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RSDFT-CIPSI-QMC
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corrections all around

This commit is contained in:
Pierre-Francois Loos 2020-08-08 14:13:03 +02:00
parent 16c822340d
commit b774c21f24
3 changed files with 17 additions and 19 deletions
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16
Data/algorithm.tex
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@ -21,14 +21,14 @@ decoration={snake,
\begin{tikzpicture}[scale=2.3]
\begin{scope}[very thick
,node distance=2cm,on grid,>=stealth'
,Op1/.style={circle,draw,fill=yellow!40}
,Ring1/.style={circle,draw,fill=red!40}
,Ring2/.style={circle,draw,fill=blue!40}
,Ring12/.style={circle,draw,fill=purple!40}
,Op1/.style={rectangle,draw,fill=yellow!40}
,Ring1/.style={rectangle,draw,fill=red!40}
,Ring2/.style={rectangle,draw,fill=blue!40}
,Ring12/.style={rectangle,draw,fill=purple!40}
,Ring1Test/.style={diamond,draw,fill=red!40}
,Ring12Test/.style={diamond,draw,fill=purple!40}
,Output/.style={ellipse,draw,fill=orange!40}
,Input/.style={rectangle,draw,fill=green!40}
,Input/.style={circle,draw,fill=green!40}
]
\node [Input, align=center] (H) at (-3.052250,2.221211) { $\Psi^{(0)}$ };
\node [Op1, align=center] (He) at (-2.258396,0.948498) { Compute \\ one-$e$ \\ density };
@ -50,18 +50,18 @@ decoration={snake,
\path
(H) edge [->,color=black ] node [above,black] {} (He)
(He) edge [->,color=black ] node [above,black] { $n^{(0)}$ } (Li)
(Li) edge [->,color=black ] node [below,black,sloped,align=left] { $H^{\mu\,(k)}$ }
(Li) edge [->,color=black ] node [below,black,sloped,align=left] { $\hat{H}^{\mu\,(k)}$ }
node [above,black,sloped] { $k\leftarrow 0$ }(Be)
(Be) edge [->,color=black ] node [above,sloped,black] { $\Psi^{\mu\,(k)}$ } (B)
(Al) edge [->,color=black ] node [above,sloped,black] { no} (Be)
(B) edge [->,color=black ] node [below,sloped,black] { $n^{(k)}$ } (C)
(C) edge [->,color=black ] node [below,sloped,black] { $\tilde{n}^{(k)}$ } (N)
(N) edge [->,color=black ] node [below,sloped,black] { $H^{\mu\,(k,l)}$ }
(N) edge [->,color=black ] node [below,sloped,black] { $\hat{H}^{\mu\,(k)}$ }
node [above,sloped,black] { $l\leftarrow 0$ } (O)
(O) edge [->,color=black ] node [below,sloped,black] { $\Psi^{\mu\,(k,l)}$ }(F)
(F) edge [->,color=black ] node [below,sloped,black] { $n^{(k,l)}$ } (Ne)
(Ne) edge [->,color=black ] node [above,sloped,black] { $\tilde{n}^{(k,l)}$ } (Na)
(Na) edge [->,color=black ] node [above,sloped,black] { $E^{(k,l)}$ } (Mg)
(Na) edge [->,color=black ] node [above,sloped,black] { $\hat{H}^{\mu\,(k,l)}$ } (Mg)
(Mg) edge [->,color=black ] node [right,black] { yes } (Al)
(Mg) edge [->,color=black ] node [right,black] { no } (O)
(Al) edge [->,color=black ] node [above,sloped,black] {yes} (Si)

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Manuscript/algorithm.pdf
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20
Manuscript/rsdft-cipsi-qmc.tex
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@ -245,8 +245,8 @@ Considering that the perturbatively corrected energy is a reliable
estimate of the FCI energy, using a fixed value of the PT2 correction
as a stopping criterion enforces a constant distance of all the
calculations to the FCI energy. In this work, we target the chemical
accuracy so all the CIPSI selections were made such that $|\EPT| <
1$~m\hartree{}.
accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
1$ millihartree.
@ -357,15 +357,15 @@ use the CIPSI algorithm to perform approximate FCI calculations
with the RS-DFT Hamiltonian $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
This provides a multi-determinant trial wave function $\Psi^{\mu}$ that one can ``feed'' to DMC.
In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selection is performed
to obtain $\Psi^{\mu\,(k)}$ with the RS Hamiltonian $\hat{H}^{\mu\,(k)}$
to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT Hamiltonian $\hat{H}^{\mu\,(k)}$
parameterized using the current one-electron density $n^{(k)}$.
At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
One exits the outer loop when the absolute energy difference between two successive macro-iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to \titou{???} in the present study.
One exits the outer loop when the absolute energy difference between two successive macro-iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to $10^{-3}$ hartree in the present study which is consistent with the CIPSI threshold (see Sec.~\ref{sec:comp-details}).
An inner (micro-iteration) loop (blue) is introduced to accelerate the
convergence of the self-consistent calculation, in which the set of
determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
$\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
The inner loop is exited when the absolute energy difference between two successive micro-iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been here set to \titou{???}.
The inner loop is exited when the absolute energy difference between two successive micro-iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been here set to $10^{-2} \times \tau_1$ hartree.
The convergence of the algorithm was further improved
by introducing a direct inversion in the iterative subspace (DIIS)
step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
@ -391,12 +391,12 @@ CCSD(T) and KS-DFT energies have been computed with
All the CIPSI calculations have been performed with \emph{Quantum
Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
of the Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
of \titou{the local-density approximation (LDA)} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
and correlation functionals defined in
Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
The convergence criterion for stopping the CIPSI calculations
has been set to $\EPT < 1$~m\hartree{} or $ \Ndet > 10^7$.
has been set to $\EPT < 10^{-3}$ hartree or $ \Ndet > 10^7$.
All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
described in Ref.~\onlinecite{Applencourt_2018}.
@ -413,8 +413,6 @@ algorithm developed by Assaraf \textit{et al.}, \cite{Assaraf_2000}
with a time step of $5 \times 10^{-4}$ a.u.
\titou{All-electron move DMC?}
\titou{Missing details and references about srLDA and srPBE functionals.}
\section{Influence of the range-separation parameter on the fixed-node
error}
\label{sec:mu-dmc}
@ -526,7 +524,7 @@ of $\Psi^J$ may be better than that of the FCI wave function, and therefore, we
To do so, we have made the following numerical experiment.
First, we extract the 200 determinants with the largest weights in the FCI wave
function out of a large CIPSI calculation. Within this set of determinants,
we solve the self-consistent equations of RS-DFT (see Eq.~\eqref{rs-dft-eigen-equation})
we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
with different values of $\mu$. This gives the CI expansions $\Psi^\mu$.
Then, within the same set of determinants we optimize the CI coefficients in the presence of
a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$.
@ -541,7 +539,7 @@ $\Psi^\mu$ together with that of $\Psi^J$.
\centering
\includegraphics[width=\columnwidth]{overlap.pdf}
\caption{\ce{H2O}, double-zeta basis set, 200 most important
determinants of the FCI expansion (see \ref{sec:rsdft-j}).
determinants of the FCI expansion (see Sec.~\ref{sec:rsdft-j}).
Overlap of the RS-DFT CI expansions $\Psi^\mu$ with the CI
expansion optimized in the presence of a Jastrow factor $\Psi^J$.}
\label{fig:overlap}