Manu: first global polishing done
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Emmanuel Fromager
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@ -559,7 +559,7 @@ c}\left[n_{\bm\gamma^{\bw}}\right]
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\eeq
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The minimizing ensemble density matrix fulfills the following
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stationarity condition
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\beq
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\beq\label{eq:commut_F_AO}
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{\bm F}^{\bw\sigma}{\bm \Gamma}^{\bw\sigma}{\bm S}={\bm S}{\bm
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\Gamma}^{\bw\sigma}{\bm F}^{\bw\sigma},
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\eeq
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@ -704,7 +704,7 @@ Note that this approximation, where the ensemble density matrix is
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optimized from a non-local exchange potential [rather than a local one,
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as expected from Eq.~(\ref{eq:var_ener_gokdft})] is applicable to real
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(three-dimension) systems. As readily seen from
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Eq.~(\ref{eq:eHF-dens_mat_func}), {\it ghost interactions}~\cite{}
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Eq.~(\ref{eq:eHF-dens_mat_func}), {\it ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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and curvature~\cite{} will be
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introduced in the Hx energy:
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\beq
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@ -760,6 +760,8 @@ We decompose the weight-dependent functional 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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\manu{Well, at the end, we only develop functionals for the correlation
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part. Should be updated.}\\
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where $\be{Hx}{\bw}(\n{}{})$ is a weight-dependent Hartree-exchange functional designed to correct the ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} [see Subsec.~\ref{sec:GIC}] and $\be{c}{\bw}(\n{}{})$ is a weight-dependent correlation functional [see Subsec.~\ref{sec:Ec}].
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The construction of these two functionals is described below.
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Note that, because we consider strict 1D systems, one cannot decompose further the Hartree-exchange contribution as each component diverges independently but their sum is finite. \cite{Astrakharchik_2011, Lee_2011a, Loos_2012, Loos_2013, Loos_2013a}
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@ -781,33 +783,6 @@ All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$
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We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Ghost-interaction correction}
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\label{sec:GIC}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\manu{I guess this subsection should be removed(?)}
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\color{purple}
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The GIC weight-dependent Hartree-exchange functional is defined as
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\begin{multline}
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\be{Hx}{\bw}(\n{}{\bw}) = (1-\sum_{I>0} \ew{I}) \be{Hx}{}(\n{}{(0)}) + \sum_{I>0} \ew{I} \be{Hx}{}(\n{}{(I)})
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\\
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- \be{Hx}{(I)}(\n{}{\bw}),
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\end{multline}
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where
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\begin{equation}
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\be{Hx}{}(\n{}{}) = \iint \frac{\n{}{}(\br_1) \n{}{}(\br_2) - \n{}{}(\br_1,\br_2)^2}{r_{12}} d\br_1 d\br_2,
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\end{equation}
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and
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\begin{equation}
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\n{}{(I)}(\omega) = (\pi R)^{-1} \cos[(I+1) \omega/2]
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\end{equation}
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is the first-order density matrix with $\omega$ the interelectronic angle.
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It yields
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\begin{equation}
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\be{Hx}{}(\n{}{}) = \n{}{} \qty[ a_1 \ew{1} (\ew{1} - 1) + a_2 \ew{1} \ew{2} + a_3 \ew{2} (\ew{2} - 1)],
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\end{equation}
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with $a_1 = 2 \ln 2 - 1/3$, $a_2 = 8/3$ and $a_3 = 32/15$.
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\color{black}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent correlation functional}
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\label{sec:Ec}
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@ -895,108 +870,6 @@ Finally, we note that, by construction,
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\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
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%%%%%%%%%%%%%%%%
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\color{purple}
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\section{Implementation}
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\manu{I think that this section can be removed (especially Sec.~\ref{sec:E_I}). Many points
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discussed in Sec.~\ref{sec:KS-eDFT} are now mentioned in the theory
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section. If we want to keep some material of Sec.~\ref{sec:KS-eDFT}, it
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should be moved to
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Secs.~\ref{subsec:gokdft} or~\ref{subsec:approx} (or maybe Sec.~\ref{sec:compdetails}).}
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%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{KS-eDFT for excited states}
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\label{sec:KS-eDFT}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here, we explain how to perform a self-consistent KS calculation for ensembles (KS-eDFT) in the context of excited states.
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In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including:
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(i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$).
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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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\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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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 KS-eDFT calculation [see Subsec.~\ref{sec:E_I}], 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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\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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\MO{p}{\bw}(\br) = \sum_{\mu=1}^{\Nbas} \cMO{\mu p}{\bw} \, \AO{\mu}(\br).
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\end{equation}
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Within the self-consistent KS-eDFT process, one is looking for the following weight-dependent density matrix:
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\begin{equation}
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\label{eq:Gamma}
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\bGamma{\bw} = (1-\ew{1}-\ew{2}) \bGamma{(0)} - \ew{1} \bGamma{(1)} - \ew{2} \bGamma{(2)},
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\end{equation}
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where $\bw = (\ew{1},\ew{2})$ and $\bGamma{(I)}$ is the $I$th-state density matrix with elements
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\begin{equation}
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\label{eq:eGamma}
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\eGamma{\mu\nu}{(I)} = \sum_{i=1}^{\Nel-I} \cMO{\mu i}{\bw} \cMO{\nu i}{\bw} + \sum_{a=\Nel+1}^{\Nel+I} \cMO{\mu a}{\bw} \cMO{\nu a}{\bw}.
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\end{equation}
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The coefficients $\cMO{\mu p}{\bw}$ used to construct the density matrix $\bGamma{\bw}$ in Eq.~\eqref{eq:Gamma} are obtained by diagonalizing the following Fock matrix
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\begin{multline}
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\label{eq:F}
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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{\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}, $\hHc$ 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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$\bE{Hxc}{\bw}[\n{}{}(\br)] = \n{}{}(\br) \be{Hxc}{\bw}[\n{}{}(\br)]$ and $\be{Hxc}{\bw}[\n{}{}(\br)]$ is the weight-dependent Hartree-exchange-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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\end{equation}
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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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= \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 \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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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 (GIC) in order to minimize this error.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Extracting individual energies}
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\label{sec:E_I}
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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 [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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\\
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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} KS-eDFT calculation is required to extract the three individual energies.
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\alert{Mention LIM?}
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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{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{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 Hartree-exchange-correlation functional $\be{Hxc}{\bw}(\n{}{})$.
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\color{black}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}\label{sec:compdetails}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -1025,11 +898,24 @@ To get reference excitation energies for both the single and double excitations,
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For the single excitations, we have also performed time-dependent HF (TDHF), configuration interaction singles (CIS) and TDLDA calculations. \cite{Dreuw_2005}
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For TDLDA, the validity of the Tamm-Dancoff approximation (TDA) has been also tested.
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Concerning the eKS calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
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Concerning the eKS calculations, two sets of weight have been tested:
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the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or
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state-averaged) limit where $\bw = (1/3,1/3)$.\\
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\manu{might be re-used}
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\color{purple}
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The self-consistent process described above is carried on until $\max
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\abs{\bF{\bw} \, \bGamma{\bw} \, \bS - \bS \, \bGamma{\bw} \, \bF{\bw}}
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< \delta$ [see Eq.~(\ref{eq:commut_F_AO})], where $\delta$ 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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\color{black}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results and discussion}\label{sec:results}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\manu{might be re-used}
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\color{purple}
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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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\color{black}
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In Fig.~\ref{fig:EvsL}, we report the error (in \%) in excitation energies (compared to FCI) for various methods and box sizes in the case of 5-boxium (i.e., $\Nel = 5$).
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Similar graphs are obtained for the other $\Nel$ values and they can be found --- alongside the numerical data associated with each method --- in the {\SI}.
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In the weakly correlated regime (i.e., small $L$), all methods provide accurate estimates of the excitation energies.
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