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Pierre-Francois Loos 2020-02-15 16:17:44 +01:00
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Manuscript/eDFT.tex
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@ -28,6 +28,8 @@
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\la}{\lambda} \newcommand{\la}{\lambda}
\newcommand{\si}{\sigma} \newcommand{\si}{\sigma}
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
% numbers % numbers
\newcommand{\nEl}{N} \newcommand{\nEl}{N}
@ -159,8 +161,8 @@ The simplest and most widespread approximation in state-of-the-art electronic st
In other words, within this so-called adiabatic approximation, the xc functional is assumed to be local in time. In other words, within this so-called adiabatic approximation, the xc functional is assumed to be local in time.
As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014} As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014}
When affordable (i.e., for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above. When affordable (\ie, for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above.
The basic idea is to describe a finite ensemble of states (ground and excited) altogether, i.e., with the same set of orbitals. The basic idea is to describe a finite ensemble of states (ground and excited) altogether, \ie, with the same set of orbitals.
Interestingly, a similar approach exists in DFT. Interestingly, a similar approach exists in DFT.
Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988a, Oliveira_1988, Gross_1988b} and is a generalization of Theophilou's variational principle for equi-ensembles. \cite{Theophilou_1979} Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988a, Oliveira_1988, Gross_1988b} and is a generalization of Theophilou's variational principle for equi-ensembles. \cite{Theophilou_1979}
In eDFT, the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest. In eDFT, the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest.
@ -197,7 +199,7 @@ where the $K$th energy level $\E{}{(K)}$ [$K=0$ refers to the ground state] is t
\hh = \sum_{i=1}^\nEl \qty[ -\frac{1}{2} \nabla_{\br{i}}^2 + \vne(\br{i}) ] \hh = \sum_{i=1}^\nEl \qty[ -\frac{1}{2} \nabla_{\br{i}}^2 + \vne(\br{i}) ]
\eeq \eeq
is the one-electron operator describing kinetic and nuclear attraction energies, and $\hat{W}_{\rm ee}$ is the electron repulsion operator. 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 $w_K$ decrease with increasing index $K$. They are normalized, i.e., The (positive) ensemble weights $\ew{K}$ decrease with increasing index $K$. They are normalized, \ie,
\beq \beq
\ew{0} = 1 - \sum_{K>0} \ew{K}, \ew{0} = 1 - \sum_{K>0} \ew{K},
\eeq \eeq
@ -205,23 +207,23 @@ so that only the weights $\bw \equiv \qty( \ew{1}, \ew{2}, \ldots, \ew{K}, \ldot
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{}. In GOK-DFT, the ensemble energy is determined variationally as follows: 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{}. In GOK-DFT, the ensemble energy is determined variationally as follows:
\beq\label{eq:var_ener_gokdft} \beq\label{eq:var_ener_gokdft}
\E{}{\bw} \E{}{\bw}
= \min_{\opGam{\bw}}\qty{ \Tr[\opGam{\bw} \hh] = \min_{\opGam{\bw}}
+ \E{Hx}{\bw} \qty[\n{\opGam{\bw}}{}] \qty{
+ \E{c}{\bw} \qty[\n{\opGam{\bw}}{}] \Tr[\opGam{\bw} \hh] + \E{Hx}{\bw} \qty[\n{\opGam{\bw}}{}] + \E{c}{\bw} \qty[\n{\opGam{\bw}}{}]
}, },
\eeq \eeq
where $\Tr$ denotes the trace and the trial ensemble density matrix operator reads where $\Tr$ denotes the trace and the trial ensemble density matrix operator reads
\beq \beq
\opGam{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}. \opGam{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}.
\eeq \eeq
The determinants (or configuration state functions) $\Phi^{(K)}$ are all constructed from the same set of (ensemble Kohn--Sham) orbitals that is optimized variationally and the trial ensemble density is simply the weighted sum of the individual densities: The determinants (or configuration state functions) $\Det{(K)}$ are all constructed from the same set of (ensemble Kohn--Sham) orbitals that is optimized variationally and the trial ensemble density is simply the weighted sum of the individual densities:
\beq\label{eq:KS_ens_density} \beq\label{eq:KS_ens_density}
\n{\opGam{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Phi^{(K)}}{}(\br{}). \n{\opGam{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K)}}{}(\br{}).
\eeq \eeq
As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange and correlation energies are described with density functionals that are \textit{weight dependent}. As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange and correlation energies are described with density functionals that are \textit{weight dependent}.
We focus here on the (exact) Hx part which is defined as follows: We focus here on the (exact) Hx part which is defined as follows:
\beq\label{eq:exact_ens_Hx} \beq\label{eq:exact_ens_Hx}
\E{Hx}{\bw}[\n{}{}]=\sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}[\n{}{}]}{\hWee}{\Det{(K)}[\n{}{}]} \E{Hx}{\bw}[\n{}{}]=\sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}[\n{}{}]}{\hWee}{\Det{(K)}[\n{}{}]},
\eeq \eeq
where the KS wavefunctions fulfill the ensemble density constraint where the KS wavefunctions fulfill the ensemble density constraint
\beq \beq
@ -233,7 +235,7 @@ In practice, one is not much interested in ensemble energies but rather in excit
\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} ) \pdv{\E{}{\bw}}{\ew{K}}, \E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \pdv{\E{}{\bw}}{\ew{K}},
\eeq \eeq
where, according to the {\it variational} ensemble energy expression of Eq.~\eqref{eq:var_ener_gokdft}, the derivative in $w_K$ can be evaluated from the minimizing KS wavefunctions $\Det{(K)} = \Det{(K),\bw}$ as follows: where, 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 KS wavefunctions $\Det{(K)} = \Det{(K),\bw}$ as follows:
\beq\label{eq:deriv_Ew_wk} \beq\label{eq:deriv_Ew_wk}
\begin{split} \begin{split}
\pdv{\E{}{\bw}}{\ew{K}} \pdv{\E{}{\bw}}{\ew{K}}
@ -510,7 +512,7 @@ stationarity condition
\eeq \eeq
where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the metric and the ensemble Fock-like matrix reads where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the metric and the ensemble Fock-like matrix reads
\beq \beq
\eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} + \sum_{\la\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw} \bF{\bw} \equiv \eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} + \sum_{\la\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw}
\eeq \eeq
with with
\beq \beq
@ -695,7 +697,7 @@ Moreover, because the infinite UEG model is a metal, it is gapless, which means
From this point of view, using finite UEGs \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs can be seen as more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a} From this point of view, using finite UEGs \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs can be seen as more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems. Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems.
As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e., a circle). \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). \cite{Loos_2012, Loos_2013a, Loos_2014b}
The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform. The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform.
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary. 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 derivative discontinuities. This is a necessary condition for being able to model derivative discontinuities.
@ -710,7 +712,7 @@ We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_
\label{sec:Ec} \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 (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant 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} \begin{equation}
\label{eq:ec} \label{eq:ec}
\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}\,\n{}{}}{\n{}{} + a_2^{(I)} \sqrt{\n{}{}} + a_3^{(I)}}, \e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}\,\n{}{}}{\n{}{} + a_2^{(I)} \sqrt{\n{}{}} + a_3^{(I)}},
@ -801,7 +803,7 @@ In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ an
These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length. 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} 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, i.e. We use as basis functions the (orthonormal) orbitals of the one-electron system, \ie,
\begin{equation} \begin{equation}
\AO{\mu}(x) = \AO{\mu}(x) =
\begin{cases} \begin{cases}
@ -823,14 +825,14 @@ Concerning the KS-eDFT calculations, two sets of weight have been tested: the ze
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion} \section{Results and discussion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In Fig.~\ref{fig:EvsL}, we report the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (i.e., $\Nel = 5$). In Fig.~\ref{fig:EvsL}, we report 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. 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.
In the weakly correlated regime (i.e., small $L$), all methods provide accurate estimates of the excitation energies. In the weakly correlated regime (\ie, small $L$), all methods provide accurate estimates of the excitation energies.
When the box gets larger, they start to deviate. When the box gets larger, they start to deviate.
For the single excitation, TDHF is extremely accurate over the whole range of $L$ values, while CIS is slightly less accurate and starts to overestimate the excitation energy by a few percent at $L=8\pi$. For the single excitation, TDHF is extremely accurate over the whole range of $L$ values, while CIS is slightly less accurate and starts to overestimate the excitation energy by a few percent at $L=8\pi$.
TDLDA yields larger errors at large $L$ by underestimating the excitation energies. TDLDA yields larger errors at large $L$ by underestimating the excitation energies.
TDA-TDLDA slightly corrects this trend thanks to error compensation. TDA-TDLDA slightly corrects this trend thanks to error compensation.
Concerning the eLDA functional, our results clearly evidences that the equi-weights [i.e., $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [i.e., $\bw = (0,0)$]. Concerning the eLDA functional, our results clearly evidences that the equi-weights [\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 for the single excitation which is significantly improved by using state-averaged weights. This is especially true for the single excitation which is significantly improved by using state-averaged weights.
The effect on the double excitation is less pronounced. The effect on the double excitation is less pronounced.
Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations. Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations.