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

starting writing He and H2st

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-04-09 22:55:00 +02:00
parent 1324723e4f
commit bbb2ce3bcc
2 changed files with 51 additions and 10 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

28
Manuscript/FarDFT.bib
View File

@ -1,13 +1,35 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-04-09 21:27:50 +0200
%% Created for Pierre-Francois Loos at 2020-04-09 22:26:01 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Becke_1988a,
Author = {A. D. Becke},
Date-Added = {2020-04-09 22:22:05 +0200},
Date-Modified = {2020-04-09 22:24:01 +0200},
Doi = {10.1103/PhysRevA.38.3098},
Journal = {Phys. Rev. A},
Pages = {3098},
Title = {Density-functional exchange-energy approximation with correct asymptotic behavior},
Volume = {38},
Year = {1988}}
@article{Lee_1988,
Author = {C. Lee and W. Yang and R. G. Parr},
Date-Added = {2020-04-09 22:20:57 +0200},
Date-Modified = {2020-04-09 22:21:44 +0200},
Doi = {10.1103/PhysRevB.37.785},
Journal = {Phys. Rev. B},
Pages = {785},
Title = {Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density},
Volume = {37},
Year = {1988}}
@article{Burges_1995,
Author = {A. Burgers and D. Wintgen and J.-M. Rost},
Date-Added = {2020-04-09 14:56:36 +0200},
@ -177,10 +199,10 @@
Year = {2001},
Bdsk-Url-1 = {https://doi.org/10.1007/s002140100263}}
@article{Becke_1988,
@article{Becke_1988b,
Author = {A. D. Becke},
Date-Added = {2020-03-30 09:58:02 +0200},
Date-Modified = {2020-03-30 09:59:13 +0200},
Date-Modified = {2020-04-09 22:24:05 +0200},
Doi = {10.1063/1.454033},
Journal = {J. Chem. Phys.},
Pages = {2547},

33
Manuscript/FarDFT.tex
View File

@ -311,15 +311,15 @@ where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-depen
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
The self-consistent GOK-DFT calculations have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented.
For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found.
For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family of atomic basis sets. \cite{Dunning_1989,Kendall_1992,Woon_1994}
Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988,Lindh_2001}
Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988b,Lindh_2001}
This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered.
Although one should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint.
Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
\titou{Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
@ -390,7 +390,7 @@ and
\end{align}
\end{subequations}
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.
As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw}, 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.
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.
@ -400,7 +400,7 @@ We shall come back to this point later on.
\includegraphics[width=\linewidth]{Cxw}
\caption{
$\Cx{\ew{}}/\Cx{\ew{}=0}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}] computed with the aug-cc-pVTZ basis set for the \ce{He} atom (blue) and the \ce{H2} molecule at $\RHH = 1.4$ bohr (red) and $\RHH = 3.7$ bohr (green).
\label{fig:Cx_H2}
\label{fig:Cxw}
}
\end{figure}
@ -689,6 +689,18 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\label{sec:H2st}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To investigate the weight dependence of the xc functional in the strongly correlated regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr).
For this particular geometry, the doubly-excited state becomes the lowest excited state.
We then follow the same protocol as in Sec.~\ref{sec:H2}, and designed a GIC-S functional for this system and the aug-cc-pVTZ basis set.
The weight-dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw}. One clearly sees that the correction brought by GIC-S is much more subtile than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the Slater-Dirac functional is much more linear at $\RHH = 3.7$ bohr and the curvature more gentle.
Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers converged results with respect to the size of the basis set), the same set of calculations as in Table \ref{tab:BigTab_H2}.
As a reference value, we have 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 best match being obtained with HF exchange.
The GIC-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.
The fundamental theoretical reason of such a poor agreement is not clear.
The fact that HF exchange yields better excitation energy hints at the effect of self-interaction error.
%%% TABLE I %%%
\begin{table}
\caption{
@ -702,8 +714,8 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\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\\
HF & eVWN5 & 19.59 & 6.72 & 13.11 & \\
HF & VWN5 & 19.40 & 6.54 & 13.02 & 6.49 \\
HF & eVWN5 & 19.59 & 6.72 & 13.11 & \\
S & & 5.31 & 5.60 & 5.46 & 5.56 \\
S & VWN5 & 5.34 & 5.57 & 5.46 & 5.52 \\
S & eVWN5 & 5.53 & 5.76 & 5.56 & 5.72 \\
@ -715,7 +727,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
B3 & LYP & & & & 5.55 \\
HF & LYP & & & & 6.68 \\
\hline
\mc{5}{l}{Accurate\fnm[1]} & 8.69 \\
\mc{5}{l}{Accurate\fnm[1]} & 8.69 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}}
@ -727,6 +739,13 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\label{sec:He}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.
In \ce{He}, the lowest doubly-excited state is extremely high in energy and lies in the continuum. \cite{Burges_1995}
In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimates an excitation energy of $2.126$ hartree.
Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions.
This is why we have considered 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 functions are gathered in Table \ref{tab:BigTab_He}.
%%% TABLE I %%%
\begin{table}
\caption{