From 9b03c58962f89599a0b7397f122acebe9fb40985 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 26 Feb 2020 23:31:10 +0100 Subject: [PATCH] titou just corrected part of his mistakes --- Manuscript/eDFT.tex | 118 +- Notebooks/eDFT_FUEG.nb | 9878 ++++++++++++++++++++++++++++++---------- 2 files changed, 7468 insertions(+), 2528 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 13b2f71..4c25061 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -62,6 +62,7 @@ % matrices/operator \newcommand{\br}[1]{\boldsymbol{r}_{#1}} +\newcommand{\bx}[1]{\boldsymbol{x}_{#1}} \newcommand{\bw}{{\boldsymbol{w}}} \newcommand{\bG}{\boldsymbol{G}} \newcommand{\bS}{\boldsymbol{S}} @@ -134,6 +135,7 @@ We report a first generation of local, weight-dependent correlation density-func These density-functional approximations for ensembles (eDFAs) are specially designed for the computation of single and double excitations within eDFT, and can be seen as a natural extension of the ubiquitous local-density approximation for ensemble (eLDA). The resulting eDFAs, based on both finite and infinite uniform electron gas models, automatically incorporate the infamous derivative discontinuity contributions to the excitation energies through their explicit ensemble weight dependence. Their accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes. +\titou{Although the present weight-dependent functional has been specifically designed for one-dimensional systems, the methodology proposed here is directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.} \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -394,18 +396,29 @@ For implementation purposes, we will use in the rest of this work (one-electron reduced) density matrices as basic variables, rather than Slater determinants. If we expand the ensemble KS (spin) orbitals [from which the latter determinants are constructed] in an atomic orbital (AO) basis, -\beq - \SO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}), +\titou{\beq + \SO{p}{}(\bx{}) = s(\omega) \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}), \eeq -then the density matrix of the +where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatial degrees of freedom, and +\beq + s(\omega) + = + \begin{cases} + \alpha(\omega), & \text{for spin-up electrons,} \\ + \text{or} \\ + \beta(\omega), & \text{for spin-down electrons,} + \end{cases} +\eeq +}then the density matrix of the determinant $\Det{(K)}$ can be expressed as follows in the AO basis: \beq \bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{}, \eeq -where the summation runs over the spinorbitals that are occupied in -$\Det{(K)}$. Note that, as the theory is applied later on to spin-polarized -systems, we drop spin indices in the density matrices, for convenience. +where the summation runs over the spinorbitals that are occupied in $\Det{(K)}$. +\trashPFL{Note that, as the theory is applied later on to spin-polarized +systems, we drop spin indices in the density matrices, for convenience.} \manu{Is the latter sentence ok with you?} +\titou{I don't think we need it anymore. What do you think?} The electron density of the $K$th KS determinant can then be evaluated as follows: \beq @@ -849,45 +862,45 @@ This is a necessary condition for being able to model the ensemble correlation derivatives with respect to the weights [last term on the right-hand side of Eq.~(\ref{eq:exact_ener_level_dets})]. Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous IUEG paradigm. \cite{Loos_2013,Loos_2013a} -\manu{Let us stress that, in a FUEG like ringium, the interacting and +Let us stress that, in a FUEG like ringium, the interacting and noninteracting densities match individually for all the states within the ensemble -[these densities are all equal to the uniform density], which means that +(these densities are all equal to the uniform density), which means that so-called density-driven correlation -effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model.} +effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model. Here, we will consider the most simple ringium system featuring electronic correlation effects, \ie, the two-electron ringium model. The present weight-dependent eDFA is specifically designed for the calculation of excited-state energies within GOK-DFT. In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including: (i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system. -All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined. +All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R$ is the radius of the ring where the electrons are confined. We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm. Generalization to a larger number of states is straightforward and is left for future work. To ensure the GOK variational principle, \cite{Gross_1988a} the -tri-ensemble weights must fulfil the following conditions: -\manu{\titou{$0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$}. The -constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed -to consider an equi-bi-ensemble -for which $\ew{1}=1/2$. This possibility is excluded with your -inequalities. The correct constraints are given in Ref.~\cite{Deur_2019} -and are the ones you also mentioned, \ie, $0 \le \ew{2} \le 1/3$ and -$\ew{2} \le \ew{1} \le (1-\ew{2})/2$.} -\manu{ -Just in case, starting from -\beq -\begin{split} -0\leq \ew{2}\leq \ew{1}\leq (1-\ew{1}-\ew{2}) -\\ -\end{split} -\eeq -we obtain -\beq -0\leq \ew{2}\leq \ew{1}\leq (1-\ew{2})/2 -\eeq -which implies $\ew{2}\leq(1-\ew{2})/2$ or, equivalently, $\ew{2}\leq -1/3$. -} +tri-ensemble weights must fulfil the following conditions: \cite{Deur_2019} +\titou{$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$}. +%The constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed +%to consider an equi-bi-ensemble +%for which $\ew{1}=1/2$. This possibility is excluded with your +%inequalities. The correct constraints are given in Ref.~\cite{Deur_2019} +%and are the ones you also mentioned, \ie, $0 \le \ew{2} \le 1/3$ and +%$\ew{2} \le \ew{1} \le (1-\ew{2})/2$.} +%\manu{ +%Just in case, starting from +%\beq +%\begin{split} +%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{1}-\ew{2}) +%\\ +%\end{split} +%\eeq +%we obtain +%\beq +%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{2})/2 +%\eeq +%which implies $\ew{2}\leq(1-\ew{2})/2$ or, equivalently, $\ew{2}\leq +%1/3$. +%} %%% TABLE 1 %%% \begin{table*} \caption{ @@ -936,7 +949,7 @@ The weight-dependence of the correlation functional is then carried exclusively Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows: \begin{equation} \label{eq:becw} - \manu{\e{c}{\bw}(\n{}{})\rightarrow}\be{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}), + \titou{\e{c}{\bw}(\n{}{})} = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}), \end{equation} where \begin{equation} @@ -959,14 +972,14 @@ where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and a_3^\text{LDA} & = 2.408779. \end{align} \end{subequations} -\manu{Note that the strategy described in Eq.~(\ref{eq:becw}) is general and -can be applied to real (higher-dimension) systems.} In order to make the +Note that the strategy described in Eq.~(\ref{eq:becw}) is general and +can be applied to real (higher-dimensional) systems. In order to make the connection with the GACE formalism \cite{Franck_2014,Deur_2017} more explicit, one may recast Eq.~\eqref{eq:becw} as \begin{equation} \label{eq:eLDA} \begin{split} - \be{c}{\bw}(\n{}{}) + \titou{\e{c}{\bw}(\n{}{})} & = \e{c}{\text{LDA}}(\n{}{}) \\ & + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})], @@ -975,30 +988,26 @@ recast Eq.~\eqref{eq:becw} as or, equivalently, \begin{equation} \label{eq:eLDA_gace} -\begin{split} - \be{c}{\bw}(\n{}{}) - & = \e{c}{\text{LDA}}(\n{}{}) - \\ - & + \sum_{K>0}\int_0^{\ew{K}} + \titou{\e{c}{\bw}(\n{}{})} + = \e{c}{\text{LDA}}(\n{}{}) + + \sum_{K>0}\int_0^{\ew{K}} \qty[\e{c}{(K)}(\n{}{})-\e{c}{(0)}(\n{}{})]d\xi_K, -\end{split} \end{equation} where the $K$th correlation excitation energy (per electron) is integrated over the -ensemble weight $\xi_K$ at fixed (uniform) density $n$. -Eq.~(\ref{eq:eLDA_gace}) nicely highlights the centrality of the LDA in the present eDFA. -In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$. +ensemble weight $\xi_K$ at fixed (uniform) density $\n{}{}$. +Equation \eqref{eq:eLDA_gace} nicely highlights the centrality of the LDA in the present eDFA. +In particular, $\titou{\e{c}{(0,0)}(\n{}{})} = \e{c}{\text{LDA}}(\n{}{})$. Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles. Finally, we note that, by construction, -\manu{ \begin{equation} - \pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \be{c}{(J)}(\n{}{}(\br{})) - \be{c}{(0)}(\n{}{}(\br{})). + \titou{\pdv{\e{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).} \end{equation} -Manu: I guess that the "overlines" and the dependence in $\bf r$ of the -densities on the RHS should be removed. The final expression should be -\beq -\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}). -\eeq -} +%Manu: I guess that the "overlines" and the dependence in $\bf r$ of the +%densities on the RHS should be removed. The final expression should be +%\beq +%\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}). +%\eeq +%} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:comp_details} @@ -1113,6 +1122,7 @@ This is definitely a very pleasing outcome, which additionally shows that, even \end{figure} %%% %%% %%% +\titou{T2: there is a micmac with the derivative discontinuity as it is only defined at zero weight. We should clean up this.} It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations. To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage (with respect to FCI) on the excitation energies obtained at the KS-eLDA and eHF levels [see Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:EI-eHF}, respectively] as a function of the box length $L$ in the case of 5-boxium. The influence of the derivative discontinuity is clearly more important in the strong correlation regime. diff --git a/Notebooks/eDFT_FUEG.nb b/Notebooks/eDFT_FUEG.nb index 193b933..dad7310 100644 --- a/Notebooks/eDFT_FUEG.nb +++ b/Notebooks/eDFT_FUEG.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 10278195, 222405] -NotebookOptionsPosition[ 10135414, 220071] -NotebookOutlinePosition[ 10135804, 220088] -CellTagsIndexPosition[ 10135761, 220085] +NotebookDataLength[ 10558319, 227335] +NotebookOptionsPosition[ 10417776, 225041] +NotebookOutlinePosition[ 10418167, 225058] +CellTagsIndexPosition[ 10418124, 225055] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -37,7 +37,8 @@ mathpazo,xcolor,bm,mhchem}\>\"", "}"}]}]}], "]"}], ";"}]}], "Input", InitializationCell->True, CellChangeTimes->{{3.7288240181604652`*^9, 3.728824027007351*^9}, { 3.733131339213026*^9, 3.733131352923026*^9}}, - 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