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Pierre-Francois Loos 2020-03-11 22:26:38 +01:00
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Manuscript/eDFT.tex
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@ -819,7 +819,7 @@ Note that, within the approximation of Eq.~\eqref{eq:min_with_HF_ener_fun}, the
optimized with a non-local exchange potential rather than a optimized with a non-local exchange potential rather than a
density-functional local one, as expected from density-functional local one, as expected from
Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie, Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie,
applicable to not-necessarily spin polarized and real (higher-dimensional) systems. applicable to not-necessarily spin-polarized and real (higher-dimensional) systems.
As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the
ensemble density matrix into the HF interaction energy functional 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} introduces unphysical \textit{ghost-interaction} errors \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
@ -834,9 +834,8 @@ as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
\eeq \eeq
The ensemble energy is of course expected to vary linearly with the ensemble The ensemble energy is of course expected to vary linearly with the ensemble
weights [see Eq.~\eqref{eq:exact_GOK_ens_ener}]. 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 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}.} levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.
Turning to the density-functional ensemble correlation energy, the Turning to the density-functional ensemble correlation energy, the
following ensemble local-density approximation (eLDA) will be employed following ensemble local-density approximation (eLDA) will be employed
@ -1031,9 +1030,9 @@ gaps, can be seen as more relevant in this context. \cite{Loos_2014a, Loos_2014b
However, an obvious drawback of using finite uniform electron gases is However, an obvious drawback of using finite uniform electron gases is
that the resulting density-functional approximation for ensembles that the resulting density-functional approximation for ensembles
will inexorably depend on the number of electrons in the finite uniform electron gas (see below). 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 Here, we propose to construct a weight-dependent LDA functional for the
calculation of excited states in 1D systems by combining finite uniform electron gases with the calculation of excited states in 1D systems by combining finite uniform electron gases with the
usual infinite uniform electron gas. usual infinite uniform electron gas paradigm.
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} 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 The most appealing feature of ringium regarding the development of
@ -1209,9 +1208,9 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
\end{equation} \end{equation}
with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations. with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
The convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} 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 \bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~\eqref{eq:commut_F_AO}] of the KS-DFT self-consistent calculation is set
to $10^{-5}$. For comparison, regular HF and KS-DFT calculations to $10^{-5}$.
are performed with the same threshold. %For comparison, regular HF and KS-DFT calculations are performed with the same threshold.
In order to compute the various density-functional In order to compute the various density-functional
integrals that cannot be performed in closed form, integrals that cannot be performed in closed form,
a 51-point Gauss-Legendre quadrature is employed. a 51-point Gauss-Legendre quadrature is employed.