making progress on the ensemble energies
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@ -40,7 +40,7 @@
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\hHc}{\Hat{h}}
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\bH}{\bm{H}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\hVext}{\Hat{V}_\text{ext}}
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\newcommand{\vext}{v_\text{ext}}
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\newcommand{\hWee}{\Hat{W}_\text{ee}}
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@ -54,6 +54,7 @@
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\newcommand{\kin}[2]{t_\text{#1}^{#2}}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\bE}[2]{\overline{E}_{#1}^{#2}}
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\newcommand{\tE}[2]{\widetilde{E}_{#1}^{#2}}
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\newcommand{\be}[2]{\overline{\epsilon}_{#1}^{#2}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\Cx}[1]{C_\text{x}^{#1}}
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@ -61,11 +62,6 @@
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% energies
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\Ecat}{E_\text{cat}}
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\newcommand{\Eneu}{E_\text{neu}}
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\newcommand{\Eani}{E_\text{ani}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\EFCI}{E_\text{FCI}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\LDA}{\text{LDA}}
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\newcommand{\eLDA}{\text{eLDA}}
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@ -77,14 +73,15 @@
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\newcommand{\xc}{\text{xc}}
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% matrices
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\newcommand{\br}{\bm{r}}
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\newcommand{\bw}{\bm{w}}
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\newcommand{\bG}{\bm{G}}
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\newcommand{\bS}{\bm{S}}
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\newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}}
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\newcommand{\bHc}{\bm{h}}
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\newcommand{\bF}[1]{\bm{F}^{#1}}
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\newcommand{\br}{\boldsymbol{r}}
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\newcommand{\bw}{\boldsymbol{w}}
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\newcommand{\bG}{\boldsymbol{G}}
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\newcommand{\bS}{\boldsymbol{S}}
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\newcommand{\bGamma}[1]{\boldsymbol{\Gamma}^{#1}}
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\newcommand{\bHc}{\boldsymbol{h}}
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\newcommand{\bF}[1]{\boldsymbol{F}^{#1}}
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\newcommand{\Ex}[2]{\Omega_{#1}^{#2}}
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\newcommand{\tEx}[2]{\widetilde{\Omega}_{#1}^{#2}}
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% elements
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\newcommand{\ew}[1]{w_{#1}}
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@ -231,7 +228,8 @@ and
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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{}'+ \int \e{\xc}{\bw}[\n{}{}(\br{})] \n{}{}(\br{}) d\br{}.
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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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are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively.
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@ -242,15 +240,35 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
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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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= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}},
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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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\end{equation}
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where $\Eps{I}{\bw} = \sum_{p}^{\Norb} \ON{p}{(I)} \eps{p}{\bw}$, $\eps{p}{\bw}$ is the $p$th KS orbital energy given by the ensemble KS equation
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where
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\begin{equation}
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\n{}{\bw}(\br{}) = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}(\br{})
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\end{equation}
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is the ensemble density,
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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 weight-dependent KS energy, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ ($\ON{p}{(I)}$ being its occupancy for the state $I$) given by 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{}{}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
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\end{equation}
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where $\hHc(\br{}) = -\frac{\nabla^2}{2} + \vext(\br{})$, $\MO{p}{\bw}(\br{})$ is a KS orbital, $\ON{p}{(I)}$ its occupancy for the state $I$, and $\n{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}$ is the ensemble density.
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Equation \eqref{eq:dEdw} is our working equation for computing excitation energies.
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where $\hHc(\br{}) = -\nabla^2/2 + \vext(\br{})$, and
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\begin{equation}
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\begin{split}
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\fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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& = \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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\\
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& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}'
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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{split}
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\end{equation}
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is the Hxc potential.
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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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%%%%%%%%%%%%%%%%%%
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%%% FUNCTIONAL %%%
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%%%%%%%%%%%%%%%%%%
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@ -324,12 +342,9 @@ and we then obtain
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We can now combine these two exchange functionals to create a weight-dependent exchange functional for a two-state ensemble
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\begin{equation}
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\label{eq:exw}
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\begin{split}
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\e{\ex}{\ew{}}(\n{}{})
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& = (1-\ew{}) \e{\ex}{(0)}(\n{}{}) + \ew{} \e{\ex}{(1)}(\n{}{})
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\\
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& = \Cx{\ew{}} \n{}{1/3}
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\end{split}
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= (1-\ew{}) \e{\ex}{(0)}(\n{}{}) + \ew{} \e{\ex}{(1)}(\n{}{})
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= \Cx{\ew{}} \n{}{1/3}
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\end{equation}
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with
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\begin{equation}
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@ -487,6 +502,9 @@ In the case of a homogeneous system (or equivalently within the LDA), substituti
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\label{sec:res}
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Here, we consider as testing ground the minimal-basis \ce{H2} molecule.
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We select STO-3G as minimal basis, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr).
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This minimal-basis example is quite pedagogical as the molecular orbitals are fixed by symmetry.
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Therefore, there is no density-driven error and the only error that we are going to see is the functional-driven error (and this is what we want to study).
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The bonding and antibonding orbitals of the \ce{H2} molecule are given by
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\begin{subequations}
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\begin{align}
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@ -519,14 +537,14 @@ with
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\end{subequations}
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Note that, in the HF case, there is no self-interaction error as $\eJ{pp} = \eK{pp}$.
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We also define the HF excitation energy as $\Ex{\HF}{(1)} = \E{\HF}{(1)} - \E{\HF}{(0)}$.
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The HF orbital energies are
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\begin{subequations}
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\begin{align}
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\eps{1}{\HF} & = \eHc{1} + 2\eJ{11} - \eK{11},
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\\
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\eps{2}{\HF} & = \eHc{2} + 2\eJ{12} - \eK{12}.
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\end{align}
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\end{subequations}
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%The HF orbital energies are
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%\begin{subequations}
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%\begin{align}
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% \eps{1}{\HF} & = \eHc{1} + 2\eJ{11} - \eK{11},
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% \\
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% \eps{2}{\HF} & = \eHc{2} + 2\eJ{12} - \eK{12}.
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%\end{align}
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%\end{subequations}
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As reference results, we consider CID (configuration interaction with doubles) computed in the same (minimal) basis set.
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The CID energies of the ground state and doubly-excited states are provided by the eigenvalues of the following CID matrix:
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@ -568,14 +586,17 @@ with
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\n{}{(1)}(\br{}) & = 2 \MO{2}{2}(\br{}),
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\end{align}
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Note that, contrary to the HF case, self-interaction is present in LDA.
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The KS orbital energies are given by
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\begin{subequations}
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\begin{align}
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\eps{1}{\LDA} & = \eHc{1} + 2\eJ{11} + \ldots,
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\\
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\eps{2}{\LDA} & = \eHc{2} + 2\eJ{12} + \ldots.
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\end{align}
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\end{subequations}
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%The KS orbital energies are given by
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%\begin{subequations}
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%\begin{align}
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% \eps{1}{\LDA}
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% & = \eHc{1} + 2\eJ{11}
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% + \frac{1}{2} \int \left. \fdv{\E{\xc}{\LDA}[\n{}{}]}{\n{}{}} \right|_{\n{}{} = \n{}{(0)}(\br{})} \n{}{(0)}(\br{}) d\br{},
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% \\
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% \eps{2}{\LDA} & = \eHc{2} + 2\eJ{12}
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% + \frac{1}{2} \int \left. \fdv{\E{\xc}{\LDA}[\n{}{}]}{\n{}{}} \right|_{\n{}{} = \n{}{(0)}(\br{})} \n{}{(1)}(\br{}) d\br{}.
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%\end{align}
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%\end{subequations}
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At the eLDA, we have
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\begin{subequations}
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@ -599,13 +620,13 @@ Interestingly here, there is a strong connection between the LDA and eLDA excita
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\end{split}
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\end{equation}
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The KS orbital energies are given by
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\begin{subequations}
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\begin{align}
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\eps{1}{\eLDA} & = \eHc{1} + 2\eJ{11} + \ldots,
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\\
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\eps{2}{\eLDA} & = \eHc{2} + 2\eJ{12} + \ldots.
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\end{align}
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\end{subequations}
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%\begin{subequations}
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%\begin{align}
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% \eps{1}{\eLDA} & = \eHc{1} + 2\eJ{11} + \ldots,
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% \\
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% \eps{2}{\eLDA} & = \eHc{2} + 2\eJ{12} + \ldots.
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%\end{align}
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%\end{subequations}
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These equations can be combined to define three ensemble energies
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@ -629,13 +650,13 @@ Similar energies than the ones given in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} a
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\n{}{\ew{}} = (1-\ew{}) \n{}{(0)} + \ew{} \n{}{(1)}.
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\end{equation}
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(This is what one would do in practice, \ie, by performing a KS ensemble calculation.)
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We will label these energies as $\bE{}{\ew{}}$ to avoid confusion.
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We will label these energies as $\tE{}{\ew{}}$ to avoid confusion.
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\begin{widetext}
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For HF, we have
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\begin{equation}
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\label{eq:bEwHF}
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\begin{split}
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\bE{\HF}{\ew{}}
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\tE{\HF}{\ew{}}
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& = \titou{\int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{}}
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+ \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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\\
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@ -648,7 +669,7 @@ In the case of the LDA, it reads
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\begin{equation}
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\label{eq:bEwLDA}
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\begin{split}
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\bE{\LDA}{\ew{}}
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\tE{\LDA}{\ew{}}
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& = \titou{\int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{}}
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+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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+ \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{}
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@ -665,7 +686,7 @@ For eLDA, the ensemble energy can be decomposed as
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\begin{equation}
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\label{eq:bEweLDA}
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\begin{split}
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\bE{\eLDA}{\ew{}}
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\tE{\eLDA}{\ew{}}
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& = \titou{\int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{}}
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+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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+ \int \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{}
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@ -694,13 +715,64 @@ This would be, for example, the case with the exact xc functional.
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Extracting excitation energies from Eqs.~\eqref{eq:bEwHF}, \eqref{eq:bEwLDA} and \eqref{eq:bEweLDA} is more tricky.
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To do so, we will employ Eq.~\eqref{eq:dEdw}.
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The derivative discontinuity, modelled by the last term of the RHS of Eq.~\eqref{eq:dEdw} and only non-zero in the case of an explicitly weight-dependent functional, is straightforward to compute in our case [see Eq.~\eqref{eq:dexcdw}].
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The two first terms are
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\begin{align}
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\Eps{0}{\ew{}} & = 2 \eHc{1} + \ldots,
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\\
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\Eps{1}{\ew{}} & = 2 \eHc{2} + \ldots.
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\Eps{0}{\ew{}} & = 2(1-\ew{}) \eps{1}{\ew{}},
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&
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\Eps{1}{\ew{}} & = 2 \ew{} \eps{2}{\ew{}},
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\end{align}
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where the HF, LDA and eLDA weight-dependent orbital energies are
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\begin{subequations}
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\begin{align}
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\eps{1}{\ew{},\HF}
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& = \eHc{1} + (1-\ew{})(2\eJ{11} - \eK{11}) + \ew{}(2\eJ{12} - \eK{12}),
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\\
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\eps{2}{\ew{},\HF}
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& = \eHc{2} + (1-\ew{})(2\eJ{12} - \eK{12}) + \ew{}(2\eJ{22} - \eK{22}),
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\end{align}
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\end{subequations}
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\begin{subequations}
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\begin{align}
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\begin{split}
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\eps{1}{\ew{},\LDA}
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& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
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\\
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& + \frac{1}{2} \int \left. \fdv{\E{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(0)}(\br{}) d\br{},
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\end{split}
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\\
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\begin{split}
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\eps{2}{\ew{},\LDA}
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& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22}
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\\
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& + \frac{1}{2} \int \left. \fdv{\E{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(1)}(\br{}) d\br{},
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\end{split}
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\end{align}
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\end{subequations}
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\begin{subequations}
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\begin{align}
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\begin{split}
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\eps{1}{\ew{},\eLDA}
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& = \eHc{1} + (1-\ew{})(2\eJ{11} - \eK{11}) + \ew{}(2\eJ{12} - \eK{12})
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\\
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& + \frac{1}{2} \int \left. \fdv{\bE{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(0)}(\br{}) d\br{},
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\end{split}
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\\
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\begin{split}
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\eps{2}{\ew{},\eLDA} & = \eHc{2} + (1-\ew{})(2\eJ{12} - \eK{12}) + \ew{}(2\eJ{22} - \eK{22})
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\\
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& + \frac{1}{2} \int \left. \fdv{\bE{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{(1)}(\br{}) d\br{},
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\end{split}
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\end{align}
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\end{subequations}
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respectively.
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The derivative discontinuity is modelled by the last term of the RHS of Eq.~\eqref{eq:dEdw}.
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Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}].
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%%%%%%%%%%%%%%%%%%
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%%% CONCLUSION %%%
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