functional

Manuscript/SI/eDFT-SI.tex

@@ -1,18 +1,16 @@
 \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem}
 
-\usepackage{mathpazo,libertine}
+\usepackage[utf8]{inputenc}
-\newcommand{\alert}[1]{\textcolor{red}{#1}}
+\usepackage[T1]{fontenc}
-\definecolor{darkgreen}{RGB}{0, 180, 0}
+\usepackage{txfonts}
 
-\usepackage{hyperref}
+\usepackage[
-\hypersetup{
+	colorlinks=true,
-    colorlinks=true,
+    citecolor=blue,
-    linkcolor=blue,
+    breaklinks=true
-    filecolor=blue,      
+	]{hyperref}
-    urlcolor=blue,	
+\urlstyle{same}
-    citecolor=blue
-}
 
 %useful stuff
 \newcommand{\cdash}{\multicolumn{1}{c}{---}}

Manuscript/eDFT.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-02-15 17:43:19 +0100 
+%% Created for Pierre-Francois Loos at 2020-02-15 22:24:41 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -59,74 +59,90 @@
 @article{Boblest_2014,
 	Author = {S. Boblest and C. Schimeczek and G. Wunner},
 	Date-Added = {2019-09-09 09:42:01 +0200},
-	Date-Modified = {2019-09-09 09:43:17 +0200},
+	Date-Modified = {2020-02-15 21:00:05 +0100},
+	Doi = {10.1103/PhysRevA.89.012505},
 	Journal = {Phys. Rev. A},
 	Pages = {012505},
 	Volume = {89},
-	Year = {2014}}
+	Year = {2014},
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.89.012505}}
 
 @article{Stopkowicz_2015,
 	Author = {S. Stopkowicz and J. Gauss and K. K. Lange and E. I. Tellgren and T. Helgaker},
 	Date-Added = {2019-09-09 09:41:46 +0200},
-	Date-Modified = {2019-09-09 09:43:56 +0200},
+	Date-Modified = {2020-02-15 20:59:21 +0100},
+	Doi = {10.1063/1.4928056},
 	Journal = {J. Chem. Phys.},
 	Pages = {074110},
 	Volume = {143},
-	Year = {2015}}
+	Year = {2015},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.4928056}}
 
 @article{Tellgren_2008,
 	Author = {E. I. Tellgren and A. Soncini and T. Helgaker},
 	Date-Added = {2019-09-09 09:41:39 +0200},
-	Date-Modified = {2019-09-09 09:44:14 +0200},
+	Date-Modified = {2020-02-15 21:00:47 +0100},
+	Doi = {10.1063/1.2996525},
 	Journal = {J. Chem. Phys.},
 	Pages = {154114},
 	Volume = {129},
-	Year = {2008}}
+	Year = {2008},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.2996525}}
 
 @article{Tellgren_2009,
 	Author = {E. I. Tellgren and T. Helgaker and A. Soncini},
 	Date-Added = {2019-09-09 09:41:39 +0200},
-	Date-Modified = {2019-09-09 09:44:18 +0200},
+	Date-Modified = {2020-02-15 21:02:41 +0100},
+	Doi = {10.1039/B822262B},
 	Journal = {Phys. Chem. Chem. Phys..},
 	Pages = {5489},
 	Volume = {11},
-	Year = {2009}}
+	Year = {2009},
+	Bdsk-Url-1 = {https://doi.org/10.1039/B822262B}}
 
 @article{Schmelcher_1997,
 	Author = {P. Schmelcher and L. S. Cederbaum},
 	Date-Added = {2019-09-09 09:41:33 +0200},
-	Date-Modified = {2019-09-09 09:45:27 +0200},
+	Date-Modified = {2020-02-15 21:01:32 +0100},
+	Doi = {10.1002/(SICI)1097-461X(1997)64:5<501::AID-QUA3>3.0.CO;2-%23},
 	Journal = {Int. J. Quantum Chem.},
 	Pages = {501},
 	Volume = {64},
-	Year = {1997}}
+	Year = {1997},
+	Bdsk-Url-1 = {https://doi.org/10.1002/(SICI)1097-461X(1997)64:5%3C501::AID-QUA3%3E3.0.CO;2-%2523}}
 
 @article{Schmelcher_1990,
 	Author = {P. Schmelcher and L. S. Cederbaum},
 	Date-Added = {2019-09-09 09:41:22 +0200},
-	Date-Modified = {2019-09-09 09:45:20 +0200},
+	Date-Modified = {2020-02-15 21:04:23 +0100},
+	Doi = {10.1103/PhysRevA.41.4936},
 	Journal = {Phys. Rev. A},
 	Pages = {4936},
 	Volume = {41},
-	Year = {1990}}
+	Year = {1990},
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.41.4936}}
 
 @article{Lange_2012,
 	Author = {K. K. Lange and E. I. Tellgren and M. R. Hoffmann and T. Helgaker},
 	Date-Added = {2019-09-09 09:41:15 +0200},
-	Date-Modified = {2019-09-09 09:44:10 +0200},
+	Date-Modified = {2020-02-15 21:03:33 +0100},
+	Doi = {10.1126/science.1219703},
 	Journal = {Science},
 	Pages = {327},
 	Volume = {337},
-	Year = {2012}}
+	Year = {2012},
+	Bdsk-Url-1 = {https://doi.org/10.1126/science.1219703}}
 
 @article{Schmelcher_2012,
 	Author = {P. Schmelcher},
 	Date-Added = {2019-09-09 09:41:08 +0200},
-	Date-Modified = {2019-09-09 09:45:15 +0200},
+	Date-Modified = {2020-02-15 21:02:03 +0100},
+	Doi = {10.1126/science.1224869},
 	Journal = {Science},
 	Pages = {302},
 	Volume = {337},
-	Year = {2012}}
+	Year = {2012},
+	Bdsk-Url-1 = {https://doi.org/10.1126/science.1224869}}
 
 @article{Patil_2001,
 	Author = {S. H. Patil},
@@ -194,7 +210,8 @@
 @article{Perdew_1983,
 	Author = {J. P. Perdew and M. Levy},
 	Date-Added = {2019-09-05 12:04:19 +0200},
-	Date-Modified = {2019-09-05 12:13:34 +0200},
+	Date-Modified = {2020-02-15 22:24:15 +0100},
+	Doi = {10.1103/PhysRevLett.51.1884},
 	Journal = {Phys. Rev. Lett.},
 	Pages = {1884},
 	Title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities},
@@ -358,14 +375,16 @@
 	Year = {2018},
 	Bdsk-Url-1 = {https://doi.org/10.1007/s00214-018-2352-7}}
 
-@article{Deur_2018b,
+@article{Deur_2019,
 	Author = {K. Deur and E. Fromager},
 	Date-Added = {2018-12-08 18:03:14 +0100},
-	Date-Modified = {2018-12-09 14:19:11 +0100},
+	Date-Modified = {2020-02-15 22:22:38 +0100},
-	Journal = {arXiv},
+	Doi = {10.1063/1.5084312},
+	Journal = {J. Chem. Phys.},
+	Pages = {094106},
 	Title = {Ground and excited energy levels can be extracted exactly from a single ensemble density-functional theory calculation},
-	Volume = {1812.02461},
+	Volume = {150},
-	Year = {2018}}
+	Year = {2019}}
 
 @book{NISTbook,
 	Address = {New York},
@@ -1299,14 +1318,16 @@
 @article{Rogers_2017,
 	Author = {Rogers, Fergus JM and Loos, Pierre-Fran{\c c}ois},
 	Date-Added = {2018-10-24 22:57:04 +0200},
-	Date-Modified = {2018-10-24 22:57:04 +0200},
+	Date-Modified = {2020-02-15 20:58:25 +0100},
+	Doi = {10.1063/1.4974839},
 	File = {/Users/loos/Zotero/storage/EJULV3C4/53.pdf},
 	Journal = {J. Chem. Phys.},
 	Number = {4},
 	Pages = {044114},
 	Title = {Excited-State {{Wigner}} Crystals},
 	Volume = {146},
-	Year = {2017}}
+	Year = {2017},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.4974839}}
 
 @article{Tognetti_2016,
 	Author = {Tognetti, Vincent and Loos, Pierre-Fran{\c c}ois},
@@ -2775,12 +2796,12 @@
 	Year = {2017},
 	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.95.035120}}
 
-@article{Deur_2018a,
+@article{Deur_2018,
 	Abstract = {Gross\textendash{}Oliveira\textendash{}Kohn density-functional theory (GOK-DFT) is an extension of DFT to excited states where the basic variable is the ensemble density, i.e. the weighted sum of ground- and excitedstate densities. The ensemble energy (i.e. the weighted sum of ground- and excited-state energies) can be obtained variationally as a functional of the ensemble density. Like in DFT, the key ingredient to model in GOK-DFT is the exchange-correlation functional. Developing density-functional approximations (DFAs) for ensembles is a complicated task as both density and weight dependencies should in principle be reproduced. In a recent paper [Phys. Rev. B 95, 035120 (2017)], the authors applied exact GOK-DFT to the simple but nontrivial Hubbard dimer in order to investigate (numerically) the importance of weight dependence in the calculation of excitation energies. In this work, we derive analytical DFAs for various density and correlation regimes by means of a Legendre\textendash{}Fenchel transform formalism. Both functional and density driven errors are evaluated for each DFA. Interestingly, when the ensemble exact-exchange-only functional is used, these errors can be large, in particular if the dimer is symmetric, but they cancel each other so that the excitation energies obtained by linear interpolation are always accurate, even in the strongly correlated regime.},
 	Archiveprefix = {arXiv},
 	Author = {Deur, Killian and Mazouin, Laurent and Senjean, Bruno and Fromager, Emmanuel},
 	Date-Added = {2018-10-24 22:38:52 +0200},
-	Date-Modified = {2018-12-11 14:07:04 +0100},
+	Date-Modified = {2020-02-15 22:22:42 +0100},
 	Doi = {10.1140/epjb/e2018-90124-7},
 	File = {/Users/loos/Zotero/storage/2398CIXN/Deur et al. - 2018 - Exploring weight-dependent density-functional appr.pdf},
 	Journal = {Eur. Phys. J. B},
@@ -8236,6 +8257,7 @@
 
 @article{Gidopoulos_2002,
 	Author = {Gidopoulos, N. I. and Papaconstantinou, P. G. and Gross, E. K. U.},
+	Date-Modified = {2020-02-15 20:57:42 +0100},
 	Doi = {10.1103/PhysRevLett.88.033003},
 	File = {/Users/loos/Zotero/storage/RRB3BXVQ/Gidopoulos et al. - 2002 - Spurious Interactions, and Their Correction, in th.pdf},
 	Issn = {0031-9007, 1079-7114},
@@ -8243,6 +8265,7 @@
 	Language = {en},
 	Month = jan,
 	Number = {3},
+	Pages = {033003},
 	Title = {Spurious {{Interactions}}, and {{Their Correction}}, in the {{Ensemble}}-{{Kohn}}-{{Sham Scheme}} for {{Excited States}}},
 	Volume = {88},
 	Year = {2002},

Manuscript/eDFT.tex

@@ -170,9 +170,9 @@ This weight dependence of the xc functional plays a crucial role in the calculat
 It actually accounts for the infamous derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
 %\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
 
-Despite its formal beauty and the fact that GOK-DFT 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}
+Despite its formal beauty and the fact that GOK-DFT 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_2018,Deur_2019,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}
 The main reason is simply the absence of density-functional approximations (DFAs) for ensembles in the literature. 
-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}
+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_2018,Deur_2019,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}
 In both cases, the key problem, namely the design of weight-dependent DFAs for ensembles (eDFAs), remains open. 
 A first step towards this goal is presented in the present manuscript with the ambition to turn, in the forthcoming future, GOK-DFT into a practical computational method for modeling excited states in molecules and extended systems.
 The present eDFA is specially designed for the computation of single and double excitations within GOK-DFT, and can be seen as a natural extension of the ubiquitous local-density approximation (LDA) for ensemble. 
@@ -695,25 +695,32 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
 	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
 \end{split}
 \eeq
+\titou{T2: I think we should specify what those terms are physically...}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}
 \label{sec:eDFA}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-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}
+Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (IUEG) 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.
+Moreover, because the IUEG 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}
+From this point of view, using finite UEGs (FUEGs), \cite{Loos_2011b, Gill_2012} which have, like an atom, discrete energy levels and non-zero gaps, 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. 
+However, one of the drawbacks of using FUEGs is that the resulting eDFA will inexorably depend on the number of electrons (see below).
+Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems by combining these FUEGs with the usual IUEG to construct a weigh-dependent LDA functional for ensembles (eLDA).
 
-As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
+As a FUEG, we consider the ringium model in which electrons move on a perfect ring (\ie, 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.
+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 ($I=2$) of the (spin-polarized) two-electron ringium system.
+Generalization to a larger number of states is straightforward and is left for future work.
+To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions: 
+$0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.
+
 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.
 
@@ -758,7 +765,11 @@ Combining these, one can build a three-state weight-dependent correlation eDFA:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{LDA-centered functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
+One of the main driving force behind the popularity of DFT is its ``universal'' nature, as xc functionals can be applied to any electronic system.
+The two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the FUEG.
+However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
+The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in Eq.~\eqref{eq:ecw}], while the remaining correlation effects are provided by the bath (\ie, the usual LDA correlation functional).
+Following this simple strategy, which is further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows:
 \begin{equation}
 \label{eq:becw}
 	\be{c}{\bw}(\n{}{}) =  (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
@@ -792,11 +803,6 @@ Equation \eqref{eq:becw} can be recast
 which nicely highlights the centrality of the LDA in the present eDFA. 
 In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$.
 Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
-
-This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) which was originally derived by Franck and Fromager. \cite{Franck_2014} 
-Within this in-principle-exact formalism, the (weight-dependent) correlation energy of the ensemble is constructed from the (weight-independent) ground-state functional (such as the LDA), yielding Eq.~\eqref{eq:eLDA}.
-This is a crucial point as we intend to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons).
-
 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{})].