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Pierre-Francois Loos 2019-11-15 10:41:07 +01:00
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Manuscript/FarDFT.tex
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@ -37,9 +37,12 @@
\newcommand{\hT}{\Hat{T}}
\newcommand{\bH}{\Hat{T}}
\newcommand{\hVext}{\Hat{V}_\text{ext}}
\newcommand{\vext}{v_\text{ext}}
\newcommand{\hWee}{\Hat{W}_\text{ee}}
% functionals, potentials, densities, etc
\newcommand{\F}[2]{F_{#1}^{#2}}
\newcommand{\Ts}[1]{T_\text{s}^{#1}}
\newcommand{\eps}{\epsilon}
\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
\newcommand{\kin}[2]{t_\text{#1}^{#2}}
@ -61,6 +64,9 @@
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\EFCI}{E_\text{FCI}}
\newcommand{\LDA}{\text{LDA}}
\newcommand{\Hxc}{\text{Hxc}}
\newcommand{\Ha}{\text{H}}
\newcommand{\xc}{\text{xc}}
% matrices
\newcommand{\br}{\bm{r}}
@ -82,6 +88,7 @@
\newcommand{\eF}[2]{F_{#1}^{#2}}
% Numbers
\newcommand{\Nens}{M}
\newcommand{\Nel}{N}
\newcommand{\Nbas}{K}
@ -146,8 +153,6 @@ However, spin contamination might be an issue. \cite{Huix-Rotllant_2010}
In order to go beyond the adiabatic approximation, a dressed TD-DFT approach has been proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} (see also Refs.~\onlinecite{Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012}).
In this approach the exchange-correlation kernel is made frequency dependent \cite{Romaniello_2009a,Sangalli_2011}, which allows to treat doubly-excited states.
Maybe surprisingly, a possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
Density-functional theory for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988,Gross_1988a,Oliveira_1988} is a viable alternative currently under active development which follow such a strategy. \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,Smith_2016,Senjean_2018}
In the assumption of monotonically decreasing weights, eDFT has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, \cite{Gross_1988a} and excitation energies can be quite easily extracted from the total ensemble energy.
@ -168,7 +173,39 @@ Unless otherwise stated, atomic units are used throughout.
%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theo}
Here is the theory.
As mentioned above, eDFT is based on the so-called Gross-Oliveria-Kohn (GOK) variational principle \cite{Gross_1988a} which states that a variational principle holds for the ensemble energy
\begin{equation}
\E{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \E{}{(I)}
\end{equation}
built from an ensemble of $\Nens$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\Nens-1)}$, and normalized, monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie,
\begin{align}
& \sum_{I=0}^{\Nens-1} \ew{I} = 1,
&
& \ew{0} \ge \ldots \ge \ew{\Nens-1}.
\end{align}
One of the key feature of eDFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
\begin{equation}
\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{(I)}
\end{equation}
where we used the fact that $\ew{0} = 1 - \sum_{I \ne 0} \ew{I}$.
For such an ensemble, one can define a universal ensemble functional $\F{}{\bw}[\n{}{}]$ such that
\begin{equation}
\E{}{\bw} = \min_{\n{}{}} \qty[ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} ]
\end{equation}
where $\vext(\br{})$ is the external potential.
In the KS formulation of eDFT, the universal ensemble functional (the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles) is decomposed as
\begin{equation}
\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
\end{equation}
where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}[\n{}{}]$ are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively with
\begin{equation}
\begin{split}
\E{\Hxc}{\bw}[\n{}{}]
& = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}]
\\
& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'+ \E{\xc}{\bw}[\n{}{}]
\end{split}
\end{equation}
%%%%%%%%%%%%%%%%%%
%%% FUNCTIONAL %%%