{}(\br{}),\n{2}{}(\br{},\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{},
\end{multline}
with with
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\label{eq:epsilon_cmdpbe} \label{eq:epsilon_cmdpbe}
\be{\text{c,md}}{\sr,\PBEot}(\n{}{},s,\zeta,\n{2}{\Bas},\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta,\n{2}{}) \rsmu{}{3} }, \be{\text{c,md}}{\sr,\PBEot}(\n{}{},\n{2}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBEot(\n{}{},\n{2}{},s,\zeta) \rsmu{}{3} },
\\ \\
\label{eq:beta_cmdpbe} \label{eq:beta_cmdpbe}
\beta(\n{}{},s,\zeta,\n{2}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\text{extr}}(\n{2}{\Bas},\rsmu{}{\Bas})/\n{}{}}. \beta^\PBEot(\n{}{},\n{2}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\tn{2}{}(\n{2}{},\rsmu{}{})/\n{}{}}.
\end{gather} \end{gather}
\end{subequations} \end{subequations}
@ -540,7 +544,7 @@ However, these results also clearly evidence that special care has to be taken f
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{ \caption{
Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, carbon monoxyde, ethylene and water for various methods and basis sets. Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, ethylene and water for various methods and basis sets.
The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} on the same geometries. The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} on the same geometries.
See the {\SI} for raw data.} See the {\SI} for raw data.}
\label{tab:Mol} \label{tab:Mol}
@ -598,12 +602,12 @@ However, these results also clearly evidence that special care has to be taken f
& 0.11 & 0.02 & 0.00 & 0.11 & 0.02 & 0.00
\\ \\
\\ \\
Carbon monoxide & $1\,^{1}\Sigma^+ \ra 1\,^{1}\Pi$ & Val. & 8.48\fnm[1] & 0.09 & 0.01 & 0.02 % Carbon monoxide & $1\,^{1}\Sigma^+ \ra 1\,^{1}\Pi$ & Val. & 8.48\fnm[1] & 0.09 & 0.01 & 0.02
& 0.05 & 0.00 & 0.00 % & 0.05 & 0.00 & 0.00
& 0.07 & 0.01 & 0.02 % & 0.07 & 0.01 & 0.02
& 0.07 & 0.00 & 0.02 % & 0.07 & 0.00 & 0.02
\\ % \\
\\ % \\
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43\fnm[3] & -0.12 & -0.04 & Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43\fnm[3] & -0.12 & -0.04 &
& -0.05 & -0.01 & & -0.05 & -0.01 &
& -0.04 & -0.01 & & -0.04 & -0.01 &
@ -733,13 +737,10 @@ iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen ato
%%% FIG 4 %%% %%% FIG 4 %%%
\begin{figure*} \begin{figure*}
\includegraphics[height=0.45\linewidth]{C2_mu} \includegraphics[height=0.35\linewidth]{C2_mu}
\hspace{0.5cm} \includegraphics[height=0.35\linewidth]{C2_PBEot}
\includegraphics[height=0.45\linewidth]{C2_PBEot} \includegraphics[height=0.35\linewidth]{C2_n2}
\includegraphics[height=0.45\linewidth]{C2_PBE.pdf} \caption{$\rsmu{}{\Bas}$ (left), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBEot}$ (center) and $\n{2}{\Bas}$ (right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce{C2} for various basis sets $\Bas$.
\hspace{0.5cm}
\includegraphics[height=0.45\linewidth]{C2_n2.pdf}
\caption{$\rsmu{}{\Bas}$ (top left), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBEot}$ (top right), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBE}$ (bottom left) and $\n{2}{\Bas}$ (bottom right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce{C2} for various basis sets $\Bas$.
The two electronic states are both of $\Sigma_g^+$ symmetry. The two electronic states are both of $\Sigma_g^+$ symmetry.
The carbon nuclei are located at $z= \pm 1.180$ bohr and represented by the thin black lines.} The carbon nuclei are located at $z= \pm 1.180$ bohr and represented by the thin black lines.}
\label{fig:C2_mu} \label{fig:C2_mu}