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