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Pierre-Francois Loos 2019-11-13 23:48:28 +01:00
parent 52d2970a29
commit ba6fc13e41
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FarDFT/FarDFT.nb
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FarDFT/Manuscript/FarDFT.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-11-12 22:50:55 +0100
%% Created for Pierre-Francois Loos at 2019-11-13 20:40:48 +0100
%% Saved with string encoding Unicode (UTF-8)

237
FarDFT/Manuscript/FarDFT.tex
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,pifont,wrapfig,txfonts,multirow}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=blue,
urlcolor=blue,
citecolor=blue
}
\usepackage[
colorlinks=true,
citecolor=blue,
breaklinks=true
]{hyperref}
\urlstyle{same}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
@ -22,15 +18,6 @@
\newcommand{\cmark}{\color{green}{\text{\ding{51}}}}
\newcommand{\xmark}{\color{red}{\text{\ding{55}}}}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=blue,
urlcolor=blue,
citecolor=blue
}
%useful stuff
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
@ -55,6 +42,7 @@
% functionals, potentials, densities, etc
\newcommand{\eps}{\epsilon}
\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
\newcommand{\kin}[2]{t_\text{#1}^{#2}}
\newcommand{\E}[2]{E_\text{#1}^{#2}}
\newcommand{\bE}[2]{\overline{E}_\text{#1}^{#2}}
\newcommand{\be}[2]{\overline{\eps}_\text{#1}^{#2}}
@ -62,6 +50,7 @@
\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}}
@ -71,6 +60,7 @@
\newcommand{\Eani}{E_\text{ani}}
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\EFCI}{E_\text{FCI}}
\newcommand{\LDA}{\text{LDA}}
% matrices
\newcommand{\br}{\bm{r}}
@ -132,29 +122,36 @@ Their accuracy is illustrated by computing on the prototypical H$_2$ molecule.
%%% INTRODUCTION %%%
%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Here is a nice introduction.
%Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (UEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
%One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states cannot be easily identified like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
%Moreover, because the infinite UEG model is a metal, it is gapless, which means that both the fundamental and optical gaps are zero.
%From this point of view, using finite UEGs \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs can be seen as more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
%
%As a finite uniform electron gas, we consider the glomium model in which electrons move on the surface of a four-dimensional sphere (also known as a glome). \cite{Loos_2011b, Agboola_2015}
%The most appealing feature of glomium (regarding the development of functionals in the context of eDFT) is the fact that both the ground- and excited-state densities have uniform densities.
%As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
%This is a necessary condition for being able to model derivative discontinuities.
Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.
At a relatively low computational cost (at least compared to the other excited-state methods), TD-DFT can provide accurate transition energies for low-lying excited states in organic molecules.
Importantly, setting up a TD-DFT calculation for a given system is an almost pain-free process from the user perspective as the only (yet essential) input variable is the choice of the so-called exchange-correlation (xc) functional.
Indeed, TD-DFT is a in-principle exact theory which recast the many-body problem by transferring its complexity to the xc functional.
However, TD-DFT is far from being perfect, and, in practice, approximations must be made for the xc functional.
One of its issues actually originates directly from the choice of the xc functional, and more specifically, the possible substantial variations in the quality of the excitation energy for two different choices of xc functionals.
Moreover, because it was so popular, it has been studied in excruciated details, and researchers have quickly unveiled various theoretical and practical deficiencies of approximate TD-DFT.
Practically, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
One key consequence of this so-called adiabatic approximation is that double excitations are completely absent from the TD-DFT spectra.
Moreover, TD-DFT has problems with charge-transfer and Rydberg excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the xc functional.
The paper is organised as follows.
In Sec.~\ref{sec:theo}, ...
Section \ref{sec:func} provides details about the construction of the weight-dependent exchange-correlation functional.
The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:res}.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
Unless otherwise stated, atomic units are used throughout.
%%%%%%%%%%%%%%%%%%%%
%%% THEORY %%%
%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theo}
Here is the theory.
%%%%%%%%%%%%%%%%%%
%%% FUNCTIONAL %%%
%%%%%%%%%%%%%%%%%%
\section{Functional}
\label{sec:func}
We adopt the usual decomposition, and write down the weight-dependent exchange-correlation functional as
\begin{equation}
@ -162,30 +159,85 @@ We adopt the usual decomposition, and write down the weight-dependent exchange-c
\end{equation}
where $\e{x}{\ew{}}(\n{}{})$ and $\e{c}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively.
The construction of these two functionals is described below.
Here, we restrict our study to spin-unpolarized systems, i.e., closed-shell systems.
Here, we restrict our study to spin-unpolarized systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$.
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 $\n{}{} = 2/(4\pi/3 R^3)$ 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent exchange functional}
\label{sec:Ex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The reduced (\ie, per electron) HF energy for these two states is
\begin{subequations}
\begin{align}
\e{HF}{(0)}(\n{}{}) & = \frac{4}{3\pi^{1/3}} \n{}{1/3},
\\
\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
\begin{subequations}
\begin{align}
\kin{s}{(0)}(\n{}{}) & = 0,
&
\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3}.
\\
\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}.
\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 (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
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
\begin{equation}
\label{eq:ec}
\e{xc}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}},
\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
\end{equation}
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}
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}
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}
@ -193,46 +245,11 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
\e{c}{\ew{}}(\n{}{}) = (1-\ew{}) \e{c}{(0)}(\n{}{}) + \ew{} \e{c}{(1)}(\n{}{}).
\end{equation}
The reduced (i.e.~per electron) HF energy for these two states is:
\begin{subequations}
\begin{align}
\e{HF}{(0)}(n) & = \ldots,
\\
\e{HF}{(1)}(n) & = \ldots.
\end{align}
\end{subequations}
All these states have the same (uniform) density $n = 2/(2\pi R)$ where $R$ is the radius of the ring on which the electrons are confined.
The total energy of the ground and doubly-excited states are given by the two lowest eigenvalues of the Hamiltonian $\bH$ with elements
\begin{equation}
\begin{split}
H_{ij}
& = \int_0^\pi \qty[ \frac{\psi_i(\omega)}{R} \frac{\psi_j(\omega)}{R} + \frac{\psi_i(\omega)\psi_j(\omega)}{2R\sin(\omega/2)} ] \sin^2 \omega \, d\omega
\\
& = \ldots,
\end{split}
\end{equation}
where $\omega$ is the interelectronic angle, $\Gamma(x)$ is the Gamma function, \cite{NISTbook} and
\begin{equation}
\psi_i(\omega) = \sin^{i-1}(\omega/2), \quad i=1,\ldots,M
\end{equation}
are (non-orthogonal) explicitly-correlated basis functions with overlap matrix elements
\begin{equation}
S_{ij}
= \int_0^\pi \psi_i(\omega)\psi_j(\omega) \sin^2 \omega \, d\omega
= \ldots.
\end{equation}
Thanks to this explicitly-correlated basis, the convergence rate of the energy is exponential with respect to $M$.
Therefore, high accuracy is reached with a very small number of basis functions.
Here, we typically use $M=10$.
For the singly-excited state, one has to modify the basis functions as
The numerical values of the correlation energy for various $R$ are reported in Table \ref{tab:Ref} for the two states of interest.
%%% FIG 1 %%%
\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 $R = 1/(\pi n)$ 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.
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.
The data gathered in Table \ref{tab:Ref} are also reported.
}
\label{fig:Ec}
@ -240,32 +257,36 @@ The numerical values of the correlation energy for various $R$ are reported in T
%%% %%% %%%
%%% TABLE I %%%
\begin{turnpage}
\begin{squeezetable}
\begin{table*}
\begin{table}
\caption{
\label{tab:Ref}
$-\e{c}{(I)}$ 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.
$-\e{c}{(I)}$ 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.
}
\begin{ruledtabular}
\begin{tabular}{lcddddddddddd}
State & $I$ & \mc{11}{c}{Ring's radius $R = 1/(\pi n)$} \\
\cline{3-13}
& & \tabc{$0$} & \tabc{$1/10$} & \tabc{$1/5$} & \tabc{$1/2$} & \tabc{$1$} & \tabc{$2$} & \tabc{$5$} & \tabc{$10$} & \tabc{$20$} & \tabc{$50$} & \tabc{$100$} \\
\begin{tabular}{ldd}
$R$ & \mc{2}{c}{State } \\
\cline{2-3}
& \tabc{Ground state} & \tabc{Doubly-excited state} \\
\hline
Ground state & $0$ & 0.013708 & 0.012859 & 0.012525 & 0.011620 & 0.010374 & 0.008558 & 0.005673 & 0.003697 & 0.002226 & 0.001046 & 0.000567 \\
Singly-excited state & $1$ & 0.0238184 & 0.023392 & 0.022979 & 0.021817 & 0.020109 & 0.017371 & 0.012359 & 0.008436 & 0.005257 & 0.002546 & 0.001399 \\
Doubly-excited state & $2$ & 0.018715 & 0.018653 & 0.018576 & 0.018300 & 0.017743 & 0.016491 & 0.013145 & 0.009670 & 0.006365 & 0.003231 & 0.001816 \\
$0$ & & \\
$1/10$ & & \\
$1/5$ & & \\
$1/2$ & & \\
$1$ & & \\
$2$ & & \\
$5$ & & \\
$10$ & & \\
$20$ & & \\
$50$ & & \\
$100$ & & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
\end{turnpage}
\end{table}
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{c_1^{(I)}\,n}{n + c_2^{(I)} \sqrt{n} + c_3^{(I)}},
\e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}},
\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}
@ -274,19 +295,19 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
%%% TABLE 1 %%%
\begin{table}
\begin{table*}
\caption{
\label{tab:OG_func}
Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
\begin{ruledtabular}
\begin{tabular}{lcddd}
State & $I$ & \tabc{$c_1^{(I)}$} & \tabc{$c_2^{(I)}$} & \tabc{$c_3^{(I)}$} \\
State & $I$ & \tabc{$a_1^{(I)}$} & \tabc{$a_2^{(I)}$} & \tabc{$a_3^{(I)}$} \\
\hline
Ground state & $0$ & & & \\
Doubly-excited state & $1$ & & & \\
Ground state & $0$ & -0.0238184 & +0.00575719 & +0.0830576 \\
Doubly-excited state & $1$ & -0.0144633 & -0.0504501 & +0.0331287 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\end{table*}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -299,21 +320,30 @@ 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}{\text{LDA}}(\n{}{}) - \e{xc}{(0)}(\n{}{}).
\be{xc}{(I)}(\n{}{}) = \e{xc}{(I)}(\n{}{}) + \e{xc}{\LDA}(\n{}{}) - \e{xc}{(0)}(\n{}{}).
\end{equation}
The local-density approximation (LDA) exchange-correlation functional is
\begin{equation}
\e{xc}{\text{LDA}}(\n{}{}) = \e{x}{\text{LDA}}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}).
\e{xc}{\LDA}(\n{}{}) = \e{x}{\LDA}(\n{}{}) + \e{c}{\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}{\text{LDA}}(\n{}{}) + \ew{} \qty[\e{xc}{(1)}(\n{}{})-\e{xc}{(0)}(\n{}{})],
= \e{xc}{\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, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\text{LDA}}(\n{}{})$.
In particular, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\LDA}(\n{}{})$.
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}
@ -324,18 +354,19 @@ Finally, we note that, by construction,
\begin{equation}
\left. \pdv{\be{xc}{\ew{}}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{xc}{(J)}[\n{}{\ew{}}(\br)] - \be{xc}{(0)}[\n{}{\ew{}}(\br)].
\end{equation}
\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
The density-functional approximations designed in this manuscript are based on highly-accurate energies 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 glomium model.
We refer the interested reader to Refs.~\onlinecite{} for more details about this paradigm.
%%%%%%%%%%%%%%%
%%% RESULTS %%%
%%%%%%%%%%%%%%%
\section{Results}
\label{sec:res}
Here, we do \ce{H2} because \ce{H2} is very interesting.
%%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
As concluding remarks, we would like to say that, what we have done is awesome.
%%%%%%%%%%%%%%%%%%%%%%%%