From aa3c4132dc17910a6cc476cf10524805ca12a816 Mon Sep 17 00:00:00 2001 From: Anthony Scemama Date: Fri, 7 Aug 2020 19:23:56 +0200 Subject: [PATCH] Small changes --- Data/RSDFT-CIPSI.org | 110 ++++++++++-- Manuscript/h2o-dmc.pdf | Bin 7745 -> 7740 bytes Manuscript/rsdft-cipsi-qmc.tex | 294 +++++++++++++-------------------- 3 files changed, 216 insertions(+), 188 deletions(-) diff --git a/Data/RSDFT-CIPSI.org b/Data/RSDFT-CIPSI.org index 51b6b6b..6ec2091 100644 --- a/Data/RSDFT-CIPSI.org +++ b/Data/RSDFT-CIPSI.org @@ -131,7 +131,7 @@ $$ set xrange [0.0:6] set xtics 0.5 set xlabel "1/$\mu$" - set ylabel "E(sr-PBE)-E(FCI)" + set ylabel "E(srPBE)-E(FCI)" plot data u (1./$1):($3+17.164669) w p notitle , data u (1./$1):($3+17.164669) sm cs title "H_{2}O", \ data u (1./$1):($7+2.*$5+15.83002472+2.*0.49904523) w p notitle , data u (1./$1):($7+2.*$5+15.83002472+2.*0.49904523) sm cs title "O" #+END_SRC @@ -308,7 +308,7 @@ $$ set xrange [0.0:6] set xtics 0.5 set xlabel "1/$\mu$" - set ylabel "E(sr-PBE)-E(FCI)" + set ylabel "E(srPBE)-E(FCI)" plot data u (1./$1):($3+17.23217115) w p notitle , data u (1./$1):($3+17.23217115) sm cs title "H_{2}O", \ data u (1./$1):($7+2.*$5+15.88257866+2.*0.499042917477163) w p notitle , data u (1./$1):($7+2.*$5+15.88257866+2.*0.499042917477163) sm cs title "O" #+END_SRC @@ -523,7 +523,7 @@ $$ set xrange [0.0:6] set xtics 0.5 set xlabel "1/$\mu$" - set ylabel "E(sr-PBE)-E(FCI)" + set ylabel "E(srPBE)-E(FCI)" plot data u (1./$1):($3+17.25606450) w p notitle , data u (1./$1):($3+17.25606450) sm cs title "H_{2}O", \ data u (1./$1):($7+2.*$5+2.*0.499915948575730+15.89763971) w p notitle , data u (1./$1):($7+2.*$5+2.*0.499915948575730+15.89763971) sm cs title "O" @@ -5065,7 +5065,7 @@ D 1 set xrange [0.0:6] set xtics 0.5 set xlabel "1/$\mu$" - set ylabel "E(sr-PBE)-E(FCI)" + set ylabel "E(srPBE)-E(FCI)" plot data u (1./$1):($3+11.02347855) w p notitle , data u (1./$1):($3+11.02347855) sm cs title "C_{2}", \ data u (1./$1):(2.*$5+2.*5.40960365) w p notitle , data u (1./$1):(2.*$5+2.*5.40960365) sm cs title "C" #+END_SRC @@ -5199,7 +5199,7 @@ F 1 set xrange [0.0:6] set xtics 0.5 set xlabel "1/$\mu$" - set ylabel "E(sr-PBE)-E(FCI)" + set ylabel "E(srPBE)-E(FCI)" plot data u (1./$1):($3+11.02347855) w p notitle , data u (1./$1):($3+11.02347855) sm cs title "C_{2}", \ data u (1./$1):(2.*$5+2.*5.42800915) w p notitle , data u (1./$1):(2.*$5+2.*5.42800915) sm cs title "C" #+END_SRC @@ -5630,21 +5630,21 @@ vdz = data["X"] vdz$"E" = data$"E.DMC." vdz$"E.lo" = data$"E.DMC." - data$"Error" vdz$"E.hi" = data$"E.DMC." + data$"Error" -vdz$Basis = "DZ, sr-PBE" +vdz$Basis = "VDZ-BFD, srPBE" vdz.spline_int <- as.data.frame(spline(vdz$X, vdz$E)) vtz = data["X"] vtz$"E" = data$"E.DMC..1" vtz$"E.lo" = data$"E.DMC..1" - data$"Error.1" vtz$"E.hi" = data$"E.DMC..1" + data$"Error.1" -vtz$Basis = "TZ, sr-PBE" +vtz$Basis = "VTZ-BFD, srPBE" vtz.spline_int <- as.data.frame(spline(vtz$X, vtz$E)) lda = data["X"] lda$"E" = data$"E.DMC..2" lda$"E.lo" = data$"E.DMC..2" - data$"Error.2" lda$"E.hi" = data$"E.DMC..2" + data$"Error.2" -lda$Basis = "DZ, sr-LDA" +lda$Basis = "VDZ-BFD, srLDA" lda.spline_int <- as.data.frame(spline(lda$X, lda$E)) d = rbind(lda,vdz,vtz) @@ -5662,13 +5662,13 @@ p <- p + geom_point() #p <- p + geom_line(data = vdz.spline_int, aes(x = x, y = y)) p <- p + scale_x_continuous(name=TeX("$\\mu$ (bohr$^{-1}$)"), breaks=breaks, labels=labels) p <- p + scale_y_continuous(name = TeX("Energy (a.u.)"), breaks=seq(-17.266,-17.252,0.002)) -p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.15, .15)) +p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.20, .20), legend.title = element_blank()) p #+end_src #+RESULTS: - [[file:/tmp/babel-eZHQur/figureN5OKd9.png]] + [[file:/tmp/babel-eZHQur/figureu3DiHX.png]] #+begin_src R :results output :session *R* :exports both pdf("../Manuscript/h2o-dmc.pdf", family="Times", width=8, height=5) @@ -5818,6 +5818,59 @@ H2O 1.0000000 #+end_example +**** DMC energies + + #+NAME:tab-h2o-200 + | Mu | E | Error | + |------+----------------+--------------| + | 0.00 | -17.2537075442 | 0.0003272719 | + | 0.25 | -17.2540586404 | 0.0002976313 | + | 0.50 | -17.2550687726 | 0.0002976167 | + | 1.00 | -17.2552742178 | 0.0002854467 | + | 2.00 | -17.2529369035 | 0.0002926223 | + | 5.00 | -17.2501268640 | 0.0003236742 | + | Inf | -17.2492453661 | 0.0003067415 | + | | | | +17.2550782462 +/- 0.0002578873 + #+begin_src R :results output :session *R* :exports both :var data=tab-h2o-200 + data + #+end_src + + #+RESULTS: + : + : Mu E Error + : 1 0.00 -17.25371 0.0003272719 + : 2 0.25 -17.25406 0.0002976313 + : 3 0.50 -17.25507 0.0002976167 + : 4 1.00 -17.25527 0.0002854467 + : 5 2.00 -17.25294 0.0002926223 + : 6 5.00 -17.25013 0.0003236742 + : 7 Inf -17.24925 0.0003067415 + +#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R* :var data=tab-h2o-200 +Mu <- c(0., 0.25, 0.5, 1., 2., 5., Inf) +data$X <- Mu/(Mu+1.) +data$X[7] <- 1. +Mu <- as.character(Mu) +Mu[7] <- TeX("$\\infty$") + +H2O.spline_int <- as.data.frame(spline(data$X, data$E, n=101)) + +p <- ggplot() +p <- p + geom_point(aes(x=X,y=data$E), color='blue') +p <- p + geom_line(data = H2O.spline_int, aes(x=x, y=y)) +p <- p + geom_errorbar(aes(x=X, y=data$E, ymin=data$E - data$Error, ymax=data$E + data$Error), width=.2 + , position=position_dodge(0.05)) +p <- p + scale_x_continuous(name=TeX("$\\mu$ (bohr$^{-1}$)"), breaks=X, labels=Mu) +p <- p + scale_y_continuous(name = TeX("E_{DMC}")) +p <- p + theme(text = element_text(size = 20, family="Times") ) +p + +#+end_src + +#+RESULTS: +[[file:/tmp/babel-eZHQur/figureDoO3ml.png]] + *** F2 Parameters of the Jastrow: a r_{12} / (1 + b r_{12}) @@ -6065,6 +6118,41 @@ dev.off() : png : 2 +** On-top pair density + #+begin_src gnuplot :file output.png +#set terminal pdf +#set output "on-top-mu.pdf" +set xlabel "distance from O" +set ylabel "on-top pair density" +plot "H2O_1.e-6.density" w l title "n2(r,r), {/Symbol m}=0", \ + "H2O_0.25.density" w l title "n2(r,r), {/Symbol m}=0.25", \ + "H2O_0.5.density" w l title "n2(r,r), {/Symbol m}=0.5", \ + "H2O_1.0.density" w l title "n2(r,r), {/Symbol m}=1.0", \ + "H2O_1e6.density" w l title "n2(r,r), {/Symbol m}=10^6", \ + "H2O.density" w l title "n2(r,r), Jastrow" + #+end_src + + #+RESULTS: + [[file:output.png]] + +** One-body density + + #+begin_src gnuplot :file output.png +#set terminal pdf +#set output "density.pdf" +set xlabel "distance from O" +set ylabel "Density" +plot "H2O_1.e-6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0", \ + "H2O_0.25.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0.25", \ + "H2O_0.5.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0.5", \ + "H2O_1.0.density" u 1:3 w l title "n2(r,r), {/Symbol m}=1.0", \ + "H2O_1e6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=10^6", \ + "H2O.density" u 1:3 w l title "n2(r,r), Jastrow" + #+end_src + + #+RESULTS: + [[file:output.png]] + ** Optimal mu *** Table Find the optimal $\mu$ in VDZ-BFD for each molecule and atom, making @@ -6582,3 +6670,5 @@ dev.off() : null device : 1 + + diff --git a/Manuscript/h2o-dmc.pdf b/Manuscript/h2o-dmc.pdf index fd7b0236c5e1c90e6f65c8424928f0c7ea9a5edb..1dea63cf5678cec4c17e81f90e091ce8b9b4ab14 100644 GIT binary patch delta 2898 zcmZXWYdq5p8^@om#fC5?o<^2sBr?aD)5hdbPAO3;bGnr?Ij!(t$YJEr7D9|lM3m8; z^O;ekoJOK#&Qar*VS2jn7tf2k>w5G1{I2iydvU$GS`8Wu(sc!~>bPI6nSq!5zjU4i zYhY>mAk@K}52=^U;qj|;$3`BjW^jxuR+>7JWN9Iwqrl2_hrb+$8}&}Os08tXcyoPn zM&I5t!-1V09=ESP_f@?3!JpeJxdr8$gN|NQm#8iM-2ms^sI}qI(@wk5`?J)(&2{HW z+qv?--q}&W*ZA!m50G!7l1^Sg$+9wOE@f7^WZ9~Kn?u!X?5$|i898)$ z@4W{B9d$ZbL7L}DHjKR3VcI;CKRWTpUdhIQp*zQoB8ewmT-UYznK9H!?0cuzGHs?V zKw+A8pJmRxs_;Ehy-cRVJI4~gzQK0ub;;)yx`q>`tB51KU*qHSCjY@()*+IlAF67* zS&`tkG6iA$xtE#p8znxg^1?xG)S}C*+}>#wI>I~&w=kP<$#f;;LdF98#Y`x3!~ROV zbS$4{o-vpv?^P<6uc#{KFr$pn4;TSllB?2=cwMudyY0q27b|ilOF6h#J{-2cmc5Cp 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a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -99,7 +99,7 @@ to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017} Diffusion Monte Carlo (DMC) is another numerical scheme to obtain the exact solution of the Schr\"odinger equation with a different -constraint. In DMC, the solution is imposed to have the same nodes (or zeroes) +constraint. In DMC, the solution is imposed to have the same nodes (or zeroes) as a given trial (approximate) wave function. Within this so-called \emph{fixed-node} (FN) approximation, the FN-DMC energy associated with a given trial wave function is an upper @@ -205,7 +205,7 @@ Jastrow factor can be interpreted as a self-consistent field procedure with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT. So KS-DFT can be viewed as a very cheap way of introducing the effect of correlation in the orbital parameters determining the nodal surface -of a single Slater determinant. +of a single Slater determinant. Nevertheless, even when using the exact exchange correlation potential at the CBS limit, a fixed-node error necessarily remains because the single-determinant ansätz does not have enough flexibility to describe the @@ -256,9 +256,9 @@ accuracy so all the CIPSI selections were made such that $|\EPT| < Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004} the Coulomb operator entering the interelectronic repulsion is split into two pieces: \begin{equation} - \frac{1}{r} - = w_{\text{ee}}^{\text{sr}, \mu}(r) - + w_{\text{ee}}^{\text{lr}, \mu}(r) + \frac{1}{r} + = w_{\text{ee}}^{\text{sr}, \mu}(r) + + w_{\text{ee}}^{\text{lr}, \mu}(r) \end{equation} where \begin{align} @@ -266,7 +266,7 @@ where & w_{\text{ee}}^{\text{lr},\mu}(r) & = \frac{\erf \qty( \mu\, r)}{r} \end{align} -are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version. +are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version. The main idea behind RS-DFT is to treat the short-range part of the interaction within KS-DFT, and the long range part within a WFT method like FCI in the present case. @@ -291,15 +291,15 @@ electronic energy \begin{equation} \label{min_rsdft} E_0= \min_{\Psi} \qty{ \mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi} - + \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi] + + \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi] }, \end{equation} -with $\hat{T}$ the kinetic energy operator, +with $\hat{T}$ the kinetic energy operator, $\hat{W}_\text{ee}^{\text{lr}}$ the long-range electron-electron interaction, $n_\Psi$ the one-electron density associated with $\Psi$, and $\hat{V}_{\text{ne}}$ the electron-nucleus potential. -The minimizing multideterminant wave function $\Psi^\mu$ +The minimizing multideterminant wave function $\Psi^\mu$ can be determined by the self-consistent eigenvalue equation \begin{equation} \label{rs-dft-eigen-equation} @@ -345,12 +345,12 @@ will always provide reduced fixed-node errors compared to the path connecting HF to FCI which consists in increasing the number of determinants. We follow the KS-to-FCI path by performing FCI calculations using the -RS-DFT Hamiltonian with different values of $\mu$. -Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a +RS-DFT Hamiltonian with different values of $\mu$. +Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a single- and multi-determinant wave function $\Psi^{(0)}$. One of the particularity of the present work is that we have used the CIPSI algorithm to perform approximate FCI calculations -with the RS-DFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18} +with the RS-DFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18} In the outer loop (red), a CIPSI selection is performed with a RS Hamiltonian parameterized using the current one-electron density. An inner loop (blue) is introduced to accelerate the @@ -364,7 +364,7 @@ Note that any range-separated post-HF method can be implemented using this scheme by just replacing the CIPSI step by the post-HF method of interest. \titou{T2: introduce $\tau_1$ and $\tau_2$. More description of the algorithm needed.} -Note that, thanks to the self-consistent nature of the algorithm, +Note that, thanks to the self-consistent nature of the algorithm, the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$. @@ -420,7 +420,7 @@ All-electron move DMC.} \cline{3-4} \cline{5-6} System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\ \hline - \ce{H2O} + \ce{H2O} & $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\ & $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\ & $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\ @@ -434,10 +434,10 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED & $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\ & $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\ \\ - \ce{F2} + \ce{F2} & $0.00$ & $23$ & $-48.419\,5(4)$ \\ & $0.25$ & $8$ & $-48.421\,9(4)$ \\ - & $0.50$ & $1743$ & $-48.424\,8(8)$ \\ + & $0.50$ & $1743$ & $-48.424\,8(8)$ \\ & $1.00$ & $11952$ & $-48.432\,4(3)$ \\ & $2.00$ & $829438$ & $-48.441\,0(7)$ \\ & $5.00$ & $5326459$ & $-48.445(2)$ \\ @@ -452,8 +452,7 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED \includegraphics[width=\columnwidth]{h2o-dmc.pdf} \caption{Fixed-node energies of \ce{H2O} for different values of $\mu$, using the srLDA or srPBE density - functionals to build the trial wave function. - \titou{Toto: please remove hyphens in sr-LDA and sr-PBE.}} + functionals to build the trial wave function.} \label{fig:h2o-dmc} \end{figure} @@ -482,35 +481,35 @@ wave functions ($\mu = \infty$) gives FN-DMC energies which are lower than the energies obtained with a single Kohn-Sham determinant ($\mu=0$): a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm 0.3$~m\hartree{} at the triple-zeta level are obtained for water, and -a gain of $18 \pm 3$~m\hartree{} for F$_2$. -Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$, -the figures~\ref{fig:h2o-dmc} show that a smooth behaviour is obtained: -starting from $\mu=0$ (\textit{i.e.} the KS determinant), -the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$, -and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function). -For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set, -with the optimal value of $\mu$, one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{} -and $8 \pm 3$~m\hartree{} for the water and difluorine molecules, respectively. -The optimal value of $\mu$ is $\mu=1.75$~bohr$^{-1}$ and $\mu=5$~bohr$^{-1}$ for the water and fluorine dimer, respectively. -When the basis set is increased, the gain in FN-DMC energy with respect to the FCI trial wave function is reduced, and the optimal value of $\mu$ is shifted towards large $\mu$. -Eventually, the nodes of the wave functions $\Psi^\mu$ obtained with short-range -LDA exchange-correlation functional give very similar FN-DMC energy with respect -to those obtained with the short-range PBE functional, even if the RS-DFT energies obtained -with these two functionals differ by several tens of m\hartree{}. +a gain of $18 \pm 3$~m\hartree{} for F$_2$. +Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$, +the figures~\ref{fig:h2o-dmc} show that a smooth behaviour is obtained: +starting from $\mu=0$ (\textit{i.e.} the KS determinant), +the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$, +and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function). +For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set, +with the optimal value of $\mu$, one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{} +and $8 \pm 3$~m\hartree{} for the water and difluorine molecules, respectively. +The optimal value of $\mu$ is $\mu=1.75$~bohr$^{-1}$ and $\mu=5$~bohr$^{-1}$ for the water and fluorine dimer, respectively. +When the basis set is increased, the gain in FN-DMC energy with respect to the FCI trial wave function is reduced, and the optimal value of $\mu$ is shifted towards large $\mu$. +Eventually, the nodes of the wave functions $\Psi^\mu$ obtained with short-range +LDA exchange-correlation functional give very similar FN-DMC energy with respect +to those obtained with the short-range PBE functional, even if the RS-DFT energies obtained +with these two functionals differ by several tens of m\hartree{}. \subsection{Link between RS-DFT and Jastrow factors } \label{sec:rsdft-j} -The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients +The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients giving trial wave functions with better nodes than FCI wave functions. -Such behaviour can be compared to the common practice of -re-optimizing the Slater part of a trial wave function in the presence of the Jastrow factor. -In the present paragraph, we would like therefore to elaborate some more on the link between RS-DFT -and wave function optimization within the presence of a jastrow factor. +Such behaviour can be compared to the common practice of +re-optimizing the Slater part of a trial wave function in the presence of the Jastrow factor. +In the present paragraph, we would like therefore to elaborate some more on the link between RS-DFT +and wave function optimization within the presence of a Jastrow factor. -Let us assume a fixed jastrow factor $J({\bf{r}_1}, \hdots , {\bf{r}_N})$, -and a corresponding Slater-jatrow wave function $\Phi = e^J \Psi$, -where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determinants $D_I$. -The only remaining variational parameters in $\Phi$ are therefore the Slater part $\Psi$. +Let us assume a fixed Jastrow factor $J({\bf{r}_1}, \hdots , {\bf{r}_N})$, +and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$, +where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determinants $D_I$. +The only remaining variational parameters in $\Phi$ are therefore the Slater part $\Psi$. Let us call $\Psi^J$ the linear combination of Slater determinant minimizing the variational energy \begin{equation} \Psi^J = \text{argmin}_{\Psi}\frac{ \langle \Psi | e^{J} H e^{J} |\Psi \rangle}{\langle \Psi | e^{2J} |\Psi \rangle}. @@ -519,29 +518,29 @@ Such a wave function $\Psi^J$ satisfies the generalized hermitian eigenvalue equ \begin{equation} e^{J} H e^{J} \Psi^J = E e^{2J} \Psi^J, \end{equation} -but also the non-hermitian transcorrelated eigenvalue problem\cite{many_things} +but also the non-hermitian transcorrelated eigenvalue problem\cite{many_things} \begin{equation} \label{eq:transcor} e^{-J} H e^{J} \Psi^J = E \Psi^J, \end{equation} -which is much easier to handle despite its non-hermicity. -Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$. -In a finite basis set and with a quite accurate jastrow factor, it is known that the nodes -of $\Psi^J$ are better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$. +which is much easier to handle despite its non-hermicity. +Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$. +In a finite basis set and with a quite accurate Jastrow factor, it is known that the nodes +of $\Psi^J$ are better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$. -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}) -with different values of $\mu$. This gives the CI expansions $\Psi^\mu$. +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}) +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 $\Psi^J$. -Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed -on the same Slater determinant basis. +a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$. +Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed +on the same Slater determinant basis. In figure~\ref{fig:overlap}, we plot the overlaps $\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer, -and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions -$\Psi^\mu$ together with that of $\Psi^J$. +and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions +$\Psi^\mu$ together with that of $\Psi^J$. \begin{figure} \centering @@ -561,54 +560,57 @@ $\Psi^\mu$ together with that of $\Psi^J$. In the case of H$_2$O, there is a clear maximum of overlap at -$\mu=1$~bohr$^{-1}$, which coincide with the minimum of the FN-DMC energy of $\Psi^\mu$. -???? hypothetic: Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is very close to that of $\Psi^\mu$ with $\mu=1$~bohr$^{-1}$. -This confirms that introducing short-range correlation with DFT has -an impact on the CI coefficients similar to the Jastrow factor. +$\mu=1$~bohr$^{-1}$, which coincide with the minimum of the FN-DMC energy of $\Psi^\mu$. +Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is very +close to that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that +introducing short-range correlation with DFT has +an impact on the CI coefficients similar to the Jastrow factor. In the case of F$_2$, the Jastrow factor has very little effect on the CI coefficients, as the overlap $\braket{\Psi^J}{\Psi^{\mu=\infty}}$ is very close to -$1$. +$1$. Nevertheless, a slight maximum is obtained for -$\mu=5$~bohr$^{-1}$. -In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, -we report, in the case of the water molecule in the double-zeta basis set, -several quantities related to the one- and two-body density of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. -First, we report in table~\ref{table_on_top} the integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$ +$\mu=5$~bohr$^{-1}$. +In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, +we report, in the case of the water molecule in the double-zeta basis set, +several quantities related to the one- and two-body density of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. +First, we report in table~\ref{table_on_top} the integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$ \begin{equation} \langle n_2({\bf r},{\bf r}) \rangle = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r}) \end{equation} -where $n_2({\bf r}_1,{\bf r}_2)$ is the two-body density (normalized to $N(N-1)$ where $N$ is the number of electrons) -obtained for both $\Psi^\mu$ and $\Psi^J$. -Then, in order to have a pictorial representation of both the on-top pair density and the density, we report in figures~\ref{fig:n1} and ~\ref{fig:n2} -the plots of the total density $n({\bf r})$ and on-top pair density $n_2({\bf r},{\bf r})$ along the OH axis of the water molecule. -From these data, one can clearly observe several trends. -First, from table~\ref{table_on_top}, we can observe that the overall on-top pair density decreases -when one increases $\mu$, which is expected as the two-electron interaction increases in $H^\mu[n]$. -Second, the relative variation of the on-top pair density with $\mu$ 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. -??? Hypothetic: the value of the on-top pair density in $\Psi^\mu$ are closer for certain values of $\mu$ to that of $\Psi^J$ than the -FCI wave function. +where $n_2({\bf r}_1,{\bf r}_2)$ is the two-body density (normalized to $N(N-1)$ where $N$ is the number of electrons) +obtained for both $\Psi^\mu$ and $\Psi^J$. +Then, in order to have a pictorial representation of both the on-top pair density and the density, we report in figures~\ref{fig:n1} and ~\ref{fig:n2} +the plots of the total density $n({\bf r})$ and on-top pair density $n_2({\bf r},{\bf r})$ along the OH axis of the water molecule. +From these data, one can clearly observe several trends. +First, from table~\ref{table_on_top}, we can observe that the overall on-top pair density decreases +when one increases $\mu$, which is expected as the two-electron interaction increases in $H^\mu[n]$. +Second, the relative variation of the on-top pair density with $\mu$ 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 +The value of the on-top pair density in $\Psi^\mu$ are closer for +certain values of $\mu$ to that of $\Psi^J$ than the FCI wave +function. -These data suggest that the wave functions $\Psi^\mu$ and $\Psi^J$ are similar, -and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ 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 -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 role 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 the plots in figures ~\ref{fig:n1} and~\ref{fig:n2} in the case of the water molecule. -Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No\cite{Ten-no2000Nov}, -the effective two-body interaction induced by the presence of a jastrow factor -can be non-divergent when a proper Jastrow factor is chosen. -Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the jastrow-Slater optimization: -they both deal with an effective non-divergent interaction but still produce reasonable one-body density. +These data suggest that the wave functions $\Psi^\mu$ and $\Psi^J$ are similar, +and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ 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 +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 role 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 the plots in figures ~\ref{fig:n1} and~\ref{fig:n2} in the case of the water molecule. +Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No\cite{Ten-no2000Nov}, +the effective two-body interaction induced by the presence of a jastrow factor +can be non-divergent when a proper Jastrow factor is chosen. +Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Jastrow-Slater optimization: +they both deal with an effective non-divergent interaction but still produce reasonable one-body density. \begin{table} - \caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$ + \caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$ for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. } \label{table_on_top} \begin{ruledtabular} @@ -622,7 +624,7 @@ they both deal with an effective non-divergent interaction but still produce rea 2.00 & 1.325 \\ 5.00 & 1.288 \\ $\infty$ & 1.277 \\ - \hline + \hline $\Psi^J$ & \\ \end{tabular} \end{ruledtabular} @@ -642,17 +644,17 @@ they both deal with an effective non-divergent interaction but still produce rea \end{figure} -As a conclusion of the first part of this study, we can notice that: -i) 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, ii) the value of the optimal $\mu$ depends -on the system and the basis set, and the larger the basis set, the larger the optimal value of $\mu$, -iii) numerical experiments (such as computation of overlap, FN-DMC energies) indicates -that the RS-DFT scheme essentially plays the role of a simple Jastrow factor, -\textit{i.e.} mimicking short-range correlation effects. -The latter statement can be qualitatively understood by noticing that both RS-DFT -and transcorrelated approach deal with an effective non-divergent electron-electron interaction, while keeping the density constant. +As a conclusion of the first part of this study, we can notice that: +i) 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, ii) the value of the optimal $\mu$ depends +on the system and the basis set, and the larger the basis set, the larger the optimal value of $\mu$, +iii) numerical experiments (such as computation of overlap, FN-DMC energies) indicates +that the RS-DFT scheme essentially plays the role of a simple Jastrow factor, +\textit{i.e.} mimicking short-range correlation effects. +The latter statement can be qualitatively understood by noticing that both RS-DFT +and transcorrelated approach deal with an effective non-divergent electron-electron interaction, while keeping the density constant. @@ -695,7 +697,7 @@ functions. But ultimately the exact energy will be reached. In the context of selected CI calculations, when the variational energy is extrapolated to the FCI energy\cite{Holmes_2017} there is no size-consistency error. But when the truncated SCI wave function is used -as a reference for post-Hartree-Fock methods such as SCI+PT2 +as a reference for post-Hartree-Fock methods such as SCI+PT2 or for QMC calculations, there is a residual size-consistency error originating from the truncation of the wave function. @@ -721,7 +723,7 @@ The parameter $a$ is determined by cusp conditions, and $b$ is obtained by energy or variance minimization.\cite{Coldwell_1977,Umrigar_2005} One can easily see that this parameterization of the two-body -interation is not size-consistent: the dissociation of a +interaction is not size-consistent: the dissociation of a diatomic molecule $AB$ with a parameter $b_{AB}$ will lead to two different two-body Jastrow factors, each with its own optimal value $b_A$ and $b_B$. To remove the @@ -788,7 +790,7 @@ closed-shell systems. A good test is to check that all the components of a spin multiplet are degenerate. FCI wave functions have this property and give degenrate energies with respect to the spin quantum number $m_s$, but the multiplication by a -Jastrow factor introduces spin contamination if the parameters +Jastrow factor introduces spin contamination if the parameters for the same-spin electron pairs are different from those for the opposite-spin pairs.\cite{Tenno_2004} Again, when pseudo-potentials are used this tiny error is transferred @@ -909,7 +911,7 @@ accuracy. Thanks to the single-reference character of these systems, the CCSD(T) energy is an excellent estimate of the FCI energy, as shown by the very good agreement of the MAE, MSE and RMSD of CCSDT(T) and FCI energies. -The imbalance of the quality of description of molecules compared +The imbalance of the quality of description of molecules compared to atoms is exhibited by a very negative value of the MSE for CCSD(T) and FCI/VDZ-BFD, which is reduced by a factor of two when going to the triple-zeta basis, and again by a factor of two when @@ -945,7 +947,7 @@ basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the FCI. For the largest systems, as shown in figure~\ref{fig:g2-ndet} there are many systems which did not reach the threshold $\EPT<1$~m\hartree{}, and the number of determinants exceeded -10~million so the calculation stopped. In this regime, there is a +10~million so the calculation stopped. In this regime, there is a small size-consistency error originating from the imbalanced truncation of the wave functions, which is not present in the extrapolated FCI energies. The same comment applies to @@ -1037,10 +1039,10 @@ energy via the variation of the parameter $\mu$. The second method is for the computation of energy differences, where the target is not the lowest possible FN-DMC energies but the best possible cancellation of errors. Using a fixed value of $\mu$ -increases the consistency of the trial wave functions, and we have found +increases the consistency of the trial wave functions, and we have found that $\mu=0.5$~bohr$^{-1}$ is the value where the cancellation of errors is the most effective. -Moreover, such a small value of $\mu$ gives extermely +Moreover, such a small value of $\mu$ gives extremely compact wave functions, making this recipe a good candidate for the accurate description of the whole potential energy surfaces of large systems. If the number of determinants is still too large, the @@ -1064,70 +1066,6 @@ The data that support the findings of this study are available within the articl \bibliography{rsdft-cipsi-qmc} -\begin{enumerate} - \item Total energies and nodal quality: - \begin{itemize} -% \item Facts: KS occupied orbitals closer to NOs than HF -% \item Even if exact functional, complete basis set, still approximated nodes for KS - \item KS -> exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill) -% \item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS -% \item With FCI, good limit at CBS ==> exact energy -% \item But slow convergence with basis set because of divergence of the e-e interaction not well represented in atom centered basis set -% \item Exponential increase of number of Slater determinants - \item Cite papiers RS-DFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté) - \item Question: does such a scheme provide better nodal quality ? -%<<<<<<< HEAD - \item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC - \item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI - \begin{itemize} - \item less determinants $\Rightarrow$ large systems - \item only one parameter to optimize $\Rightarrow$ deterministic - \item $\Rightarrow$ reproducible - \end{itemize} - \item with the optimal $\mu$: - \begin{itemize} - \item Direct optimization of FNDMC with one parameter - \item Do we improve energy differences ? - \item system dependent - \item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$ - \item large wave functions - \end{itemize} -% \item Invariance with m_s -%======= - \item In RS-DFT we cannot optimize energy with $\mu$ , but in FNDMC -% \item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI -% \begin{itemize} -% \item less determinants $\Rightarrow$ large systems -% \item only one parameter to optimize $\Rightarrow$ deterministic -% \item $\Rightarrow$ reproducible -% \end{itemize} -% \item with the optimal $\mu$: -% \begin{itemize} -% \item Direct optimization of FNDMC with one parameter -% \item Do we improve energy differences ? -% \item system dependent -% \item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$ -% \item large wave functions -% \end{itemize} -%>>>>>>> 44470b89936d1727b638aefd982ce83be9075cc8 - - \end{itemize} -\end{enumerate} - - - \end{document} -% * Recouvrement avec Be : Optimization tous electrons -% impossible. Abandon. On va prendre H2O. -% * Manu doit faire des programmes pour des plots de ensite a 1 et 2 -% corps le long des axes de liaison, et l'integrale de la densite a -% 2 corps a coalescence. -% 5 Toto optimise les coefs en presence e jastrow -% 6 Toto renvoie a manu psicoef -% 7 Manu fait tourner ses petits programmes avec psi_J -% 8 Toto calcule les energies FN-DMC de -% +) toutes les fonctions d'ondes psi_mu -% +) la fonction d'onde psi_J -