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@ -131,7 +131,7 @@ $$
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set xrange [0.0:6]
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set xtics 0.5
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set xlabel "1/$\mu$"
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set ylabel "E(sr-PBE)-E(FCI)"
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set ylabel "E(srPBE)-E(FCI)"
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plot data u (1./$1):($3+17.164669) w p notitle , data u (1./$1):($3+17.164669) sm cs title "H_{2}O", \
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data u (1./$1):($7+2.*$5+15.83002472+2.*0.49904523) w p notitle , data u (1./$1):($7+2.*$5+15.83002472+2.*0.49904523) sm cs title "O"
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#+END_SRC
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@ -308,7 +308,7 @@ $$
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set xrange [0.0:6]
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set xtics 0.5
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set xlabel "1/$\mu$"
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set ylabel "E(sr-PBE)-E(FCI)"
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set ylabel "E(srPBE)-E(FCI)"
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plot data u (1./$1):($3+17.23217115) w p notitle , data u (1./$1):($3+17.23217115) sm cs title "H_{2}O", \
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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"
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#+END_SRC
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@ -523,7 +523,7 @@ $$
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set xrange [0.0:6]
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set xtics 0.5
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set xlabel "1/$\mu$"
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set ylabel "E(sr-PBE)-E(FCI)"
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set ylabel "E(srPBE)-E(FCI)"
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plot data u (1./$1):($3+17.25606450) w p notitle , data u (1./$1):($3+17.25606450) sm cs title "H_{2}O", \
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data u (1./$1):($7+2.*$5+2.*0.499915948575730+15.89763971) w p notitle , data u (1./$1):($7+2.*$5+2.*0.499915948575730+15.89763971) sm cs title "O"
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@ -5065,7 +5065,7 @@ D 1
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set xrange [0.0:6]
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set xtics 0.5
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set xlabel "1/$\mu$"
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set ylabel "E(sr-PBE)-E(FCI)"
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set ylabel "E(srPBE)-E(FCI)"
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plot data u (1./$1):($3+11.02347855) w p notitle , data u (1./$1):($3+11.02347855) sm cs title "C_{2}", \
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data u (1./$1):(2.*$5+2.*5.40960365) w p notitle , data u (1./$1):(2.*$5+2.*5.40960365) sm cs title "C"
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#+END_SRC
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@ -5199,7 +5199,7 @@ F 1
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set xrange [0.0:6]
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set xtics 0.5
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set xlabel "1/$\mu$"
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set ylabel "E(sr-PBE)-E(FCI)"
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set ylabel "E(srPBE)-E(FCI)"
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plot data u (1./$1):($3+11.02347855) w p notitle , data u (1./$1):($3+11.02347855) sm cs title "C_{2}", \
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data u (1./$1):(2.*$5+2.*5.42800915) w p notitle , data u (1./$1):(2.*$5+2.*5.42800915) sm cs title "C"
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#+END_SRC
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@ -5630,21 +5630,21 @@ vdz = data["X"]
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vdz$"E" = data$"E.DMC."
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vdz$"E.lo" = data$"E.DMC." - data$"Error"
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vdz$"E.hi" = data$"E.DMC." + data$"Error"
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vdz$Basis = "DZ, sr-PBE"
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vdz$Basis = "VDZ-BFD, srPBE"
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vdz.spline_int <- as.data.frame(spline(vdz$X, vdz$E))
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vtz = data["X"]
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vtz$"E" = data$"E.DMC..1"
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vtz$"E.lo" = data$"E.DMC..1" - data$"Error.1"
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vtz$"E.hi" = data$"E.DMC..1" + data$"Error.1"
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vtz$Basis = "TZ, sr-PBE"
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vtz$Basis = "VTZ-BFD, srPBE"
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vtz.spline_int <- as.data.frame(spline(vtz$X, vtz$E))
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lda = data["X"]
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lda$"E" = data$"E.DMC..2"
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lda$"E.lo" = data$"E.DMC..2" - data$"Error.2"
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lda$"E.hi" = data$"E.DMC..2" + data$"Error.2"
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lda$Basis = "DZ, sr-LDA"
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lda$Basis = "VDZ-BFD, srLDA"
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lda.spline_int <- as.data.frame(spline(lda$X, lda$E))
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d = rbind(lda,vdz,vtz)
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@ -5662,13 +5662,13 @@ p <- p + geom_point()
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#p <- p + geom_line(data = vdz.spline_int, aes(x = x, y = y))
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p <- p + scale_x_continuous(name=TeX("$\\mu$ (bohr$^{-1}$)"), breaks=breaks, labels=labels)
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p <- p + scale_y_continuous(name = TeX("Energy (a.u.)"), breaks=seq(-17.266,-17.252,0.002))
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p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.15, .15))
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p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.20, .20), legend.title = element_blank())
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p
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#+end_src
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#+RESULTS:
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[[file:/tmp/babel-eZHQur/figureN5OKd9.png]]
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[[file:/tmp/babel-eZHQur/figureu3DiHX.png]]
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#+begin_src R :results output :session *R* :exports both
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pdf("../Manuscript/h2o-dmc.pdf", family="Times", width=8, height=5)
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@ -5818,6 +5818,59 @@ H2O
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1.0000000
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#+end_example
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**** DMC energies
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#+NAME:tab-h2o-200
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| Mu | E | Error |
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|------+----------------+--------------|
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| 0.00 | -17.2537075442 | 0.0003272719 |
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| 0.25 | -17.2540586404 | 0.0002976313 |
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| 0.50 | -17.2550687726 | 0.0002976167 |
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| 1.00 | -17.2552742178 | 0.0002854467 |
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| 2.00 | -17.2529369035 | 0.0002926223 |
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| 5.00 | -17.2501268640 | 0.0003236742 |
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| Inf | -17.2492453661 | 0.0003067415 |
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| | | |
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17.2550782462 +/- 0.0002578873
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#+begin_src R :results output :session *R* :exports both :var data=tab-h2o-200
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data
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#+end_src
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#+RESULTS:
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:
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: Mu E Error
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: 1 0.00 -17.25371 0.0003272719
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: 2 0.25 -17.25406 0.0002976313
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: 3 0.50 -17.25507 0.0002976167
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: 4 1.00 -17.25527 0.0002854467
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: 5 2.00 -17.25294 0.0002926223
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: 6 5.00 -17.25013 0.0003236742
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: 7 Inf -17.24925 0.0003067415
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#+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
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Mu <- c(0., 0.25, 0.5, 1., 2., 5., Inf)
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data$X <- Mu/(Mu+1.)
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data$X[7] <- 1.
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Mu <- as.character(Mu)
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Mu[7] <- TeX("$\\infty$")
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H2O.spline_int <- as.data.frame(spline(data$X, data$E, n=101))
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p <- ggplot()
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p <- p + geom_point(aes(x=X,y=data$E), color='blue')
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p <- p + geom_line(data = H2O.spline_int, aes(x=x, y=y))
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p <- p + geom_errorbar(aes(x=X, y=data$E, ymin=data$E - data$Error, ymax=data$E + data$Error), width=.2
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, position=position_dodge(0.05))
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p <- p + scale_x_continuous(name=TeX("$\\mu$ (bohr$^{-1}$)"), breaks=X, labels=Mu)
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p <- p + scale_y_continuous(name = TeX("E_{DMC}"))
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p <- p + theme(text = element_text(size = 20, family="Times") )
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p
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#+end_src
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#+RESULTS:
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[[file:/tmp/babel-eZHQur/figureDoO3ml.png]]
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*** F2
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Parameters of the Jastrow:
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a r_{12} / (1 + b r_{12})
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@ -6065,6 +6118,41 @@ dev.off()
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: png
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: 2
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** On-top pair density
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#+begin_src gnuplot :file output.png
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#set terminal pdf
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#set output "on-top-mu.pdf"
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set xlabel "distance from O"
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set ylabel "on-top pair density"
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plot "H2O_1.e-6.density" w l title "n2(r,r), {/Symbol m}=0", \
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"H2O_0.25.density" w l title "n2(r,r), {/Symbol m}=0.25", \
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"H2O_0.5.density" w l title "n2(r,r), {/Symbol m}=0.5", \
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"H2O_1.0.density" w l title "n2(r,r), {/Symbol m}=1.0", \
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"H2O_1e6.density" w l title "n2(r,r), {/Symbol m}=10^6", \
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"H2O.density" w l title "n2(r,r), Jastrow"
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#+end_src
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#+RESULTS:
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[[file:output.png]]
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** One-body density
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#+begin_src gnuplot :file output.png
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#set terminal pdf
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#set output "density.pdf"
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set xlabel "distance from O"
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set ylabel "Density"
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plot "H2O_1.e-6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0", \
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"H2O_0.25.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0.25", \
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"H2O_0.5.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0.5", \
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"H2O_1.0.density" u 1:3 w l title "n2(r,r), {/Symbol m}=1.0", \
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"H2O_1e6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=10^6", \
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"H2O.density" u 1:3 w l title "n2(r,r), Jastrow"
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#+end_src
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#+RESULTS:
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[[file:output.png]]
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** Optimal mu
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*** Table
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Find the optimal $\mu$ in VDZ-BFD for each molecule and atom, making
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@ -6582,3 +6670,5 @@ dev.off()
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: null device
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: 1
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Binary file not shown.
@ -99,7 +99,7 @@ to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}
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Diffusion Monte Carlo (DMC) is another numerical scheme to obtain
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the exact solution of the Schr\"odinger equation with a different
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constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)
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constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)
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as a given trial (approximate) wave function.
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Within this so-called \emph{fixed-node} (FN) approximation,
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the FN-DMC energy associated with a given trial wave function is an upper
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@ -205,7 +205,7 @@ Jastrow factor can be interpreted as a self-consistent field procedure
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with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.
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So KS-DFT can be viewed as a very cheap way of introducing the effect of
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correlation in the orbital parameters determining the nodal surface
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of a single Slater determinant.
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of a single Slater determinant.
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Nevertheless, even when using the exact exchange correlation potential at the
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CBS limit, a fixed-node error necessarily remains because the
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single-determinant ansätz does not have enough flexibility to describe the
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@ -256,9 +256,9 @@ accuracy so all the CIPSI selections were made such that $|\EPT| <
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Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004}
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the Coulomb operator entering the interelectronic repulsion is split into two pieces:
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\begin{equation}
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\frac{1}{r}
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= w_{\text{ee}}^{\text{sr}, \mu}(r)
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+ w_{\text{ee}}^{\text{lr}, \mu}(r)
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\frac{1}{r}
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= w_{\text{ee}}^{\text{sr}, \mu}(r)
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+ w_{\text{ee}}^{\text{lr}, \mu}(r)
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\end{equation}
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where
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\begin{align}
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@ -266,7 +266,7 @@ where
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&
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w_{\text{ee}}^{\text{lr},\mu}(r) & = \frac{\erf \qty( \mu\, r)}{r}
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\end{align}
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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.
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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.
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The main idea behind RS-DFT is to treat the short-range part of the
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interaction within KS-DFT, and the long range part within a WFT method like FCI in the present case.
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@ -291,15 +291,15 @@ electronic energy
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\begin{equation}
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\label{min_rsdft} E_0= \min_{\Psi} \qty{
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\mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi}
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+ \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi]
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+ \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi]
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},
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\end{equation}
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with $\hat{T}$ the kinetic energy operator,
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with $\hat{T}$ the kinetic energy operator,
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$\hat{W}_\text{ee}^{\text{lr}}$ the long-range
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electron-electron interaction,
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$n_\Psi$ the one-electron density associated with $\Psi$,
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and $\hat{V}_{\text{ne}}$ the electron-nucleus potential.
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The minimizing multideterminant wave function $\Psi^\mu$
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The minimizing multideterminant wave function $\Psi^\mu$
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can be determined by the self-consistent eigenvalue equation
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\begin{equation}
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\label{rs-dft-eigen-equation}
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@ -345,12 +345,12 @@ will always provide reduced fixed-node errors compared to the path
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connecting HF to FCI which consists in increasing the number of determinants.
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We follow the KS-to-FCI path by performing FCI calculations using the
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RS-DFT Hamiltonian with different values of $\mu$.
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Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a
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RS-DFT Hamiltonian with different values of $\mu$.
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Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a
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single- and multi-determinant wave function $\Psi^{(0)}$.
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One of the particularity of the present work is that we
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have used the CIPSI algorithm to perform approximate FCI calculations
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with the RS-DFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
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with the RS-DFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
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In the outer loop (red), a CIPSI selection is performed with a RS Hamiltonian
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parameterized using the current one-electron density.
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An inner loop (blue) is introduced to accelerate the
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@ -364,7 +364,7 @@ Note that any range-separated post-HF method can be
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implemented using this scheme by just replacing the CIPSI step by the
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post-HF method of interest.
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\titou{T2: introduce $\tau_1$ and $\tau_2$. More description of the algorithm needed.}
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Note that, thanks to the self-consistent nature of the algorithm,
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Note that, thanks to the self-consistent nature of the algorithm,
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the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$.
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@ -420,7 +420,7 @@ All-electron move DMC.}
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\cline{3-4} \cline{5-6}
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System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
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\hline
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\ce{H2O}
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\ce{H2O}
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& $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
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& $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
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& $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
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@ -434,10 +434,10 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED
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& $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
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& $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
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\\
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\ce{F2}
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\ce{F2}
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& $0.00$ & $23$ & $-48.419\,5(4)$ \\
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& $0.25$ & $8$ & $-48.421\,9(4)$ \\
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& $0.50$ & $1743$ & $-48.424\,8(8)$ \\
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& $0.50$ & $1743$ & $-48.424\,8(8)$ \\
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& $1.00$ & $11952$ & $-48.432\,4(3)$ \\
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& $2.00$ & $829438$ & $-48.441\,0(7)$ \\
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& $5.00$ & $5326459$ & $-48.445(2)$ \\
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@ -452,8 +452,7 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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\caption{Fixed-node energies of \ce{H2O} for different
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values of $\mu$, using the srLDA or srPBE density
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functionals to build the trial wave function.
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\titou{Toto: please remove hyphens in sr-LDA and sr-PBE.}}
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functionals to build the trial wave function.}
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\label{fig:h2o-dmc}
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\end{figure}
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@ -482,35 +481,35 @@ wave functions ($\mu = \infty$) gives FN-DMC energies which are lower
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than the energies obtained with a single Kohn-Sham determinant ($\mu=0$):
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a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm
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0.3$~m\hartree{} at the triple-zeta level are obtained for water, and
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a gain of $18 \pm 3$~m\hartree{} for F$_2$.
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Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$,
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the figures~\ref{fig:h2o-dmc} show that a smooth behaviour is obtained:
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starting from $\mu=0$ (\textit{i.e.} the KS determinant),
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the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,
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and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).
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For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set,
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with the optimal value of $\mu$, one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
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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
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user