diff --git a/Data/algorithm.tex b/Data/algorithm.tex index b5d519c..7dcdb9f 100644 --- a/Data/algorithm.tex +++ b/Data/algorithm.tex @@ -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) diff --git a/Manuscript/algorithm.pdf b/Manuscript/algorithm.pdf index f0b9fba..2d2437b 100644 Binary files a/Manuscript/algorithm.pdf and b/Manuscript/algorithm.pdf differ diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index a4f0566..7f4cf36 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -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}