From 6766543ef0d8348c139a15863cf19c9b42f958ce Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 6 Sep 2019 17:26:37 +0200 Subject: [PATCH] work under progress --- Manuscript/eDFT.tex | 83 ++++++++++++++++++++++-------------------- Notebooks/eDFT_FUEG.nb | 22 +++++------ 2 files changed, 55 insertions(+), 50 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 699f195..923a89e 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -24,17 +24,23 @@ \newcommand{\la}{\lambda} \newcommand{\si}{\sigma} +% operators +\newcommand{\hHc}{\Hat{h}} +\newcommand{\hT}{\Hat{T}} +\newcommand{\vne}{\Hat{v}_\text{ne}} +\newcommand{\hWee}{\Hat{W}_\text{ee}} + % functionals, potentials, densities, etc \newcommand{\eps}{\epsilon} \newcommand{\e}[2]{\eps_\text{#1}^{#2}} -\renewcommand{\v}[2]{E_\text{#1}^{#2}} +\newcommand{\E}[2]{E_\text{#1}^{#2}} +\newcommand{\bE}[2]{\bar{E}_\text{#1}^{#2}} \newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}} \newcommand{\bv}[2]{\bar{f}_\text{#1}^{#2}} \newcommand{\n}[1]{n^{#1}} \newcommand{\DD}[2]{\Delta_\text{#1}^{#2}} \newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}} - % energies \newcommand{\EHF}{E_\text{HF}} \newcommand{\Ec}{E_\text{c}} @@ -50,17 +56,17 @@ \newcommand{\bG}{\bm{G}} \newcommand{\bS}{\bm{S}} \newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}} -\newcommand{\bHc}{\bm{H}^\text{c}} +\newcommand{\bHc}{\bm{h}} \newcommand{\bF}[1]{\bm{F}^{#1}} \newcommand{\Ex}[1]{\Omega^{#1}} -\newcommand{\E}[1]{E^{#1}} % elements \newcommand{\ew}[1]{w_{#1}} \newcommand{\eG}[1]{G_{#1}} \newcommand{\eS}[1]{S_{#1}} \newcommand{\eGamma}[2]{\Gamma_{#1}^{#2}} -\newcommand{\eHc}[1]{H_{#1}^\text{c}} +\newcommand{\hGamma}[2]{\Hat{\Gamma}_{#1}^{#2}} +\newcommand{\eHc}[1]{h_{#1}} \newcommand{\eF}[2]{F_{#1}^{#2}} % Numbers @@ -72,7 +78,6 @@ \newcommand{\cMO}[2]{c_{#1}^{#2}} \newcommand{\AO}[1]{\chi_{#1}} - % units \newcommand{\IneV}[1]{#1~eV} \newcommand{\InAU}[1]{#1~a.u.} @@ -159,18 +164,7 @@ Atomic units are used throughout. \label{sec:geKS} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\alert{In eDFT, the ensemble energy -\begin{equation} - \E{\bw} = (1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)} -\end{equation} -is obtained variationally as follows, -In analogy with ground-state generalized KS-DFT, we consider the following partitioning of the ensemble Levy-Lieb functional -\begin{equation} - 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\} -\end{equation} -\begin{equation} - 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\} -\end{equation}} +\alert{Manu, you might want to add here details about the general KS-eDFT procedure.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{KS-eDFT for excited states} @@ -182,16 +176,16 @@ In order to take into account both single and double excitations simultaneously, 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)}. + \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. +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 eKS calculation (see below), the weights must remain independent. +Note that, in order to extract individual energies from a single KS-eDFT calculation (see below), 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)}. + \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} @@ -214,15 +208,15 @@ The coefficients $\cMO{\mu p}{\bw}$ used to construct the density matrix $\bGamm \eF{\mu\nu}{\bw} = \eHc{\mu\nu} + \sum_{\la\si} \eGamma{\la\si}{\bw} \eG{\mu\nu\la\si} \\ - + \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \AO{\mu}(\br) \AO{\nu}(\br) d\br, + + \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}, $\bHc$ is the core Hamiltonian (including kinetic and electron-nucleus attraction terms), $\eG{\mu\nu\la\si} = (\mu\nu|\la\si) - (\mu\si|\la\nu)$, +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, -$\v{c}{\bw}[\n{}(\br)] = \n{}(\br) \e{c}{\bw}[\n{}(\br)]$ and $\e{c}{\bw}[\n{}(\br)]$ is the weight-dependent correlation functional to be built in the present study. +$\bE{Hxc}{\bw}[\n{}(\br)] = \n{}(\br) \be{Hxc}{\bw}[\n{}(\br)]$ and $\be{Hxc}{\bw}[\n{}(\br)]$ is the weight-dependent correlation functional to be built in the present study. The one-electron ensemble density is \begin{equation} \n{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br), @@ -230,37 +224,37 @@ The one-electron ensemble density is with a similar expression for $\n{(I)}(\br)$, while the ensemble energy reads \begin{equation} \label{eq:Ew} - \E{\bw} + \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 \v{c}{\bw}[\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 $\bS$. -\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. -Below, we propose a ghost-interaction correction in order to minimize this error.} +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 in order to minimize this error. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Extracting individual energies} \label{sec:ind_energy} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -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: +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)}) + \E{}{(I)} = \Tr(\bGamma{(I)} \, \bHc) + \frac{1}{2} \Tr(\bGamma{(I)} \, \bG \, \bGamma{(I)}) \\ - + \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{(I)}(\br) d\br - + \LZ{c}{} + \DD{c}{(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} eKS calculation is required to extract the three individual energies. +Note that a \emph{single} KS-eDFT calculation is required to extract the three individual energies. The correlation part of the (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by \begin{align} - \LZ{c}{} & = - \int \left. \fdv{\e{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br, + \LZ{Hxc}{} & = - \int \left. \fdv{\be{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br, \\ - \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. + \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 correlation functional $\e{c}{\bw}(\n{})$. +The only remaining piece of information to define at this stage is the weight-dependent correlation functional $\be{Hxc}{\bw}(\n{})$. \alert{Mention LIM?} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -301,10 +295,21 @@ This is a necessary condition for being able to model derivative discontinuities \section{Density-functional approximations for ensembles} \label{sec:eDFA} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +We decompose the weight-dependent functional $\be{Hxc}{\bw}(\n{})$ as +\begin{equation} + \be{Hxc}{\bw}(\n{}) = \be{Hx}{\bw}(\n{}) + \be{c}{\bw}(\n{}), +\end{equation} +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. +The construction of these two functionals is described below. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Weight-dependent DFAs} -\label{sec:wDFAs} +\subsection{Ghost-interaction correction} +\label{sec:GIC} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Weight-dependent correlation functional} +\label{sec:Ec} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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. diff --git a/Notebooks/eDFT_FUEG.nb b/Notebooks/eDFT_FUEG.nb index 79b9b3e..ea35754 100644 --- a/Notebooks/eDFT_FUEG.nb +++ b/Notebooks/eDFT_FUEG.nb @@ -37296,7 +37296,7 @@ Cell[BoxData[ 3.7533751885350037`*^9}, {3.753427363238578*^9, 3.753427443878433*^9}}, CellLabel-> "In[206]:=",ExpressionUUID->"06cbc894-5531-4a6e-b39f-fb7110953aad"] -}, Open ]] +}, Closed]] }, Open ]], Cell[CellGroupData[{ @@ -43535,7 +43535,7 @@ oq8yS/cZzbNsHBmFxJ2u+6uZFplja5uW9u4HHE/0nX+P0OL/DABCymxFw4we CellLabel-> "Out[196]=",ExpressionUUID->"2960c5a9-2484-4db9-93ab-7b90a8e7a40e"] }, Open ]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ @@ -96275,7 +96275,7 @@ Cell[BoxData[ "Out[367]//TableForm=",ExpressionUUID->"c28bfbbf-aaba-4edf-86df-\ c936781415f2"] }, Open ]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ @@ -96520,7 +96520,7 @@ Cell[BoxData[ 7ae551e60491"] }, Open ]] }, Closed]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ @@ -146837,7 +146837,7 @@ Cell[1533272, 35734, 65579, 1371, 3516, "Input",ExpressionUUID->"11221944-5033-4 InitializationCell->True], Cell[1598854, 37107, 10749, 190, 202, "Input",ExpressionUUID->"06cbc894-5531-4a6e-b39f-fb7110953aad", InitializationCell->True] -}, Open ]] +}, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[1609652, 37303, 216, 4, 67, "Section",ExpressionUUID->"7f116dc0-d26d-4a3b-8240-0f41c8301a9c"], @@ -146903,9 +146903,9 @@ Cell[CellGroupData[{ Cell[1743670, 40437, 5694, 130, 502, "Input",ExpressionUUID->"1a883756-8ff1-4d3f-9563-4a084568ffa3"], Cell[1749367, 40569, 164535, 2966, 569, "Output",ExpressionUUID->"2960c5a9-2484-4db9-93ab-7b90a8e7a40e"] }, Open ]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ -Cell[1913951, 43541, 192, 3, 54, "Subsection",ExpressionUUID->"19680b9a-e6ed-4fc2-aa15-72b80fcbd0ab"], +Cell[1913951, 43541, 192, 3, 38, "Subsection",ExpressionUUID->"19680b9a-e6ed-4fc2-aa15-72b80fcbd0ab"], Cell[1914146, 43546, 457, 11, 89, "Input",ExpressionUUID->"75c0e938-eb67-4d7c-b553-62fa6f39140b", InitializationCell->True], Cell[1914606, 43559, 1512, 29, 215, "Input",ExpressionUUID->"5df0a514-6a2c-4f9a-a7f6-01a7a981b874", @@ -147323,9 +147323,9 @@ Cell[4503287, 96060, 2881, 68, 180, "Output",ExpressionUUID->"7c406f18-a662-45c6 Cell[4506171, 96130, 2995, 71, 180, "Output",ExpressionUUID->"5b1573bc-2217-4d7d-8f1c-11dc04d4cc56"], Cell[4509169, 96203, 3034, 72, 180, "Output",ExpressionUUID->"c28bfbbf-aaba-4edf-86df-c936781415f2"] }, Open ]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ -Cell[4512252, 96281, 157, 3, 45, "Subsubsection",ExpressionUUID->"dbf537e7-1c5d-45dc-af40-a300842cd416"], +Cell[4512252, 96281, 157, 3, 37, "Subsubsection",ExpressionUUID->"dbf537e7-1c5d-45dc-af40-a300842cd416"], Cell[CellGroupData[{ Cell[4512434, 96288, 1151, 27, 79, "Input",ExpressionUUID->"e88cb52c-6f10-4cf4-b957-9da7481b69a2"], Cell[4513588, 96317, 3919, 88, 170, "Output",ExpressionUUID->"33eb60cc-17fb-4ff9-bab1-65bfa6d27c54"] @@ -147335,9 +147335,9 @@ Cell[4517544, 96410, 1150, 27, 79, "Input",ExpressionUUID->"d5ed830c-adee-47b5-b Cell[4518697, 96439, 3696, 80, 170, "Output",ExpressionUUID->"bbf784c3-4234-4ee9-8c97-7ae551e60491"] }, Open ]] }, Closed]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ -Cell[4522454, 96526, 155, 3, 54, "Subsection",ExpressionUUID->"42370903-32f2-402b-ae7f-df3a3b6bf3aa"], +Cell[4522454, 96526, 155, 3, 38, "Subsection",ExpressionUUID->"42370903-32f2-402b-ae7f-df3a3b6bf3aa"], Cell[CellGroupData[{ Cell[4522634, 96533, 216, 4, 45, "Subsubsection",ExpressionUUID->"d67b63c0-2997-4cc2-afde-fb7d5b4cd7ca"], Cell[CellGroupData[{