update
This commit is contained in:
parent
ba6fc13e41
commit
b6360b6868
931
FarDFT/FarDFT.nb
931
FarDFT/FarDFT.nb
File diff suppressed because it is too large
Load Diff
@ -1,5 +1,6 @@
|
||||
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
|
||||
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem}
|
||||
\usepackage{libertine}
|
||||
|
||||
\usepackage[
|
||||
colorlinks=true,
|
||||
@ -50,7 +51,6 @@
|
||||
\newcommand{\n}[2]{n_{#1}^{#2}}
|
||||
\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
|
||||
\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
|
||||
\newcommand{\Cx}[1]{C_\text{x}^{#1}}
|
||||
|
||||
% energies
|
||||
\newcommand{\EHF}{E_\text{HF}}
|
||||
@ -60,7 +60,6 @@
|
||||
\newcommand{\Eani}{E_\text{ani}}
|
||||
\newcommand{\EPT}{E_\text{PT2}}
|
||||
\newcommand{\EFCI}{E_\text{FCI}}
|
||||
\newcommand{\LDA}{\text{LDA}}
|
||||
|
||||
% matrices
|
||||
\newcommand{\br}{\bm{r}}
|
||||
@ -97,6 +96,7 @@
|
||||
\newcommand{\kcal}{kcal/mol}
|
||||
|
||||
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
|
||||
|
||||
\begin{document}
|
||||
|
||||
@ -104,8 +104,11 @@
|
||||
|
||||
\author{Clotilde \surname{Marut}}
|
||||
\affiliation{\LCPQ}
|
||||
\author{Emmanuel Fromager}
|
||||
\email{fromagere@unistra.fr}
|
||||
\affiliation{\LCQ}
|
||||
\author{Pierre-Fran\c{c}ois \surname{Loos}}
|
||||
\email{loos@irsamc.ups-tlse.fr}
|
||||
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
|
||||
\affiliation{\LCPQ}
|
||||
|
||||
\begin{abstract}
|
||||
@ -163,7 +166,7 @@ Here, we restrict our study to spin-unpolarized systems, \ie, $\n{\uparrow}{} =
|
||||
|
||||
The present weight-dependent eDFA is specifically designed for the calculation of double excitations within eDFT.
|
||||
As mentioned previously, we consider a two-state ensemble including the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the two-electron glomium system.
|
||||
All these states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$ where $R$ is the radius of the glome where the electrons are confined.
|
||||
All these states have the same (uniform) density $\n{}{} = 2/(2\pi/2 R^3)$ where $R$ is the radius of the glome where the electrons are confined.
|
||||
We refer the interested reader to Refs.~\onlinecite{Loos_2011b} for more details about this paradigm.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -178,66 +181,27 @@ The reduced (\ie, per electron) HF energy for these two states is
|
||||
\e{HF}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}.
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
These two energies can be conveniently decomposed as
|
||||
\begin{equation}
|
||||
\e{HF}{(I)}(\n{}{}) = \kin{s}{(I)}(\n{}{}) + \e{H}{(0)}(\n{}{}) + \e{x}{(I)}(\n{}{}),
|
||||
\end{equation}
|
||||
with
|
||||
These two energies can be conveniently decomposed as
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\kin{s}{(0)}(\n{}{}) & = 0,
|
||||
&
|
||||
\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3}.
|
||||
\kin{s}{(0)}(\n{}{}) & = \frac{4}{3\pi^{1/3}} \n{}{1/3},
|
||||
\\
|
||||
\e{H}{(0)}(\n{}{}) & = \frac{8}{3\pi^{1/3}} \n{}{1/3},
|
||||
&
|
||||
\e{H}{(1)}(\n{}{}) & = \frac{352}{105\pi^{1/3}} \n{}{1/3}.
|
||||
\\
|
||||
\e{x}{(0)}(\n{}{}) & = - \frac{4}{3\pi^{1/3}} \n{}{1/3},
|
||||
&
|
||||
\e{x}{(1)}(\n{}{}) & = - \frac{176}{105\pi^{1/3}} \n{}{1/3}.
|
||||
\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}.
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
Knowing that the exchange functional has the following form
|
||||
\begin{equation}
|
||||
\e{x}{(I)}(\n{}{}) = \Cx{(I)} \n{}{1/3}
|
||||
\end{equation}
|
||||
we obtain
|
||||
\begin{align}
|
||||
\Cx{(0)} & = - \frac{4}{3} \qty( \frac{2}{\pi} )^{1/3},
|
||||
&
|
||||
\Cx{(1)} & = - \frac{176}{105} \qty( \frac{2}{\pi} )^{1/3}
|
||||
\end{align}
|
||||
We can now combine these two exchange functionals to create a weight-dependent exchange functional
|
||||
\begin{equation}
|
||||
\begin{split}
|
||||
\e{x}{\ew{}}(\n{}{})
|
||||
& = (1-\ew{}) \e{x}{(0)}(\n{}{}) + \ew{} \e{x}{(1)}(\n{}{})
|
||||
\\
|
||||
& = \Cx{\ew{}} \n{}{1/3}
|
||||
\end{split}
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}
|
||||
\end{equation}
|
||||
Amazingly, the weight dependence of the exchange functional can be transfered to the Subscript[C, x] coefficient.
|
||||
This is obvious but kind of nice.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Weight-dependent correlation functional}
|
||||
\label{sec:Ec}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Based on highly-accurate calculations, 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
|
||||
Based on highly-accurate calculations (see below), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
|
||||
\begin{equation}
|
||||
\label{eq:ec}
|
||||
\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
|
||||
\e{xc}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}},
|
||||
\end{equation}
|
||||
where the $a_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
|
||||
The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
|
||||
where the $c_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
|
||||
The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
|
||||
Equation \eqref{eq:ec} provides two state-specific correlation DFAs based on a two-electron system.
|
||||
Combining these, one can build a two-state weight-dependent correlation eDFA:
|
||||
\begin{equation}
|
||||
@ -249,7 +213,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
|
||||
\begin{figure}
|
||||
% \includegraphics[width=\linewidth]{Ec}
|
||||
\caption{
|
||||
Reduced (i.e., per electron) correlation energy $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = ...$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
|
||||
Reduced (i.e., per electron) correlation energy $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi n)$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
|
||||
The data gathered in Table \ref{tab:Ref} are also reported.
|
||||
}
|
||||
\label{fig:Ec}
|
||||
@ -260,7 +224,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
|
||||
\begin{table}
|
||||
\caption{
|
||||
\label{tab:Ref}
|
||||
$-\e{c}{(I)}$ as a function of the radius of the glome $R$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
|
||||
$-\e{c}{(I)}$ as a function of the radius of the ring $R$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{ldd}
|
||||
@ -279,6 +243,8 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
|
||||
$20$ & & \\
|
||||
$50$ & & \\
|
||||
$100$ & & \\
|
||||
$150$ & & \\
|
||||
$200$ & & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
@ -286,7 +252,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
|
||||
Based on these highly-accurate calculations, 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}
|
||||
\label{eq:ec}
|
||||
\e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}},
|
||||
\e{c}{(I)}(\n{}{}) = \frac{c_1^{(I)}}{1 + c_2^{(I)} \n{}{-1/6} + c_3^{(I)} \n{}{-1/3}},
|
||||
\end{equation}
|
||||
where $c_2^{(I)}$ and $c_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
|
||||
The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
|
||||
@ -301,10 +267,10 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
|
||||
Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lcddd}
|
||||
State & $I$ & \tabc{$a_1^{(I)}$} & \tabc{$a_2^{(I)}$} & \tabc{$a_3^{(I)}$} \\
|
||||
State & $I$ & \tabc{$c_1^{(I)}$} & \tabc{$c_2^{(I)}$} & \tabc{$c_3^{(I)}$} \\
|
||||
\hline
|
||||
Ground state & $0$ & -0.0238184 & +0.00575719 & +0.0830576 \\
|
||||
Doubly-excited state & $1$ & -0.0144633 & -0.0504501 & +0.0331287 \\
|
||||
Ground state & $0$ & & & \\
|
||||
Doubly-excited state & $1$ & & & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
@ -320,30 +286,21 @@ In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more
|
||||
\end{equation}
|
||||
where
|
||||
\begin{equation}
|
||||
\be{xc}{(I)}(\n{}{}) = \e{xc}{(I)}(\n{}{}) + \e{xc}{\LDA}(\n{}{}) - \e{xc}{(0)}(\n{}{}).
|
||||
\be{xc}{(I)}(\n{}{}) = \e{xc}{(I)}(\n{}{}) + \e{xc}{\text{LDA}}(\n{}{}) - \e{xc}{(0)}(\n{}{}).
|
||||
\end{equation}
|
||||
The local-density approximation (LDA) exchange-correlation functional is
|
||||
\begin{equation}
|
||||
\e{xc}{\LDA}(\n{}{}) = \e{x}{\LDA}(\n{}{}) + \e{c}{\LDA}(\n{}{}).
|
||||
\e{xc}{\text{LDA}}(\n{}{}) = \e{x}{\text{LDA}}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}).
|
||||
\end{equation}
|
||||
where we use here the Dirac exchange functional and the VWN5 correlation functional
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\e{x}{\LDA}(\n{}{}) & = \Cx{\LDA} \n{}{1/3}
|
||||
\\
|
||||
\e{c}{\LDA}(\n{}{}) & \equiv \e{c}{\text{VWN5}}(\n{}{}).
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
with $\Cx{\LDA} = -\frac{3}{2} \qty(\frac{3}{4\pi})^{1/3}$.
|
||||
|
||||
|
||||
Equation \eqref{eq:becw} can be recast
|
||||
\begin{equation}
|
||||
\label{eq:eLDA}
|
||||
\be{xc}{\ew{}}(\n{}{})
|
||||
= \e{xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{xc}{(1)}(\n{}{})-\e{xc}{(0)}(\n{}{})],
|
||||
= \e{xc}{\text{LDA}}(\n{}{}) + \ew{} \qty[\e{xc}{(1)}(\n{}{})-\e{xc}{(0)}(\n{}{})],
|
||||
\end{equation}
|
||||
which nicely highlights the centrality of the LDA in the present eDFA.
|
||||
In particular, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\LDA}(\n{}{})$.
|
||||
In particular, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\text{LDA}}(\n{}{})$.
|
||||
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
|
||||
|
||||
This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) which was originally derived by Franck and Fromager. \cite{Franck_2014}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user