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Pierre-Francois Loos 2020-02-11 21:39:45 +01:00
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Manuscript/FarDFT.tex
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@ -187,9 +187,9 @@ Unless otherwise stated, atomic units are used throughout.
\label{sec:theo}
As mentioned above, eDFT for excited states is based on the 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)},
\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, $\sum_{I=0}^{\Nens-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\Nens-1}$.
built from an ensemble of $\Nens$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\Nens-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\Nens-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\Nens-1}$.
Multiplet degeneracies can be easily handled by assigning the same weight to the degenerate states.
One of the key feature of GOK-DFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
@ -280,10 +280,10 @@ The construction of these two functionals is described below.
Here, we restrict our study to spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Extension to spin-polarised systems will be reported in future work.
Although this choice is far from being unique, we consider here the singlet ground state and the first singlet doubly-excited state of a two-electron FUEG which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
These two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalized hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}
As mentioned above, we confine our attention to paramagnetic (or unpolarized) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere.
To build our weight-dependent xc functional, we propose to consider the singlet ground state and the first singlet doubly-excited state of a two-electron FUEG which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
Notably, these two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}
As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere.
Note that the present paradigm is equivalent to the IUEG model in the thermodynamic limit. \cite{Loos_2011b}
We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
@ -342,7 +342,7 @@ with
\begin{equation}
\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}.
\end{equation}
Quite remarkably, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient, which is expected from a theoretical point of view, yet a nice property from a more practical aspect.
Conveniently, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient; an expected feature from a theoretical point of view, yet a nice property from a more practical aspect.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -369,7 +369,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
\begin{figure}
\includegraphics[width=\linewidth]{fig/fig1}
\caption{
Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron FUEG.
Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarised) two-electron FUEG.
The data gathered in Table \ref{tab:Ref} are also reported.
}
\label{fig:Ec}
@ -380,7 +380,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
\begin{table}
\caption{
\label{tab:Ref}
$-\e{\co}{(I)}$ as a function of the radius of the glome $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron FUEG.
$-\e{\co}{(I)}$ as a function of the radius of the glome $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarised) two-electron FUEG.
}
\begin{ruledtabular}
\begin{tabular}{lcc}
@ -428,7 +428,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
Our intent is 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).
Hence, we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
The weight-dependence of the xc functional then carried exclusively by the impurity [\ie, the functionals defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA xc functional).
The weight-dependence of the xc functional is then carried exclusively by the impurity [\ie, the functionals defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA xc functional).
Consistently with such a strategy, Eqs.~\eqref{eq:exw} and \eqref{eq:ecw} are ``centred'' on their corresponding jellium reference
\begin{equation}
@ -464,12 +464,12 @@ Equation \eqref{eq:becw} can be recast
\be{\xc}{\ew{}}(\n{}{})
& = \e{\xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})]
\\
& = \e{\xc}{\LDA}(\n{}{}) + \ew{} \pdv{\e{\xc}{\ew{}}(\n{}{})}{\ew{}}
& = \e{\xc}{\LDA}(\n{}{}) + \ew{} \pdv{\e{\xc}{\ew{}}(\n{}{})}{\ew{}},
\end{split}
\end{equation}
which nicely highlights the centrality of the LDA in the present eDFA.
In particular, $\be{\xc}{(0)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{})$.
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
Consequently, in the following, we name this weight-dependent xc functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
Also, we note that, by construction,
\begin{equation}
\label{eq:dexcdw}
@ -803,9 +803,6 @@ The derivative discontinuity is modelled by the last term of the RHS of Eq.~\eqr
Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}].
\end{widetext}
%%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%%
@ -818,7 +815,8 @@ As concluding remarks, we would like to say that what we have done, we think, is
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
CM thanks the \textit{Universit\'e Paul Sabatier} (Toulouse, France) for a PhD scholarship.
PFL acknowledges funding from the \textit{Centre National de la Recherche Scientifique}.
PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
\end{acknowledgements}
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