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Pierre-Francois Loos 2020-04-28 21:41:24 +02:00
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@ -599,8 +599,9 @@ Indeed, the pure-state limits (\ie, $\ew{1} = 1 \land \ew{2} = 0$ or $\ew{1} = 0
\section{Results and Discussion}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this Section, we propose a two-step procedure to design, first, a weight-dependent, system-dependent local exchange functional in order to remove the curvature of the ensemble energy.
In this Section, we propose a two-step procedure to design, first, a weight- and system-dependent local exchange functional in order to remove the curvature of the ensemble energy.
Second, we describe the construction of a universal, weight-dependent local correlation functional based on FUEGs.
This procedure is applied to various two-electron systems in order to extract excitation energies associated with doubly-excited states.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hydrogen molecule at equilibrium}
@ -611,7 +612,7 @@ Second, we describe the construction of a universal, weight-dependent local corr
\subsubsection{Weight-independent exchange functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bond length, \ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent LDA Slater exchange functional (\ie, no correlation functional is employed), \cite{Dirac_1930, Slater_1951} which is explicitly given by
First, we compute the ensemble energy of the \ce{H2} molecule at equilibrium bond length (\ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent LDA Slater exchange functional (\ie, no correlation functional is employed), \cite{Dirac_1930, Slater_1951} which is explicitly given by
\begin{align}
\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
&
@ -632,7 +633,7 @@ The ensemble energy $\E{}{\eW}$ is depicted in Fig.~\ref{fig:Ew_H2} as a functio
Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
As anticipated, $\E{}{\eW}$ is far from being linear, which means that the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\eW$ (see Fig.~\ref{fig:Om_H2}).
Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\eW = 0$ to $1/2$.
Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\eW$.
Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of the ensemble weights.
\begin{figure}
\includegraphics[width=\linewidth]{Ew_H2}
@ -696,8 +697,7 @@ The parameters $\alpha$, $\beta$, and $\gamma$ entering Eq.~\eqref{eq:Cxw} have
Note that this procedure makes, by construction, the ensemble energy also linear with respect to $\ew{1}$ and $\ew{2}$.
As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the three ghost-interaction-free limits at $\ew{1} = \ew{2} = 0$, $\ew{1} = 1 \land \ew{2} = 0$, and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\eW}$ reduces to $\Cx{}$ in these three limits thanks to the factor $\eW (1 - \eW)$.
%Maybe surprisingly, one would have noticed that we are not only using data from $0 \le \ew{} \le 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is deterred by the GOK variational principle. \cite{Gross_1988a}
However, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which is a genuine saddle point of the KS equations, as mentioned above.
Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which are genuine saddle points of the KS equations, as mentioned above.
Finally, let us mention that, around $\eW = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
We shall come back to this point later on.
@ -729,10 +729,10 @@ The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex
%previous results.}
Fourth, in the spirit of our recent work, \cite{Loos_2020} we design a universal, weight-dependent correlation functional.
To build this correlation functional, we consider the singlet ground state, the first singly-excited state as well as the first doubly-excited state of a two-electron FUEGs 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.
To build this correlation functional, we consider the singlet ground state, the first singly-excited state, as well as the first doubly-excited state of a two-electron FUEGs 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 three 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 in its ground state, thus yielding an electron density that is uniform on the 3-sphere.
Note that the present paradigm is equivalent to the conventional 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.
@ -857,13 +857,11 @@ We note also that, by construction, we have
= \e{\co}{(I)}(n) - \e{\co}{(0)}(n),
\end{equation}
showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model.
Note that, in the correlation case, one cannot define straightforwardly a composite weight like in the exchange case [see Eq.~\eqref{eq:Cxw}].
As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is slightly less concave than its CC-SVWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
In particular, we report the excitation energies obtained with GOK-DFT
in the zero-weight limit (\ie, $\eW = 0$) and for the equi-ensemble for which $\ew{1} = \ew{2} = 1/3$ (\ie, $\eW = 1/2$).
in the zero-weight limit (\ie, $\eW = 0$) and for equi-weights, \ie, $\ew{1} = \ew{2} = 1/3$ (or $\eW = 1/2$).
These excitation energies are computed using Eq.~\eqref{eq:dEdw}.
For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016}
@ -883,20 +881,17 @@ which require three independent calculations, as well as the MOM excitation ener
\Ex{\MOM}{(2)} & = \E{}{\bw{}=(0,1)} - \E{}{\bw{}=(0,0)}.
\end{align}
\end{subequations}
which also require three separate calculations.
which also require three separate calculations at a different set of ensemble weights.
%We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
%They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
The CC-SeVWN5 excitation energies at equi-weights (\ie, $\eW = 1/2$) are less satisfactory, but still remain in good agreement with FCI, with again a small improvement as compared to CC-SVWN5.
It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{2} = 1$ (\textit{vide supra}).
Note that by construction,
for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper),
LIM and MOM can be reduced to a single calculation
at $\ew{} = 1/4$ and $\ew=1/2$, respectively, instead of performing an interpolation between two different calculations.
Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between $\ew{} = 0$ and $1$.
It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\eW = 1$ (\textit{vide supra}).
\bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\eW = 1/4$ and $\eW = 1/2$, respectively, instead of performing an interpolation between two different calculations.}
Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between $\eW = 0$ and $\eW = 1/2$.
\titou{The present protocol can be related to optimally-tuned range-separated hybrid functionals. T2: more to come.}
\titou{This two-step procedure can be related to optimally-tuned range-separated hybrid functionals. T2: more to come.}
%%% TABLE III %%%
\begin{table}
@ -908,7 +903,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\begin{tabular}{llccccc}
\mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\
\cline{1-2} \cline{4-5}
\tabc{x} & \tabc{c} & Basis & $\bw{} = \bO$ & $\bw{} = \bthird$ & LIM & MOM \\
\tabc{x} & \tabc{c} & Basis & $\eW = 0$ & $\eW = 1/2$ & LIM & MOM \\
\hline
HF & & aug-cc-pVDZ & 38.52 & 30.86 & 34.00 & 28.65 \\
& & aug-cc-pVTZ & 38.58 & 35.82 & 35.80 & 28.65 \\
@ -968,24 +963,24 @@ To investigate the weight dependence of the xc functional in the strong correlat
lowest excited state with the same symmetry as
the ground state.}
We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH = 3.7$ bohr.
It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve).
It yields $\alpha = +0.019\,182$, $\beta = -0.015\,453$, and $\gamma = -0.012\,720$ [see Eq.~\eqref{eq:Cxw}].
The weight dependence of $\Cx{\eW}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve).
One clearly sees that the correction brought by CC-S is much more gentle than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the LDA exchange functional is much more linear at $\RHH = 3.7$ bohr.
In other words, the \titou{curvature ``hole''} depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
%In other words, the \titou{curvature ``hole''} depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
Note that this linearity at $\RHH = 3.7$ bohr was also observed using weight-independent xc functionals in Ref.~\onlinecite{Senjean_2015}.
Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers basis set converged results), the same set of calculations as in Table \ref{tab:BigTab_H2}.
As a reference value, we computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015}
For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the closest match being reached with HF exchange and eVWN5 correlation at equi-ensemble.
\manu{We did not mention HF exchange neither in the theory section nor
in the computational details. We should be clear about this. Is this an
ad-hoc correction, like in our previous work on ringium? Is HF exchange
used for the full ensemble energy (i.e. the HF interaction energy is
computed with the ensemble density matrix and therefore with
ghost-interaction errors) or for
individual energies (that you state-average then), like in our previous
work. I guess the latter option is what you did. We need to explain more
what we do!!!}
For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the closest match being reached with HF exchange and eVWN5 correlation at equi-weights.
%\manu{We did not mention HF exchange neither in the theory section nor
%in the computational details. We should be clear about this. Is this an
%ad-hoc correction, like in our previous work on ringium? Is HF exchange
%used for the full ensemble energy (i.e. the HF interaction energy is
%computed with the ensemble density matrix and therefore with
%ghost-interaction errors) or for
%individual energies (that you state-average then), like in our previous
%work. I guess the latter option is what you did. We need to explain more
%what we do!!!}
%\bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}.
As expected from the linearity of the ensemble energy, the CC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
Nonetheless, the excitation energy is still off by $3$ eV.
@ -1006,7 +1001,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\begin{tabular}{llcccc}
\mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
\cline{1-2} \cline{3-4}
\tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
\tabc{x} & \tabc{c} & $\eW = 0$ & $\eW = 1/2$ & LIM & MOM \\
\hline
HF & & 19.09 & 6.59 & 12.92 & 6.52 \\
HF & VWN5 & 19.40 & 6.54 & 13.02 & 6.49 \\
@ -1040,28 +1035,21 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As a final example, we consider the \ce{He} atom which can be seen as the limiting form of the \ce{H2} molecule for very short bond lengths.
Similar to \ce{H2}, our ensemble contains the ground state of configuration $1s^2$, the lowest singlet excited state of configuration $1s2s$, and the first doubly-excited state of configuration $2s^2$.
In \ce{He}, the lowest doubly-excited state is an auto-ionising resonance state, extremely high in energy and lies in the continuum. \cite{Madden_1963}
In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimate an excitation energy of $2.126$ hartree for this $1s^2 \rightarrow 2s^2$ transition.
\manu{same comment as for H$_2$ at equilibrium. I would expect the
singly-excited configuration $1s2s$ to be considered in the ensemble
with the doubly-excited one. We need to know if the former has an impact
(I guess it does)
on the computations.}
Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions.
Consequently, we consider for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
The excitation energies associated with this double excitation computed with various methods and combinations of xc functionals are gathered in Table \ref{tab:BigTab_He}.
The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.912\,574$, $\beta = +2.715\,267$, and $\gamma = +2.163\,422$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
In other words, the \titou{curvature hole} is deeper.
The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.901\,572$, $\beta = +2.523\,660$, and $\gamma = +1.665\,228$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.
%\bruno{But also with CC-SVWN5, as in the rest of this article, so one could wonder about the usefulness of the eVWN5 functional...}
The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy for $\eW = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.
As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}.
This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy.
As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
%%% TABLE V %%%
\begin{table}
\caption{
@ -1072,7 +1060,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
\begin{tabular}{llcccc}
\mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
\cline{1-2} \cline{3-4}
\tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
\tabc{x} & \tabc{c} & $\eW = 0$ & $\eW = 1/2$ & LIM & MOM \\
\hline
HF & & 1.874 & 2.212 & 2.080 & 2.142 \\
HF & VWN5 & 1.988 & 2.260 & 2.153 & 2.193 \\
@ -1134,7 +1122,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
\label{sec:ccl}
In the present article, we have discussed the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron systems (\ce{He} and \ce{H2}) specifically designed for the computation of double excitations within GOK-DFT, a time-\textit{independent} formalism thanks to which one can extract excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
\titou{In the spirit of optimally-tuned range-separated hybrid functionals,} we have found that the construction of a system-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy (by removing most of the curvature of the ensemble energy).
In the spirit of optimally-tuned range-separated hybrid functionals, we have found that the construction of a system-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy (by removing most of the curvature of the ensemble energy).
Although the weight-dependent correlation functional developed in this paper (eVWN5) performs systematically better than their weight-independent counterpart (VWN5), the improvement remains rather small.
To better understand the reasons behind this, it would be particularly interesting to investigate the influence of the self-consistent procedure,
\ie, the variation in excitation energy when the \textit{exact} ensemble density (built with the exact individual densities) is used instead