1606 lines
75 KiB
TeX
1606 lines
75 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
|
|
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem}
|
|
|
|
\usepackage[utf8]{inputenc}
|
|
\usepackage[T1]{fontenc}
|
|
\usepackage{txfonts}
|
|
\usepackage{mathrsfs}
|
|
|
|
\usepackage[
|
|
colorlinks=true,
|
|
citecolor=blue,
|
|
breaklinks=true
|
|
]{hyperref}
|
|
\urlstyle{same}
|
|
|
|
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
|
\usepackage[normalem]{ulem}
|
|
\newcommand{\titou}[1]{\textcolor{red}{#1}}
|
|
\newcommand{\manu}[1]{\textcolor{blue}{#1}}
|
|
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
|
|
\newcommand{\trashEF}[1]{\textcolor{blue}{\sout{#1}}}
|
|
|
|
%useful stuff
|
|
\newcommand{\cdash}{\multicolumn{1}{c}{---}}
|
|
\newcommand{\mc}{\multicolumn}
|
|
\newcommand{\mr}{\multirow}
|
|
\newcommand{\fnm}{\footnotemark}
|
|
\newcommand{\fnt}{\footnotetext}
|
|
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
|
\newcommand{\la}{\lambda}
|
|
\newcommand{\si}{\sigma}
|
|
\newcommand{\ie}{\textit{i.e.}}
|
|
\newcommand{\eg}{\textit{e.g.}}
|
|
|
|
% operators
|
|
\newcommand{\hH}{\Hat{H}}
|
|
\newcommand{\hh}{\Hat{h}}
|
|
\newcommand{\hT}{\Hat{T}}
|
|
\newcommand{\vne}{v_\text{ne}}
|
|
\newcommand{\hWee}{\Hat{W}_\text{ee}}
|
|
\newcommand{\WHF}{W_\text{HF}}
|
|
|
|
% functionals, potentials, densities, etc
|
|
\newcommand{\eps}{\epsilon}
|
|
\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
|
|
\newcommand{\E}[2]{E_\text{#1}^{#2}}
|
|
\newcommand{\bE}[2]{\overline{E}_\text{#1}^{#2}}
|
|
\newcommand{\be}[2]{\overline{\eps}_\text{#1}^{#2}}
|
|
\newcommand{\bv}[2]{\overline{f}_\text{#1}^{#2}}
|
|
\newcommand{\n}[2]{n_{#1}^{#2}}
|
|
\newcommand{\dn}[2]{\Delta n_{#1}^{#2}}
|
|
\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}}
|
|
\newcommand{\Ecat}{E_\text{cat}}
|
|
\newcommand{\Eneu}{E_\text{neu}}
|
|
\newcommand{\Eani}{E_\text{ani}}
|
|
\newcommand{\EPT}{E_\text{PT2}}
|
|
\newcommand{\EFCI}{E_\text{FCI}}
|
|
|
|
% matrices/operator
|
|
\newcommand{\br}[1]{\boldsymbol{r}_{#1}}
|
|
\newcommand{\bx}[1]{\boldsymbol{x}_{#1}}
|
|
\newcommand{\bw}{{\boldsymbol{w}}}
|
|
\newcommand{\bG}{\boldsymbol{G}}
|
|
\newcommand{\bS}{\boldsymbol{S}}
|
|
\newcommand{\bGam}[1]{\boldsymbol{\Gamma}^{#1}}
|
|
\newcommand{\bgam}[1]{\boldsymbol{\gamma}^{#1}}
|
|
\newcommand{\opGam}[1]{\hat{\Gamma}^{#1}}
|
|
\newcommand{\bh}{\boldsymbol{h}}
|
|
\newcommand{\bF}[1]{\boldsymbol{F}^{#1}}
|
|
\newcommand{\Ex}[2]{\Omega_\text{#1}^{#2}}
|
|
|
|
|
|
% elements
|
|
\newcommand{\ew}[1]{w_{#1}}
|
|
\newcommand{\eG}[1]{G_{#1}}
|
|
\newcommand{\eS}[1]{S_{#1}}
|
|
\newcommand{\eGam}[2]{\Gamma_{#1}^{#2}}
|
|
\newcommand{\hGam}[2]{\Hat{\Gamma}_{#1}^{#2}}
|
|
\newcommand{\eh}[2]{h_{#1}^{#2}}
|
|
\newcommand{\eF}[2]{F_{#1}^{#2}}
|
|
\newcommand{\ERI}[2]{(#1|#2)}
|
|
\newcommand{\dbERI}[2]{(#1||#2)}
|
|
|
|
% Numbers
|
|
\newcommand{\nEl}{N}
|
|
\newcommand{\nBas}{K}
|
|
|
|
% AO and MO basis
|
|
\newcommand{\Det}[1]{\Phi^{#1}}
|
|
\newcommand{\MO}[2]{\phi_{#1}^{#2}}
|
|
\newcommand{\SO}[2]{\varphi_{#1}^{#2}}
|
|
\newcommand{\cMO}[2]{c_{#1}^{#2}}
|
|
\newcommand{\AO}[1]{\chi_{#1}}
|
|
|
|
% units
|
|
\newcommand{\IneV}[1]{#1~eV}
|
|
\newcommand{\InAU}[1]{#1~a.u.}
|
|
\newcommand{\InAA}[1]{#1~\AA}
|
|
|
|
\newcommand{\SI}{\textcolor{blue}{supplementary material}}
|
|
|
|
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
|
\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
|
|
|
|
%%% added by Manu %%%
|
|
\newcommand{\beq}{\begin{equation}}
|
|
\newcommand{\eeq}{\end{equation}}
|
|
\newcommand{\bmk}{\boldsymbol{\kappa}} % orbital rotation vector
|
|
\newcommand{\bmg}{\boldsymbol{\Gamma}} % orbital rotation vector
|
|
\newcommand{\bxi}{\boldsymbol{\xi}}
|
|
\newcommand{\bfx}{{\bf{x}}}
|
|
\newcommand{\bfr}{{\bf{r}}}
|
|
\DeclareMathOperator*{\argmax}{arg\,max}
|
|
\DeclareMathOperator*{\argmin}{arg\,min}
|
|
%%%%
|
|
|
|
\begin{document}
|
|
|
|
\title{Weight-dependent local density-functional approximations for ensembles}
|
|
|
|
\author{Pierre-Fran\c{c}ois Loos}
|
|
\email{loos@irsamc.ups-tlse.fr}
|
|
\affiliation{\LCPQ}
|
|
\author{Emmanuel Fromager}
|
|
\email{fromagere@unistra.fr}
|
|
\affiliation{\LCQ}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{abstract}
|
|
We report a local, weight-dependent correlation density-functional approximation that incorporates information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
|
|
This density-functional approximation for ensembles is specially
|
|
designed for the computation of single and double excitations within
|
|
Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for neutral
|
|
excitations), and can be seen as a natural extension of the ubiquitous local-density approximation in the context of ensembles.
|
|
The resulting density-functional approximation, based on both finite and infinite uniform electron gas models, automatically incorporates the infamous derivative discontinuity contributions to the excitation energies through its explicit ensemble weight dependence.
|
|
Its accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
|
|
Although the present weight-dependent functional has been specifically
|
|
designed for one-dimensional systems, the methodology proposed here is
|
|
general, \ie, directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.
|
|
\end{abstract}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\maketitle
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Introduction}
|
|
\label{sec:intro}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
Over the last two decades, density-functional theory (DFT)
|
|
\cite{Hohenberg_1964,Kohn_1965,ParrBook} has become the method of choice for
|
|
modeling the electronic structure of large molecular systems and
|
|
materials.
|
|
The main reason is that, within DFT, the quantum contributions to the
|
|
electronic repulsion energy --- the so-called exchange-correlation (xc)
|
|
energy --- is rewritten as a functional of the electron density $\n{}{} \equiv \n{}{}(\br{})$, the latter being a much simpler quantity than the many-electron wave function.
|
|
The complexity of the many-body problem is then transferred to the xc
|
|
density functional.
|
|
Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} (KS-DFT) suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003,Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tapavicza_2008,Levine_2006}
|
|
The description of strongly multiconfigurational ground states (often
|
|
referred to as ``strong correlation problem'') still remains a
|
|
challenge. \cite{Gori-Giorgi_2010,Fromager_2015,Gagliardi_2017}
|
|
Another issue, which is partly connected to the previous one, is the
|
|
description of low-lying quasi-degenerate states.
|
|
|
|
The standard approach for modeling excited states in a DFT framework is
|
|
linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,Casida,Casida_2012}
|
|
In this case, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, which may break down when electron correlation is strong.
|
|
Moreover, in exact TDDFT, the xc energy is in fact an xc {\it action} \cite{Vignale_2008} which is a
|
|
functional of the time-dependent density $\n{}{} \equiv \n{}{}(\br,t)$ and, as
|
|
such, it should incorporate memory effects. Standard implementations of TDDFT rely on
|
|
the adiabatic approximation where these effects are neglected. \cite{Dreuw_2005} In other
|
|
words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012}
|
|
As a result, double electronic excitations (where two electrons are simultaneously promoted by a single photon) are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
|
|
|
|
When affordable (\ie, for relatively small molecules), time-independent
|
|
state-averaged wave function methods
|
|
\cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002,Helgakerbook} can be employed to fix the various issues mentioned above.
|
|
The basic idea is to describe a finite (canonical) ensemble of ground
|
|
and excited states altogether, \ie, with the same set of orbitals.
|
|
Interestingly, a similar approach exists in DFT. Referred to as
|
|
Gross--Oliveira--Kohn (GOK) DFT, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} it was proposed at the end of the 80's as a generalization
|
|
of Theophilou's DFT for equiensembles. \cite{Theophilou_1979}
|
|
In GOK-DFT, the ensemble xc energy is a functional of the
|
|
density {\it and} a
|
|
function of the ensemble weights. Note that, unlike in conventional
|
|
Boltzmann ensembles, \cite{Pastorczak_2013} the ensemble weights (each state in the ensemble
|
|
is assigned a given and fixed weight) are allowed to vary
|
|
independently in a GOK ensemble.
|
|
The weight dependence of the xc functional plays a crucial role in the
|
|
calculation of excitation energies.
|
|
\cite{Gross_1988b,Yang_2014,Deur_2017,Deur_2019,Senjean_2018,Senjean_2020}
|
|
It actually accounts for the 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?}
|
|
|
|
Even though GOK-DFT is in principle able to
|
|
describe near-degenerate situations and multiple-electron excitation
|
|
processes, it has not
|
|
been given much attention until quite 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}
|
|
One of the reason is the lack, not to say the absence, of reliable
|
|
density-functional approximations for ensembles.
|
|
The most recent works dealing with this particular issue are still fundamental and
|
|
exploratory, as they rely either on simple (but nontrivial) model
|
|
systems
|
|
\cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018,Senjean_2020,Fromager_2020,Gould_2019}
|
|
or atoms. \cite{Yang_2014,Yang_2017,Gould_2019_insights}
|
|
Despite all these efforts, it is still unclear how weight dependencies
|
|
can be incorporated into density-functional approximations. This problem is actually central not
|
|
only in GOK-DFT but also in conventional (ground-state) DFT as the infamous derivative
|
|
discontinuity problem that occurs when crossing an integral number of
|
|
electrons can be recast into a weight-dependent ensemble
|
|
one. \cite{Senjean_2018,Senjean_2020}
|
|
|
|
The present work is an attempt to address the ensemble weight dependence problem
|
|
in GOK-DFT,
|
|
with the ambition to turn the theory, in the forthcoming future, into a
|
|
(low-cost) practical computational method for modeling excited states in molecules and extended systems.
|
|
Starting from the ubiquitous local-density approximation (LDA), we
|
|
design a weight-dependent ensemble correction based on a finite uniform
|
|
electron gas from which density-functional excitation energies can be
|
|
extracted. The present density-functional approximation for ensembles, which can be seen as a natural
|
|
extension of the LDA, will be referred to as eLDA in the remaining of this paper.
|
|
As a proof of concept, we apply this general strategy to
|
|
ensemble correlation energies (that we combine with
|
|
ensemble exact exchange energies) in the particular case of
|
|
\emph{strict} one-dimensional (1D)
|
|
spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
|
|
In other words, the Coulomb interaction used in this work corresponds to
|
|
particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
|
|
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}
|
|
%Early models of 1D atoms using this interaction have been used to study the effects of external fields upon Rydberg atoms \cite{Burnett_1993, Mayle_2007} and the dynamics of surface-state electrons in liquid helium. \cite{Nieto_2000, Patil_2001}
|
|
This description of 1D systems also has interesting connections with the exotic chemistry of ultra-high magnetic fields (such as those in white dwarf stars), where the electronic cloud is dramatically compressed perpendicular to the magnetic field. \cite{Schmelcher_1990, Lange_2012, Schmelcher_2012}
|
|
In these extreme conditions, where magnetic effects compete with Coulombic forces, entirely new bonding paradigms emerge. \cite{Schmelcher_1990, Schmelcher_1997, Tellgren_2008, Tellgren_2009, Lange_2012, Schmelcher_2012, Boblest_2014, Stopkowicz_2015}
|
|
|
|
The paper is organized as follows.
|
|
Exact and approximate formulations of GOK-DFT are discussed in Sec.~\ref{sec:eDFT},
|
|
with a particular emphasis on the extraction of individual energy levels.
|
|
In Sec.~\ref{sec:eDFA}, we detail the construction of the
|
|
weight-dependent local correlation functional specially designed for the
|
|
computation of single and double excitations within GOK-DFT.
|
|
Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
|
|
In Sec.~\ref{sec:res}, we illustrate the accuracy of the present eLDA functional by computing single and double excitations in 1D many-electron systems in the weak, intermediate and strong correlation regimes.
|
|
Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
|
|
Atomic units are used throughout.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Theory}
|
|
\label{sec:eDFT}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{GOK-DFT}\label{subsec:gokdft}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
In this section we give a brief review of GOK-DFT and discuss the
|
|
extraction of individual energy levels \cite{Deur_2019,Fromager_2020} with a particular focus on exact
|
|
individual exchange energies.
|
|
Let us start by introducing the GOK ensemble energy \cite{Gross_1988a}
|
|
\beq\label{eq:exact_GOK_ens_ener}
|
|
\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
|
|
\eeq
|
|
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
|
|
\beq
|
|
\hh = \sum_{i=1}^\nEl \qty[ -\frac{1}{2} \nabla_{i}^2 + \vne(\br{i}) ]
|
|
\eeq
|
|
is the one-electron operator describing kinetic and nuclear attraction energies, and $\hat{W}_{\rm ee}$ is the electron repulsion operator.
|
|
The (positive) ensemble weights $\ew{K}$ decrease with increasing index $K$.
|
|
They are normalized, \ie,
|
|
\beq\label{eq:weight_norm_cond}
|
|
\ew{0} = 1 - \sum_{K>0} \ew{K},
|
|
\eeq
|
|
so that only the weights $\bw \equiv \qty( \ew{1}, \ew{2}, \ldots, \ew{K}, \ldots )$ assigned to the excited states can vary independently.
|
|
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{Gross_1988b}
|
|
In the KS formulation of GOK-DFT, {which is simply referred to as
|
|
KS ensemble DFT (KS-eDFT) in the following}, the ensemble energy is determined variationally as follows:\cite{Gross_1988b}
|
|
\beq\label{eq:var_ener_gokdft}
|
|
\E{}{\bw}
|
|
= \min_{\opGam{\bw}}
|
|
\qty{
|
|
\Tr[\opGam{\bw} \hh] + \E{Hx}{\bw} \qty[\n{\opGam{\bw}}{}] + \E{c}{\bw} \qty[\n{\opGam{\bw}}{}]
|
|
},
|
|
\eeq
|
|
where $\Tr$ denotes the trace and the trial ensemble density matrix operator reads
|
|
\beq
|
|
\opGam{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}.
|
|
\eeq
|
|
The KS determinants [or configuration state functions~\cite{Gould_2017}]
|
|
$\Det{(K)}$ are all constructed from the same set of ensemble KS
|
|
orbitals that are variationally optimized.
|
|
The trial ensemble density in Eq.~\eqref{eq:var_ener_gokdft} is simply
|
|
the weighted sum of the individual KS densities, \ie,
|
|
\beq\label{eq:KS_ens_density}
|
|
\n{\opGam{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K)}}{}(\br{}).
|
|
\eeq
|
|
As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange (Hx) and correlation (c) energies are described with density functionals that are \textit{weight dependent}.
|
|
We focus in the following on the (exact) Hx part, which is defined as~\cite{Gould_2017}
|
|
\beq\label{eq:exact_ens_Hx}
|
|
\E{Hx}{\bw}[\n{}{}]=\sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bw}[\n{}{}]}{\hWee}{\Det{(K),\bw}[\n{}{}]},
|
|
\eeq
|
|
where the KS wavefunctions fulfill the ensemble density constraint
|
|
\beq
|
|
\sum_{K\geq 0} \ew{K} \n{\Det{(K),\bw}[\n{}{}]}{}(\br{}) = \n{}{}(\br{}).
|
|
\eeq
|
|
The (approximate) description of the correlation part is discussed in
|
|
Sec.~\ref{sec:eDFA}.
|
|
|
|
In practice, the ensemble energy is not the most interesting quantity, and one is more concerned with excitation energies or individual energy levels (for geometry optimizations, for example).
|
|
As pointed out recently in Ref.~\onlinecite{Deur_2019}, the latter can be extracted
|
|
exactly from a single ensemble calculation as follows:
|
|
\beq\label{eq:indiv_ener_from_ens}
|
|
\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} )
|
|
\pdv{\E{}{\bw}}{\ew{K}},
|
|
\eeq
|
|
where, according to the normalization condition of Eq.~\eqref{eq:weight_norm_cond},
|
|
\beq
|
|
\pdv{\E{}{\bw}}{\ew{K}}= \E{}{(K)} -
|
|
\E{}{(0)}\equiv\Ex{}{(K)}
|
|
\eeq
|
|
corresponds to the $K$th excitation energy.
|
|
According to the {\it variational} ensemble energy expression of
|
|
Eq.~\eqref{eq:var_ener_gokdft}, the derivative with respect to $\ew{K}$
|
|
can be evaluated from the minimizing weight-dependent KS wavefunctions
|
|
$\Det{(K)} \equiv \Det{(K),\bw}$ as follows:
|
|
\beq\label{eq:deriv_Ew_wk}
|
|
\begin{split}
|
|
\pdv{\E{}{\bw}}{\ew{K}}
|
|
& = \mel*{\Det{(K)}}{\hh}{\Det{(K)}}-\mel*{\Det{(0)}}{\hh}{\Det{(0)}}
|
|
\\
|
|
& + \Bigg\{\int \fdv{\E{Hx}{\bw}[\n{}{}]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ] d\br{}
|
|
+ \pdv{\E{Hx}{\bw} [\n{}{}]}{\ew{K}}
|
|
\\
|
|
& + \int \fdv{\E{c}{\bw}[n]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ] d\br{}
|
|
+ \pdv{\E{c}{\bw}[n]}{\ew{K}}
|
|
\Bigg\}_{\n{}{} = \n{\opGam{\bw}}{}}.
|
|
\end{split}
|
|
\eeq
|
|
The Hx contribution from Eq.~\eqref{eq:deriv_Ew_wk} can be recast as
|
|
\beq\label{eq:_deriv_wk_Hx}
|
|
\left.
|
|
\pdv{}{\xi_K} \qty(\E{Hx}{\bxi} [\n{}{\bxi,\bxi}]
|
|
- \E{Hx}{\bw}[\n{}{\bw,\bxi}] )
|
|
\right|_{\bxi=\bw},
|
|
\eeq
|
|
where $\bxi \equiv (\xi_1,\xi_2,\ldots,\xi_K,\ldots)$ and the
|
|
auxiliary double-weight ensemble density reads
|
|
\beq
|
|
\n{}{\bw,\bxi}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K),\bxi}}{}(\br{}).
|
|
\eeq
|
|
Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble
|
|
densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are obtained from the \textit{same} KS potential (which is unique up to a constant), it comes
|
|
from the exact expression in Eq.~\eqref{eq:exact_ens_Hx} that
|
|
\beq
|
|
\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
|
|
\eeq
|
|
and
|
|
\beq
|
|
\E{Hx}{\bw}[\n{}{\bw,\bxi}] = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}.
|
|
\eeq
|
|
This yields, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_Hx}, the simplified expression
|
|
\beq\label{eq:deriv_Ew_wk_simplified}
|
|
\begin{split}
|
|
\pdv{\E{}{\bw}}{\ew{K}}
|
|
& = \mel*{\Det{(K)}}{\hH}{\Det{(K)}}
|
|
- \mel*{\Det{(0)}}{\hH}{\Det{(0)}}
|
|
\\
|
|
& + \qty{
|
|
\int \fdv{\E{c}{\bw}[\n{}{}]}{\n{}{}({\br{}})}
|
|
\qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
|
|
+
|
|
\pdv{\E{c}{\bw} [\n{}{}]}{\ew{K}}
|
|
}_{\n{}{} = \n{\opGam{\bw}}{}} d\br{}.
|
|
\end{split}
|
|
\eeq
|
|
Since, according to Eqs.~\eqref{eq:var_ener_gokdft} and \eqref{eq:exact_ens_Hx}, the ensemble energy can be evaluated as
|
|
\beq
|
|
\E{}{\bw} = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}}{\hH}{\Det{(K)}} + \E{c}{\bw}[\n{\opGam{\bw}}{}],
|
|
\eeq
|
|
with $\Det{(K)} = \Det{(K),\bw}$ [note that, when the minimum is reached in Eq.~\eqref{eq:var_ener_gokdft}, $\n{\opGam{\bw}}{} = \n{}{\bw,\bw}$],
|
|
we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
|
|
\eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\onlinecite{Fromager_2020} for the $I$th energy level:
|
|
\beq\label{eq:exact_ener_level_dets}
|
|
\begin{split}
|
|
\E{}{(I)}
|
|
& = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{{\bw}}[\n{\opGam{\bw}}{}]
|
|
\\
|
|
& + \int \fdv{\E{c}{\bw}[\n{\opGam{\bw}}{}]}{\n{}{}(\br{})}
|
|
\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGam{\bw}}{}(\br{}) ] d\br{}
|
|
\\
|
|
&+
|
|
\sum_{K>0} \qty(\delta_{IK} - \ew{K} )
|
|
\left.
|
|
\pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}
|
|
\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
|
|
\end{split}
|
|
\eeq
|
|
Note that, when $\bw=0$, the ensemble correlation functional reduces to the
|
|
conventional (ground-state) correlation functional $E_{\rm c}[n]$. As a
|
|
result, the regular KS-DFT expression is recovered from
|
|
Eq.~\eqref{eq:exact_ener_level_dets} for the ground-state energy:
|
|
\beq
|
|
\E{}{(0)}=\mel*{\Det{(0)}}{\hH}{\Det{(0)}} +
|
|
\E{c}{}[\n{\Det{(0)}}{}],
|
|
\eeq
|
|
or, equivalently,
|
|
\beq\label{eq:gs_ener_level_gs_lim}
|
|
\E{}{(0)}=\mel*{\Det{(0)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(0)}}
|
|
,
|
|
\eeq
|
|
where the density-functional Hamiltonian reads
|
|
\beq\label{eq:dens_func_Hamilt}
|
|
\hat{H}[n]=\hH+
|
|
\sum^N_{i=1}\left(\fdv{\E{c}{}[n]}{\n{}{}(\br{i})}
|
|
+C_{\rm c}[n]
|
|
\right),
|
|
\eeq
|
|
and
|
|
\beq\label{eq:corr_LZ_shift}
|
|
C_{\rm c}[n]=\dfrac{\E{c}{}[n]
|
|
-\int
|
|
\fdv{\E{c}{}[n]}{\n{}{}(\br{})}n(\br{})d\br{}}{\int n(\br{})d\br{}}
|
|
\eeq
|
|
is the correlation component of
|
|
Levy--Zahariev's constant shift in potential.\cite{Levy_2014}
|
|
Similarly, the excited-state ($I>0$) energy level expressions
|
|
can be recast as follows:
|
|
\beq\label{eq:excited_ener_level_gs_lim}
|
|
\E{}{(I)}
|
|
= \mel*{\Det{(I)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(I)}}
|
|
+
|
|
\left.
|
|
\pdv{\E{c}{\bw}[\n{\Det{(0)}}{}]}{\ew{I}}
|
|
\right|_{\bw=0}.
|
|
\eeq
|
|
As readily seen from Eqs.~\eqref{eq:dens_func_Hamilt} and
|
|
\eqref{eq:corr_LZ_shift}, introducing any constant shift $\delta
|
|
\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})\rightarrow \delta
|
|
\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})+C$ into the correlation
|
|
potential leaves the density-functional Hamiltonian $\hat{H}[n]$ (and
|
|
therefore the individual energy levels) unchanged. As a result, in
|
|
this context,
|
|
the correlation derivative discontinuities induced by the
|
|
excitation process~\cite{Levy_1995} will be fully described by the
|
|
correlation ensemble derivatives [second term on the right-hand side of
|
|
Eq.~\eqref{eq:excited_ener_level_gs_lim}].
|
|
|
|
%%%%%%%%%%%%%%%%
|
|
\subsection{One-electron reduced density matrix formulation}
|
|
%%%%%%%%%%%%%%%%
|
|
For implementation purposes, we will use in the rest of this work
|
|
(one-electron reduced) density matrices
|
|
as basic variables, rather than Slater determinants.
|
|
As the theory is applied later on to \textit{spin-polarized}
|
|
systems, we drop spin indices in the density matrices, for convenience.
|
|
If we expand the
|
|
ensemble KS orbitals (from which the determinants are constructed) in an atomic orbital (AO) basis,
|
|
\beq
|
|
\MO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
|
|
\eeq
|
|
\iffalse%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\titou{\beq
|
|
\SO{p}{}(\bx{}) = s(\omega) \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
|
|
\eeq
|
|
where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatial degrees of freedom, and
|
|
\beq
|
|
s(\omega)
|
|
=
|
|
\begin{cases}
|
|
\alpha(\omega), & \text{for spin-up electrons,} \\
|
|
\text{or} \\
|
|
\beta(\omega), & \text{for spin-down electrons,}
|
|
\end{cases}
|
|
\eeq
|
|
}
|
|
\fi%%%%%%%%%%%%%%%%%%%%%
|
|
then the density matrix of the
|
|
determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
|
|
\beq
|
|
\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
|
|
\eeq
|
|
where the summation runs over the orbitals that are occupied in $\Det{(K)}$.
|
|
The electron density of the $K$th KS determinant can then be evaluated
|
|
as follows:
|
|
\beq
|
|
\n{\bGam{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}),
|
|
\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)}
|
|
\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)}
|
|
\nonumber\\
|
|
&=&\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{})
|
|
\nonumber\\
|
|
&=&\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
|
|
}
|
|
\fi%%%
|
|
%%%% end Manu
|
|
while the ensemble density matrix
|
|
and the ensemble density read
|
|
\beq
|
|
\bGam{\bw}
|
|
= \sum_{K\geq 0} \ew{K} \bGam{(K)}
|
|
\equiv \eGam{\mu\nu}{\bw}
|
|
= \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)},
|
|
\eeq
|
|
and
|
|
\beq
|
|
\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
|
|
\eeq
|
|
respectively.
|
|
The exact individual energy expression in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as
|
|
\beq\label{eq:exact_ind_ener_rdm}
|
|
\begin{split}
|
|
\E{}{(I)}
|
|
& =\Tr[\bGam{(I)} \bh]
|
|
+ \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
|
|
+ \E{c}{{\bw}}[\n{\bGam{\bw}}{}]
|
|
\\
|
|
& + \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})}
|
|
\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ] d\br{}
|
|
\\
|
|
& + \sum_{K>0} \qty(\delta_{IK} - \ew{K})
|
|
\left. \pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}\right|_{\n{}{} = \n{\bGam{\bw}}{}}
|
|
,
|
|
\end{split}
|
|
\eeq
|
|
where
|
|
\beq
|
|
\bh \equiv h_{\mu\nu} = \mel*{\AO{\mu}}{\hh}{\AO{\nu}}
|
|
\eeq
|
|
denotes the matrix of the one-electron integrals.
|
|
The exact individual Hx energies are obtained from the following trace formula
|
|
\beq
|
|
\Tr[\bGam{(K)} \bG \bGam{(L)}]
|
|
= \sum_{\mu\nu\la\si} \eGam{\mu\nu}{(K)} \eG{\mu\nu\la\si} \eGam{\la\si}{(L)},
|
|
\eeq
|
|
where the antisymmetrized two-electron integrals read
|
|
\beq
|
|
\bG
|
|
\equiv G_{\mu\nu\la\si}
|
|
= \dbERI{\mu\nu}{\la\si}
|
|
= \ERI{\mu\nu}{\la\si} - \ERI{\mu\si}{\la\nu},
|
|
\eeq
|
|
with
|
|
\beq
|
|
\ERI{\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}.
|
|
\eeq
|
|
%Note that, in Sec.~\ref{sec:results}, the theory is applied to (1D) spin
|
|
%polarized systems in which $\eGam{\mu\nu}{(K)\beta}=0$ and
|
|
%$G_{\mu\nu\lambda\omega}^{\alpha\alpha}\equiv G_{\mu\nu\lambda\omega}=({\mu}{\nu}\vert{\lambda}{\omega})
|
|
%-(\mu\omega\vert\lambda\nu)$.
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%%%%%%%%%%%%%%% Hx energy ...
|
|
%%% Manu's derivation
|
|
\iffalse%%%%
|
|
\blue{
|
|
\beq
|
|
&&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert
|
|
RS\rangle\eGam{PR}^{(K)}\eGam{QS}^{(L)}
|
|
\nonumber\\
|
|
&&
|
|
=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}RS}
|
|
\nonumber\\
|
|
&&\Big(\langle p^\sigma\sigma q^\tau\tau\vert RS\rangle -\langle
|
|
p^\sigma\sigma q^\tau\tau
|
|
\vert SR\rangle
|
|
\Big)\Gamma^{(K)}_{p^\sigma\sigma,R}\Gamma^{(L)}_{q^\tau\tau, S}
|
|
\nonumber\\
|
|
&&
|
|
=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
|
|
\nonumber\\
|
|
&&\Big(\sum_{r^\sigma s^\tau}\langle p^\sigma q^\tau\vert r^\sigma s^\tau\rangle
|
|
\Gamma^{(K)\sigma}_{p^\sigma r^\sigma}\Gamma^{(L)\tau}_{q^\tau s^\tau}
|
|
\nonumber\\
|
|
&& -\sum_{s^\sigma r^\tau}\langle
|
|
p^\sigma q^\tau
|
|
\vert s^\sigma r^\tau\rangle
|
|
\delta_{\sigma\tau}\Gamma^{(K)\sigma}_{p^\sigma
|
|
r^\sigma}\Gamma^{(L)\sigma}_{q^\sigma s^\sigma}\Big)
|
|
\nonumber\\
|
|
&&=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
|
|
\nonumber\\
|
|
&&\left(\langle p^\sigma q^\tau\vert p^\sigma q^\tau\rangle
|
|
n_{p^\sigma}^{(K)\sigma}n_{q^\tau}^{(L)\tau}
|
|
-\delta_{\sigma\tau}\langle p^\sigma q^\sigma\vert q^\sigma p^\sigma \rangle
|
|
n_{p^\sigma}^{(K)\sigma}n_{q^\sigma}^{(L)\sigma}\right)
|
|
\nonumber\\
|
|
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
|
|
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
|
|
-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\sigma}
|
|
\Big)
|
|
\nonumber\\
|
|
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
|
|
-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
|
|
\Big)
|
|
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
|
|
\nonumber\\
|
|
&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big[({\mu}{\nu}\vert{\lambda}{\omega})
|
|
-\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu)
|
|
\Big]
|
|
\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
|
|
\eeq
|
|
}
|
|
\fi%%%%%%%
|
|
%%%%
|
|
%%%%%%%%%%%%%%%%%%%%%
|
|
\iffalse%%%% Manu's derivation ...
|
|
\blue{
|
|
\beq
|
|
n^{\bw}({\br{}})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
|
|
w}_Kn^{(K)}({\bfx})
|
|
\nonumber\\
|
|
&=&
|
|
\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
|
|
w}_K\sum_{pq}\varphi_p({\bfx})\varphi_q({\bfx})\Gamma_{pq}^{(K)}
|
|
\nonumber\\
|
|
&=&
|
|
\sum_{\sigma=\alpha,\beta}
|
|
\sum_{K\geq 0}
|
|
{\tt
|
|
w}_K\sum_{p\in (K)}\varphi^2_p({\bfx})
|
|
\nonumber\\
|
|
&=&
|
|
\sum_{\sigma=\alpha,\beta}
|
|
\sum_{K\geq 0}
|
|
{\tt
|
|
w}_K
|
|
\sum_{\mu\nu}
|
|
\sum_{p\in (K)}c_{\mu p}c_{\nu p}\AO{\mu}({\bfx})\AO{\nu}({\bfx})
|
|
\nonumber\\
|
|
&=&\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\bfx})\AO{\nu}({\bfx}){\Gamma}^{\bw}_{\mu\nu}
|
|
\eeq
|
|
}
|
|
\fi%%%%%%%% end
|
|
%%%%%%%%%%%%%%%
|
|
%\subsection{Hybrid GOK-DFT}
|
|
%%%%%%%%%%%%%%%
|
|
|
|
|
|
%%%%%%%%%%%%%%%
|
|
\subsection{Approximations}\label{subsec:approx}
|
|
%%%%%%%%%%%%%%%
|
|
|
|
|
|
In the following, GOK-DFT will be applied
|
|
to 1D
|
|
spin-polarized systems where
|
|
Hartree and exchange energies cannot be separated.
|
|
For that reason, we will substitute the Hartree--Fock (HF) density-matrix-functional interaction energy,
|
|
\beq\label{eq:eHF-dens_mat_func}
|
|
\WHF[\bGam{}] = \frac{1}{2} \Tr[\bGam{} \bG \bGam{}],
|
|
\eeq
|
|
for the Hx density-functional energy in the variational energy
|
|
expression of Eq.~\eqref{eq:var_ener_gokdft}, thus leading to the
|
|
following approximation:
|
|
\beq\label{eq:min_with_HF_ener_fun}
|
|
\bGam{\bw}
|
|
\rightarrow \argmin_{\bgam{\bw}}
|
|
\qty{
|
|
\Tr[\bgam{\bw} \bh ] + \WHF[ \bgam{\bw}] + \E{c}{\bw}[\n{\bgam{\bw}}{}]
|
|
}.
|
|
\eeq
|
|
The minimizing ensemble density matrix in Eq.~\eqref{eq:min_with_HF_ener_fun} fulfills the following
|
|
stationarity condition
|
|
\beq\label{eq:commut_F_AO}
|
|
\bF{\bw} \bGam{\bw} \bS = \bS \bGam{\bw} \bF{\bw},
|
|
\eeq
|
|
where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the
|
|
overlap matrix and the ensemble Fock-like matrix reads
|
|
\beq
|
|
\bF{\bw} \equiv \eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} +
|
|
\sum_{\la\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw},
|
|
\eeq
|
|
with
|
|
\beq
|
|
\eh{\mu\nu}{\bw}
|
|
= \eh{\mu\nu}{} + \int \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}) d\br{}.
|
|
\eeq
|
|
%%%%%%%%%%%%%%%
|
|
\iffalse%%%%%%
|
|
% Manu's derivation %%%%
|
|
\color{blue}
|
|
I am teaching myself ...\\
|
|
Stationarity condition
|
|
\beq
|
|
&&0=\sum_{K\geq 0}w_K\sum_{t^\sigma}\Big(f_{p^\sigma\sigma,t^\sigma\sigma}\Gamma^{(K)\sigma}_{t^\sigma
|
|
q^\sigma}-\Gamma^{(K)\sigma}_{p^\sigma
|
|
t^\sigma}f_{t^\sigma\sigma,q^\sigma\sigma}\Big)
|
|
\nonumber\\
|
|
&&=\sum_{K\geq 0}w_K
|
|
\Big(f_{p^\sigma\sigma,q^\sigma\sigma}n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}f_{p^\sigma\sigma,q^\sigma\sigma}\Big)
|
|
\nonumber\\
|
|
&&
|
|
=\sum_{\mu\nu}\sum_{K\geq 0}w_KF_{\mu\nu}^\sigma c^\sigma_{\mu
|
|
p}c^\sigma_{\nu q}\left(n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}\right)
|
|
\eeq
|
|
thus leading to
|
|
\beq
|
|
&&0=\sum_{p^\sigma q^\sigma}c^\sigma_{\lambda
|
|
p}c^\sigma_{\omega q}\left(\sum_{\mu\nu}\sum_{K\geq 0}w_KF_{\mu\nu}^\sigma c^\sigma_{\mu
|
|
p}c^\sigma_{\nu q}\left(n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}\right)\right)
|
|
\nonumber\\
|
|
&&=\sum_{\mu\nu}\sum_{K\geq 0}w_K
|
|
F_{\mu\nu}^\sigma\left(\Gamma^{(K)\sigma}_{\nu\omega}\sum_{p^\sigma}c^\sigma_{\lambda
|
|
p}c^\sigma_{\mu
|
|
p}-\Gamma^{(K)\sigma}_{\mu\lambda}\sum_{q^\sigma}c^\sigma_{\omega q}c^\sigma_{\nu q}\right)
|
|
\nonumber\\
|
|
\eeq
|
|
If we denote $M^\sigma_{\lambda\mu}=\sum_{p^\sigma}c^\sigma_{\lambda
|
|
p}c^\sigma_{\mu
|
|
p}$ it comes
|
|
\beq
|
|
S_{\mu\nu}=\sum_{\lambda\omega}S_{\mu\lambda}M^\sigma_{\lambda\omega}S_{\omega\nu}
|
|
\eeq
|
|
which simply means that
|
|
\beq
|
|
{\bm S}={\bm S}{\bm M}{\bm S}
|
|
\eeq
|
|
or, equivalently,
|
|
\beq
|
|
{\bm M}={\bm S}^{-1}.
|
|
\eeq
|
|
The stationarity condition simply reads
|
|
\beq
|
|
\sum_{\mu\nu}F_{\mu\nu}^\sigma\left(\Gamma^{\bw\sigma}_{\nu\omega}
|
|
\left[{\bm S}^{-1}\right]_{\lambda\mu}
|
|
-\Gamma^{\bw\sigma}_{\mu\lambda}\left[{\bm S}^{-1}\right]_{\omega\nu}\right)
|
|
=0
|
|
\eeq
|
|
thus leading to
|
|
\beq
|
|
{\bm S}^{-1}{{\bm F}^\sigma}{\bm \Gamma}^{\bw\sigma}={\bm \Gamma}^{\bw\sigma}{{\bm F}^\sigma}{\bm S}^{-1}
|
|
\eeq
|
|
or, equivalently,
|
|
\beq
|
|
{{\bm F}^\sigma}{\bm \Gamma}^{\bw\sigma}{\bm S}={\bm S}{\bm
|
|
\Gamma}^{\bw\sigma}{{\bm F}^\sigma}.
|
|
\eeq
|
|
%%%%%
|
|
|
|
Fock operator:\\
|
|
\beq
|
|
&&f_{p^\sigma\sigma,q^\sigma\sigma}-\langle\varphi_p^\sigma\vert\hat{h}\vert\varphi_q^\sigma\rangle
|
|
\nonumber\\
|
|
&&=\sum_{L\geq 0}w_L\sum_{\tau}\sum_{r^\tau s^\tau}
|
|
\nonumber\\
|
|
&&
|
|
\Big(\langle p^\sigma r^\tau\vert
|
|
q^\sigma s^\tau\rangle
|
|
-\delta_{\sigma\tau}\langle p^\sigma r^\sigma\vert
|
|
s^\sigma q^\sigma\rangle
|
|
\Big)
|
|
\Gamma^{(L)\tau}_{r^\tau
|
|
s^\tau}
|
|
\nonumber\\
|
|
&&
|
|
=\sum_{L\geq 0}w_L\sum_{\tau}\sum_{r^\tau}\Big(\langle p^\sigma r^\tau\vert
|
|
q^\sigma r^\tau\rangle
|
|
-\delta_{\sigma\tau}\langle p^\sigma r^\tau\vert
|
|
r^\tau q^\sigma\rangle
|
|
\Big)
|
|
n^{(L)\tau}_{r^\tau}
|
|
\nonumber\\
|
|
&&=\sum_{L\geq 0}w_L
|
|
\sum_{\lambda\omega}\sum_{\tau}\Big[\langle
|
|
p^\sigma\lambda\vert q^\sigma\omega\rangle
|
|
-\delta_{\sigma\tau}
|
|
\langle
|
|
p^\sigma\lambda\vert \omega q^\sigma\rangle\Big]
|
|
\Gamma^{(L)\tau}_{\lambda\omega}
|
|
\nonumber\\
|
|
&&=
|
|
\sum_{\lambda\omega}\sum_{\tau}\Big[\langle
|
|
p^\sigma\lambda\vert q^\sigma\omega\rangle
|
|
-\delta_{\sigma\tau}
|
|
\langle
|
|
p^\sigma\lambda\vert \omega q^\sigma\rangle\Big]
|
|
\Gamma^{\bw\tau}_{\lambda\omega}
|
|
\nonumber\\
|
|
&&=\sum_{\mu\nu\lambda\omega}\sum_{\tau}
|
|
\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
|
|
-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
|
|
\Big)\Gamma^{\bw\tau}_{\lambda\omega}c^\sigma_{\mu p}c^\sigma_{\nu q}
|
|
\nonumber\\
|
|
\eeq
|
|
or, equivalently,
|
|
\beq
|
|
f_{p^\sigma\sigma,q^\sigma\sigma}=\sum_{\mu\nu}F_{\mu\nu}^\sigma c^\sigma_{\mu p}c^\sigma_{\nu q}
|
|
\eeq
|
|
where
|
|
\beq
|
|
F_{\mu\nu}^\sigma=h_{\mu\nu}+\sum_{\lambda\omega}\sum_\tau
|
|
G_{\mu\nu\lambda\omega}^{\sigma\tau}\Gamma^{\bw\tau}_{\lambda\omega}
|
|
\eeq
|
|
and
|
|
\color{black}
|
|
\\
|
|
\fi%%%%%%%%%%%
|
|
%%%%% end Manu
|
|
%%%%%%%%%%%%%%%%%%%%
|
|
Note that, within the approximation of Eq.~\eqref{eq:min_with_HF_ener_fun}, the ensemble density matrix is
|
|
optimized with a non-local exchange potential rather than a
|
|
density-functional local one, as expected from
|
|
Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie,
|
|
applicable to not-necessarily spin polarized and real (higher-dimensional) systems.
|
|
As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the
|
|
ensemble density matrix into the HF interaction energy functional
|
|
introduces unphysical \textit{ghost-interaction} errors \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
|
|
as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
|
|
\beq\label{eq:WHF}
|
|
\begin{split}
|
|
\WHF[\bGam{\bw}]
|
|
& = \frac{1}{2} \sum_{K\geq 0} \ew{K}^2 \Tr[\bGam{(K)} \bG \bGam{(K)}]
|
|
\\
|
|
& + \sum_{L>K\geq 0} \ew{K} \ew{L}\Tr[\bGam{(K)} \bG \bGam{(L)}].
|
|
\end{split}
|
|
\eeq
|
|
The ensemble energy is of course expected to vary linearly with the ensemble
|
|
weights [see Eq.~\eqref{eq:exact_GOK_ens_ener}].
|
|
\manu{
|
|
The explicit linear weight dependence of the ensemble Hx energy is actually restored when evaluating the individual energy
|
|
levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.}
|
|
|
|
Turning to the density-functional ensemble correlation energy, the
|
|
following ensemble local-density approximation (eLDA) will be employed
|
|
\beq\label{eq:eLDA_corr_fun}
|
|
\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
|
|
\eeq
|
|
where the ensemble correlation energy per particle
|
|
\beq\label{eq:decomp_ens_correner_per_part}
|
|
\e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{})
|
|
\eeq
|
|
is explicitly \textit{weight dependent}.
|
|
As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed
|
|
from a finite uniform electron gas model.
|
|
%\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
|
|
%What do you think?}
|
|
|
|
The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}
|
|
reads
|
|
\beq\label{eq:Ew-GIC-eLDA}
|
|
\E{eLDA}{\bw}=\Tr[\bGam{\bw}\bh] + \WHF[\bGam{\bw}] +\int
|
|
\e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{\bw}}{}(\br{}) d\br{}.
|
|
\eeq
|
|
%Manu, would it be useful to add this equation and the corresponding text?
|
|
%I think it is useful for the discussion later on when we talk about the different contributions to the excitation energies.
|
|
%This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative
|
|
Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with
|
|
Eq.~\eqref{eq:eLDA_corr_fun} leads to our final expression of the
|
|
KS-eLDA energy levels
|
|
\beq\label{eq:EI-eLDA}
|
|
\begin{split}
|
|
\E{{eLDA}}{(I)}
|
|
=
|
|
\E{HF}{(I)}
|
|
+ \Xi_\text{c}^{(I)}
|
|
+ \Upsilon_\text{c}^{(I)},
|
|
\end{split}
|
|
\eeq
|
|
where
|
|
\beq\label{eq:ind_HF-like_ener}
|
|
\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
|
|
\eeq
|
|
is the analog for ground and excited states (within an ensemble) of the HF energy, and
|
|
\begin{gather}
|
|
\begin{split}
|
|
\Xi_\text{c}^{(I)}
|
|
& = \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
|
|
\\
|
|
&
|
|
+ \int \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
|
|
\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} =
|
|
\n{\bGam{\bw}}{}(\br{})} d\br{},
|
|
\\
|
|
\end{split}
|
|
\\
|
|
\Upsilon_\text{c}^{(I)}
|
|
= \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
|
|
\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
|
|
\end{gather}
|
|
|
|
If, for analysis purposes, we Taylor expand the density-functional
|
|
correlation contributions
|
|
around the $I$th KS state density
|
|
$\n{\bGam{(I)}}{}(\br{})$, the
|
|
second term on the right-hand side
|
|
of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
|
|
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
|
|
\beq\label{eq:Taylor_exp_ind_corr_ener_eLDA}
|
|
\Xi_\text{c}^{(I)}
|
|
= \int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
|
|
+ \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]^2}.
|
|
\eeq
|
|
Therefore, it can be identified as
|
|
an individual-density-functional correlation energy where the density-functional
|
|
correlation energy per particle is approximated by the ensemble one for
|
|
all the states within the ensemble.
|
|
|
|
|
|
Let us stress that, to the best of our knowledge, eLDA is the first
|
|
density-functional approximation that incorporates ensemble weight
|
|
dependencies explicitly, thus allowing for the description of derivative
|
|
discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
|
|
comment that follows] {\it via} the third term on the right-hand side
|
|
of Eq.~\eqref{eq:EI-eLDA}. According to the decomposition of
|
|
the ensemble
|
|
correlation energy per particle in Eq.
|
|
\eqref{eq:decomp_ens_correner_per_part}, the latter can be recast
|
|
\begin{equation}
|
|
\Upsilon_\text{c}^{(I)}
|
|
%&=
|
|
%\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
|
|
%\Big(\be{c}{(K)}(\n{\bGam{\bw}}{}(\br{}))
|
|
%-
|
|
%\be{c}{(0)}(\n{\bGam{\bw}}{}(\br{}))
|
|
%\Big)
|
|
%d\br{}
|
|
%\\
|
|
=\int
|
|
\qty[\be{c}{(I)}(\n{\bGam{\bw}}{}(\br{}))
|
|
-
|
|
\e{c}{\bw}(\n{\bGam{\bw}}{}(\br{}))
|
|
] \n{\bGam{\bw}}{}(\br{})
|
|
d\br{},
|
|
%\sum_{K>0}\delta_{IK}\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})}
|
|
\end{equation}
|
|
thus leading to the following Taylor expansion through first order in
|
|
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
|
|
\beq\label{eq:Taylor_exp_DDisc_term}
|
|
\begin{split}
|
|
\Upsilon_\text{c}^{(I)}
|
|
%& = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
|
|
% \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
|
|
%\\
|
|
&=
|
|
\int \qty[ \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) - \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) ] \n{\bGam{(I)}}{}(\br{}) d\br{}
|
|
\\
|
|
&+\int \Bigg[
|
|
\n{\bGam{(I)}}{}(\br{})
|
|
\left.\left(
|
|
\pdv{\be{c}{{(I)}}(\n{}{})}{\n{}{}}
|
|
-
|
|
\pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}}
|
|
\right)\right|_{\n{}{} =
|
|
\n{\bGam{(I)}}{}(\br{})}
|
|
\\
|
|
&+\be{c}{(I)}(\n{\bGam{(I)}}{}(\br{}))
|
|
-
|
|
\e{c}{\bw}(\n{\bGam{(I)}}{}(\br{}))\Bigg]
|
|
\qty[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]
|
|
d\br{}
|
|
\\
|
|
&
|
|
+ \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]^2}.
|
|
\end{split}
|
|
\eeq
|
|
As readily seen from Eqs. \eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}, the
|
|
role of the correlation ensemble derivative contribution $\Upsilon_\text{c}^{(I)}$ is, through zeroth order, to substitute the expected
|
|
individual correlation energy per particle for the ensemble one.
|
|
|
|
Let us finally mention that, while the weighted sum of the
|
|
individual KS-eLDA energy levels delivers a \textit{ghost-interaction-corrected} (GIC) version of
|
|
the KS-eLDA ensemble energy, \ie,
|
|
\beq\label{eq:Ew-eLDA}
|
|
\begin{split}
|
|
\E{GIC-eLDA}{\bw}&=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)}
|
|
\\
|
|
&=
|
|
\E{eLDA}{\bw}
|
|
-\WHF[\bGam{\bw}]+\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}],
|
|
\end{split}
|
|
\eeq
|
|
the excitation energies computed from the KS-eLDA individual energy level
|
|
expressions in Eq. \eqref{eq:EI-eLDA} can be simplified as follows:
|
|
\beq\label{eq:Om-eLDA}
|
|
\begin{split}
|
|
\Ex{eLDA}{(I)}
|
|
&=
|
|
\Ex{HF}{(I)}
|
|
\\
|
|
&+ \int
|
|
\qty[\e{c}{{\bw}}(\n{}{})+n\pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}}]
|
|
_{\n{}{} =
|
|
\n{\bGam{\bw}}{}(\br{})}
|
|
\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{}
|
|
\\ & + \DD{c}{(I)},
|
|
\end{split}
|
|
\eeq
|
|
where the HF-like excitation energies, $\Ex{HF}{(I)} = \E{HF}{(I)} -
|
|
\E{HF}{(0)}$, are determined from a single set of ensemble KS orbitals and
|
|
\beq\label{eq:DD-eLDA}
|
|
\DD{c}{(I)}
|
|
= \int \n{\bGam{\bw}}{}(\br{})
|
|
\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
|
|
\eeq
|
|
is the eLDA correlation ensemble derivative contribution to the $I$th excitation energy.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Density-functional approximations for ensembles}
|
|
\label{sec:eDFA}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Paradigm}
|
|
\label{sec:paradigm}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Most of the standard local and semi-local density-functional approximations rely on the infinite uniform electron gas model (also known as jellium). \cite{ParrBook, Loos_2016}
|
|
One major drawback of the jellium paradigm, when it comes to develop density-functional approximations for ensembles, is that the ground and excited states are not easily accessible like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
|
|
Moreover, because the infinite uniform electron gas 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 uniform electron gases, \cite{Loos_2011b,
|
|
Gill_2012} which have, like an atom, discrete energy levels and non-zero
|
|
gaps, can be seen as more relevant in this context. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
|
|
However, an obvious drawback of using finite uniform electron gases is
|
|
that the resulting density-functional approximation for ensembles
|
|
will inexorably depend on the number of electrons in the finite uniform electron gas (see below).
|
|
Here, we propose to construct a weight-dependent eLDA for the
|
|
calculation of excited states in 1D systems by combining finite uniform electron gases with the
|
|
usual infinite uniform electron gas.
|
|
|
|
As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle) but interact \textit{through} the ring. \cite{Loos_2012, Loos_2013a, Loos_2014b}
|
|
The most appealing feature of ringium regarding the development of
|
|
functionals in the context of GOK-DFT is the fact that both ground- and
|
|
excited-state densities are uniform, and therefore {\it equal}.
|
|
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 the
|
|
correlation ensemble derivatives [last term
|
|
on the right-hand side of Eq.~\eqref{eq:exact_ener_level_dets}].
|
|
Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous infinite uniform electron gas paradigm. \cite{Loos_2013,Loos_2013a}
|
|
Let us stress that, in a finite uniform electron gas like ringium, the interacting and
|
|
noninteracting densities match individually for all the states within the
|
|
ensemble
|
|
(these densities are all equal to the uniform density), which means that
|
|
so-called density-driven correlation
|
|
effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model.
|
|
Here, we will consider the most simple ringium system featuring electronic correlation effects, \ie, the two-electron ringium model.
|
|
|
|
The present weight-dependent density-functional approximation is specifically designed for the
|
|
calculation of excited-state energies within GOK-DFT.
|
|
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.
|
|
To ensure the GOK variational principle, \cite{Gross_1988a} the
|
|
triensemble weights must fulfil the following conditions: \cite{Deur_2019}
|
|
$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$, where $\ew{1}$ and $\ew{2}$ are the weights associated with the singly- and doubly-excited states, respectively.
|
|
All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R$ is the radius of the ring on which the electrons are confined.
|
|
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
|
|
Generalization to a larger number of states is straightforward and is left for future work.
|
|
%The constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed
|
|
%to consider an equi-bi-ensemble
|
|
%for which $\ew{1}=1/2$. This possibility is excluded with your
|
|
%inequalities. The correct constraints are given in Ref.~\cite{Deur_2019}
|
|
%and are the ones you also mentioned, \ie, $0 \le \ew{2} \le 1/3$ and
|
|
%$\ew{2} \le \ew{1} \le (1-\ew{2})/2$.}
|
|
%\manu{
|
|
%Just in case, starting from
|
|
%\beq
|
|
%\begin{split}
|
|
%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{1}-\ew{2})
|
|
%\\
|
|
%\end{split}
|
|
%\eeq
|
|
%we obtain
|
|
%\beq
|
|
%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{2})/2
|
|
%\eeq
|
|
%which implies $\ew{2}\leq(1-\ew{2})/2$ or, equivalently, $\ew{2}\leq
|
|
%1/3$.
|
|
%}
|
|
%%% TABLE 1 %%%
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:OG_func}
|
|
Parameters of the weight-dependent correlation density-functional approximations defined in Eq.~\eqref{eq:ec}.}
|
|
% \begin{ruledtabular}
|
|
\begin{tabular}{lcddd}
|
|
\hline\hline
|
|
State & $I$ & \tabc{$a_1^{(I)}$} & \tabc{$a_2^{(I)}$} & \tabc{$a_3^{(I)}$} \\
|
|
\hline
|
|
Ground state & $0$ & -0.0137078 & 0.0538982 & 0.0751740 \\
|
|
Singly-excited state & $1$ & -0.0238184 & 0.00413142 & 0.0568648 \\
|
|
Doubly-excited state & $2$ & -0.00935749 & -0.0261936 & 0.0336645 \\
|
|
\hline\hline
|
|
\end{tabular}
|
|
% \end{ruledtabular}
|
|
\end{table*}
|
|
%%% %%% %%% %%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Weight-dependent correlation functional}
|
|
\label{sec:Ec}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (\ie, per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
|
|
\begin{equation}
|
|
\label{eq:ec}
|
|
\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}\,\n{}{}}{\n{}{} + a_2^{(I)} \sqrt{\n{}{}} + a_3^{(I)}},
|
|
\end{equation}
|
|
where the $a_k^{(I)}$'s are state-specific fitting parameters provided in Table \ref{tab:OG_func}.
|
|
The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
|
|
Equation \eqref{eq:ec} provides three state-specific correlation density-functional approximations based on a two-electron system.
|
|
Combining these, one can build the following three-state weight-dependent correlation density-functional approximation:
|
|
\begin{equation}
|
|
\label{eq:ecw}
|
|
%\e{c}{\bw}(\n{}{})
|
|
\Tilde{\epsilon}_{\rm c}^\bw(n)= (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
|
|
\end{equation}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{LDA-centered functional}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
One of the main driving force behind the popularity of DFT is its ``universal'' nature, as xc density functionals can be applied to any electronic system.
|
|
Obviously, the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the finite uniform electron gas.
|
|
However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (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 can be 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 density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} as follows:
|
|
\begin{equation}
|
|
\label{eq:becw}
|
|
\Tilde{\epsilon}_{\rm c}^\bw(n)\rightarrow{\e{c}{\bw}(\n{}{})} = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\be{c}{(I)}(\n{}{}) = \e{c}{(I)}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}) - \e{c}{(0)}(\n{}{}).
|
|
\end{equation}
|
|
In the following, we will use the LDA correlation functional that has been specifically designed for 1D systems in
|
|
Ref.~\onlinecite{Loos_2013}:
|
|
\begin{equation}
|
|
\label{eq:LDA}
|
|
\e{c}{\text{LDA}}(\n{}{})
|
|
= a_1^\text{LDA} F\qty[1,\frac{3}{2},a_3^\text{LDA}, \frac{a_1^\text{LDA}(1-a_3^\text{LDA})}{a_2^\text{LDA}} {\n{}{}}^{-1}],
|
|
\end{equation}
|
|
where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
|
|
\begin{subequations}
|
|
\begin{align}
|
|
a_1^\text{LDA} & = - \frac{\pi^2}{360},
|
|
\\
|
|
a_2^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
|
|
\\
|
|
a_3^\text{LDA} & = 2.408779.
|
|
\end{align}
|
|
\end{subequations}
|
|
Note that the strategy described in Eq.~\eqref{eq:becw} is general and
|
|
can be applied to real (higher-dimensional) systems. In order to make the
|
|
connection with the GACE formalism \cite{Franck_2014,Deur_2017} more explicit, one may
|
|
recast Eq.~\eqref{eq:becw} as
|
|
\begin{equation}
|
|
\label{eq:eLDA}
|
|
\begin{split}
|
|
{\e{c}{\bw}(\n{}{})}
|
|
& = \e{c}{\text{LDA}}(\n{}{})
|
|
\\
|
|
& + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})],
|
|
\end{split}
|
|
\end{equation}
|
|
or, equivalently,
|
|
\begin{equation}
|
|
\label{eq:eLDA_gace}
|
|
{\e{c}{\bw}(\n{}{})}
|
|
= \e{c}{\text{LDA}}(\n{}{})
|
|
+ \sum_{K>0}\int_0^{\ew{K}}
|
|
\qty[\e{c}{(K)}(\n{}{})-\e{c}{(0)}(\n{}{})]d\xi_K,
|
|
\end{equation}
|
|
where the $K$th correlation excitation energy (per electron) is integrated over the
|
|
ensemble weight $\xi_K$ at fixed (uniform) density $\n{}{}$.
|
|
Equation \eqref{eq:eLDA_gace} nicely highlights the centrality of the
|
|
LDA in the present density-functional approximation for ensembles.
|
|
In particular, ${\e{c}{(0,0)}(\n{}{})} = \e{c}{\text{LDA}}(\n{}{})$.
|
|
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
|
|
Finally, we note that, by construction,
|
|
\begin{equation}
|
|
{\pdv{\e{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}).}
|
|
\end{equation}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Computational details}
|
|
\label{sec:comp_details}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
|
|
Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\nEl$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\nEl$-boxium in the following.
|
|
In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \nEl \le 7$.
|
|
These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
|
|
For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime. \cite{Rogers_2017,Rogers_2016}
|
|
|
|
We use as basis functions the (orthonormal) orbitals of the one-electron system, \ie,
|
|
\begin{equation}
|
|
\AO{\mu}(x) =
|
|
\begin{cases}
|
|
\sqrt{2/L} \cos(\mu \pi x/L), & \mu \text{ is odd,}
|
|
\\
|
|
\sqrt{2/L} \sin(\mu \pi x/L), & \mu \text{ is even,}
|
|
\end{cases}
|
|
\end{equation}
|
|
with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
|
|
The convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw}
|
|
\bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~\eqref{eq:commut_F_AO}] is set
|
|
to $10^{-5}$. For comparison, regular HF and KS-DFT calculations
|
|
are performed with the same threshold.
|
|
In order to compute the various density-functional
|
|
integrals that cannot be performed in closed form,
|
|
a 51-point Gauss-Legendre quadrature is employed.
|
|
|
|
In order to test the present eLDA functional we perform various sets of calculations.
|
|
To get reference excitation energies for both the single and double excitations, we compute full configuration interaction (FCI) energies with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
|
|
For the single excitations, we also perform time-dependent LDA (TDLDA)
|
|
calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}].
|
|
Its Tamm-Dancoff approximation version (TDA-TDLDA) is also considered. \cite{Dreuw_2005}
|
|
|
|
Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
|
|
(ground-state) limit where $\bw = (0,0)$ and the
|
|
equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Results and discussion}
|
|
\label{sec:res}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{EvsW_n5}
|
|
\caption{
|
|
\label{fig:EvsW}
|
|
Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost-interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
|
|
}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
First, we discuss the linearity of the computed (approximate)
|
|
ensemble energies.
|
|
To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
|
|
The deviation from linearity of the three-state ensemble energy
|
|
$\E{}{(\ew{1},\ew{2})}$ (\ie, the deviation from the
|
|
linearly-interpolated ensemble energy) is represented
|
|
in Fig.~\ref{fig:EvsW} as a function of $\ew{1}$ or $\ew{2}$ while
|
|
fulfilling the restrictions on the ensemble weights to ensure the GOK
|
|
variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
|
|
To illustrate the magnitude of the ghost-interaction error, we report the KS-eLDA ensemble energy with and without GIC as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
|
|
As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
|
|
ensemble energy becomes less and less linear as $L$
|
|
gets larger, while the GIC reduces the curvature of the ensemble energy
|
|
drastically.
|
|
%\manu{This
|
|
%is a strong statement I am not sure about. The nature of the excitation
|
|
%should also be invoked I guess (charge transfer or not, etc ...). If we look at the GIE:
|
|
%\beq
|
|
%\WHF[
|
|
%\bGam{\bw}]-\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}]
|
|
%\eeq
|
|
%For a bi-ensemble ($w_1=w$) it can be written as
|
|
%\beq
|
|
%\dfrac{1}{2}\left[(w^2-1)W_0+w(w-2)W_1\right]+w(1-w)W_{01}
|
|
%\eeq
|
|
%If, for some reason, $W_0\approx W_1\approx W_{01}=W$, then the error
|
|
%reduces to $-W/2$, which is weight-independent (it fits for example with
|
|
%what you see in the weakly correlated regime). Such an assumption depends on the nature of the
|
|
%excitation, not only on the correlation strength, right? Nevertheless,
|
|
%when looking at your curves, this assumption cannot be made when the
|
|
%correlation is strong. It is not clear to me which integral ($W_{01}?$)
|
|
%drives the all thing.\\}
|
|
It is important to note that, even though the GIC removes the explicit
|
|
quadratic terms from the ensemble energy, a non-negligible curvature
|
|
remains in the GIC-eLDA ensemble energy. \manu{This might be due to
|
|
\textit{(i)} the correlation eLDA
|
|
functional, which induces linear or quadratic weight dependencies of the individual
|
|
energies [see Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
|
|
\eqref{eq:Taylor_exp_DDisc_term}], and \textit{(ii)} the optimization of the
|
|
ensemble KS orbitals in the presence of ghost-interaction errors {[see
|
|
Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:WHF}]}}.
|
|
%However, this orbital-driven error is small (in our case at
|
|
%least) \trashEF{as the correlation part of the ensemble KS potential $\delta
|
|
%\E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared
|
|
%to the Hx contribution}.\manu{Manu: well, I guess that the problem arises
|
|
%from the density matrices (or orbitals) that are used to compute
|
|
%individual Coulomb-exchange energies (I would not expect the DFT
|
|
%correlation part to have such an impact, as you say). The best way to check is to plot the
|
|
%ensemble energy without the correlation functional.}\\
|
|
%\\
|
|
%\manu{Manu: another idea. As far as I can see we do
|
|
%not show any individual energies (excitation energies are plotted in the
|
|
%following). Plotting individual energies (to be compared with the FCI
|
|
%ones) would immediately show if there is some curvature (in the ensemble
|
|
%energy). The latter would
|
|
%be induced by any deviation from the expected horizontal straight lines.}
|
|
|
|
%%% FIG 2 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{EIvsW_n5}
|
|
\caption{
|
|
\label{fig:EIvsW}
|
|
KS-eLDA individual energies, $\E{eLDA}{(0)}$ (black), $\E{eLDA}{(1)}$ (red), and $\E{eLDA}{(2)}$ (blue), as functions of the weights $\ew{1}$ (solid) and $\ew{2}$ (dashed) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual energies as functions of the weights.
|
|
Unlike in the exact theory, we do not obtain
|
|
straight horizontal lines when plotting these
|
|
energies, which is in agreement with
|
|
the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
|
|
individual energies do not vary in the same way depending on the state
|
|
considered and the value of the weights.
|
|
We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
|
|
the ground and first excited-state increase with respect to the
|
|
first-excited-state weight $\ew{1}$, thus showing that, in this
|
|
case, we
|
|
``deteriorate'' these states by optimizing the orbitals for the
|
|
ensemble, rather than for each state individually. The reverse actually occurs for the ground state in the triensemble
|
|
as $\ew{2}$ increases. The variations in the ensemble
|
|
weights are essentially linear or quadratic. They are induced by the
|
|
eLDA functional, as readily seen from
|
|
Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
|
|
\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
|
|
excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
|
|
systematically enhances the weight dependence, due to the lowering of the
|
|
ground-state energy, as $\ew{2}$ increases.
|
|
The reverse is observed for the second excited state.
|
|
|
|
%%% FIG 3 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{EvsL_5}
|
|
\caption{
|
|
\label{fig:EvsL}
|
|
Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) of 5-boxium for various methods and box length $L$.
|
|
Graphs for additional values of $\nEl$ can be found as {\SI}.
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%%
|
|
|
|
Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
|
|
Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
|
|
For small $L$, the single and double excitations can be labeled as
|
|
``pure'', as revealed by a thorough analysis of the FCI wavefunctions.
|
|
In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
|
|
However, when the box gets larger (\ie, as $L$ increases), there is a strong mixing between the different excitation degrees.
|
|
In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more disputable. \cite{Loos_2019}
|
|
This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
|
|
% TITOU: shall we keep the paragraph below?
|
|
%Therefore, it is paramount to construct a two-weight correlation functional
|
|
%(\ie, a triensemble functional, as we have done here) which
|
|
%allows the mixing of singly- and doubly-excited configurations.
|
|
%Using a single-weight (\ie, a biensemble) functional where only the ground state and the lowest singly-excited states are taken into account, one would observe a neat deterioration of the excitation energies (as compared to FCI) when the box gets larger.
|
|
%\titou{Titou might add results for the biensemble to illustrate this.}
|
|
%\manu{Well, neglecting the second excited state is not the same as
|
|
%considering the $w_2=0$ limit. I thought you were referring to an
|
|
%approximation where the triensemble calculation is performed with
|
|
%the biensemble functional. This is not the same as taking $w_2=0$
|
|
%because, in this limit, you may still have a derivative discontinuity
|
|
%correction. The latter is absent if you truly neglect the second excited
|
|
%state in your ensemble functional. This should be clarified.}\\
|
|
%\manu{Are the results in the supp mat? We could just add "[not
|
|
%shown]" if not. This is fine as long as you checked that, indeed, the
|
|
%results deteriorate ;-)}
|
|
%\manu{Should we add that, in the bi-ensemble case, the ensemble
|
|
%correlation derivative $\partial \epsilon^\bw_{\rm c}(n)/\partial w_2$
|
|
%is neglected (if this is really what you mean (?)). I guess that this is the reason why
|
|
%the second excitation energy would not be well described (?)}
|
|
|
|
As shown in Fig.~\ref{fig:EvsL}, all methods provide accurate estimates of the excitation energies in the weak correlation regime (\ie, small $L$).
|
|
When the box gets larger, they start to deviate.
|
|
For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies.
|
|
TDA-TDLDA slightly corrects this trend thanks to error compensation.
|
|
Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
|
|
This is especially true, in the strong correlation regime, for the single excitation
|
|
which is significantly improved by using equal weights.
|
|
The effect on the double excitation is less pronounced.
|
|
Overall, one clearly sees that, with
|
|
equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
|
|
This conclusion is verified for smaller and larger numbers of electrons
|
|
(see {\SI}).
|
|
%\\
|
|
%\manu{Manu: now comes the question that is, I believe, central in this
|
|
%work. How important are the
|
|
%ensemble correlation derivatives $\partial \epsilon^\bw_{\rm
|
|
%c}(n)/\partial w_I$ that, unlike any functional
|
|
%in the literature, the eLDA functional contains. We have to discuss this
|
|
%point... I now see, after reading what follows that this question is
|
|
%addressed later on. We should say something here and then refer to the
|
|
%end of the section, or something like that ...}
|
|
|
|
|
|
%%% FIG 4 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{EvsN}
|
|
\caption{
|
|
\label{fig:EvsN}
|
|
Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and electron numbers $\nEl$ at $L=\pi/8$ (left), $L=\pi$ (center), and $L=8\pi$ (right).
|
|
}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
For the same set of methods, Fig.~\ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI) as a function of $\nEl$ for three values of $L$ ($\pi/8$, $\pi$, and $8\pi$).
|
|
We draw similar conclusions as above: irrespectively of the number of
|
|
electrons, the eLDA functional with equal
|
|
weights is able to accurately model single and double excitations, with
|
|
a very significant improvement brought by the
|
|
equiensemble KS-eLDA orbitals as compared to their zero-weight
|
|
(\ie, conventional ground-state) analogs.
|
|
As a rule of thumb, in the weak and intermediate correlation regimes, we
|
|
see that the single
|
|
excitation obtained from equiensemble KS-eLDA is of
|
|
the same quality as the one obtained in the linear response formalism
|
|
(such as TDLDA). On the other hand, the double
|
|
excitation energy only deviates
|
|
from the FCI value by a few tenth of percent.
|
|
Moreover, we note that, in the strong correlation regime
|
|
(right graph of Fig.~\ref{fig:EvsN}), the single excitation
|
|
energy obtained at the equiensemble KS-eLDA level remains in good
|
|
agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
|
|
This also applies to the double excitation, the discrepancy
|
|
between FCI and equiensemble KS-eLDA remaining of the order of a few percents in the strong correlation regime.
|
|
These observations nicely illustrate the robustness of the
|
|
GOK-DFT scheme in any correlation regime for both single and double excitations.
|
|
This is definitely a very pleasing outcome, which additionally shows
|
|
that, even though we have designed the eLDA functional based on a
|
|
two-electron model system, the present methodology is applicable to any
|
|
1D electronic system, \ie, a system that has more than two
|
|
electrons.
|
|
|
|
%%% FIG 5 %%%
|
|
\begin{figure*}
|
|
\includegraphics[width=\linewidth]{EvsL_DD}
|
|
\caption{
|
|
\label{fig:EvsL_DD}
|
|
Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 3-boxium (left), 5-boxium (center), and 7-boxium (right) at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.
|
|
Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and equiweight (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported.
|
|
}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
It is also interesting to investigate the influence of the
|
|
correlation ensemble derivative contribution $\DD{c}{(I)}$
|
|
to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
|
|
In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
|
|
To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, for $\nEl = 3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$
|
|
on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
|
|
%\manu{Manu: there is something I do not understand. If you want to
|
|
%evaluate the importance of the ensemble correlation derivatives you
|
|
%should only remove the following contribution from the $K$th KS-eLDA
|
|
%excitation energy:
|
|
%\beq\label{eq:DD_term_to_compute}
|
|
%\int \n{\bGam{\bw}}{}(\br{})
|
|
% \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
|
|
%\eeq
|
|
%%rather than $E^{(I)}_{\rm HF}$
|
|
%}
|
|
We first stress that although for $\nEl=3$ both single and double excitation energies are
|
|
systematically improved, as the strength of electron correlation
|
|
increases, when
|
|
taking into account
|
|
the correlation ensemble derivative, this is not
|
|
always the case for larger numbers of electrons.
|
|
The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime.
|
|
For 3-boxium, in the zero-weight limit, its contribution is
|
|
significantly larger for the single
|
|
excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
|
|
case.
|
|
However, for 5- and 7-boxium, the correlation ensemble derivative hardly
|
|
influences the double excitation (except when the correlation is strong), and slightly deteriorates the single excitation in the intermediate and strong correlation regimes.
|
|
This non-systematic behavior in terms of the number of electrons might
|
|
be a consequence of how we constructed eLDA.
|
|
Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of
|
|
the eLDA functional is based on a \textit{two-electron} finite uniform electron gas.
|
|
Incorporating an $\nEl$-dependence in the functional through the
|
|
curvature of the Fermi hole, in the spirit of Ref.~\onlinecite{Loos_2017a}, would be
|
|
valuable in this respect. This is left for future work.
|
|
%\\
|
|
%\manu{Manu: I am sorry to insist but I have a real problem with what follows. If
|
|
%we look at the N=3 results, one has the impression that, indeed, for the
|
|
%single excitation, a zero-weight calculation with the ensemble derivative
|
|
%is almost equivalent to an equal-weight calculation without the
|
|
%derivative. This is not the case for $N=5$ or 7, maybe because our
|
|
%derivative is based on two electrons. }\\
|
|
%{\it
|
|
%Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble
|
|
%derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
|
|
%compared to the zero-weight calculations).
|
|
%%\manu{Manu: well, this is not
|
|
%%really the case for the double excitation, right? I would remove this
|
|
%%sentence or mention the single excitation explicitly.}
|
|
%This could explain why equiensemble calculations are clearly more
|
|
%accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
|
|
%for a given method, equiensemble orbitals partially remove the burden
|
|
%of modelling properly the ensemble correlation derivative.
|
|
%}\\
|
|
%\manu{Manu: I propose to rephrase this part as follows:}\\
|
|
%\\
|
|
\titou{
|
|
Interestingly, for the single excitation in 3-boxium, the magnitude of the correlation ensemble
|
|
derivative is substantially reduced when switching from a zero-weight to
|
|
an equal-weight calculation, while giving similar excitation energies,
|
|
even in the strongly correlated regime. A possible interpretation is
|
|
that, at least for the single excitation, equiensemble orbitals partially remove the burden
|
|
of modelling properly the correlation ensemble derivative.
|
|
This conclusion does not hold for larger
|
|
numbers of electrons ($N=5$ or $7$), possibly because eLDA extracts density-functional correlation ensemble
|
|
derivatives from a two-electron uniform electron gas, as mentioned previously.
|
|
For the double excitation, the ensemble derivative remains important, even in
|
|
the equiensemble case.
|
|
To summarize, in all cases, the equiensemble calculation
|
|
is always more accurate than a zero-weight
|
|
(\ie, a conventional ground-state DFT) one, with or without including the ensemble
|
|
derivative correction.
|
|
}
|
|
\\
|
|
Note that the second term in
|
|
Eq.~\eqref{eq:Om-eLDA}, which involves the weight-dependent correlation
|
|
potential and the density difference between ground and excited states,
|
|
has a negligible effect on the excitation energies (results not shown).
|
|
%\manu{Manu: Is this
|
|
%something that you checked but did not show? It feels like we can see
|
|
%this in the Figure but we cannot, right?}
|
|
%\manu{Manu: well, we
|
|
%would need the exact derivative value to draw such a conclusion. We can
|
|
%only speculate. Let us first see how important the contribution in
|
|
%Eq.~\eqref{eq:DD_term_to_compute} is. What follows should also be
|
|
%updated in the light of the new results.}
|
|
|
|
%%% FIG 6 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{EvsN_DD}
|
|
\caption{
|
|
\label{fig:EvsN_DD}
|
|
Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.
|
|
Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and equiweight (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported.
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%%
|
|
|
|
Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime).
|
|
The difference between the solid and dashed curves
|
|
undoubtedly show that, even in the strong correlation regime, the
|
|
ensemble correlation derivative has a rather significant impact on the double
|
|
excitations (around $10\%$) with a slight tendency of worsening the excitation energies
|
|
in the case of equal weights, as the number of electrons
|
|
increases. It has a rather large influence on the single
|
|
excitation energies obtained in the zero-weight limit, showing once
|
|
again that the usage of equal weights has the benefit of significantly reducing the magnitude of the ensemble correlation derivative.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Concluding remarks}
|
|
\label{sec:conclusion}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
A local and ensemble-weight-dependent correlation density-functional approximation
|
|
(eLDA) has been constructed in the context of GOK-DFT for spin-polarized
|
|
triensembles in
|
|
1D. The approach is actually general and can be extended to real
|
|
(three-dimensional)
|
|
systems~\cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a}
|
|
and larger ensembles in order to
|
|
model excited states in molecules and solids. Work is currently in
|
|
progress in this direction.
|
|
|
|
Unlike any standard functional, eLDA incorporates derivative
|
|
discontinuities through its weight dependence. The latter originates
|
|
from the finite uniform electron gas \titou{on which} eLDA is
|
|
(partially) based on. The KS-eLDA scheme, where exact exchange is
|
|
combined with eLDA, delivers accurate excitation energies for both
|
|
single and double excitations, especially when an equiensemble is used.
|
|
In the latter case, the same weights are assigned to each state belonging to the ensemble.
|
|
The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.
|
|
We have observed that, although the ensemble correlation discontinuity has a
|
|
non-negligible effect on the excitation energies (especially for the
|
|
single excitations), its magnitude can be significantly reduced by
|
|
performing equiweight calculations instead of zero-weight
|
|
calculations.
|
|
|
|
Let us finally stress that the present methodology can be extended
|
|
straightforwardly to other types of ensembles like, for example, the
|
|
$\nEl$-centered ones, \cite{Senjean_2018,Senjean_2020} thus allowing for the design of a LDA-type functional for the
|
|
calculation of ionization potentials, electron affinities, and
|
|
fundamental gaps.
|
|
Like in the present
|
|
eLDA, such a functional would incorporate the infamous derivative
|
|
discontinuity contribution to the fundamental gap through its explicit weight
|
|
dependence. We hope to report on this in the near future.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section*{Supplementary material}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
See {\SI} for the additional details about the construction of the functionals, raw data and additional graphs.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{acknowledgements}
|
|
The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.
|
|
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
|
|
\end{acknowledgements}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\bibliography{eDFT}
|
|
|
|
\end{document}
|