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minor corrections

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Pierre-Francois Loos 2020-04-08 13:23:45 +02:00
parent fa3b19c112
commit dd2e672a59
5 changed files with 578 additions and 327 deletions
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@ -10,10 +10,10 @@
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40
Manuscript/FarDFT.bib
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@ -1,13 +1,48 @@
%% 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 2020-04-07 20:33:37 +0200 %% Created for Pierre-Francois Loos at 2020-04-08 13:05:55 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Bottcher_1974,
Author = {C. Bottcher and K. Docken},
Date-Added = {2020-04-08 13:03:40 +0200},
Date-Modified = {2020-04-08 13:05:55 +0200},
Doi = {10.1088/0022-3700/7/1/002},
Journal = {J. Phys. B: At. Mol. Phys.},
Pages = {L5},
Title = {Autoionizing States of the Hydrogen Molecule.},
Volume = {7},
Year = {1974}}
@article{Mielke_2005,
Author = {S. L. Mielke and D. W. Schwenke and K. A. Peterson},
Date-Added = {2020-04-08 12:47:49 +0200},
Date-Modified = {2020-04-08 12:49:45 +0200},
Doi = {10.1063/1.1917838},
Journal = {J. Chem. Phys.},
Pages = {224313},
Title = {Benchmark calculations of the complete configuration-interaction limit of Born-Oppenheimer diagonal corrections to the saddle points of isotopomers of the {{H+H2}} reaction.},
Volume = {122},
Year = {2005},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1917838}}
@article{Sun_2016,
Author = {J. Sun and J. P. Perdew and Z. Yang and H. Peng},
Date-Added = {2020-04-08 10:56:23 +0200},
Date-Modified = {2020-04-08 10:56:47 +0200},
Doi = {10.1063/1.4950845},
Journal = {J. Chem. Phys.},
Pages = {191101},
Title = {Near-locality of exchange and correlation density functionals for 1- and 2-electron systems},
Volume = {144},
Year = {2016},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4950845}}
@article{Slater_1951, @article{Slater_1951,
Author = {J. C. Slater}, Author = {J. C. Slater},
Date-Added = {2020-04-07 19:53:52 +0200}, Date-Added = {2020-04-07 19:53:52 +0200},
@ -17,7 +52,8 @@
Pages = {385}, Pages = {385},
Title = {A Simplification of the Hartree-Fock Method}, Title = {A Simplification of the Hartree-Fock Method},
Volume = {81}, Volume = {81},
Year = {1981}} Year = {1981},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.81.385}}
@book{Slater_1974, @book{Slater_1974,
Date-Added = {2020-04-07 19:48:23 +0200}, Date-Added = {2020-04-07 19:48:23 +0200},

193
Manuscript/FarDFT.tex
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@ -3,7 +3,7 @@
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc} \usepackage[T1]{fontenc}
\usepackage{txfonts} %\usepackage{txfonts}
\usepackage{grffile} \usepackage{grffile}
\usepackage[ \usepackage[
@ -67,7 +67,10 @@
\newcommand{\Ec}{E_\text{c}} \newcommand{\Ec}{E_\text{c}}
\newcommand{\HF}{\text{HF}} \newcommand{\HF}{\text{HF}}
\newcommand{\LDA}{\text{LDA}} \newcommand{\LDA}{\text{LDA}}
\newcommand{\eLDA}{\text{eLDA}} \newcommand{\SD}{\text{S}}
\newcommand{\VWN}{\text{VWN5}}
\newcommand{\SVWN}{\text{SVWN5}}
\newcommand{\MSFL}{\text{MSFL}}
\newcommand{\CID}{\text{CID}} \newcommand{\CID}{\text{CID}}
\newcommand{\Hxc}{\text{Hxc}} \newcommand{\Hxc}{\text{Hxc}}
\newcommand{\Ha}{\text{H}} \newcommand{\Ha}{\text{H}}
@ -173,13 +176,12 @@ We believe that it is partly due to the lack of accurate approximations for GOK-
In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation for ensemble (eDFA) has never been developed for atoms and molecules. In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation for ensemble (eDFA) has never been developed for atoms and molecules.
The present contribution is a small step towards this goal. The present contribution is a small step towards this goal.
\titou{When one talks about constructing functionals, the local-density approximation (LDA) is never far away. When one talks about constructing functionals, the local-density approximation (LDA) is never far away.
The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016} The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016}
Although the Hohenberg--Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behaviour in a real system. \cite{Kohn_1965} Although the Hohenberg--Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behaviour in a real system. \cite{Kohn_1965}
However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct LDA functionals. \cite{Loos_2014a,Loos_2014b,Loos_2017a} However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct LDA functionals. \cite{Loos_2014a,Loos_2014b,Loos_2017a}
Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b} Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
Here, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA functional for ensembles (eLDA). In particular, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT, and automatically incorporates the infamous derivative discontinuity contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}
The present eLDA functional is specifically designed to compute double excitations within GOK-DFT, and it automatically incorporates the infamous derivative discontinuity contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}}
%The paper is organised as follows. %The paper is organised as follows.
%In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented. %In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.
@ -250,10 +252,13 @@ Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes t
From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019} From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019}
\begin{equation} \begin{equation}
\begin{split}
\label{eq:dEdw} \label{eq:dEdw}
\pdv{\E{}{\bw}}{\ew{I}} \pdv{\E{}{\bw}}{\ew{I}}
= \E{}{(I)} - \E{}{(0)} & = \E{}{(I)} - \E{}{(0)}
= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}(\br{})}, \\
& = \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}(\br{})},
\end{split}
\end{equation} \end{equation}
where where
\begin{align} \begin{align}
@ -276,15 +281,19 @@ The latters are determined by solving the ensemble KS equation
\end{equation} \end{equation}
where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and
\begin{equation} \begin{equation}
\begin{split}
\fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})} \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
& = \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} = \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
\\
& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}'
+ \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{}))
\end{split}
\end{equation} \end{equation}
is the Hxc potential. is the Hxc potential, with
\begin{subequations}
\begin{align}
\fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})}
& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}',
\\
\fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
& = \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
\end{align}
\end{subequations}
Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view. Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view.
Note that the individual densities $\n{\Det{I}{\bw}}{}(\br{})$ defined in Eq.~\eqref{eq:nI} do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density. Note that the individual densities $\n{\Det{I}{\bw}}{}(\br{})$ defined in Eq.~\eqref{eq:nI} do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density.
Nevertheless, Nevertheless,
@ -314,20 +323,26 @@ Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a g
Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error. Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results} \section{Hydrogen molecule}
\label{sec:res} \label{sec:H2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First, we compute the ensemble energy of the \ce{H2} molecule using the aug-cc-pVTZ basis set and the weight-independent Slater local exchange functional, \cite{Dirac_1930, Slater_1951} \bruno{notée S51 sur les figures?} which is explicitly given by %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-independent exchange functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First, we compute the ensemble energy of the \ce{H2} molecule using the aug-cc-pVTZ basis set and the weight-independent Slater-Dirac local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by
\begin{align} \begin{align}
\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3}, \e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
& &
\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}. \Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
\end{align} \end{align}
The ensemble energy $\E{}{w}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the weight $0 \le \ew{} \le 1$. In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state and the lowest doubly-excited state of configuration $1\sigma_u^2$, which has an autoionising resonance nature. \cite{Bottcher_1974}
The ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the weight $0 \le \ew{} \le 1$.
Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}]. Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
As anticipated, $\E{}{w}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}). As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}).
Note that the exact xc correlation ensemble functional would yield a perfectly linear energy and the same value of the excitation energy independently of $\ew{}$. Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/2$.
Note that the exact xc correlation ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\ew{}$.
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{Ew_H2} \includegraphics[width=\linewidth]{Ew_H2}
@ -345,10 +360,15 @@ Note that the exact xc correlation ensemble functional would yield a perfectly l
} }
\end{figure} \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent exchange functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Second, in order to remove this spurious curvature of the ensemble energy (which is partly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$. Second, in order to remove this spurious curvature of the ensemble energy (which is partly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
Doing so, we have found that the present weight-dependent exchange functional (denoted as MSFL in the following), represented in Fig.~\ref{fig:Cx_H2}, Doing so, we have found that the present weight-dependent exchange functional (denoted as GIC-S in the following as its main purpose is to correct for the ghost-interaction error), represented in Fig.~\ref{fig:Cx_H2},
\begin{equation} \begin{equation}
\e{\ex}{\ew{},\text{MSFL}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3}, \e{\ex}{\ew{},\text{GIC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
\end{equation} \end{equation}
with with
\begin{equation} \begin{equation}
@ -363,11 +383,11 @@ and
& &
\gamma & = - 0.367\,189, \gamma & = - 0.367\,189,
\end{align} \end{align}
makes the ensemble much more linear (see Fig.~\ref{fig:Ew_H2})\bruno{C'est celle notée ``GIC'' sur la figure ? Pourquoi pas MSFL ? A clarifier pour le lecteur}, and the excitation energy much more stable and closer to the full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (see Fig.~\ref{fig:Om_H2}) makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (see Fig.~\ref{fig:Om_H2})
As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$. As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$.
Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$. Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$.
We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limit at $\ew{} = 0$ and $1$. We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $1$.
It is interesting to note that, around $\ew{} = 0$, the behavior of Eq.~\eqref{eq:Cxw} is linear It is interesting to note that, around $\ew{} = 0$, the behavior of Eq.~\eqref{eq:Cxw} is linear.
\begin{figure} \begin{figure}
\includegraphics[width=0.8\linewidth]{Cx_H2} \includegraphics[width=0.8\linewidth]{Cx_H2}
\caption{ \caption{
@ -376,9 +396,14 @@ It is interesting to note that, around $\ew{} = 0$, the behavior of Eq.~\eqref{e
} }
\end{figure} \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-independent correlation functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Third, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980} Third, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}. For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly non-linear ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the MSFL and VWN5 functionals exhibit a small curvature and improved excitation energies, especially at small weights. The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the GIC-S and VWN5 functionals (GIC-SVWN5) exhibit a small curvature (the ensemble energy is now slightly concave) and improved excitation energies, especially at small weights.
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%% FUNCTIONAL %%% %%% FUNCTIONAL %%%
@ -391,12 +416,17 @@ The combination of the Slater and VWN5 functionals (SVWN5) yield a highly non-li
%The construction of these two functionals is described below. %The construction of these two functionals is described below.
%Extension to spin-polarised systems will be reported in future work. %Extension to spin-polarised systems will be reported in future work.
Fourth, in the spirit of our recent work \cite{Loos_2020}, we have designed a weight-dependent correlation functional.
To build this weight-dependent correlation functional, we consider the singlet ground state and the first singlet doubly-excited state of a two-electron finite UEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent correlation functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Fourth, in the spirit of our recent work, \cite{Loos_2020} we have designed a weight-dependent correlation functional.
To build this weight-dependent correlation functional, we consider the singlet ground state and the first singlet doubly-excited state of a two-electron FUEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
Notably, these two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined. Notably, these two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993} Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}
As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere. As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere.
Note that the present paradigm is equivalent to the IUEG model in the thermodynamic limit. \cite{Loos_2011b} Note that the present paradigm is equivalent to the conventional IUEG model in the thermodynamic limit. \cite{Loos_2011b}
We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm. We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -408,8 +438,10 @@ The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\e{\HF}{(0)}(\n{}{}) & = \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3}, \e{\HF}{(0)}(\n{}{}) & = \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
\label{eq:eHF_0}
\\ \\
\e{\HF}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3} + \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}. \e{\HF}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3} + \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.
\label{eq:eHF_1}
\end{align} \end{align}
\end{subequations} \end{subequations}
%These two energies can be conveniently decomposed as %These two energies can be conveniently decomposed as
@ -456,13 +488,7 @@ The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states
%\end{equation} %\end{equation}
%Conveniently, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient; an expected feature from a theoretical point of view, yet a nice property from a more practical aspect. %Conveniently, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient; an expected feature from a theoretical point of view, yet a nice property from a more practical aspect.
Thanks to highly-accurate calculations and the expressions of the HF energies provided by Eqs.~\eqref{eq:eHF_0} and \eqref{eq:eHF_1}, \cite{Loos_2009a,Loos_2009c,Loos_2010e} one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant \cite{Sun_2016,Loos_2020}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Weight-dependent correlation functional}
%\label{sec:Ec}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Based on highly-accurate calculations, \cite{Loos_2009a,Loos_2009c,Loos_2010e} one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
\begin{equation} \begin{equation}
\label{eq:ec} \label{eq:ec}
\e{\co}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}}, \e{\co}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
@ -538,69 +564,59 @@ Combining these, we build a two-state weight-dependent correlation functional:
%\subsection{LDA-centered functional} %\subsection{LDA-centered functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons). Because our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons), we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
Hence, we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath). As explained further in Ref.~\onlinecite{Loos_2020}, this embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles originally derived by Franck and Fromager. \cite{Franck_2014}
The weight-dependence of the xc functional is then carried exclusively by the impurity [\ie, the functionals defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA xc functional). The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA correlation functional).
Consistently with such a strategy, Eqs.~\eqref{eq:exw} and \eqref{eq:ecw} are ``centred'' on their corresponding jellium reference Consistently with such a strategy, Eq.~\eqref{eq:ecw} is ``centred'' on its corresponding weight-independent LDA reference
\bruno{you commented the exchange part, why ?}
\begin{equation} \begin{equation}
\label{eq:becw} \label{eq:becw}
\be{\xc}{\ew{}}(\n{}{}) = (1-\ew{}) \be{\xc}{(0)}(\n{}{}) + \ew{} \be{\xc}{(1)}(\n{}{}) \be{\co}{\ew{}}(\n{}{}) = (1-\ew{}) \be{\co}{(0)}(\n{}{}) + \ew{} \be{\co}{(1)}(\n{}{})
\end{equation} \end{equation}
via the following shift: via the following shift:
\begin{equation} \begin{equation}
\be{\xc}{(I)}(\n{}{}) = \e{\xc}{(I)}(\n{}{}) + \e{\xc}{\LDA}(\n{}{}) - \e{\xc}{(0)}(\n{}{}). \be{\co}{(I)}(\n{}{}) = \e{\co}{(I)}(\n{}{}) + \e{\co}{\VWN}(\n{}{}) - \e{\co}{(0)}(\n{}{}).
\end{equation} \end{equation}
The LDA xc functional is similarly decomposed as
\begin{equation}
\e{\xc}{\LDA}(\n{}{}) = \e{\ex}{\LDA}(\n{}{}) + \e{\co}{\LDA}(\n{}{}),
\end{equation}
where we consider here the Dirac exchange functional \cite{Dirac_1930}
\begin{equation}
\e{\ex}{\LDA}(\n{}{}) = \Cx{\LDA} \n{}{1/3},
\end{equation}
with
\begin{equation}
\Cx{\LDA} = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3},
\end{equation}
and the VWN5 correlation functional \cite{Vosko_1980}
\begin{equation}
\e{\co}{\LDA}(\n{}{}) \equiv \e{\co}{\text{VWN5}}(\n{}{}).
\end{equation}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
Equation \eqref{eq:becw} can be recast Equation \eqref{eq:becw} can be recast
\begin{equation} \begin{equation}
\label{eq:eLDA} \label{eq:eLDA}
\begin{split} \begin{split}
\be{\xc}{\ew{}}(\n{}{}) \be{\co}{\ew{}}(\n{}{})
& = \e{\xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})] & = \e{\co}{\LDA}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
\\ \\
& = \e{\xc}{\LDA}(\n{}{}) + \ew{} \pdv{\e{\xc}{\ew{}}(\n{}{})}{\ew{}}, & = \e{\co}{\LDA}(\n{}{}) + \ew{} \pdv{\e{\co}{\ew{}}(\n{}{})}{\ew{}},
\end{split} \end{split}
\end{equation} \end{equation}
which nicely highlights the centrality of the LDA in the present eDFA. which nicely highlights the centrality of the LDA in the present weight-dependent density-functional approximation for ensembles.
In particular, $\be{\xc}{(0)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{})$. In particular, $\be{\co}{(0)}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
Consequently, in the following, we name this weight-dependent xc functional ``eLDA'' as it is a natural extension of the LDA for ensembles. Consequently, in the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the LDA for ensembles.
Also, we note that, by construction, Also, we note that, by construction,
\begin{equation} \begin{equation}
\label{eq:dexcdw} \label{eq:dexcdw}
\pdv{\be{\xc}{\ew{}}(\n{}{})}{\ew{}} \pdv{\be{\co}{\ew{}}(\n{}{})}{\ew{}}
= \be{\xc}{(1)}(n(\br)) - \be{\xc}{(0)}(n(\br)). = \be{\co}{(1)}(n) - \be{\co}{(0)}(n),
\end{equation} \end{equation}
which shows that the weight correction is purely linear.
%This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
%\begin{equation}
%\label{eq:GACE}
% \E{\xc}{\bw}[\n{}{}]
% = \E{\xc}{}[\n{}{}]
% + \sum_{I=1}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi,
%\end{equation}
%(where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014}
%Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional.
%In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir
%$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}
%%%%%%%%%%%%%%%%%%
%%% DISCUSSION %%%
%%%%%%%%%%%%%%%%%%
\section{Discussion}
\label{sec:dis}
This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
\begin{equation}
\label{eq:GACE}
\E{\xc}{\bw}[\n{}{}]
= \E{\xc}{}[\n{}{}]
+ \sum_{I=1}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi,
\end{equation}
(where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014}
Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional.
In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir
$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
@ -626,26 +642,27 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
& & aug-cc-pVTZ & 38.54 & 27.81 & 24.46 & 27.17 \\ & & aug-cc-pVTZ & 38.54 & 27.81 & 24.46 & 27.17 \\
& & aug-cc-pVQZ & 38.81 & 27.81 & 24.46 & 27.17 \\ & & aug-cc-pVQZ & 38.81 & 27.81 & 24.46 & 27.17 \\
\\ \\
MSFL & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\ GIC-S & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\
& & aug-cc-pVTZ & 26.88 & 26.59 & 26.61 & 26.67 \\ & & aug-cc-pVTZ & 26.88 & 26.59 & 26.61 & 26.67 \\
& & aug-cc-pVQZ & 26.82 & 26.60 & 26.62 & 26.67 \\ & & aug-cc-pVQZ & 26.82 & 26.60 & 26.62 & 26.67 \\
\\ \\
MSFL & VWN5 & aug-cc-pVDZ & 28.54 & 26.94 & 27.48 & 27.10 \\ GIC-S & VWN5 & aug-cc-pVDZ & 28.54 & 26.94 & 27.48 & 27.10 \\
& & aug-cc-pVTZ & 28.66 & 27.00 & 27.56 & 27.17 \\ & & aug-cc-pVTZ & 28.66 & 27.00 & 27.56 & 27.17 \\
& & aug-cc-pVQZ & 28.64 & 27.00 & 27.56 & 27.17 \\ & & aug-cc-pVQZ & 28.64 & 27.00 & 27.56 & 27.17 \\
\\ \\
MSFL & MSFL & aug-cc-pVDZ & 28.78 & 27.10 & 27.56 & 27.27 \\ GIC-S & eVWN5 & aug-cc-pVDZ & 28.78 & 27.10 & 27.56 & 27.27 \\
& & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\ & & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\
& & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\ & & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\
\\ \\
B88 & LYP & aug-mcc-pV8Z & & & & 28.42\fnm[1] \\ B88 & LYP & aug-mcc-pV8Z\fnm[1] & & & & 28.42\fnm[2] \\
B3 & LYP & aug-mcc-pV8Z & & & & 27.77\fnm[1] \\ B3 & LYP & aug-mcc-pV8Z\fnm[1] & & & & 27.77\fnm[2] \\
HF & LYP & aug-mcc-pV8Z & & & & 29.18\fnm[1] \\ HF & LYP & aug-mcc-pV8Z\fnm[1] & & & & 29.18\fnm[2] \\
HF & & aug-mcc-pV8Z & & & & 28.65\fnm[1] \\ HF & & aug-mcc-pV8Z\fnm[1] & & & & 28.65\fnm[2] \\
HF & FCI & aug-mcc-pV8Z & & & & 28.75\fnm[1] \\ HF & FCI & aug-mcc-pV8Z\fnm[1] & & & & 28.75\fnm[2] \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Reference \onlinecite{Barca_2018a}.} \fnt[1]{Reference \onlinecite{Mielke_2005}.}
\fnt[2]{Reference \onlinecite{Barca_2018a}.}
\end{table*} \end{table*}
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