work under progress
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@ -24,17 +24,23 @@
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\newcommand{\la}{\lambda}
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\newcommand{\si}{\sigma}
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% operators
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\newcommand{\hHc}{\Hat{h}}
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\vne}{\Hat{v}_\text{ne}}
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\newcommand{\hWee}{\Hat{W}_\text{ee}}
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% functionals, potentials, densities, etc
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\newcommand{\eps}{\epsilon}
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\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
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\renewcommand{\v}[2]{E_\text{#1}^{#2}}
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\newcommand{\E}[2]{E_\text{#1}^{#2}}
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\newcommand{\bE}[2]{\bar{E}_\text{#1}^{#2}}
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\newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}}
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\newcommand{\bv}[2]{\bar{f}_\text{#1}^{#2}}
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\newcommand{\n}[1]{n^{#1}}
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\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
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\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
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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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@ -50,17 +56,17 @@
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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}^\text{c}}
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\newcommand{\bHc}{\bm{h}}
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\newcommand{\bF}[1]{\bm{F}^{#1}}
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\newcommand{\Ex}[1]{\Omega^{#1}}
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\newcommand{\E}[1]{E^{#1}}
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% elements
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\newcommand{\ew}[1]{w_{#1}}
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\newcommand{\eG}[1]{G_{#1}}
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\newcommand{\eS}[1]{S_{#1}}
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\newcommand{\eGamma}[2]{\Gamma_{#1}^{#2}}
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\newcommand{\eHc}[1]{H_{#1}^\text{c}}
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\newcommand{\hGamma}[2]{\Hat{\Gamma}_{#1}^{#2}}
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\newcommand{\eHc}[1]{h_{#1}}
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\newcommand{\eF}[2]{F_{#1}^{#2}}
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% Numbers
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@ -72,7 +78,6 @@
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\newcommand{\cMO}[2]{c_{#1}^{#2}}
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\newcommand{\AO}[1]{\chi_{#1}}
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% units
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\newcommand{\IneV}[1]{#1~eV}
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\newcommand{\InAU}[1]{#1~a.u.}
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@ -159,18 +164,7 @@ Atomic units are used throughout.
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\label{sec:geKS}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\alert{In eDFT, the ensemble energy
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\begin{equation}
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\E{\bw} = (1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)}
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\end{equation}
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is obtained variationally as follows,
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In analogy with ground-state generalized KS-DFT, we consider the following partitioning of the ensemble Levy-Lieb functional
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\begin{equation}
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F^{\bw}[n]=\underset{\hat{\Gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\Gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
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\end{equation}
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\begin{equation}
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F^{\mathbf{w}}[n]=\underset{\hat{\gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
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\end{equation}}
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\alert{Manu, you might want to add here details about the general KS-eDFT procedure.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{KS-eDFT for excited states}
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@ -182,16 +176,16 @@ In order to take into account both single and double excitations simultaneously,
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Generalization to a larger number of states is straightforward and is left for future work.
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By definition, the ensemble energy is
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\begin{equation}
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\E{\bw} = (1 - \ew{1} - \ew{2}) \E{(0)} + \ew{1} \E{(1)} + \ew{2} \E{(2)}.
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\E{}{\bw} = (1 - \ew{1} - \ew{2}) \E{}{(0)} + \ew{1} \E{}{(1)} + \ew{2} \E{}{(2)}.
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\end{equation}
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The $\E{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the single and double excitation, respectively.
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The $\E{}{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the single and double excitation, respectively.
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To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions:
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$0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.
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Note that, in order to extract individual energies from a single eKS calculation (see below), the weights must remain independent.
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Note that, in order to extract individual energies from a single KS-eDFT calculation (see below), the weights must remain independent.
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By construction, the excitation energies are
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\begin{equation}
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\label{eq:Ex}
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\Ex{(I)} = \pdv{\E{(I)}}{\ew{I}} = \E{(I)} - \E{(0)}.
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\Ex{(I)} = \pdv{\E{}{(I)}}{\ew{I}} = \E{}{(I)} - \E{}{(0)}.
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\end{equation}
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In the following, the orbitals $\MO{p}{\bw}(\br)$ are defined as linear combination of basis functions $\AO{\mu}(\br)$, such as
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\begin{equation}
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@ -214,15 +208,15 @@ The coefficients $\cMO{\mu p}{\bw}$ used to construct the density matrix $\bGamm
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\eF{\mu\nu}{\bw}
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= \eHc{\mu\nu} + \sum_{\la\si} \eGamma{\la\si}{\bw} \eG{\mu\nu\la\si}
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\\
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+ \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \AO{\mu}(\br) \AO{\nu}(\br) d\br,
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+ \int \left. \fdv{\bE{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \AO{\mu}(\br) \AO{\nu}(\br) d\br,
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\end{multline}
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which itself depends on $\bGamma{\bw}$.
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In Eq.~\eqref{eq:F}, $\bHc$ is the core Hamiltonian (including kinetic and electron-nucleus attraction terms), $\eG{\mu\nu\la\si} = (\mu\nu|\la\si) - (\mu\si|\la\nu)$,
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In Eq.~\eqref{eq:F}, $\hHc$ is the core Hamiltonian (including kinetic and electron-nucleus attraction terms), $\eG{\mu\nu\la\si} = (\mu\nu|\la\si) - (\mu\si|\la\nu)$,
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\begin{equation}
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(\mu\nu|\la\si) = \iint \frac{\AO\mu(\br_1) \AO\nu(\br_1) \AO\la(\br_2) \AO\si(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
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\end{equation}
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are two-electron repulsion integrals,
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$\v{c}{\bw}[\n{}(\br)] = \n{}(\br) \e{c}{\bw}[\n{}(\br)]$ and $\e{c}{\bw}[\n{}(\br)]$ is the weight-dependent correlation functional to be built in the present study.
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$\bE{Hxc}{\bw}[\n{}(\br)] = \n{}(\br) \be{Hxc}{\bw}[\n{}(\br)]$ and $\be{Hxc}{\bw}[\n{}(\br)]$ is the weight-dependent correlation functional to be built in the present study.
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The one-electron ensemble density is
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\begin{equation}
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\n{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br),
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@ -230,37 +224,37 @@ The one-electron ensemble density is
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with a similar expression for $\n{(I)}(\br)$, while the ensemble energy reads
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\begin{equation}
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\label{eq:Ew}
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\E{\bw}
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\E{}{\bw}
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= \Tr(\bGamma{\bw} \, \bHc)
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+ \frac{1}{2} \Tr(\bGamma{\bw} \, \bG \, \bGamma{\bw})
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% \\
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% + \int \e{c}{\bw}[\n{\bw}(\br)] \n{\bw}(\br) d\br.
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+ \int \v{c}{\bw}[\n{\bw}(\br)] d\br.
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+ \int \bE{Hxc}{\bw}[\n{\bw}(\br)] d\br.
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\end{equation}
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The self-consistent process described above is carried on until $\max \abs{\bF{\bw} \, \bGamma{\bw} \, \bS - \bS \, \bGamma{\bw} \, \bF{\bw}} < \tau$, where $\tau$ is a user-defined threshold and $\eS{\mu\nu} = \braket{\AO{\mu}}{\AO{\nu}}$ are elements of the overlap matrix $\bS$.
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\alert{Note that the weight-dependent energy given by Eq.~\eqref{eq:Ew} is polluted by the so-called ghost interaction. \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} which makes the ensemble energy non linear.
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Below, we propose a ghost-interaction correction in order to minimize this error.}
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Note that because the second term of the RHS of Eq.~\eqref{eq:Ew} is quadratic in $\bGamma{\bw}$, the weight-dependent energy contains the so-called ghost interaction which makes the ensemble energy non linear. \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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Below, we propose a ghost-interaction correction in order to minimize this error.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Extracting individual energies}
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\label{sec:ind_energy}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Following Deur and Fromager, \cite{Deur_2018b} it is possible to extract individual energies, $\E{(I)}$, from the ensemble energy [Eq.~\eqref{eq:Ew}] as follows:
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Following Deur and Fromager, \cite{Deur_2018b} it is possible to extract individual energies, $\E{}{(I)}$, from the ensemble energy [see Eq.~\eqref{eq:Ew}] as follows:
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\begin{multline}
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\E{(I)} = \Tr(\bGamma{(I)} \, \bHc) + \frac{1}{2} \Tr(\bGamma{(I)} \, \bG \, \bGamma{(I)})
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\E{}{(I)} = \Tr(\bGamma{(I)} \, \bHc) + \frac{1}{2} \Tr(\bGamma{(I)} \, \bG \, \bGamma{(I)})
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\\
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+ \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{(I)}(\br) d\br
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+ \LZ{c}{} + \DD{c}{(I)}.
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+ \int \left. \fdv{\bE{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{(I)}(\br) d\br
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+ \LZ{Hxc}{} + \DD{Hxc}{(I)}.
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\end{multline}
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Note that a \emph{single} eKS calculation is required to extract the three individual energies.
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Note that a \emph{single} KS-eDFT calculation is required to extract the three individual energies.
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The correlation part of the (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by
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\begin{align}
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\LZ{c}{} & = - \int \left. \fdv{\e{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br,
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\LZ{Hxc}{} & = - \int \left. \fdv{\be{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br,
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\\
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\DD{c}{(I)} & = \sum_{J>0} (\delta_{IJ} - \ew{J}) \int \left. \pdv{\e{c}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br) d\br.
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\DD{Hxc}{(I)} & = \sum_{J>0} (\delta_{IJ} - \ew{J}) \int \left. \pdv{\be{Hxc}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br) d\br.
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\end{align}
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Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}].
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The only remaining piece of information to define at this stage is the weight-dependent correlation functional $\e{c}{\bw}(\n{})$.
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The only remaining piece of information to define at this stage is the weight-dependent correlation functional $\be{Hxc}{\bw}(\n{})$.
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\alert{Mention LIM?}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -301,10 +295,21 @@ This is a necessary condition for being able to model derivative discontinuities
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\section{Density-functional approximations for ensembles}
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\label{sec:eDFA}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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We decompose the weight-dependent functional $\be{Hxc}{\bw}(\n{})$ as
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\begin{equation}
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\be{Hxc}{\bw}(\n{}) = \be{Hx}{\bw}(\n{}) + \be{c}{\bw}(\n{}),
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\end{equation}
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where $\be{Hx}{\bw}(\n{})$ is an Hartree-exchange functional designed to correct the ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} and $\be{c}{\bw}(\n{})$ is a weight-dependent correlation functional.
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The construction of these two functionals is described below.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent DFAs}
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\label{sec:wDFAs}
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\subsection{Ghost-interaction correction}
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\label{sec:GIC}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent correlation functional}
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\label{sec:Ec}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
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As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
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