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2nd screening of Sec IV done

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Pierre-Francois Loos 2020-08-18 10:07:54 +02:00
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27
Manuscript/rsdft-cipsi-qmc.bib
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@ -1,13 +1,35 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-17 12:55:12 +0200
%% Created for Pierre-Francois Loos at 2020-08-18 10:07:35 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Pack_1966,
Author = {{R. T. Pack and W. Byers-Brown}},
Date-Added = {2020-08-18 09:51:18 +0200},
Date-Modified = {2020-08-18 09:51:34 +0200},
Journal = {J. Chem. Phys.},
Pages = {556},
Title = {{Cusp conditions for molecular wavefunctions}},
Volume = {45},
Year = {1966}}
@article{Kato_1957,
Author = {T. Kato},
Date-Added = {2020-08-18 09:50:39 +0200},
Date-Modified = {2020-08-18 09:50:46 +0200},
Doi = {10.1002/cpa.3160100201},
Journal = {Comm. Pure Appl. Math.},
Pages = {151},
Title = {{On the eigenfunctions of many-particle systems in quantum mechanics}},
Volume = {10},
Year = {1957},
Bdsk-Url-1 = {https://doi.org/10.1002/cpa.3160100201}}
@article{Loos_2015b,
Author = {Loos, Pierre-Fran{\c c}ois and Bressanini, Dario},
Date-Added = {2020-08-17 12:54:30 +0200},
@ -1487,8 +1509,9 @@
Year = {2004},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1757439}}
@article{Ten-no2000Nov,
@article{Tenno_2000,
Author = {Ten-no, Seiichiro},
Date-Modified = {2020-08-18 09:50:06 +0200},
Doi = {10.1016/S0009-2614(00)01066-6},
Issn = {0009-2614},
Journal = {Chem. Phys. Lett.},

44
Manuscript/rsdft-cipsi-qmc.tex
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@ -568,7 +568,7 @@ Such a wave function satisfies the generalized Hermitian eigenvalue equation
e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E \, e^{2J} \Psi^J,
\label{eq:ci-j}
\end{equation}
but also the non-Hermitian transcorrelated eigenvalue problem\cite{BoyHan-PRSLA-69,BoyHanLin-1-PRSLA-69,BoyHanLin-2-PRSLA-69,Ten-no2000Nov,Luo-JCP-10,YanShi-JCP-12,CohLuoGutDowTewAla-JCP-19}
but also the non-Hermitian transcorrelated eigenvalue problem\cite{BoyHan-PRSLA-69,BoyHanLin-1-PRSLA-69,BoyHanLin-2-PRSLA-69,Tenno_2000,Luo-JCP-10,YanShi-JCP-12,CohLuoGutDowTewAla-JCP-19}
\begin{equation}
\label{eq:transcor}
e^{-J} \hat{H} \qty( e^{J} \Psi^J) = E \, \Psi^J,
@ -647,7 +647,7 @@ This is yet another key result of the present study.
\begin{table}
\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ P }$
for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
\titou{Please remove table and merge data in Fig. 5.}}
\titou{Please remove table and merge data in Fig. 4.}}
\label{tab:table_on_top}
\begin{ruledtabular}
\begin{tabular}{cc}
@ -697,52 +697,56 @@ From these data, one can clearly notice several trends.
First, from Table~\ref{tab:table_on_top}, we can observe that the overall
on-top pair density decreases when $\mu$ increases, which is expected
as the two-electron interaction increases in $H^\mu[n]$.
Second, the relative variations of the on-top pair density with $\mu$
Second, Fig.~\ref{fig:densities} shows that the relative variations of the on-top pair density with respect to $\mu$
are much more important than that of the one-body density, the latter
being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the
former can vary by about 10$\%$ in some regions.
%TODO TOTO
In the high-density region of the \ce{O-H} bond, the value of the on-top
pair density obtained from $\Psi^J$ is superimposed with
$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is
$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density of $\Psi^J$ is
the closest to $\mu=\infty$. The integrated on-top pair density
obtained with $\Psi^J$ lies between the values obtained with
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$ (see Table~\ref{tab:table_on_top}), consistently with the FN-DMC energies
and the overlap curve depicted in Fig.~\ref{fig:overlap}.
These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
Considering the form of $\hat{H}^\mu[n]$ [see Eq.~\eqref{H_mu}],
one can notice that the differences with respect to the usual Hamiltonian come
one can notice that the differences with respect to the usual bare Hamiltonian come
from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional.
The roles of these two terms are therefore very different: with respect
to the exact ground-state wave function $\Psi$, the non-divergent two-body interaction
increases the probability to find electrons at short distances in $\Psi^\mu$,
while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
provided that it is exact, maintains the exact one-body density.
This is clearly what has been observed from
providing that it is exact, maintains the exact one-body density.
This is clearly what has been observed in
Fig.~\ref{fig:densities}.
Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-no,\cite{Ten-no2000Nov}
Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-no,\cite{Tenno_2000}
the effective two-body interaction induced by the presence of a Jastrow factor
can be non-divergent when a proper Jastrow factor is chosen.
can be non-divergent when a proper Jastrow factor is chosen, \ie, the Jastrow factor must fulfill the so-called electron-electron cusp conditions. \cite{Kato_1957,Pack_1966}
\titou{T2: I think we are missing the point here that the one-body Jastrow mimics the effective one-body potential which makes the one-body density fixed.
The two-body Jastrow makes the interaction non-divergent like the non-divergent two-body interaction in RS-DFT.
Therefore, the one-body terms take care of the one-body properties and the two-body terms take care of the two-body properties. QED.}
Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Slater-Jastrow optimization:
they both deal with an effective non-divergent interaction but still
produce a reasonable one-body density.
As a conclusion of the first part of this study, we can notice that:
%============================
\subsection{Intermediate conclusion}
%============================
As a conclusion of the first part of this study, we can highlight the following observations:
\begin{itemize}
\item with respect to the nodes of a KS determinant or a FCI wave function,
\item With respect to the nodes of a KS determinant or a FCI wave function,
one can obtain a multi-determinant trial wave function $\Psi^\mu$ with a smaller
fixed-node error by properly choosing an optimal value of $\mu$
in RS-DFT calculations,
\item the optimal value of $\mu$ depends on the system and the
basis set, and the larger the basis set, the larger the optimal value
of $\mu$,
\item numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$,
fixed-node error by properly choosing an optimal value of $\mu$.
\item The optimal $\mu$ value is system- and basis-set-dependent, and it grows with basis set size.
\item Numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$,
one-body density, on-top pair density, and FN-DMC energy) indicate
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
\ie, mimicking short-range correlation effects. The latter
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor
by mimicking short-range correlation effects. This latter
statement can be qualitatively understood by noticing that both RS-DFT
and the trans-correlated approach deal with an effective non-divergent
electron-electron interaction, while keeping the density constant.