starting clean up
This commit is contained in:
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@ -24,10 +24,14 @@
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\newcommand{\la}{\lambda}
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\newcommand{\si}{\sigma}
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% numbers
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\newcommand{\nEl}{N}
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% operators
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\newcommand{\hHc}{\Hat{h}}
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\hh}{\Hat{h}}
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hVext}{\Hat{V}_\text{ext}}
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\newcommand{\vne}{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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@ -51,13 +55,13 @@
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\newcommand{\EFCI}{E_\text{FCI}}
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% matrices/operator
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\newcommand{\br}{\bm{r}}
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\newcommand{\br}[1]{\bm{r}_{#1}}
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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{\opGamma}[1]{\hat{\Gamma}^{#1}}
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\newcommand{\bHc}{\bm{h}}
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\newcommand{\bh}{\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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@ -75,7 +79,8 @@
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\newcommand{\Nel}{N}
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\newcommand{\Nbas}{K}
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% Ao and MO basis
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% AO and MO basis
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\newcommand{\Det}[1]{\Phi^{#1}}
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\newcommand{\MO}[2]{\phi_{#1}^{#2}}
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\newcommand{\cMO}[2]{c_{#1}^{#2}}
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\newcommand{\AO}[1]{\chi_{#1}}
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@ -93,8 +98,8 @@
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%%% added by Manu %%%
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\newcommand{\manu}[1]{{\textcolor{darkgreen}{ Manu: #1 }} }
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\newcommand{\beq}{\begin{eqnarray}}
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\newcommand{\eeq}{\end{eqnarray}}
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\newcommand{\beq}{\begin{equation}}
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\newcommand{\eeq}{\end{equation}}
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\newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
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\newcommand{\bmg}{\bm{\Gamma}} % orbital rotation vector
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\newcommand{\bxi}{\bm{\xi}}
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@ -129,8 +134,6 @@ Their accuracy is illustrated by computing single and double excitations in one-
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\manu{Sections that might be (partially) removed are in
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{\color{purple}{purple}}. See my comments in the text.}
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -151,17 +154,17 @@ As a result, double electronic excitations are completely absent from the TDDFT
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When affordable (i.e., for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above.
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The basic idea is to describe a finite ensemble of states (ground and excited) altogether, i.e., with the same set of orbitals.
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Interestingly, a similar approach exists in DFT.
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Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988, Gross_1988a, Oliveira_1988} and is a generalization of Theophilou's variational principle for equi-ensembles. \cite{Theophilou_1979}
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Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988a, Oliveira_1988, Gross_1988b} and is a generalization of Theophilou's variational principle for equi-ensembles. \cite{Theophilou_1979}
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In eDFT, the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest.
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This weight dependence of the xc functional plays a crucial role in the calculation of excitation energies.
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It actually accounts for the infamous derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
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\alert{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
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%\alert{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
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Despite its formal beauty and the fact that eDFT can in principle tackle near-degenerate situations and multiple excitations, it has not been given much attention until recently. \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018,Gould_2019,Sagredo_2018,Ayers_2018,Deur_2018a,Deur_2018b,Kraisler_2013, Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Senjean_2018,Smith_2016}
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The main reason is simply the absence of density-functional approximations (DFAs) for ensembles in the literature.
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Recent works on this topic are still fundamental and exploratory, as they rely either on simple (but nontrivial) models like the Hubbard dimer \cite{Carrascal_2015,Deur_2017,Deur_2018a,Deur_2018b,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018} or on atoms for which highly accurate or exact-exchange-only calculations have been performed. \cite{Yang_2014,Yang_2017}
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In both cases, the key problem, namely the design of weight-dependent DFAs for ensembles (eDFAs), remains open.
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A first step towards this goal is presented in this article with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
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\alert{Mention WIDFA?}
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%\alert{Mention WIDFA?}
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In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
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In other words, the Coulomb interaction used in this work describes particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
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Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires. \cite{Meyer_2009, Deshpande_2010}
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@ -177,161 +180,122 @@ Atomic units are used throughout.
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\subsection{GOK-DFT}\label{subsec:gokdft}
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The GOK ensemble energy~\cite{} is defined as follows:
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The GOK ensemble energy~\cite{Gross_1988a,Oliveira_1988,Gross_1988b} is defined as follows:
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\beq
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E^{\bw}=\sum_{K\geq0}w_KE^{(K)},
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\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
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\eeq
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where the $K$th energy level $E^{(K)}$ [$K=0$ refers to the ground state] is the eigenvalue of the electronic Hamiltonian $\hat{H}=\hat{h}
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%\sum^N_{i=1}\hat{h}(i)
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+\hat{W}_{\rm ee}$, where $\hat{h}\equiv \sum^N_{i=1}\left(-\frac{1}{2}\nabla_{\br_i}^2+v_{\rm ne}(\br_i)\right)$ is the one-electron operator describing kinetic and nuclear attraction energies, and $\hat{W}_{\rm ee}$ is the electron repulsion operator.
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The (positive) ensemble weights $w_K$ decrease with increasing index $K$. They are normalized, i.e.
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where the $K$th energy level $\E{}{(K)}$ [$K=0$ refers to the ground state] is the eigenvalue of the electronic Hamiltonian $\hH = \hh + \hWee$, where
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\beq
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w_0=1-\sum_{K>0}w_K,
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\hh = \sum_{i=1}^\nEl \qty[ -\frac{1}{2} \nabla_{\br{i}}^2 + \vne(\br{i}) ]
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\eeq
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so that only the weights $\bw\equiv\left(w_1,w_2,\ldots w_K,\ldots\right)$ assigned to the excited states can vary independently.
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is the one-electron operator describing kinetic and nuclear attraction energies, and $\hat{W}_{\rm ee}$ is the electron repulsion operator.
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The (positive) ensemble weights $w_K$ decrease with increasing index $K$. They are normalized, i.e.,
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\beq
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\ew{0} = 1 - \sum_{K>0} \ew{K},
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\eeq
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so that only the weights $\bw \equiv \qty( \ew{1}, \ew{2}, \ldots, \ew{K}, \ldots )$ assigned to the excited states can vary independently.
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For simplicity we will assume in the following that the energies are not degenerate. Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states~\cite{}. In GOK-DFT, the ensemble energy is determined variationally as follows:
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\beq\label{eq:var_ener_gokdft}
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E^{{\bw}}=
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\underset{\opGamma{{\bw}}}{\rm min}\left\{
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{\rm
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Tr}\left[\opGamma{{\bw}}\hat{h}\right]
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+
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{E}^{{\bw}}_{\rm
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Hx}\left[n_{\opGamma{\bw}}\right]
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+
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{E}^{{\bw}}_{\rm
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c}\left[n_{\opGamma{\bw}}\right]
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\right\},
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\E{}{\bw}
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= \min_{\opGamma{{\bw}}}\qty{ \Tr[\opGamma{{\bw}}\hh]
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+ \E{Hx}{\bw} \qty[\n{\opGamma{\bw}}{}]
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+ \E{c}{\bw} \qty[\n{\opGamma{\bw}}{}]
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},
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\eeq
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where ${\rm
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Tr}$ denotes the trace and the trial ensemble density matrix operator reads
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where $\Tr$ denotes the trace and the trial ensemble density matrix operator reads
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\beq
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\opGamma{{\bw}}=\sum_{K\geq 0}w_K\ket{\Phi^{(K)}}\bra{\Phi^{(K)}}.
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\opGamma{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}.
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\eeq
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The determinants (or configuration state functions) $\Phi^{(K)}$ are all constructed from the same set of (ensemble Kohn--Sham) orbitals that is optimized variationally and the trial ensemble density is simply the weighted sum of the individual densities:
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\beq\label{eq:KS_ens_density}
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n_{\opGamma{\bw}}(\br)=\sum_{K\geq 0}w_Kn_{\Phi^{(K)}}(\br).
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\n{\opGamma{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Phi^{(K)}}{}(\br{}).
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\eeq
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As readily seen from Eq.~(\ref{eq:var_ener_gokdft}), both Hartree-exchange and
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correlation energies are described with density functionals that are {\it weight-dependent}. We focus here on the (exact) Hx part which is defined as follows:
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As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange and correlation energies are described with density functionals that are \textit{weight dependent}.
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We focus here on the (exact) Hx part which is defined as follows:
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\beq\label{eq:exact_ens_Hx}
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{E}^{{\bw}}_{\rm
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Hx}[n]=\sum_{K\geq 0}w_K\bra{\Phi^{(K)}[n]}\hat{W}_{\rm
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ee}\ket{\Phi^{(K)}[n]}
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\E{Hx}{\bw}[\n{}{}]=\sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}[\n{}{}]}{\hWee}{\Det{(K)}[\n{}{}]}
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\eeq
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where the KS wavefunctions fulfill the ensemble density constraint
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\beq
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\sum_{K\geq 0}w_Kn_{\Phi^{(K)}[n]}(\br)=n(\br).
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\sum_{K\geq 0} \ew{K} \n{\Det{(K)}[\n{}{}]}{}(\br{})=n(\br{}).
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\eeq
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The (approximate) description of the correlation part is discussed in Sec.~\ref{sec:eDFA}.\\
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In practice, one is not much interested in ensemble energies but rather in excitation energies or individual energy levels (for geometry optimizations, for example). The latter can be extracted exactly as follows~\cite{}:
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\beq\label{eq:indiv_ener_from_ens}
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E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w_K\right)\dfrac{\partial
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E^{{\bw}}}{\partial w_K},
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\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \pdv{\E{}{\bw}}{\ew{K}},
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\eeq
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where, according to the {\it variational} ensemble energy expression of Eq.~(\ref{eq:var_ener_gokdft}), the derivative in $w_K$ can be evaluated from the minimizing KS wavefunctions $\Phi^{(K)}=\Phi^{(K),\bw}$ as follows:
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where, according to the {\it variational} ensemble energy expression of Eq.~\eqref{eq:var_ener_gokdft}, the derivative in $w_K$ can be evaluated from the minimizing KS wavefunctions $\Det{(K)} = \Det{(K),\bw}$ as follows:
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\beq\label{eq:deriv_Ew_wk}
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&&\dfrac{\partial
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E^{{\bw}}}{\partial w_K}=\bra{\Phi^{(K)}}\hat{h}\ket{\Phi^{(K)}}-\bra{\Phi^{(0)}}\hat{h}\ket{\Phi^{(0)}}
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\nonumber\\
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&&+\Bigg[\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
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Hx}\left[n\right]}{\delta
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n({\br})}\left(n_{\Phi^{(K)}}(\br)-n_{\Phi^{(0)}}(\br)\right)
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%\nonumber\\
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%&&
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+
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%\left.
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\dfrac{\partial {E}^{{\bw}}_{\rm
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Hx}\left[n\right]}{\partial w_K}
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%\right|_{n=n_{\opGamma{\bw}}}
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\nonumber\\
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&&+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
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c}\left[n\right]}{\delta
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n({\br})}\left(n_{\Phi^{(K)}}(\br)-n_{\Phi^{(0)}}(\br)\right)
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%\nonumber\\
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%&&
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+
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%\left.
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\dfrac{\partial {E}^{{\bw}}_{\rm
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c}\left[n\right]}{\partial w_K}
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%\right|
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\Bigg]_{n=n_{\opGamma{\bw}}}
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.
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\nonumber\\
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\begin{split}
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\pdv{\E{}{\bw}}{\ew{K}}
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& = \mel*{\Det{(K)}}{\hh}{\Det{(K)}}-\mel*{\Det{(0)}}{\hh}{\Det{(0)}}
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\\
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& + \Bigg\{\int d\br{}\,\fdv{\E{Hx}{\bw}[\n{}{}]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
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+ \pdv{\E{Hx}{\bw} [\n{}{}]}{\ew{K}}
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\\
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& + \int d\br{} \fdv{\E{c}{\bw}[n]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
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+ \pdv{\E{c}{\bw}[n]}{\ew{K}}
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\Bigg\}_{\n{}{} = \n{\opGamma{\bw}}{}}.
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\end{split}
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\eeq
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The Hx contribution to Eq.~(\ref{eq:deriv_Ew_wk}) can be rewritten as follows:
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The Hx contribution to Eq.~\eqref{eq:deriv_Ew_wk} can be rewritten as follows:
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\beq\label{eq:_deriv_wk_Hx}
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\left.\dfrac{\partial
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}{\partial \xi_K}
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\left({E}^{{\bxi}}_{\rm
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Hx}\left[n^{\bxi,\bxi}\right]
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-
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{E}^{{\bw}}_{\rm
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Hx}\left[n^{\bw,\bxi}\right]
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\right)\right|_{\bxi=\bw},
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\left.
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\pdv{}{\xi_K} \qty(\E{Hx}{\bxi} [\n{}{\bxi,\bxi}]
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- \E{Hx}{\bw}[\n{}{\bw,\bxi}] )
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\right|_{\bxi=\bw},
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\eeq
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where $\bxi \equiv (\xi_1,\xi_2,\ldots,\xi_K,\ldots)$ and
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\beq
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n^{\bw,\bxi}(\br)=\sum_{K\geq 0}w_Kn_{\Phi^{(K),\bxi}}(\br).
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\n{}{\bw,\bxi}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K),\bxi}}{}(\br{}).
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\eeq
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Since, for given ensemble weight $\bw$ and $\bxi$ values, the ensemble densities $n^{\bxi,\bxi}$ and $n^{\bw,\bxi}$ are generated from the {\it same} KS potential (which is unique up to a constant), it comes
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from the exact expression in Eq.~(\ref{eq:exact_ens_Hx}) that
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Since, for given ensemble weight $\bw$ and $\bxi$ values, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes
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from the exact expression in Eq.~\ref{eq:exact_ens_Hx} that
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\beq
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{E}^{{\bxi}}_{\rm
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Hx}\left[n^{\bxi,\bxi}\right]=\sum_{K\geq 0}\xi_K\bra{\Phi^{(K),\bxi}}\hat{W}_{\rm
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ee}\ket{\Phi^{(K),\bxi}}
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\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}
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\eeq
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with $\xi_0=1-\sum_{K>0}\xi_K$
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and
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\beq
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{E}^{{\bw}}_{\rm
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Hx}\left[n^{\bw,\bxi}\right]=\sum_{K\geq 0}w_K\bra{\Phi^{(K),\bxi}}\hat{W}_{\rm
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ee}\ket{\Phi^{(K),\bxi}},
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\E{Hx}{\bw}[\n{}{\bw,\bxi}] = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
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\eeq
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thus leading, according to Eqs.~(\ref{eq:deriv_Ew_wk}) and (\ref{eq:_deriv_wk_Hx}), to the simplified expression
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thus leading, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_Hx}, to the simplified expression
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\beq\label{eq:deriv_Ew_wk_simplified}
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&&\dfrac{\partial
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E^{{\bw}}}{\partial w_K}=\bra{\Phi^{(K)}}\hat{H}\ket{\Phi^{(K)}}-\bra{\Phi^{(0)}}\hat{H}\ket{\Phi^{(0)}}
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\nonumber\\
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&&+\Bigg[
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\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
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c}\left[n\right]}{\delta
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n({\br})}\left(n_{\Phi^{(K)}}(\br)-n_{\Phi^{(0)}}(\br)\right)
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%\nonumber\\
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%&&
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\begin{split}
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\pdv{\E{}{\bw}}{\ew{K}}
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& = \mel*{\Det{(K)}}{\hH}{\Det{(K)}}
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- \mel*{\Det{(0)}}{\hH}{\Det{(0)}}
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\\
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& + \qty{
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\int d\br{} \fdv{\E{c}{\bw}[\n{}{}]}{\n{}{}({\br{}})}
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\qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
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+
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%\left.
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\dfrac{\partial {E}^{{\bw}}_{\rm
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c}\left[n\right]}{\partial w_K}
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%\right|
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\Bigg]_{n=n_{\opGamma{\bw}}}
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.
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\nonumber\\
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\pdv{\E{c}{\bw} [\n{}{}]}{\ew{K}}
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}_{\n{}{} = \n{\opGamma{\bw}}{}}.
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\end{split}
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\eeq
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Since the ensemble energy can be evaluated as follows:
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\beq
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E^{{\bw}}=\sum_{K\geq 0}w_K\bra{\Phi^{(K)}}\hat{H}\ket{\Phi^{(K)}}+{E}^{{\bw}}_{\rm
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c}\left[n_{\opGamma{\bw}}\right],
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\E{}{\bw} = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}}{\hH}{\Det{(K)}} + \E{c}{\bw}[\n{\opGamma{\bw}}{}],
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\eeq
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with $\Phi^{(K)}=\Phi^{(K),\bw}$ [note that, when the minimum is reached
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in Eq.~(\ref{eq:var_ener_gokdft}), $n_{\opGamma{\bw}}=n^{\bw,\bw}$],
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we finally recover from Eqs.~(\ref{eq:KS_ens_density}) and
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(\ref{eq:indiv_ener_from_ens}) the {\it exact} expression of Ref.~\cite{} for the $I$th energy level:
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with $\Det{(K)} = \Det{(K),\bw}$ [note that, when the minimum is reached in Eq.~\eqref{eq:var_ener_gokdft}, $\n{\opGamma{\bw}}{} = \n{}{\bw,\bw}$],
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we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
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\eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\cite{} for the $I$th energy level:
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\beq\label{eq:exact_ener_level_dets}
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E^{(I)}&=&\bra{\Phi^{(I)}}\hat{H}\ket{\Phi^{(I)}}+{E}^{{\bw}}_{\rm
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c}\left[n_{\opGamma{\bw}}\right]
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\nonumber\\
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&&+
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\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
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c}\left[n_{\opGamma{\bw}}\right]}{\delta
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n({\br})}\left(n_{\Phi^{(I)}}(\br)-n_{\opGamma{\bw}}(\br)\right)
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\nonumber\\
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&&+
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\sum_{K>0}\left(\delta_{IK}-w_K\right)\left.\dfrac{\partial {E}^{{\bw}}_{\rm
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c}\left[n\right]}{\partial w_K}
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\right|
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_{n=n_{\opGamma{\bw}}}.
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\begin{split}
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\E{}{(I)}
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& = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{{\bw}}[\n{\opGamma{\bw}}{}]
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\\
|
||||
& + \int d\br{} \fdv{\E{c}{\bw}[\n{\opGamma{\bw}}{}]}{\n{}{}(\br{})}
|
||||
\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGamma{\bw}}{}(\br{}) ]
|
||||
\\
|
||||
&+
|
||||
\sum_{K>0} \qty(\delta_{IK} - \ew{K} )
|
||||
\left.
|
||||
\pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}
|
||||
\right|_{\n{}{} = \n{\opGamma{\bw}}{}}.
|
||||
\end{split}
|
||||
\eeq
|
||||
%%%%%%%%%%%%%%%%
|
||||
\subsection{One-electron reduced density matrix formulation}
|
||||
@ -342,10 +306,10 @@ as basic variables, rather than Slater determinants. If we expand the
|
||||
ensemble KS (spin) orbitals [from which the latter are constructed] in an atomic
|
||||
orbital (AO) basis,
|
||||
\beq
|
||||
%\varphi_{p\sigma}(\br,\tau)
|
||||
\varphi^\sigma_p(\br)=
|
||||
%\varphi_{p\sigma}(\br{},\tau)
|
||||
\varphi^\sigma_p(\br{})=
|
||||
%\sigma(\tau)
|
||||
\sum_{\mu}c^\sigma_{{\mu p}}\AO{\mu}(\br),
|
||||
\sum_{\mu}c^\sigma_{{\mu p}}\AO{\mu}(\br{}),
|
||||
\eeq
|
||||
then the density matrix elements obtained from the
|
||||
determinant $\Phi^{(K)}$ can be expressed as follows in the AO basis:
|
||||
@ -359,26 +323,26 @@ p}}c^\sigma_{{\nu p}},
|
||||
where the summation runs over the spin-orbitals that are occupied in
|
||||
$\Phi^{(K)}$. Note that the density of the $K$th KS state reads
|
||||
\beq
|
||||
n_{\bmg^{(K)}}(\br)=
|
||||
\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}(\br)\AO{\nu}(\br)\Gamma_{\mu\nu}^{(K)\sigma}.
|
||||
%\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br,
|
||||
%\sigma})\AO{\nu}(\br,\sigma){\Gamma}^{(K)}_{\mu\nu}.
|
||||
n_{\bmg^{(K)}}(\br{})=
|
||||
\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}(\br{})\AO{\nu}(\br{})\Gamma_{\mu\nu}^{(K)\sigma}.
|
||||
%\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br{},
|
||||
%\sigma})\AO{\nu}(\br{},\sigma){\Gamma}^{(K)}_{\mu\nu}.
|
||||
\eeq
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Manu's derivation %%%
|
||||
\iffalse%%
|
||||
\blue{
|
||||
\beq
|
||||
n_{\bmg^{(K)}}(\br)&=&\sum_\sigma\left\langle\hat{\Psi}^\dagger(\br\sigma)\hat{\Psi}(\br\sigma)\right\rangle^{(K)}
|
||||
n_{\bmg^{(K)}}(\br{})&=&\sum_\sigma\left\langle\hat{\Psi}^\dagger(\br{}\sigma)\hat{\Psi}(\br{}\sigma)\right\rangle^{(K)}
|
||||
\nonumber\\
|
||||
&=&\sum_\sigma\sum_{pq}\varphi^\sigma_p(\br)\varphi^\sigma_q(\br)\left\langle\hat{a}_{p^\sigma,\sigma}^\dagger\hat{a}_{q^\sigma,\sigma}\right\rangle^{(K)}
|
||||
&=&\sum_\sigma\sum_{pq}\varphi^\sigma_p(\br{})\varphi^\sigma_q(\br{})\left\langle\hat{a}_{p^\sigma,\sigma}^\dagger\hat{a}_{q^\sigma,\sigma}\right\rangle^{(K)}
|
||||
\nonumber\\
|
||||
&=&\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}\left(\varphi^\sigma_p(\br)\right)^2
|
||||
&=&\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}\left(\varphi^\sigma_p(\br{})\right)^2
|
||||
\nonumber\\
|
||||
&=&\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}\sum_{\mu\nu}c^\sigma_{{\mu
|
||||
p}}c^\sigma_{{\nu p}}\AO{\mu}(\br)\AO{\nu}(\br)
|
||||
p}}c^\sigma_{{\nu p}}\AO{\mu}(\br{})\AO{\nu}(\br{})
|
||||
\nonumber\\
|
||||
&=&\sum_{\mu\nu}\AO{\mu}(\br)\AO{\nu}(\br)\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}c^\sigma_{{\mu
|
||||
&=&\sum_{\mu\nu}\AO{\mu}(\br{})\AO{\nu}(\br{})\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}c^\sigma_{{\mu
|
||||
p}}c^\sigma_{{\nu p}}
|
||||
\eeq
|
||||
}
|
||||
@ -392,34 +356,28 @@ and the ensemble density as follows:
|
||||
\eeq
|
||||
and
|
||||
\beq
|
||||
n_{\bmg^\bw}({\br})=\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}(\br)\AO{\nu}(\br){\Gamma}^{\bw\sigma}_{\mu\nu},
|
||||
n_{\bmg^\bw}({\br{}})=\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}(\br{})\AO{\nu}(\br{}){\Gamma}^{\bw\sigma}_{\mu\nu},
|
||||
\eeq
|
||||
respectively. The exact energy level expression in Eq.~(\ref{eq:exact_ener_level_dets}) can be
|
||||
respectively. The exact energy level expression in Eq.~\eqref{eq:exact_ener_level_dets} can be
|
||||
rewritten as follows:
|
||||
\beq\label{eq:exact_ind_ener_rdm}
|
||||
E^{(I)}&&={\rm
|
||||
Tr}\left[{\bmg}^{(I)}{\bm h}\right]
|
||||
+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
|
||||
\bmg^{(I)})
|
||||
\nonumber\\
|
||||
&&+{E}^{{\bw}}_{\rm
|
||||
c}\left[n_{\bmg^{\bw}}\right]
|
||||
+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
|
||||
c}\left[n_{\bmg^{\bw}}\right]}{\delta
|
||||
n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
|
||||
\nonumber\\
|
||||
&&+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left. \dfrac{\partial {E}^{{\bw}}_{\rm
|
||||
c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
|
||||
\begin{split}
|
||||
\E{}{(I)}
|
||||
& =\Tr[\bmg^{(I)} \bh]
|
||||
+ \frac{1}{2} \Tr[\bmg^{(I)} \bG \bmg^{(I)}]
|
||||
\\
|
||||
& + \E{c}{{\bw}}[\n{\bmg^{\bw}}{}]
|
||||
+ \int d\br{} \fdv{\E{c}{\bw}[\n{\bmg^{\bw}}{}]}{\n{}{}(\br{})}
|
||||
\qty[ \n{\bmg^{(I)}}{}(\br{}) - \n{\bmg^{\bw}}{}(\br{}) ]
|
||||
\\
|
||||
& + \sum_{K>0} \qty(\delta_{IK} - \ew{K})
|
||||
\left. \pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}\right|_{\n{}{} = \n{\bmg^{\bw}}{}}
|
||||
,
|
||||
\end{split}
|
||||
\eeq
|
||||
where
|
||||
\beq
|
||||
{\bm
|
||||
h}\equiv h_{\mu\nu}=
|
||||
%\langle\AO{\mu}\vert-\frac{1}{2}\nabla_{\br}^2+v_{\rm
|
||||
%ne}(\br)\vert\AO{\nu}\rangle
|
||||
\int d\br\;\AO{\mu}(\br)\left[-\frac{1}{2}\nabla_{\br}^2+v_{\rm
|
||||
ne}(\br)\right]\AO{\nu}(\br)
|
||||
\bh \equiv h_{\mu\nu} = \int d\br{} \AO{\mu}(\br{}) \qty[-\frac{1}{2} \nabla_{\br{}}^2 + \vne(\br{}) ]\AO{\nu}(\br{})
|
||||
\eeq
|
||||
denote the one-electron integrals matrix.
|
||||
The individual Hx energy is obtained from the following trace
|
||||
@ -436,8 +394,8 @@ G_{\mu\nu\lambda\omega}^{\sigma\tau}=({\mu}{\nu}\vert{\lambda}{\omega})
|
||||
\eeq
|
||||
with
|
||||
\beq
|
||||
(\mu\nu|\la\omega) = \iint \frac{\AO\mu(\br_1) \AO\nu(\br_1)
|
||||
\AO\la(\br_2) \AO\omega(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
|
||||
(\mu\nu|\la\omega) = \iint \frac{\AO\mu(\br{1}) \AO\nu(\br{1})
|
||||
\AO\la(\br{2}) \AO\omega(\br{2})}{\abs{\br{1} - \br{2}}} d\br{1} d\br{2}
|
||||
.
|
||||
\nonumber\\
|
||||
\eeq
|
||||
@ -503,7 +461,7 @@ n_{p^\sigma}^{(K)\sigma}n_{q^\sigma}^{(L)\sigma}\right)
|
||||
\iffalse%%%% Manu's derivation ...
|
||||
\blue{
|
||||
\beq
|
||||
n^{\bw}({\br})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
|
||||
n^{\bw}({\br{}})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
|
||||
w}_Kn^{(K)}({\bfx})
|
||||
\nonumber\\
|
||||
&=&
|
||||
@ -543,7 +501,7 @@ W_{\rm
|
||||
HF}\left[{\bmg}\right]=\frac{1}{2} \Tr(\bmg \, \bG \, \bmg),
|
||||
\eeq
|
||||
for the Hx density-functional energy in the variational energy
|
||||
expression of Eq.~(\ref{eq:var_ener_gokdft}):
|
||||
expression of Eq.~\eqref{eq:var_ener_gokdft}:
|
||||
\beq
|
||||
{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
|
||||
\Big\{
|
||||
@ -573,9 +531,9 @@ with
|
||||
\beq
|
||||
h^\bw_{\mu\nu}=h_{\mu\nu}+
|
||||
%\left\langle\AO{\mu}\middle\vert\dfrac{\delta E^\bw_{\rm
|
||||
%c}[n_{\bmg^\bw}]}{\delta n(\br)}\middle\vert\AO{\nu}\right\rangle
|
||||
\int d\br\;\AO{\mu}(\br)\dfrac{\delta E^\bw_{\rm
|
||||
c}[n_{\bmg^\bw}]}{\delta n(\br)}\AO{\nu}(\br).
|
||||
%c}[n_{\bmg^\bw}]}{\delta n(\br{})}\middle\vert\AO{\nu}\right\rangle
|
||||
\int d\br{}\;\AO{\mu}(\br{})\dfrac{\delta E^\bw_{\rm
|
||||
c}[n_{\bmg^\bw}]}{\delta n(\br{})}\AO{\nu}(\br{}).
|
||||
\eeq
|
||||
|
||||
%%%%%%%%%%%%%%%
|
||||
@ -702,69 +660,60 @@ and
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
Note that this approximation, where the ensemble density matrix is
|
||||
optimized from a non-local exchange potential [rather than a local one,
|
||||
as expected from Eq.~(\ref{eq:var_ener_gokdft})] is applicable to real
|
||||
as expected from Eq.~\eqref{eq:var_ener_gokdft}] is applicable to real
|
||||
(three-dimension) systems. As readily seen from
|
||||
Eq.~(\ref{eq:eHF-dens_mat_func}), {\it ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
|
||||
Eq.~\eqref{eq:eHF-dens_mat_func}, \textit{ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
|
||||
and curvature~\cite{} will be
|
||||
introduced in the Hx energy:
|
||||
\beq
|
||||
W_{\rm
|
||||
HF}\left[{\bmg}^\bw\right]&=&\frac{1}{2}\sum_{K\geq 0}w^2_K
|
||||
\Tr(\bmg^{(K)} \,
|
||||
\bG \, \bmg^{(K)})
|
||||
\nonumber\\
|
||||
&&+\sum_{L>K\geq 0}w_Kw_L\Tr(\bmg^{(K)} \,
|
||||
\bG \, \bmg^{(L)}).
|
||||
\begin{split}
|
||||
W_{\rm HF}[{\bmg}^\bw]
|
||||
& = \frac{1}{2}\sum_{K\geq 0} \ew{K}^2
|
||||
\Tr(\bmg^{(K)} \bG \bmg^{(K)})
|
||||
\\
|
||||
& + \sum_{L>K\geq 0} \ew{K} \ew{L}\Tr(\bmg^{(K)} \bG \bmg^{(L)}).
|
||||
\end{split}
|
||||
\eeq
|
||||
These errors will be removed when computing individual energies
|
||||
according to Eq.~(\ref{eq:exact_ind_ener_rdm}).\\
|
||||
according to Eq.~\eqref{eq:exact_ind_ener_rdm}.\\
|
||||
|
||||
Turning to the density-functional ensemble correlation energy, the
|
||||
following eLDA will be employed:
|
||||
\beq\label{eq:eLDA_corr_fun}
|
||||
{E}^{{\bw}}_{\rm
|
||||
c}[n]=\int d\br\;n(\br)\;\epsilon_{c}^{\bw}(n(\br)),
|
||||
c}[n]=\int d\br{}\;n(\br{}) \epsilon_{c}^{\bw}(n(\br{})),
|
||||
\eeq
|
||||
where the correlation energy per particle is {\it weight-dependent}. Its
|
||||
construction from a finite uniform electron gas model is discussed
|
||||
in detail
|
||||
in Sec.~\ref{sec:eDFA}. Combining Eq.~(\ref{eq:exact_ind_ener_rdm}) with
|
||||
Eq.~(\ref{eq:eLDA_corr_fun}) leads to our final energy level expression
|
||||
in Sec.~\ref{sec:eDFA}. Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with
|
||||
Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression
|
||||
within eLDA:
|
||||
\beq
|
||||
E^{(I)}&&\approx{\rm
|
||||
Tr}\left[{\bmg}^{(I)}{\bm h}\right]
|
||||
+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
|
||||
\bmg^{(I)})
|
||||
\nonumber\\
|
||||
&&+\int d\br\,
|
||||
{\epsilon}^{{\bw}}_{\rm
|
||||
c}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
|
||||
\begin{split}
|
||||
\E{}{(I)} & \approx \Tr[\bmg^{(I)} \bh]
|
||||
+ \frac{1}{2} \Tr(\bmg^{(I)} \bG \bmg^{(I)})
|
||||
\\
|
||||
& +\int d\br{} \epsilon^{\bw}_{\rm
|
||||
c}(n_{\bmg^{\bw}}(\br{}))\,n_{\bmg^{(I)}}(\br{})
|
||||
\nonumber\\
|
||||
&&
|
||||
+\int d\br\,
|
||||
n_{\bmg^{\bw}}(\br)\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
|
||||
+\int d\br{}\,
|
||||
n_{\bmg^{\bw}}(\br{})\left(n_{\bmg^{(I)}}(\br{})-n_{\bmg^{\bw}}(\br{})\right)
|
||||
\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
|
||||
c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}
|
||||
c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br{})}
|
||||
\nonumber\\
|
||||
&&
|
||||
+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
|
||||
+\int d\br{}\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br{})\left.
|
||||
\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
|
||||
c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
|
||||
c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br{})}.
|
||||
\end{split}
|
||||
\eeq
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Density-functional approximations for ensembles}
|
||||
\label{sec:eDFA}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
We decompose the weight-dependent functional as
|
||||
\begin{equation}
|
||||
\be{Hxc}{\bw}(\n{}{}) = \be{Hx}{\bw}(\n{}{}) + \be{c}{\bw}(\n{}{}),
|
||||
\end{equation}
|
||||
\manu{Well, at the end, we only develop functionals for the correlation
|
||||
part. Should be updated.}\\
|
||||
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}].
|
||||
The construction of these two functionals is described below.
|
||||
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}
|
||||
|
||||
Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (UEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
|
||||
One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states cannot be easily identified like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
|
||||
@ -782,61 +731,6 @@ As mentioned previously, we consider a three-state ensemble including the ground
|
||||
All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
|
||||
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Density-functional approximations for ensembles}
|
||||
\label{sec:eDFA}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%We decompose the weight-dependent functional as
|
||||
%\begin{equation}
|
||||
% \be{Hxc}{\bw}(\n{}{}) = \be{Hx}{\bw}(\n{}{}) + \be{c}{\bw}(\n{}{}),
|
||||
%\end{equation}
|
||||
%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}].
|
||||
%The construction of these two functionals is described below.
|
||||
%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}
|
||||
|
||||
Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (UEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
|
||||
One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states cannot be easily identified like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
|
||||
Moreover, because the infinite UEG model is a metal, it is gapless, which means that both the fundamental and optical gaps are zero.
|
||||
From this point of view, using finite UEGs \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs can be seen as more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
|
||||
Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems.
|
||||
|
||||
As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e., a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
|
||||
The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform.
|
||||
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
|
||||
This is a necessary condition for being able to model derivative discontinuities.
|
||||
|
||||
The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
|
||||
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.
|
||||
All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
|
||||
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\subsection{Ghost-interaction correction}
|
||||
%\label{sec:GIC}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%
|
||||
%The GIC weight-dependent Hartree-exchange functional is defined as
|
||||
%\begin{multline}
|
||||
% \be{Hx}{\bw}(\n{}{\bw}) = (1-\sum_{I>0} \ew{I}) \be{Hx}{}(\n{}{(0)}) + \sum_{I>0} \ew{I} \be{Hx}{}(\n{}{(I)})
|
||||
% \\
|
||||
% - \be{Hx}{(I)}(\n{}{\bw}),
|
||||
%\end{multline}
|
||||
%where
|
||||
%\begin{equation}
|
||||
% \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,
|
||||
%\end{equation}
|
||||
%and
|
||||
%\begin{equation}
|
||||
% \n{}{(I)}(\omega) = (\pi R)^{-1} \cos[(I+1) \omega/2]
|
||||
%\end{equation}
|
||||
%is the first-order density matrix with $\omega$ the interelectronic angle.
|
||||
%It yields
|
||||
%\begin{equation}
|
||||
% \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)],
|
||||
%\end{equation}
|
||||
%with $a_1 = 2 \ln 2 - 1/3$, $a_2 = 8/3$ and $a_3 = 32/15$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Weight-dependent correlation functional}
|
||||
\label{sec:Ec}
|
||||
@ -919,7 +813,7 @@ This is a crucial point as we intend to incorporate into standard functionals (w
|
||||
|
||||
Finally, we note that, by construction,
|
||||
\begin{equation}
|
||||
\left. \pdv{\be{c}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br)} = \be{c}{(J)}[\n{}{\bw}(\br)] - \be{c}{(0)}[\n{}{\bw}(\br)].
|
||||
\left. \pdv{\be{c}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br{})} = \be{c}{(J)}[\n{}{\bw}(\br{})] - \be{c}{(0)}[\n{}{\bw}(\br{})].
|
||||
\end{equation}
|
||||
\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
|
||||
|
||||
@ -1030,7 +924,9 @@ See {\SI} for the additional details about the construction of the functionals,
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Grant No.~ANR-14-CE06-0014-01) for funding.
|
||||
PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
|
||||
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
|
||||
EF thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Grant No.~ANR-14-CE06-0014-01) for funding.
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
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