revised theory
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@ -161,14 +161,14 @@ In the present article, we discuss the construction of first-rung (\textit{i.e.}
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%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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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.\cite{Casida_1995,Ulrich_2012,Loos_2020a}
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\titou{At a moderate computational cost} (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein).
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At a moderate computational cost (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein).
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\titou{Importantly, within the widely-used adiabatic approximation, setting up a TD-DFT calculation for a given system is an
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almost pain-free process from a user perspective as the only (yet
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essential) input variable is the choice of the
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ground-state exchange-correlation (xc) functional.}
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Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundations relie on the Runge-Gross theorem. \cite{Runge_1984}
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The Kohn-Sham (KS) \titou{formulation} of TD-DFT transfers the
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The Kohn-Sham (KS) formulation of TD-DFT transfers the
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complexity of the many-body problem to the xc functional thanks to a
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judicious mapping between a time-dependent non-interacting reference
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system and its interacting analog \titou{which have both
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@ -235,23 +235,19 @@ Unless otherwise stated, atomic units are used throughout.
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\section{Theory}
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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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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 $\bw = (\ew{1},\ldots,\ew{M-1})$, \ie, $\ew{0}=1-\sum_{I=1}^{\nEns-1} \ew{I}$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$.
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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 \manu{can be obtained from?} the variational principle
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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,
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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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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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@ -259,7 +255,7 @@ density matrix operator of the form
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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
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weight to the degenerate states \cite{Gross_1988b}.
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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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@ -274,54 +270,49 @@ Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles al
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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 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}
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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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= \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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\manuf{$\Tr[ \hgamdens{\bw} \hT ]=\Ts{\bw}[\n{}{}]$} is the noninteracting ensemble kinetic energy functional,
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$\Tr{ \hgamdens{\bw} \hT } =\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional,
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\begin{equation}
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\hgam{\bw}[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 \manuf{KS density-functional} density matrix operator, and $\lbrace
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is the 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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wave functions (or configuration state functions).
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Their dependence on the density is determined from the ensemble density constraint
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\begin{equation}
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\sum_{I=0}^{\nEns-1} \ew{I} n_{\Det{I}{\bw}[n]}(\br)=n(\br).
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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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Note that the original decomposition \cite{Gross_1988b} shown in Eq.~(\ref{eq:FGOK_decomp}), where the
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Note that the original decomposition \cite{Gross_1988b} shown in Eq.~\eqref{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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\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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causes the infamous ghost-interaction error. \cite{Gidopoulos_2002,Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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The latter should in principle be removed by the exchange component of the ensemble xc functional
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$\E{\xc}{\bw}[\n{}{}] \equiv \E{\ex}{\bw}[\n{}{}] + \E{\co}{\bw}[\n{}{}]$,
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as readily seen from the 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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\E{\ex}{\bw}[\n{}{}]
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= \sum_{I=0}^{\nEns-1} \ew{I}\mel{\Det{I}{\bw}[\n{}{}]}{\hWee}{\Det{I}{\bw}[\n{}{}]} - \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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The minimum in Eq.~\eqref{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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@ -341,7 +332,7 @@ result, the orbitals
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$\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le
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\nOrb}$ from which the 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 can be obtained by solving the following ensemble KS equation
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I\leq \nEns-1}$ are constructed can be obtained by solving the following 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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@ -355,7 +346,7 @@ where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and
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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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approximation is made) from these orbitals, \ie,
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\beq\label{eq:ens_KS_dens}
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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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@ -363,19 +354,19 @@ approximation is made) from those orbitals:
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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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density-functional ensemble as follows [see Eqs.~\eqref{eq:diff_Ew}
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and \eqref{eq:Ew-GOK} and Refs.~\onlinecite{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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\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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is the energy of the $I$th KS state.
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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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@ -383,14 +374,14 @@ $\n{\Det{I}{\bw}\left[n^{\bw}\right]}{}(\br{})=\sum_{p}^{\nOrb}
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not necessarily match the \textit{exact} (interacting) individual-state
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densities $n_{\Psi_I}(\br)$ 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
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exactly from the KS ensemble as shown by Fromager.
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\cite{Fromager_2020}.\\
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exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020}
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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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\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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@ -403,11 +394,8 @@ We will also adopt the usual decomposition, and write down the weight-dependent
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where $\e{\ex}{\bw{}}(\n{}{})$ and $\e{\co}{\bw{}}(\n{}{})$ are the
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weight-dependent density-functional exchange and correlation energies
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per particle, respectively.
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}%%%%%% end manuf
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The explicit construction of these functionals is discussed at length in Sec.~\ref{sec:res}.
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\manu{Maybe we should say a little bit more about how we will design
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such approximations, or just say the design of these functionals will be
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presented in the following...}
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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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@ -477,8 +465,7 @@ is the Hxc potential, with
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\section{Computational details}
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\label{sec:compdet}
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The self-consistent GOK-DFT calculations \manuf{[see Eqs.~(\ref{eq:eKS})
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and (\ref{eq:ens_KS_dens})]} have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented.
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The self-consistent GOK-DFT calculations [see Eqs.~\eqref{eq:eKS} and \eqref{eq:ens_KS_dens}] have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented.
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For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found.
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For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family of atomic basis sets. \cite{Dunning_1989,Kendall_1992,Woon_1994}
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Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988b,Lindh_2001}
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@ -490,7 +477,7 @@ Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a g
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it stands a little bit beyond the theory discussed previously. What you
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are looking at in the range $1/2\leq \ew{}\leq 1$ are, indeed, other
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stationary points (than the minimizing ones) of the density matrix
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operator functional in Eq.~(\ref{eq:min_KS_DM}). I would say that we
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operator functional in Eq.~\eqref{eq:min_KS_DM}. I would say that we
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look at these solutions for analysis purposes. I personally never looked
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(formally) at these solutions and their physical meaning. One should clearly
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mention that applying GOK-DFT in this range of weights would simply
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@ -541,6 +528,10 @@ As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the exc
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Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/2$.
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Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\ew{}$.
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\titou{The eDFT purist is going to surprised to see that we left out the singly-excited states $\sigma_g \sigma_u$ from the ensemble as this state is lower in energy than the doubly-excited state of configuration $1\sigma_u^2$.
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As we wish to stick with a restricted formalism, the single excitation is naturally left out of the ensemble.
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However, as a sanity check, we have tried to introduce the single excitations as well.}
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\begin{figure}
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\includegraphics[width=\linewidth]{Ew_H2}
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\caption{
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@ -762,7 +753,7 @@ As shown in Fig.~\ref{fig:Ew_H2}, the GIC-SeVWN5 is slightly less concave than i
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For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
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In particular, we report the excitation energies obtained with GOK-DFT
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in the zero-weight limit (\ie, $\ew{} = 0$) and for the equi-ensemble
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(\ie, $\ew{} = 1/2$). \manu{Maybe we should refer to Eq.~(\ref{eq:dEdw}) for clarity.}
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(\ie, $\ew{} = 1/2$). \manu{Maybe we should refer to Eq.~\eqref{eq:dEdw} for clarity.}
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For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016}
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a pragmatic way of getting weight-independent
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excitation energies defined as
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