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Manuscript/eDFT.tex
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@ -254,7 +254,7 @@ Atomic units are used throughout.
In this section we give a brief review of GOK-DFT and discuss the 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 extraction of individual energy levels \cite{Deur_2019,Fromager_2020} with a particular focus on exact
individual exchange energies. individual exchange energies.
Let us start by introducing the GOK ensemble energy~\cite{Gross_1988a}: Let us start by introducing the GOK ensemble energy: \cite{Gross_1988a}
\beq\label{eq:exact_GOK_ens_ener} \beq\label{eq:exact_GOK_ens_ener}
\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)}, \E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
\eeq \eeq
@ -270,9 +270,9 @@ They are normalized, \ie,
\eeq \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. 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. 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}. 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 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}: 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} \beq\label{eq:var_ener_gokdft}
\E{}{\bw} \E{}{\bw}
= \min_{\opGam{\bw}} = \min_{\opGam{\bw}}
@ -305,7 +305,7 @@ The (approximate) description of the correlation part is discussed in
Sec.~\ref{sec:eDFA}. 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). 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.~\cite{Deur_2019}, the latter can be extracted As pointed out recently in Ref.~\onlinecite{Deur_2019}, the latter can be extracted
exactly from a single ensemble calculation as follows: exactly from a single ensemble calculation as follows:
\beq\label{eq:indiv_ener_from_ens} \beq\label{eq:indiv_ener_from_ens}
\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} )
@ -349,7 +349,7 @@ auxiliary double-weight ensemble density reads
Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes
from the exact expression in Eq.~(\ref{eq:exact_ens_Hx}) that from the exact expression in Eq.~(\ref{eq:exact_ens_Hx}) that
\beq \beq
\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}} \E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
\eeq \eeq
and and
\beq \beq
@ -376,7 +376,7 @@ Since, according to Eqs.~(\ref{eq:var_ener_gokdft}) and (\ref{eq:exact_ens_Hx}),
\eeq \eeq
with $\Det{(K)} = \Det{(K),\bw}$ [note that, when the minimum is reached in Eq.~\eqref{eq:var_ener_gokdft}, $\n{\opGam{\bw}}{} = \n{}{\bw,\bw}$], 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 we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
\eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\cite{Fromager_2020} for the $I$th energy level: \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} \beq\label{eq:exact_ener_level_dets}
\begin{split} \begin{split}
\E{}{(I)} \E{}{(I)}
@ -419,7 +419,7 @@ C_{\rm c}[n]=\dfrac{\E{c}{}[n]
\fdv{\E{c}{}[n]}{\n{}{}(\br{})}n(\br{})d\br{}}{\int n(\br{})d\br{}} \fdv{\E{c}{}[n]}{\n{}{}(\br{})}n(\br{})d\br{}}{\int n(\br{})d\br{}}
\eeq \eeq
is the correlation component of is the correlation component of
Levy--Zahariev's constant shift in potential~\cite{Levy_2014}. Levy--Zahariev's constant shift in potential.\cite{Levy_2014}
Similarly, the excited-state ($I>0$) energy level expressions Similarly, the excited-state ($I>0$) energy level expressions
can be recast as follows: can be recast as follows:
\beq\label{eq:excited_ener_level_gs_lim} \beq\label{eq:excited_ener_level_gs_lim}
@ -451,7 +451,7 @@ as basic variables, rather than Slater determinants.
As the theory is applied later on to {\it spin-polarized} As the theory is applied later on to {\it spin-polarized}
systems, we drop spin indices in the density matrices, for convenience. systems, we drop spin indices in the density matrices, for convenience.
If we expand the If we expand the
ensemble KS orbitals [from which the determinants are constructed] in an atomic orbital (AO) basis, ensemble KS orbitals (from which the determinants are constructed) in an atomic orbital (AO) basis,
\beq \beq
\MO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}), \MO{p}{}(\br{}) = \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
\eeq \eeq
@ -535,7 +535,7 @@ where
\beq \beq
\bh \equiv h_{\mu\nu} = \mel*{\AO{\mu}}{\hh}{\AO{\nu}} \bh \equiv h_{\mu\nu} = \mel*{\AO{\mu}}{\hh}{\AO{\nu}}
\eeq \eeq
denotes the one-electron integrals matrix. denotes the matrix of the one-electron integrals.
The exact individual Hx energies are obtained from the following trace formula The exact individual Hx energies are obtained from the following trace formula
\beq \beq
\Tr[\bGam{(K)} \bG \bGam{(L)}] \Tr[\bGam{(K)} \bG \bGam{(L)}]
@ -650,7 +650,7 @@ w}_K
In the following, GOK-DFT will be applied In the following, GOK-DFT will be applied
to one-dimension to 1D
spin-polarized systems where spin-polarized systems where
Hartree and exchange energies cannot be separated. Hartree and exchange energies cannot be separated.
For that reason, we will substitute the Hartree--Fock (HF) density-matrix-functional interaction energy, For that reason, we will substitute the Hartree--Fock (HF) density-matrix-functional interaction energy,
@ -809,11 +809,11 @@ Note that, within the approximation of Eq.~(\ref{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-dimension) 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}
as well as {\it curvature}~\cite{Alam_2016,Alam_2017}: as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
\beq\label{eq:WHF} \beq\label{eq:WHF}
\begin{split} \begin{split}
\WHF[\bGam{\bw}] \WHF[\bGam{\bw}]
@ -828,7 +828,7 @@ These errors are essentially removed 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 \textit{approximation} (eLDA) will be employed:
\beq\label{eq:eLDA_corr_fun} \beq\label{eq:eLDA_corr_fun}
\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{}, \E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
\eeq \eeq
@ -870,9 +870,9 @@ of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
\beq \beq
\int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{} \int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
+\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right), +\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right).
\eeq \eeq
and it can therefore be identified as Therefore, it can be identified as
an individual-density-functional correlation energy where the density-functional an individual-density-functional correlation energy where the density-functional
correlation energy per particle is approximated by the ensemble one for correlation energy per particle is approximated by the ensemble one for
all the states within the ensemble. all the states within the ensemble.
@ -1222,11 +1222,11 @@ However, when the box gets larger (\ie, $L$ increases), there is a strong mixing
In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019} In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
This can be clearly evidenced by the weights of the different configurations in the FCI wave function. This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
% TITOU: shall we keep the paragraph below? % TITOU: shall we keep the paragraph below?
Therefore, it is paramount to construct a two-weight correlation functional %Therefore, it is paramount to construct a two-weight correlation functional
(\ie, a triensemble functional, as we have done here) which %(\ie, a triensemble functional, as we have done here) which
allows the mixing of singly- and doubly-excited configurations. %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. %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 will add results for the biensemble to illustrate this.} %\titou{Titou might add results for the biensemble to illustrate this.}
%\manu{Well, neglecting the second excited state is not the same as %\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 %considering the $w_2=0$ limit. I thought you were referring to an
%approximation where the triensemble calculation is performed with %approximation where the triensemble calculation is performed with
@ -1314,9 +1314,9 @@ electrons.
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%%
It is also interesting to investigate the influence of the ensemble correlation derivative $\DD{c}{I}$ [defined in Eq.~\eqref{eq:DD-eLDA}] on both the single and double excitations. It is also interesting to investigate the influence of the ensemble correlation derivative $\DD{c}{(I)}$ [defined in Eq.~\eqref{eq:DD-eLDA}] on both the single and double excitations.
To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, in the case of 3-boxium, the error percentage (with respect to FCI) as a function of the box length $L$ To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, in the case of 3-boxium, 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}]. 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 %\manu{Manu: there is something I do not understand. If you want to
%evaluate the importance of the ensemble correlation derivatives you %evaluate the importance of the ensemble correlation derivatives you
%should only remove the following contribution from the $K$th KS-eLDA %should only remove the following contribution from the $K$th KS-eLDA
@ -1336,6 +1336,7 @@ This could explain why equiensemble calculations are clearly more
accurate as it reduces the influence of the ensemble correlation derivative: accurate as it reduces the influence of the ensemble correlation derivative:
for a given method, equiensemble orbitals partially remove the burden for a given method, equiensemble orbitals partially remove the burden
of modeling properly the ensemble correlation derivative. of modeling properly the ensemble correlation derivative.
\titou{Note also that, in our case, 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.}
%\manu{Manu: well, we %\manu{Manu: well, we
%would need the exact derivative value to draw such a conclusion. We can %would need the exact derivative value to draw such a conclusion. We can
%only speculate. Let us first see how important the contribution in %only speculate. Let us first see how important the contribution in
@ -1354,7 +1355,7 @@ of modeling properly the ensemble correlation derivative.
%%% %%% %%% %%% %%% %%%
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). 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 lines and KS-eLDA excitation energies The difference between the solid and dashed curves
undoubtedly show that, even in the strong correlation regime, the undoubtedly show that, even in the strong correlation regime, the
ensemble correlation derivative has a small impact on the double ensemble correlation derivative has a small impact on the double
excitations with a slight tendency of worsening the excitation energies excitations with a slight tendency of worsening the excitation energies
@ -1384,11 +1385,12 @@ from the finite uniform electron gas eLDA is
combined with eLDA, delivers accurate excitation energies for both combined with eLDA, delivers accurate excitation energies for both
single and double excitations, especially when an equiensemble is used. 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. In the latter case, the same weights are assigned to each state belonging to the ensemble.
{\it We have observed that, although the derivative discontinuity has a \titou{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 non-negligible effect on the excitation energies (especially for the
single excitations), its magnitude can be significantly reduced by single excitations), its magnitude can be significantly reduced by
performing state-averaged calculations instead of zero-weight performing equiweight calculations instead of zero-weight
calculations.}\manu{to be updated ...} calculations.}
Let us finally stress that the present methodology can be extended Let us finally stress that the present methodology can be extended
straightforwardly to other types of ensembles like, for example, the straightforwardly to other types of ensembles like, for example, the

27953
Notebooks/eDFT_FUEG.nb
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