diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 5cb7372..3627f09 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -19,6 +19,7 @@ \newcommand{\cloclo}[1]{\textcolor{purple}{#1}} \newcommand{\bruno}[1]{\textcolor{blue}{Bruno: #1}} \newcommand{\manu}[1]{\textcolor{magenta}{Manu: #1}} +\newcommand{\manuf}[1]{\textcolor{magenta}{#1}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} \newcommand{\trashCM}[1]{\textcolor{purple}{\sout{#1}}} @@ -46,6 +47,7 @@ \newcommand{\hWee}{\Hat{W}_\text{ee}} \newcommand{\hGam}[1]{\Hat{\Gamma}^{#1}} \newcommand{\hgam}[1]{\Hat{\gamma}^{#1}} +\newcommand{\hgamdens}[1]{\Hat{\gamma}^{#1}[n]} \newcommand{\bH}{\boldsymbol{H}} \newcommand{\hVne}{\Hat{V}_\text{ne}} \newcommand{\vne}{v_\text{ne}} @@ -120,6 +122,11 @@ \newcommand{\UL}{Instituut-Lorentz, Universiteit Leiden, P.O.~Box 9506, 2300 RA Leiden, The Netherlands} \newcommand{\VU}{Division of Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands} +%%% added by Manu %%% +\newcommand{\beq}{\begin{eqnarray}} +\newcommand{\eeq}{\end{eqnarray}} + + \begin{document} \title{Weight Dependence of Local Exchange-Correlation Functionals in Ensemble Density-Functional Theory: Double Excitations in Two-Electron Systems} @@ -213,13 +220,21 @@ semi-empirical as she used experimental excitation energies. It would be fair to cite her paper. There is also this paper [Journal of Molecular Structure (Theochem) 571 (2001) 153-161] that I never really understood but they tried something}. -The present contribution is a small step towards this goal. +The present contribution is a small \manu{too modest I think. ``paves +the way towards ...'' or something like that} step towards this goal. -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 \manu{too ``oral'' style I think}. 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} 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} +\manu{It goes much too fast here. One should make a clear distinction +between your previous work with Peter on the ground-state theory. Then +we should refer to our latest work where GOK-DFT is applied +to ringium. In the present work we extend the approach to glomium. As we +did in our previous work we should motivate the use of FUEGs for +developing weight-dependent functionals.} 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 ensemble derivative contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983} The paper is organised as follows. @@ -236,24 +251,33 @@ Unless otherwise stated, atomic units are used throughout. \label{sec:theo} Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\nEns-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$. +\manu{For clarity, I usually exclude $\ew{0}$ from $\bw$ so that $\bw$ +only contains the weights that are allowed to vary independently. One +should write explicitly $\ew{0}=1-\sum_{I=1}^{\nEns-1} \ew{I}$ and +define $\bw$ as $\bw = (\ew{1},\ldots,\ew{M-1})$} The corresponding ensemble energy \begin{equation} \E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)} \end{equation} -fulfils the variational principle +fulfils \manu{can be obtained from?} the variational principle as follows\cite{Gross_1988a} \begin{eqnarray}\label{eq:ens_energy} \E{}{\bw} = \min_{\hGam{\bw}} \Tr[\hGam{\bw} \hH], \end{eqnarray} -where $\hH = \hT + \hWee + \hVne$ contains the kinetic, electron-electron and nuclei-electron interaction potential operators, respectively, $\Tr$ denotes the trace and $\hGam{\bw}$ is a trial density matrix of the form +where $\hH = \hT + \hWee + \hVne$ contains the kinetic, +electron-electron and nuclei-electron interaction potential operators, +respectively, $\Tr$ denotes the trace and $\hGam{\bw}$ is a trial +density matrix operator of the form \begin{eqnarray} \hGam{\bw} = \sum_{I=0}^{\nEns - 1} \ew{I} \dyad*{\overline{\Psi}^{(I)}}, \end{eqnarray} where $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1}$ is a set of $\nEns$ orthonormal trial wave functions. The lower bound of Eq.~\eqref{eq:ens_energy} is reached when the set of wave functions correspond to the exact eigenstates of $\hH$, \ie, $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1} = \lbrace \Psi^{(I)} \rbrace_{0 \le I \le \nEns-1}$. -Multiplet degeneracies can be easily handled by assigning the same weight to the degenerate states. -One of the key feature of the GOK ensemble is that individual excitation energies are extracted from the ensemble energy via differentiation with respect to individual weights: -\begin{equation} +Multiplet degeneracies can be easily handled by assigning the same +weight to the degenerate states \cite{Gross_1988b}. +One of the key feature of the GOK ensemble is that individual excitation +energies can be extracted from the ensemble energy via differentiation with respect to individual weights: +\begin{equation}\label{eq:diff_Ew} \pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{}{(I)}. \end{equation} @@ -264,39 +288,137 @@ Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles al \end{equation} where $\vne(\br{})$ is the external potential and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional (the weight-dependent analog of the Hohenberg--Kohn universal functional for ensembles). -In the KS formulation, this functional is decomposed as +In the KS formulation, this functional can be decomposed as \begin{equation} \F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}] = \Tr[ \hgam{\bw} \hT ] + \Tr[ \hgam{\bw} \hWee ], \end{equation} +\manu{The above equation is wrong (the correlation is missing) and the +notations are ambiguous. I should also say that Tim does not like the +original separation into H and xc. I propose the following reformulation +to get everyone satisfied. I also reorganized the theory for clarity. +\begin{equation}\label{eq:FGOK_decomp} + \F{}{\bw}[\n{}{}] + = \Tr[ \hgamdens{\bw} \hT ]+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}], +\end{equation} +} where -$\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional, +\manuf{$\Tr[ \hgamdens{\bw} \hT ]=\Ts{\bw}[\n{}{}]$} is the noninteracting ensemble kinetic energy functional, \begin{equation} - \hgam{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}} + \hgam{\bw}[n] = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}[n]} \end{equation} -is the density matrix operator, $\lbrace \Det{I}{\bw} \rbrace_{0 \le I \le \nEns-1}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le \nOrb}$, and +is the \manuf{KS density-functional} density matrix operator, and $\lbrace +\Det{I}{\bw}[n] \rbrace_{0 \le I \le \nEns-1}$ are single-determinant +wave functions (or configuration state functions). \manuf{Their +dependence on the density +is determined from the ensemble density +constraint \begin{equation} -\label{eq:exc_def} -\begin{split} - \E{\Hxc}{\bw}[\n{}{}] - & = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}] - \\ - & = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' - + \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{} -\end{split} +\sum_{I=0}^{\nEns-1} \ew{I} n_{\Det{I}{\bw}[n]}(\br)=n(\br). \end{equation} -is the ensemble Hartree-exchange-correlation (Hxc) functional. -Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$. +Note that the original decomposition \cite{Gross_1988b} shown in Eq.~(\ref{eq:FGOK_decomp}), where the +conventional (weight-independent) Hartree functional +\beq +\E{\Ha}{}[\n{}{}]=\frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' +\eeq +is separated +from the (weight-dependent) exchange-correlation (xc) functional, is +formally exact. In practice, the use of such a decomposition might be +problematic as inserting an ensemble density into $\E{\Ha}{}[\n{}{}]$ +causes the infamous ghost-interaction error \cite{Gidopoulos_2002, +Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}. The latter should in +principle be removed by the exchange component of the ensemble xc +functional $\E{\xc}{\bw}[\n{}{}]\equiv +\E{\ex}{\bw}[\n{}{}]+\E{\co}{\bw}[\n{}{}]$, as readily seen from the +exact expression +\beq +\E{\ex}{\bw}[\n{}{}]=\sum_{I=0}^{\nEns-1} \ew{I}\bra{\Det{I}{\bw}[n]}\hat{W}_{\rm ee}\ket{\Det{I}{\bw}[n]} +-\E{\Ha}{}[\n{}{}]. +\eeq +The minimum in Eq.~(\ref{eq:Ew-GOK}) is reached when the density $n$ +equals the exact ensemble one +\beq\label{eq:nw} +n^{\bw}(\br)=\sum_{I=0}^{\nEns-1} +\ew{I}n_{\Psi_I}(\br). +\eeq + The minimizing density-functional KS +wavefunctions $\left\{\Det{I}{\bw}\left[n^{\bw}\right]\right\}_{0\leq +I\leq M-1}$ are constructed from (weight-dependent) KS orbitals +$\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le +\nOrb}$. The latters are determined by solving the ensemble KS equation +\begin{equation} +\label{eq:eKS} + \qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}), +\end{equation} +where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and +\begin{equation} + \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})} + = +\int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}' +%\fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})} ++ \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}. +\end{equation} +The ensemble density can be obtained directly (and exactly, if no +approximation is made) from those orbitals: +\beq +\n{}{\bw}(\br{})=\sum_{I=0}^{\nEns-1} \ew{I}\left(\sum_{p}^{\nOrb} +\ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2\right), +\eeq +where $\ON{p}{(I)}$ denotes the occupation of $\MO{p}{\bw}(\br{})$ in +the $I$th KS wave function $\Det{I}{\bw}\left[n^{\bw}\right]$. Turning +to the excitation energies, they can be extracted from the +density-functional ensemble as follows [see Eqs. ({\ref{eq:diff_Ew}}) +and ({\ref{eq:Ew-GOK}}) and Refs. +\cite{Gross_1988b,Deur_2019}]: +\beq +\label{eq:dEdw} +\Omega^{(I)}= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}}, +\eeq +where +\begin{equation} +\label{eq:KS-energy} + \Eps{I}{\bw} = \sum_{p}^{\nOrb} \ON{p}{(I)} \eps{p}{\bw} +\end{equation} +is the energy of the $I$th KS state. +}%%%%%% end manuf -From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019} +Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view. +Note that the individual KS densities +$\n{\Det{I}{\bw}\left[n^{\bw}\right]}{}(\br{})=\sum_{p}^{\nOrb} +\ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2$ do +not necessarily match the \textit{exact} (interacting) 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, these densities can still be extracted in principle exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020} + +\manuf{ +In the following, we will work at the (weight-dependent) LDA +level of approximation, \ie +\beq +\E{\xc}{\bw}[\n{}{}] +&\overset{\rm LDA}{\approx}& +\int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{} +\\ + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} +&\overset{\rm LDA}{\approx}& +\left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})). +\eeq +In the following, we adopt the usual decomposition, and write down the weight-dependent xc functional as +\begin{equation} + \e{\xc}{\ew{}}(\n{}{}) = \e{\ex}{\ew{}}(\n{}{}) + \e{\co}{\ew{}}(\n{}{}), +\end{equation} +where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively. +} + +%%%%%%%%%%%%%%%% +%%%%%%% Manu: stuff that I removed from the first version %%%%% +\iffalse%%%% \begin{equation} \begin{split} \label{eq:dEdw} \pdv{\E{}{\bw}}{\ew{I}} & = \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}}, \end{split} \end{equation} where @@ -308,16 +430,30 @@ where \n{\Det{I}{\bw}}{}(\br{}) & = \sum_{p}^{\nOrb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2 \end{align} are the ensemble and individual one-electron densities, respectively, + +and \begin{equation} -\label{eq:KS-energy} - \Eps{I}{\bw} = \sum_{p}^{\nOrb} \ON{p}{(I)} \eps{p}{\bw} +\label{eq:exc_def} +\begin{split} + \E{\Hxc}{\bw}[\n{}{}] + & = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}] + \\ + & = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' + + \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{} +\end{split} \end{equation} +is the ensemble Hartree-exchange-correlation (Hxc) functional. +Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ + is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$]. The latters are determined by solving the ensemble KS equation \begin{equation} \label{eq:eKS} \qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}), \end{equation} +built with KS orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le +\nOrb}$, + where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and \begin{equation} \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})} @@ -333,16 +469,8 @@ is the Hxc potential, with & = \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. -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, these densities can still be extracted in principle exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020} - -In the following, we adopt the usual decomposition, and write down the weight-dependent xc functional as -\begin{equation} - \e{\xc}{\ew{}}(\n{}{}) = \e{\ex}{\ew{}}(\n{}{}) + \e{\co}{\ew{}}(\n{}{}), -\end{equation} -where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively. - +%%%%% end stuff removed by Manu %%%%%% +\fi%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% COMPUTATIONAL DETAILS %%%