diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 0f1ec22..60ec2a6 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -559,7 +559,7 @@ c}\left[n_{\bm\gamma^{\bw}}\right] \eeq The minimizing ensemble density matrix fulfills the following stationarity condition -\beq +\beq\label{eq:commut_F_AO} {\bm F}^{\bw\sigma}{\bm \Gamma}^{\bw\sigma}{\bm S}={\bm S}{\bm \Gamma}^{\bw\sigma}{\bm F}^{\bw\sigma}, \eeq @@ -704,7 +704,7 @@ 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 (three-dimension) systems. As readily seen from -Eq.~(\ref{eq:eHF-dens_mat_func}), {\it ghost interactions}~\cite{} +Eq.~(\ref{eq:eHF-dens_mat_func}), {\it ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} and curvature~\cite{} will be introduced in the Hx energy: \beq @@ -760,6 +760,8 @@ 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} @@ -781,33 +783,6 @@ All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ 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} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\manu{I guess this subsection should be removed(?)} -\color{purple} -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 $a_3 = 32/15$. -\color{black} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Weight-dependent correlation functional} \label{sec:Ec} @@ -895,108 +870,6 @@ Finally, we note that, by construction, \alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}} -%%%%%%%%%%%%%%%% -\color{purple} -\section{Implementation} - -\manu{I think that this section can be removed (especially Sec.~\ref{sec:E_I}). Many points -discussed in Sec.~\ref{sec:KS-eDFT} are now mentioned in the theory -section. If we want to keep some material of Sec.~\ref{sec:KS-eDFT}, it -should be moved to -Secs.~\ref{subsec:gokdft} or~\ref{subsec:approx} (or maybe Sec.~\ref{sec:compdetails}).} -%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{KS-eDFT for excited states} -\label{sec:KS-eDFT} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -Here, we explain how to perform a self-consistent KS calculation for ensembles (KS-eDFT) in the context of excited states. -In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including: -(i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$). -Generalization to a larger number of states is straightforward and is left for future work. -By definition, the ensemble energy is -\begin{equation} - \E{}{\bw} = (1 - \ew{1} - \ew{2}) \E{}{(0)} + \ew{1} \E{}{(1)} + \ew{2} \E{}{(2)}. -\end{equation} -The $\E{}{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the single and double excitation, respectively. -To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions: -$0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$. -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. -By construction, the excitation energies are -\begin{equation} -\label{eq:Ex} - \Ex{(I)} = \pdv{\E{}{(I)}}{\ew{I}} = \E{}{(I)} - \E{}{(0)}. -\end{equation} -In the following, the orbitals $\MO{p}{\bw}(\br)$ are defined as linear combination of basis functions $\AO{\mu}(\br)$, such as -\begin{equation} - \MO{p}{\bw}(\br) = \sum_{\mu=1}^{\Nbas} \cMO{\mu p}{\bw} \, \AO{\mu}(\br). -\end{equation} - -Within the self-consistent KS-eDFT process, one is looking for the following weight-dependent density matrix: -\begin{equation} -\label{eq:Gamma} - \bGamma{\bw} = (1-\ew{1}-\ew{2}) \bGamma{(0)} - \ew{1} \bGamma{(1)} - \ew{2} \bGamma{(2)}, -\end{equation} -where $\bw = (\ew{1},\ew{2})$ and $\bGamma{(I)}$ is the $I$th-state density matrix with elements -\begin{equation} -\label{eq:eGamma} - \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}. -\end{equation} -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 -\begin{multline} -\label{eq:F} - \eF{\mu\nu}{\bw} - = \eHc{\mu\nu} + \sum_{\la\si} \eGamma{\la\si}{\bw} \eG{\mu\nu\la\si} - \\ - + \int \left. \fdv{\bE{Hxc}{\bw}[\n{}{}]}{\n{}{}(\br)} \right|_{\n{}{} = \n{}{\bw}(\br)} \AO{\mu}(\br) \AO{\nu}(\br) d\br, -\end{multline} -which itself depends on $\bGamma{\bw}$. -In Eq.~\eqref{eq:F}, $\hHc$ is the core Hamiltonian (including kinetic and electron-nucleus attraction terms), $\eG{\mu\nu\la\si} = (\mu\nu|\la\si) - (\mu\si|\la\nu)$, -\begin{equation} - (\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 -\end{equation} -are two-electron repulsion integrals, -$\bE{Hxc}{\bw}[\n{}{}(\br)] = \n{}{}(\br) \be{Hxc}{\bw}[\n{}{}(\br)]$ and $\be{Hxc}{\bw}[\n{}{}(\br)]$ is the weight-dependent Hartree-exchange-correlation functional to be built in the present study. -The one-electron ensemble density is -\begin{equation} - \n{}{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br), -\end{equation} -with a similar expression for $\n{}{(I)}(\br)$, while the ensemble energy reads -\begin{equation} -\label{eq:Ew} - \E{}{\bw} - = \Tr(\bGamma{\bw} \, \bHc) - + \frac{1}{2} \Tr(\bGamma{\bw} \, \bG \, \bGamma{\bw}) -% \\ -% + \int \e{c}{\bw}[\n{}{\bw}(\br)] \n{}{\bw}(\br) d\br. - + \int \bE{Hxc}{\bw}[\n{}{\bw}(\br)] d\br. -\end{equation} -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 $\bS$. -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} -Below, we propose a ghost-interaction correction (GIC) in order to minimize this error. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Extracting individual energies} -\label{sec:E_I} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -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: -\begin{multline} - \E{}{(I)} = \Tr(\bGamma{(I)} \, \bHc) + \frac{1}{2} \Tr(\bGamma{(I)} \, \bG \, \bGamma{(I)}) - \\ - + \int \left. \fdv{\bE{Hxc}{\bw}[\n{}{}]}{\n{}{}(\br)} \right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{(I)}(\br) d\br - + \LZ{Hxc}{} + \DD{Hxc}{(I)}. -\end{multline} -Note that a \emph{single} KS-eDFT calculation is required to extract the three individual energies. -\alert{Mention LIM?} -The (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by -\begin{align} - \LZ{Hxc}{} & = - \int \left. \fdv{\be{Hxc}{\bw}[\n{}{}]}{\n{}{}(\br)} \right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{\bw}(\br)^2 d\br, - \\ - \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. -\end{align} -Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}]. -The only remaining piece of information to define at this stage is the weight-dependent Hartree-exchange-correlation functional $\be{Hxc}{\bw}(\n{}{})$. -\color{black} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details}\label{sec:compdetails} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -1025,11 +898,24 @@ To get reference excitation energies for both the single and double excitations, For the single excitations, we have also performed time-dependent HF (TDHF), configuration interaction singles (CIS) and TDLDA calculations. \cite{Dreuw_2005} For TDLDA, the validity of the Tamm-Dancoff approximation (TDA) has been also tested. -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 $\bw = (1/3,1/3)$. +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 $\bw = (1/3,1/3)$.\\ +\manu{might be re-used} +\color{purple} +The self-consistent process described above is carried on until $\max +\abs{\bF{\bw} \, \bGamma{\bw} \, \bS - \bS \, \bGamma{\bw} \, \bF{\bw}} +< \delta$ [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 $\bS$. +\color{black} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results and discussion}\label{sec:results} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\manu{might be re-used} +\color{purple} +To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions: +$0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.\\ +\color{black} 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., $\Nel = 5$). 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}. In the weakly correlated regime (i.e., small $L$), all methods provide accurate estimates of the excitation energies.