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

clean up Bruno comments

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-05-06 11:22:03 +02:00
parent 29d803aa49
commit a72d7e9b88
1 changed files with 32 additions and 37 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

69
Manuscript/FarDFT.tex
View File

@ -481,8 +481,7 @@ Numerical quadratures are performed with the \texttt{numgrid} library \cite{numg
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 three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), a singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered.
Assuming that the singly-excited state is lower in energy than the doubly-excited state, one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle.
If the doubly-excited state is lower in energy than the singly-excited one (which can be the case as one would notice later), then one has to swap $w_1$ and $w_2$ in the
above inequalities.
If the doubly-excited state is lower in energy than the singly-excited state (which can be the case as one would notice later), then one has to swap $\ew{1}$ and $\ew{2}$ in the above inequalities.
Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$).
In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$.
The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$) are considered in the following.
@ -750,16 +749,16 @@ a pragmatic way of getting weight-independent excitation energies, defined as
\Ex{\LIM}{(2)} & = 3 \qty[\E{}{\bw{}=(1/3,1/3)} - \E{}{\bw{}=(1/2,0)}] + \frac{1}{2} \Ex{\LIM}{(1)}, \label{eq:LIM2}
\end{align}
\end{subequations}
\manu{
$\frac{1}{2}\Ex{\LIM}{(1)}=\frac{1}{2}\left(E_1-E_0\right)$\\
$\E{}{\bw{}=(1/3,1/3)}=\frac{1}{3}\left(E_0+E_1+E_2\right)$\\
$\E{}{\bw{}=(1/2,0)}=\frac{1}{2}\left(E_0+E_1\right)$\\
$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
\E{}{\bw{}=(1/2,0)}]=-\frac{1}{2}\left(E_0+E_1\right)+E_2$
\\
$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$
}\\
%\manu{
%$\frac{1}{2}\Ex{\LIM}{(1)}=\frac{1}{2}\left(E_1-E_0\right)$\\
%$\E{}{\bw{}=(1/3,1/3)}=\frac{1}{3}\left(E_0+E_1+E_2\right)$\\
%$\E{}{\bw{}=(1/2,0)}=\frac{1}{2}\left(E_0+E_1\right)$\\
%$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
%\E{}{\bw{}=(1/2,0)}]=-\frac{1}{2}\left(E_0+E_1\right)+E_2$
%\\
%$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
%\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$
%}\\
which require three independent calculations, as well as the MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
\begin{subequations}
\begin{align}
@ -769,29 +768,26 @@ which require three independent calculations, as well as the MOM excitation ener
\end{align}
\end{subequations}
which also require three separate calculations at a different set of ensemble weights.
As readily seen in Eqs.~(\ref{eq:LIM1}) and (\ref{eq:LIM2}), LIM is a recursive strategy where the first excitation energy has to be determined
For a general expression with multiple (and possibly degenerate) states, we refer the reader to Eq.~(106) of Ref.~\onlinecite{Senjean_2015}, where LIM is shown to interpolate linearly the ensemble energy between equi-ensembles.
Note that two calculations are needed to get the first LIM excitation energy, but only one is required for each higher excitation.
As readily seen in Eqs.~\eqref{eq:LIM1} and \eqref{eq:LIM2}, LIM is a recursive strategy where the first excitation energy has to be determined
in order to compute the second one.
In the above equations, we
assumed that the singly-excited state (with weight $w_1$) was lower
in energy compared to the doubly-excited one (with weight $w_2$).
If the ordering changes, then one should read $\E{}{\bw{}=(0,1/2)}$
instead of $\E{}{\bw{}=(1/2,0)}$ in Eqs.~(\ref{eq:LIM1}) and (\ref{eq:LIM2}) which then correspond to the excitation energies of the
doubly-excited state and the singly-excited one, respectively.
assumed that the singly-excited state (with weight $\ew{1}$) is lower
in energy than the doubly-excited state (with weight $\ew{2}$).
If the ordering changes (like in the case of the stretched \ce{H2} molecule, see below), then one should substitute $\E{}{\bw{}=(0,1/2)}$
by $\E{}{\bw{}=(1/2,0)}$ in Eqs.~\eqref{eq:LIM1} and \eqref{eq:LIM2} which then correspond to the excitation energies of the
doubly-excited and singly-excited states, respectively.
The same holds for the MOM excitation energies in
Eqs.~\ref{eq:MOM1} and \ref{eq:MOM2}.
For a general expression with multiple (and possibly degenerate)
states, we
refer the reader to Eq.~106 of Ref.~\onlinecite{Senjean_2015}, where
LIM is shown to interpolate linearly the
ensemble energy between equi-ensembles.
Note that two calculations are needed for the first excitation energy
within LIM, but only one is required for each higher excitation
energies. By construction, for ensemble energies that are quadratic with
respect to the weight (which is almost always the case in this paper), the
first excitation energy within LIM
and MOM can actually be obtained in a single calculation at
$\ew{} = 1/4$ and
$\ew{} = 1/2$, respectively.
Eqs.~\eqref{eq:MOM1} and \eqref{eq:MOM2}.
%By construction, for ensemble energies that are quadratic with
%respect to the weight (which is almost always the case in this paper), the
%first excitation energy within LIM
%and MOM can actually be obtained in a single calculation at
%$\ew{} = 1/4$ and
%$\ew{} = 1/2$, respectively.
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights 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, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI.
@ -854,7 +850,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\mc{6}{l}{Accurate\fnm[2]} & 28.75 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Eqs.~(\ref{eq:LIM2}) and (\ref{eq:MOM2}) are used where the first weight corresponds to the singly-excited state.}
\fnt[1]{Equations \eqref{eq:LIM2} and \eqref{eq:MOM2} are used where the first weight corresponds to the singly-excited state.}
\fnt[2]{FCI/aug-mcc-pV8Z calculation from Ref.~\onlinecite{Barca_2018a}.}
\end{table}
%%% %%% %%% %%%
@ -867,8 +863,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
To investigate the weight dependence of the xc functional in the strong correlation regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr).
Note that, for this particular geometry, the doubly-excited state becomes the lowest excited state with the same symmetry as the ground state.
Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble defined in Sec.~\ref{sec:H2}.
One should just be careful when reading the equations, as they correspond to the case where the
singly-excited state is lower in energy than the doubly-excited one.
Nonetheless, one should just be careful when reading the equations reported above, as they correspond to the case where the singly-excited state is lower in energy than the doubly-excited 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).
@ -927,7 +922,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
\mc{5}{l}{Accurate\fnm[4]} & 8.69 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Eqs.~(\ref{eq:LIM1}) and (\ref{eq:MOM1}) are used where the first weight corresponds to the doubly-excited state.}
\fnt[1]{Equations \eqref{eq:LIM1} and \eqref{eq:MOM1} are used where the first weight corresponds to the doubly-excited state.}
\fnt[2]{KS calculation does not converge.}
\fnt[3]{Short-range multiconfigurational DFT/aug-cc-pVQZ calculations from Ref.~\onlinecite{Senjean_2015}.}
\fnt[4]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}}
@ -990,7 +985,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
\mc{2}{l}{Accurate\fnm[2]} & & & & 2.126 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Eqs.~(\ref{eq:LIM2}) and (\ref{eq:MOM2}) are used where the first weight corresponds to the singly-excited state.}
\fnt[1]{Equations \eqref{eq:LIM2} and \eqref{eq:MOM2} are used where the first weight corresponds to the singly-excited state.}
\fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.}
\end{table}