cleaning up the git mess
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@ -1,6 +1,5 @@
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem}
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\usepackage{libertine}
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\usepackage[
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colorlinks=true,
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@ -51,6 +50,7 @@
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
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\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
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\newcommand{\Cx}[1]{C_\text{x}^{#1}}
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% energies
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\newcommand{\EHF}{E_\text{HF}}
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@ -60,6 +60,7 @@
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\newcommand{\Eani}{E_\text{ani}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\EFCI}{E_\text{FCI}}
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\newcommand{\LDA}{\text{LDA}}
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% matrices
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\newcommand{\br}{\bm{r}}
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@ -166,7 +167,7 @@ Here, we restrict our study to spin-unpolarized systems, \ie, $\n{\uparrow}{} =
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The present weight-dependent eDFA is specifically designed for the calculation of double excitations within eDFT.
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As mentioned previously, we consider a two-state ensemble including the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the two-electron glomium system.
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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.
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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.
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We refer the interested reader to Refs.~\onlinecite{Loos_2011b} for more details about this paradigm.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -181,27 +182,66 @@ The reduced (\ie, per electron) HF energy for these two states is
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\e{HF}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}.
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\end{align}
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\end{subequations}
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These two energies can be conveniently decomposed as
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These two energies can be conveniently decomposed as
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\begin{equation}
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\e{HF}{(I)}(\n{}{}) = \kin{s}{(I)}(\n{}{}) + \e{H}{(0)}(\n{}{}) + \e{x}{(I)}(\n{}{}),
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\end{equation}
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with
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\begin{subequations}
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\begin{align}
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\kin{s}{(0)}(\n{}{}) & = \frac{4}{3\pi^{1/3}} \n{}{1/3},
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\kin{s}{(0)}(\n{}{}) & = 0,
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&
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\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3}.
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\\
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\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}.
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\e{H}{(0)}(\n{}{}) & = \frac{8}{3\pi^{1/3}} \n{}{1/3},
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&
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\e{H}{(1)}(\n{}{}) & = \frac{352}{105\pi^{1/3}} \n{}{1/3}.
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\\
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\e{x}{(0)}(\n{}{}) & = - \frac{4}{3\pi^{1/3}} \n{}{1/3},
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&
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\e{x}{(1)}(\n{}{}) & = - \frac{176}{105\pi^{1/3}} \n{}{1/3}.
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\end{align}
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\end{subequations}
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Knowing that the exchange functional has the following form
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\begin{equation}
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\e{x}{(I)}(\n{}{}) = \Cx{(I)} \n{}{1/3}
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\end{equation}
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we obtain
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\begin{align}
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\Cx{(0)} & = - \frac{4}{3} \qty( \frac{2}{\pi} )^{1/3},
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&
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\Cx{(1)} & = - \frac{176}{105} \qty( \frac{2}{\pi} )^{1/3}
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\end{align}
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We can now combine these two exchange functionals to create a weight-dependent exchange functional
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\begin{equation}
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\begin{split}
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\e{x}{\ew{}}(\n{}{})
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& = (1-\ew{}) \e{x}{(0)}(\n{}{}) + \ew{} \e{x}{(1)}(\n{}{})
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\\
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& = \Cx{\ew{}} \n{}{1/3}
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\end{split}
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\end{equation}
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with
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\begin{equation}
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\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}
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\end{equation}
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Amazingly, the weight dependence of the exchange functional can be transfered to the Subscript[C, x] coefficient.
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This is obvious but kind of nice.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent correlation functional}
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\label{sec:Ec}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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
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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
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\begin{equation}
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\label{eq:ec}
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\e{xc}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}},
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\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
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\end{equation}
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where the $c_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
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The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
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where the $a_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
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The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
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Equation \eqref{eq:ec} provides two state-specific correlation DFAs based on a two-electron system.
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Combining these, one can build a two-state weight-dependent correlation eDFA:
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\begin{equation}
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@ -213,7 +253,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
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\begin{figure}
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% \includegraphics[width=\linewidth]{Ec}
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\caption{
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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.
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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.
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The data gathered in Table \ref{tab:Ref} are also reported.
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}
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\label{fig:Ec}
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@ -224,7 +264,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
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\begin{table}
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\caption{
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\label{tab:Ref}
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$-\e{c}{(I)}$ 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.
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$-\e{c}{(I)}$ 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.
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}
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\begin{ruledtabular}
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\begin{tabular}{ldd}
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@ -243,8 +283,6 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
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$20$ & & \\
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$50$ & & \\
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$100$ & & \\
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$150$ & & \\
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$200$ & & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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@ -252,7 +290,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
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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
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\begin{equation}
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\label{eq:ec}
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\e{c}{(I)}(\n{}{}) = \frac{c_1^{(I)}}{1 + c_2^{(I)} \n{}{-1/6} + c_3^{(I)} \n{}{-1/3}},
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\e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}},
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\end{equation}
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where $c_2^{(I)}$ and $c_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
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The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
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@ -267,10 +305,10 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
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Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
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\begin{ruledtabular}
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\begin{tabular}{lcddd}
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State & $I$ & \tabc{$c_1^{(I)}$} & \tabc{$c_2^{(I)}$} & \tabc{$c_3^{(I)}$} \\
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State & $I$ & \tabc{$a_1^{(I)}$} & \tabc{$a_2^{(I)}$} & \tabc{$a_3^{(I)}$} \\
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\hline
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Ground state & $0$ & & & \\
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Doubly-excited state & $1$ & & & \\
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Ground state & $0$ & -0.0238184 & +0.00575719 & +0.0830576 \\
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Doubly-excited state & $1$ & -0.0144633 & -0.0504501 & +0.0331287 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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@ -286,21 +324,30 @@ In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more
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\end{equation}
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where
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\begin{equation}
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\be{xc}{(I)}(\n{}{}) = \e{xc}{(I)}(\n{}{}) + \e{xc}{\text{LDA}}(\n{}{}) - \e{xc}{(0)}(\n{}{}).
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\be{xc}{(I)}(\n{}{}) = \e{xc}{(I)}(\n{}{}) + \e{xc}{\LDA}(\n{}{}) - \e{xc}{(0)}(\n{}{}).
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\end{equation}
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The local-density approximation (LDA) exchange-correlation functional is
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\begin{equation}
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\e{xc}{\text{LDA}}(\n{}{}) = \e{x}{\text{LDA}}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}).
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\e{xc}{\LDA}(\n{}{}) = \e{x}{\LDA}(\n{}{}) + \e{c}{\LDA}(\n{}{}).
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\end{equation}
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where we use here the Dirac exchange functional and the VWN5 correlation functional
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\begin{subequations}
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\begin{align}
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\e{x}{\LDA}(\n{}{}) & = \Cx{\LDA} \n{}{1/3}
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\\
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\e{c}{\LDA}(\n{}{}) & \equiv \e{c}{\text{VWN5}}(\n{}{}).
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\end{align}
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\end{subequations}
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with $\Cx{\LDA} = -\frac{3}{2} \qty(\frac{3}{4\pi})^{1/3}$.
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Equation \eqref{eq:becw} can be recast
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\begin{equation}
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\label{eq:eLDA}
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\be{xc}{\ew{}}(\n{}{})
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= \e{xc}{\text{LDA}}(\n{}{}) + \ew{} \qty[\e{xc}{(1)}(\n{}{})-\e{xc}{(0)}(\n{}{})],
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= \e{xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{xc}{(1)}(\n{}{})-\e{xc}{(0)}(\n{}{})],
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\end{equation}
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which nicely highlights the centrality of the LDA in the present eDFA.
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In particular, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\text{LDA}}(\n{}{})$.
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In particular, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\LDA}(\n{}{})$.
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Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
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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}
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@ -330,8 +377,7 @@ As concluding remarks, we would like to say that, what we have done is awesome.
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%%% ACKNOWLEDGEMENTS %%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{acknowledgements}
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PFL would like to thank Emmanuel Fromager for enlightening discussions.
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He also acknowledges funding from the \textit{Centre National de la Recherche Scientifique}.
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PFL acknowledges funding from the \textit{Centre National de la Recherche Scientifique}.
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CM thanks the \textit{Universit\'e Paul Sabatier} (Toulouse, France) for a PhD scholarship.
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\end{acknowledgements}
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