loos/FarDFT
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

H2 eq done

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-04-09 12:23:26 +02:00
parent 6257ccc1b4
commit d90f73a3d9
2 changed files with 28 additions and 36 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

9
Manuscript/FarDFT.bib
View File

@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-04-09 10:05:15 +0200
%% Created for Pierre-Francois Loos at 2020-04-09 12:04:38 +0200
%% Saved with string encoding Unicode (UTF-8)
@ -54,9 +54,7 @@
Publisher = {Zenodo},
Title = {numgrid: numerical integration grid for molecules},
Url = {https://github.com/dftlibs/numgrid},
Year = {2019},
Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package},
Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}}
Year = {2019}}
@misc{QuAcK,
Author = {P. F. Loos},
@ -439,7 +437,8 @@
Pages = {1884},
Title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities},
Volume = {51},
Year = {1983}}
Year = {1983},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.51.1884}}
@article{Schulz_1993,
Author = {H. J. Schulz},

55
Manuscript/FarDFT.tex
View File

@ -69,6 +69,7 @@
\newcommand{\LDA}{\text{LDA}}
\newcommand{\SD}{\text{S}}
\newcommand{\VWN}{\text{VWN5}}
\newcommand{\eVWN}{\text{eVWN5}}
\newcommand{\SVWN}{\text{SVWN5}}
\newcommand{\LIM}{\text{LIM}}
\newcommand{\MOM}{\text{MOM}}
@ -373,7 +374,7 @@ Doing so, we have found that the present weight-dependent exchange functional (d
with
\begin{equation}
\label{eq:Cxw}
\frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ]
\frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ],
\end{equation}
and
\begin{subequations}
@ -385,11 +386,12 @@ and
\gamma & = - 0.367\,189,
\end{align}
\end{subequations}
makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable (with respect to $\ew{}$) and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cx_H2}, the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$, as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits.
Note 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 strictly forbidden by the GOK variational principle. \cite{Gross_1988a}
However, it is important to ensure that the weight-dependent functional does not alter the $\ew{} = 1$, which corresponds to a genuine saddle point of the KS equations, as mentioned above.
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 the $\ew{} = 1$ limit, which is a genuine saddle point 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.
\begin{figure}
\includegraphics[width=\linewidth]{Cxw}
@ -404,9 +406,9 @@ Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq
\subsection{Weight-independent correlation functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Third, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980}
Third, we add up correlation effects via the VWN5 local correlation functional. \cite{Vosko_1980}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the GIC-S and VWN5 functionals (GIC-SVWN5) exhibit a small curvature (the ensemble energy is now slightly concave) and improved excitation energies, especially at small weights, where the SVWN5 excitation energy is almost spot on.
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the GIC-S and VWN5 functionals (GIC-SVWN5) exhibit a small curvature (the ensemble energy is slightly concave) and improved excitation energies, especially at small weights, where the SVWN5 excitation energy is almost spot on.
%%%%%%%%%%%%%%%%%%
%%% FUNCTIONAL %%%
@ -573,33 +575,27 @@ The weight-dependence of the correlation functional is then carried exclusively
Consistently with such a strategy, Eq.~\eqref{eq:ecw} is ``centred'' on its corresponding weight-independent VWN5 LDA reference
\begin{equation}
\label{eq:becw}
\be{\co}{\ew{}}(\n{}{}) = (1-\ew{}) \be{\co}{(0)}(\n{}{}) + \ew{} \be{\co}{(1)}(\n{}{})
\e{\co}{\ew{},\eVWN}(\n{}{}) = (1-\ew{}) \be{\co}{(0)}(\n{}{}) + \ew{} \be{\co}{(1)}(\n{}{})
\end{equation}
via the following shift:
via the following global, state-independent shift:
\begin{equation}
\be{\co}{(I)}(\n{}{}) = \e{\co}{(I)}(\n{}{}) + \e{\co}{\VWN}(\n{}{}) - \e{\co}{(0)}(\n{}{}).
\end{equation}
Equation \eqref{eq:becw} can be recast
In the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the VWN5 local correlation functional for ensembles.
Also, Eq.~\eqref{eq:becw} can be recast
\begin{equation}
\label{eq:eLDA}
\begin{split}
\be{\co}{\ew{}}(\n{}{})
& = \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
\\
& = \e{\co}{\VWN}(\n{}{}) + \ew{} \pdv{\e{\co}{\ew{}}(\n{}{})}{\ew{}},
\end{split}
\e{\co}{\ew{},\eVWN}(\n{}{}) = \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
\end{equation}
which nicely highlights the centrality of the LDA in the present weight-dependent density-functional approximation for ensembles.
In particular, $\be{\co}{(0)}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
Consequently, in the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the VWN5 local correlation functional for ensembles.
Also, we note that, by construction,
In particular, $\e{\co}{\ew{}=0,\eVWN}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
We note also that, by construction, we have
\begin{equation}
\label{eq:dexcdw}
\pdv{\be{\co}{\ew{}}(\n{}{})}{\ew{}}
= \be{\co}{(1)}(n) - \be{\co}{(0)}(n),
\pdv{\e{\co}{\ew{},\eVWN}(\n{}{})}{\ew{}}
= \e{\co}{(1)}(n) - \e{\co}{(0)}(n),
\end{equation}
which shows that the weight correction is purely linear in eVWN5.
showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model.
As shown in Fig.~\ref{fig:Ew_H2}, the SGIC-eVWN5 is slightly less concave than its SGIC-VWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
@ -617,23 +613,22 @@ As shown in Fig.~\ref{fig:Ew_H2}, the SGIC-eVWN5 is slightly less concave than i
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 (\ie, $\ew{} = 1/2$).
For comparison purposes, we also report the linear interpolation method (LIM), \cite{Senjean_2015,Senjean_2016} which is
For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016}
a pragmatic way of getting weight-independent
excitation energies defined as
\begin{equation}
\Ex{\LIM}{(1)} = 2 (\E{}{\ew{}=1/2} - \E{}{\ew{}=0}),
\Ex{\LIM}{} = 2 (\E{}{\ew{}=1/2} - \E{}{\ew{}=0}),
\end{equation}
as well as the MOM excitation energies. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
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 $\ew{} = 0$.
MOM excitation energies can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
\begin{equation}
\Ex{\MOM}{(1)} = \E{}{\ew{}=1} - \E{}{\ew{}=0}.
\Ex{\MOM}{} = \E{}{\ew{}=1} - \E{}{\ew{}=0}.
\end{equation}
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the GIC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to GIC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remains in good agreement with FCI, with again a small improvement as compared to GIC-SVWN5.
The GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$.
Finally, note that the GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
%%% TABLE I %%%
\begin{table*}
@ -736,14 +731,12 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\end{table*}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
We have studied the weight dependence of the ensemble energy in the framework of GOK-DFT.
\titou{We have studied the weight dependence of the ensemble energy in the framework of GOK-DFT.}
%%%%%%%%%%%%%%%%%%%%%%%%
%%% ACKNOWLEDGEMENTS %%%