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This commit is contained in:
Anthony Scemama 2020-07-20 19:05:15 +02:00
parent b5aa67c092
commit 04594fab4f
18 changed files with 2504 additions and 334 deletions
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814
Data/RSDFT-CIPSI.org
View File

@ -5055,9 +5055,6 @@ loadfonts()
#+RESULTS:
#+begin_example
RStudio Community is a great place to get help:
https://community.rstudio.com/c/tidyverse
Registering fonts with R
More than one version of regular/bold/italic found for Cannam Codec Fixed. Skipping setup for this font.
@ -5239,328 +5236,7 @@ summary(raw_data)
NA's :1139 NA's :408
#+end_example
*** Overlap with Jastrow optimized wf
Parameters of the Jastrow:
a r_{12} / (1 + b r_{12})
p r_{12}^2 / (1 + p r_{12}^2)
a = 1/2 b = 1/2(sqrt(pi)) = 0.886226925453
p_H = 0.3467 p_O = 1.1485 (energy minimization)
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
fun.0 <- function(x) (1.1485 * x^2) / (1 + 1.1485* x^2)
fun.1 <- function(x) (0.3467 * x^2) / (1 + 0.3467* x^2)
fun.2 <- function(x) (0.5 * x) / (1 + 0.5*sqrt(pi) * x)
p <- p + stat_function(fun = fun.0) + xlim(0,5)
p <- p + stat_function(fun = fun.1) + xlim(0,5)
p <- p + stat_function(fun = fun.2) + xlim(0,5)
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-0OtcKh/figurePGHfN9.png]]
#+begin_src R :results output :session *R* :exports both
normalize <- function(x) {x / sqrt(sum(x^2))}
FCI <- normalize(scan("fci.dat"))
d = data.frame(FCI)
d$mu_0 <- normalize(scan("mu00.dat"))
d$mu_025 <- normalize(scan("mu25.dat"))
d$mu_05 <- normalize(scan("mu05.dat"))
d$mu_1 <- normalize(scan("mu10.dat"))
d$mu_2 <- normalize(scan("mu20.dat"))
d$mu_5 <- normalize(scan("mu50.dat"))
d$mu_inf<- normalize(scan("diag.dat"))
d$jast <- normalize(scan("jast.dat"))
overlap <- function(x) {x <- abs(crossprod(x,jast)[1]) }
ov <- apply(d,2,overlap)
ov
#+end_src
#+RESULTS:
#+begin_example
Read 200 items
Read 200 items
Read 200 items
Read 200 items
Read 200 items
Read 200 items
Read 200 items
Read 200 items
Read 200 items
Named num [1:9] 0.991 0.988 0.989 0.992 0.994 ...
- attr(*, "names")= chr [1:9] "FCI" "mu_0" "mu_025" "mu_05" ...
#+end_example
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
ov <- subset(ov, names(ov) != "jast" & names(ov) != "FCI")
mu <- c(0., 0.25, 0.5, 1., 2., 5., 1.e6)
X <- mu/(mu+1.)
mu <- as.character(mu)
mu[7] <- TeX("$\\infty$")
p <- ggplot(data.frame(ov, X), aes(y=ov, x=X))
p <- p + geom_point()
p <- p + scale_x_continuous(name=TeX("$\\mu$"), breaks=X, labels=mu)
p <- p + scale_y_continuous(name = TeX("Overlap") )
p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.15, .15))
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-0OtcKh/figure6cb0ql.png]]
*** Optimal mu
**** Table
Find the optimal $\mu$ in cc-pVDZ for each molecule and atom, making
a spline interpolation of the total energy wrt $\mu$.
#+begin_src R :results output :exports both :session *R*
rows <- list()
j <- 0
for (mol in levels(ref$"Molecule") )
{
data <- subset(raw_data, (Method == "DMC" & Basis == "cc-pVDZ" & Molecule == mol & Mu != "opt"))
data$"Mu" <- as.numeric(data$"Mu")
vdz <- data["Mu"]
vdz$"X" <- data$"Mu"/(data$"Mu"+1.)
vdz$"E" <- data$"TotalEnergy"
vdz$"E.lo" <- data$"TotalEnergy" - data$"TotalEnergyErr"
vdz$"E.hi" <- data$"TotalEnergy" + data$"TotalEnergyErr"
vdz$"Basis" <- "VDZ-BFD"
vdz$"X"[7] <- 1.
vdz$"Mu"[7] <- TeX("$\\infty$")
vdz.spline_int <- as.data.frame(spline(vdz$"X", vdz$"E", n=101))
vdz.spline_int_lo <- as.data.frame(spline(vdz$"X", vdz$"E.lo", n=101))
vdz.spline_int_hi <- as.data.frame(spline(vdz$"X", vdz$"E.hi", n=101))
Emin = min(vdz.spline_int)
i <- which(vdz.spline_int$y == Emin)
x <- vdz.spline_int$x[i]
Eerr <- (vdz.spline_int_hi$y - vdz.spline_int_lo$y)[i]
mu <- x / (1-x)
dat <- data.frame("Molecule"=mol, "Mu"=mu, "TotalEnergy"=Emin, "TotalEnergyErr"=Eerr)
j <- j + 1
rows[[j]] <- dat
}
mu_opt = do.call(rbind, rows)
mu_opt
#+end_src
#+RESULTS:
#+begin_example
Molecule Mu TotalEnergy TotalEnergyErr
1 Be 22.076923 -1.0104925 5.769640e-06
2 BeH 3.918033 -1.5804522 -3.092773e-05
3 C 6.500000 -5.4320500 1.999488e-04
4 C2H2 5.976744 -12.4977890 1.631542e-03
5 C2H4 5.976744 -13.7496692 3.475880e-03
6 C2H6 5.976744 -14.9862763 3.649831e-03
7 CH 6.142857 -6.0641447 4.604460e-04
8 CH2_1A1 6.500000 -6.7168723 9.408143e-04
9 CH2_3B1 6.500000 -6.7315386 8.420592e-04
10 CH3 5.250000 -7.4161384 4.575831e-04
11 CH3Cl 6.692308 -22.5123351 4.313597e-03
12 CH4 5.818182 -8.0950439 6.943745e-04
13 Cl Inf -14.9623070 5.120000e-04
14 Cl2 6.317073 -30.0149662 4.286545e-03
15 ClF Inf -39.2471030 2.942000e-03
16 ClO 6.317073 -30.9553639 3.379987e-03
17 CN 6.142857 -15.5135815 1.475441e-03
18 CO 6.317073 -21.7422093 1.746556e-03
19 CO2 Inf -37.8388280 3.554000e-03
20 CS 8.090909 -15.8227883 2.261302e-03
21 F 6.692308 -24.1947898 7.291550e-04
22 F2 10.538462 -48.4486772 1.191581e-02
23 H 2.000000 -0.5000220 3.200000e-05
24 H2CO 6.317073 -22.9170478 3.280498e-03
25 H2O 6.142857 -17.2591198 1.451158e-03
26 H2O2 19.000000 -33.2358899 -4.071474e-03
27 H2S 6.317073 -11.4106220 4.174444e-04
28 H3COH Inf -24.1332880 2.934000e-03
29 H3CSH 6.142857 -18.2991121 4.432337e-03
30 HCl 6.692308 -15.6303653 1.821596e-03
31 HCN 6.142857 -16.2206750 1.801664e-03
32 HCO 5.976744 -22.2672833 3.238134e-03
33 HF 6.142857 -24.9103839 1.130866e-03
34 HOCl 8.677419 -31.6179815 2.879019e-03
35 Li 22.076923 -0.1963761 3.904561e-05
36 Li2 22.076923 -0.4317381 1.040406e-04
37 LiF 6.317073 -24.6087084 1.505761e-03
38 LiH 6.142857 -0.7887410 3.993582e-05
39 N 6.500000 -9.7986241 2.993923e-04
40 N2 6.142857 -19.9524462 1.089443e-03
41 N2H4 5.976744 -22.2697720 4.213610e-03
42 Na 26.272727 -0.1822304 1.180745e-04
43 Na2 22.076923 -0.3915674 2.174183e-04
44 NaCl 19.000000 -15.3131353 9.113055e-03
45 NH 6.142857 -10.4264783 7.713270e-04
46 NH2 6.142857 -11.0809754 1.375113e-03
47 NH3 5.818182 -11.7605110 3.663185e-04
48 NO 6.142857 -25.9343787 2.038331e-03
49 O 6.317073 -15.9016039 5.110339e-04
50 O2 8.090909 -31.9863532 -1.189911e-03
51 OH 6.500000 -16.5656422 1.038452e-03
52 P 6.317073 -6.4727574 3.574169e-04
53 P2 6.317073 -13.1211638 2.071087e-03
54 PH2 Inf -7.7124550 8.060000e-04
55 PH3 Inf -8.3503240 1.756000e-03
56 S Inf -10.1249220 3.820000e-04
57 S2 16.647059 -20.4112519 -1.963031e-03
58 Si Inf -3.7651370 1.260000e-04
59 Si2 10.538462 -7.6463418 1.778160e-04
60 Si2H6 8.677419 -11.3730775 3.703778e-04
61 SiH2_1A1 17.750000 -5.0122525 5.573410e-04
62 SiH2_3B1 19.000000 -4.9762802 -8.064654e-04
63 SiH3 22.076923 -5.6262336 -6.690674e-03
64 SiH4 6.142857 -6.2776229 9.405144e-04
65 SiO 7.571429 -19.9632204 2.175365e-03
66 SO Inf -26.2122290 4.822000e-03
67 SO2 6.500000 -42.2992438 6.376208e-03
#+end_example
**** Atoms
Average percentage of the difference E_{DMC}(KS) - E_{DMC}(FCI) for all
atoms.
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
atoms = c("Be", "C", "Cl", "F", "N", "O", "P", "S", "Si")
mid <- rep(0.,7)
hi <- rep(0.,7)
lo <- rep(0.,7)
x <- rep(0.,7)
for (atom in atoms) {
data <- subset(raw_data, (Method == "DMC" & Basis == "cc-pVDZ" & Molecule == atom))
vdz = data["Mu"]
data$"Mu" <- as.numeric(data$"Mu")
vdz$"X" = data$"Mu"/(data$"Mu"+1.)
vdz$"E" = data$"TotalEnergy"
vdz$"E.lo" = data$"TotalEnergy" - data$"TotalEnergyErr"
vdz$"E.hi" = data$"TotalEnergy" + data$"TotalEnergyErr"
vdz$"Basis" = "VDZ-BFD"
vdz$"X"[7] <- 1.
vdz$"Mu"[7] <- TeX("$\\infty$")
# vdz.spline_int <- as.data.frame(spline(vdz$"X", vdz$"E", n=101))
# vdz.spline_int_lo <- as.data.frame(spline(vdz$"X", vdz$"E.lo", n=101))
# vdz.spline_int_hi <- as.data.frame(spline(vdz$"X", vdz$"E.hi", n=101))
# vdz.spline_int_lo$y <- (vdz.spline_int_lo$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
# vdz.spline_int_hi$y <- (vdz.spline_int_hi$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
# vdz.spline_int$y <- (vdz.spline_int$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
# mid <- mid + vdz.spline_int$y
# hi <- hi + vdz.spline_int_hi$y
# lo <- lo + vdz.spline_int_lo$y
# x <- vdz.spline_int$x
mid <- mid + (vdz$"E" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
hi <- hi + (vdz$"E.hi" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
lo <- lo + (vdz$"E.lo" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
x <- vdz$"X"
}
out = data.frame(mid)
out$mid <- mid / length(atoms)
out$hi <- hi / length(atoms)
out$lo <- lo / length(atoms)
out$x <- x
breaks <- vdz$"X"
labels <- vdz$"Mu"
labels[7] <- TeX("$\\infty$")
p <- ggplot(out, aes(x=x, y=mid, ymin=lo, ymax=hi) )
p <- p + geom_point()
p <- p + geom_ribbon(alpha=0.5)
p <- p + scale_x_continuous(name=TeX("$\\mu$"), breaks=breaks, labels=labels)
p <- p + scale_y_continuous(name = TeX("\\frac{E_{DMC} - E_{DMC}(FCI)}{E_{DMC}(KS) - E_{DMC}(FCI)}"), breaks=seq(0.,1.,0.2))
p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.15, .15))
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-8WVd9w/figureBMs1Mr.png]]
**** Molecules
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
molecules <- c("BeH", "C2H2", "C2H4", "C2H6", "CH", "CH2_1A1", "CH2_3B1", "CH3", "CH3Cl", "CH4", "CN", "CO", "CO2", "CS", "Cl2", "ClF", "ClO", "F2", "H2CO", "H2O", "H2O2", "H2S", "H3COH", "H3CSH", "HCN", "HCO", "HCl", "HF", "HOCl", "Li2", "LiF", "LiH", "N2", "N2H4", "NH", "NH2", "NH3", "NO", "Na2", "NaCl", "O2", "OH", "P2", "PH2", "PH3", "S2", "SO", "SO2", "Si2", "Si2H6", "SiH2_1A1", "SiH2_3B1", "SiH3", "SiH4", "SiO")
molecules <- c("H2O")
mid <- rep(0.,101)
hi <- rep(0.,101)
lo <- rep(0.,101)
x <- rep(0.,101)
#mid <- rep(0.,7)
#hi <- rep(0.,7)
#lo <- rep(0.,7)
#x <- rep(0.,7)
for (molecule in molecules) {
data <- subset(raw_data, (Method == "DMC" & Basis == "cc-pVDZ" & Molecule == molecule & Mu != "opt"))
vdz = data["Mu"]
data$"Mu" <- na.omit(as.numeric(data$"Mu"))
vdz$"X" = data$"Mu"/(data$"Mu"+1.)
vdz$"E" = data$"TotalEnergy"
vdz$"E.lo" = data$"TotalEnergy" - data$"TotalEnergyErr"
vdz$"E.hi" = data$"TotalEnergy" + data$"TotalEnergyErr"
vdz$"Basis" = "VDZ-BFD"
vdz$"X"[7] <- 1.
vdz$"Mu"[7] <- TeX("$\\infty$")
vdz.spline_int <- as.data.frame(spline(vdz$"X", vdz$"E", n=101))
vdz.spline_int_lo <- as.data.frame(spline(vdz$"X", vdz$"E.lo", n=101))
vdz.spline_int_hi <- as.data.frame(spline(vdz$"X", vdz$"E.hi", n=101))
vdz.spline_int_lo$y <- (vdz.spline_int_lo$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
vdz.spline_int_hi$y <- (vdz.spline_int_hi$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
vdz.spline_int$y <- (vdz.spline_int$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
mid <- mid + vdz.spline_int$y
hi <- hi + vdz.spline_int_hi$y
lo <- lo + vdz.spline_int_lo$y
x <- vdz.spline_int$x
# mid <- mid + (vdz$"E" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
# hi <- hi + (vdz$"E.hi" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
# lo <- lo + (vdz$"E.lo" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
# x <- vdz$"X"
}
out = data.frame(mid)
out$mid <- mid / length(molecules)
out$hi <- hi / length(molecules)
out$lo <- lo / length(molecules)
out$x <- x
breaks <- vdz$"X"
labels <- vdz$"Mu"
labels[7] <- TeX("$\\infty$")
p <- ggplot(out, aes(x=x, y=mid, ymin=lo, ymax=hi) )
p <- p + geom_line()
p <- p + geom_ribbon(alpha=0.5)
p <- p + scale_x_continuous(name=TeX("$\\mu$"), breaks=breaks, labels=labels)
p <- p + scale_y_continuous(name = TeX("\\frac{E_{DMC} - E_{DMC}(FCI)}{E_{DMC}(KS) - E_{DMC}(FCI)}"))
p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.15, .15))
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-8WVd9w/figureRuNz9o.png]]
*** Figure H2O R
*** Figure energy H2O
#+begin_src R :var data=tab1 :results output :session *R* :exports both
str(data)
data$X
@ -5609,14 +5285,14 @@ labels <- data$"X..mu"
labels[9] <- ""
labels[11] <- TeX("$\\infty$")
spline_int1 <- as.data.frame(spline(d$X, d$E))
#vdz.spline_int <- as.data.frame(spline(d$X, d$E))
p <- ggplot(d, aes(x=X, y=E, ymin=E.lo, ymax=E.hi, fill=Basis ))
p <- p + geom_point()
p <- p + geom_smooth(span=.5, se=FALSE, color='darkgrey')
p <- p + geom_ribbon(alpha=0.5)
#p <- p + geom_smooth(span=.7, se=FALSE)
#p <- p + geom_line(data = vdz.spline_int, aes(x = x, y = y))
p <- p + scale_x_continuous(name=TeX("$\\mu$"), breaks=breaks, labels=labels)
p <- p + geom_point()
#p <- p + geom_line(data = vdz.spline_int, aes(x = x, y = y))
p <- p + scale_x_continuous(name=TeX("$\\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
@ -5624,7 +5300,7 @@ p
#+end_src
#+RESULTS:
[[file:/tmp/babel-cHKMq8/figureXKea51.png]]
[[file:/tmp/babel-RBQjE9/figurelVzAZK.png]]
#+begin_src R :results output :session *R* :exports both
pdf("../Manuscript/h2o-dmc.pdf", family="Times", width=8, height=5)
@ -5637,6 +5313,482 @@ dev.off()
: null device
: 1
*** Figure energy F2
#+begin_src R :var data=tab1 :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
data <- subset(raw_data, (Method == "DMC" & Basis == "cc-pVDZ" & Molecule == "F2" & Mu != "opt"))
vdz = data["Mu"]
vdz$"Mu" <- as.numeric(levels(factor(data$"Mu")))
vdz$"X" = vdz$"Mu"/(vdz$"Mu"+1.)
vdz$"E" = data$"TotalEnergy"
vdz$"E.lo" = data$"TotalEnergy" - data$"TotalEnergyErr"
vdz$"E.hi" = data$"TotalEnergy" + data$"TotalEnergyErr"
vdz$"Basis" = "VDZ-BFD"
vdz$"X"[7] <- 1.
vdz$"Mu"[7] <- TeX("$\\infty$")
vdz.spline_int <- as.data.frame(spline(vdz$"X", vdz$"E", n=101))
breaks <- vdz$"X"
labels <- vdz$"Mu"
labels[7] <- TeX("$\\infty$")
d <- vdz
p <- ggplot(d, aes(x=X, y=E, ymin=E.lo, ymax=E.hi, fill="red"))
p <- p + geom_smooth(span=.75, se=FALSE, color='darkgrey', show.legend=FALSE )
p <- p + geom_ribbon(alpha=0.5, show.legend = FALSE)
p <- p + geom_point(show.legend=FALSE )
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(-48.45,-48.42,0.005))
p <- p + theme(text = element_text(size = 20, family="Times"))
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-RBQjE9/figurewct2z7.png]]
#+begin_src R :results output :session *R* :exports both
pdf("../Manuscript/f2-dmc.pdf", family="Times", width=8, height=5)
p
dev.off()
#+end_src
#+RESULTS:
:
: `geom_smooth()` using method = 'loess' and formula 'y ~ x'
:
: png
: 2
*** Overlap with Jastrow optimized wf
**** H2O
Parameters of the Jastrow:
a r_{12} / (1 + b r_{12})
p r_{12}^2 / (1 + p r_{12}^2)
a = 1/2 b = 1/2(sqrt(pi)) = 0.886226925453
p_H = 0.3467 p_O = 1.1485 (energy minimization)
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
fun.0 <- function(x) ((1.1485 * x) / (1 + 1.1485* x))**2
fun.1 <- function(x) ((0.3467 * x) / (1 + 0.3467* x))**2
fun.2 <- function(x) (0.5 * x) / (1 + 0.5*sqrt(pi) * x)
p <- p + stat_function(fun = fun.0) + xlim(0,5)
p <- p + stat_function(fun = fun.1) + xlim(0,5)
p <- p + stat_function(fun = fun.2) + xlim(0,5)
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-RBQjE9/figureuAYjXv.png]]
#+begin_src R :results output :session *R* :exports both
normalize <- function(x) {x / sqrt(sum(x^2))}
FCI <- normalize(scan("fci.dat"))
dH2O <- data.frame(FCI)
dH2O$mu_0 <- normalize(scan("mu00.dat"))
dH2O$mu_025 <- normalize(scan("mu25.dat"))
dH2O$mu_05 <- normalize(scan("mu05.dat"))
dH2O$mu_1 <- normalize(scan("mu10.dat"))
dH2O$mu_2 <- normalize(scan("mu20.dat"))
dH2O$mu_5 <- normalize(scan("mu50.dat"))
dH2O$mu_inf<- normalize(scan("diag.dat"))
dH2O$jast <- normalize(scan("jast.dat"))
overlap <- function(x) {x <- abs(crossprod(x,dH2O$jast)[1]) }
H2O <- apply(dH2O,2,overlap)
H2O
#+end_src
#+RESULTS:
#+begin_example
FCI mu_0 mu_025 mu_05 mu_1 mu_2 mu_5 mu_inf
0.9908628 0.9875157 0.9885532 0.9918674 0.9943390 0.9926737 0.9906157 0.9899355
jast
1.0000000
#+end_example
**** F2
Parameters of the Jastrow:
a r_{12} / (1 + b r_{12})
p r_{12}^2 / (1 + p r_{12}^2)
a = 1/2 b = 1/2(sqrt(pi)) = 0.886226925453
p_F = 1.1900 (energy minimization)
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
fun.0 <- function(x) ((1.1900 * x) / (1 + 1.1900* x))**2
fun.2 <- function(x) (0.5 * x) / (1 + 0.8862 * x)
p <- p + stat_function(fun = fun.0) + xlim(0,5)
p <- p + stat_function(fun = fun.2) + xlim(0,5)
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-RBQjE9/figureFQ9sdT.png]]
#+begin_src R :results output :session *R* :exports both
normalize <- function(x) {x / sqrt(sum(x^2))}
FCI <- normalize(scan("F2_fci.dat"))
dF2 = data.frame(FCI)
dF2$mu_0 <- normalize(scan("F2_mu00.dat"))
dF2$mu_025 <- normalize(scan("F2_mu02.dat"))
dF2$mu_05 <- normalize(scan("F2_mu05.dat"))
dF2$mu_1 <- normalize(scan("F2_mu10.dat"))
dF2$mu_2 <- normalize(scan("F2_mu20.dat"))
dF2$mu_5 <- normalize(scan("F2_mu50.dat"))
dF2$mu_inf<- normalize(scan("F2_diag.dat"))
dF2$jast <- normalize(scan("F2_jast.dat"))
overlap <- function(x) {x <- abs(crossprod(x,dF2$jast)[1]) }
F2 <- apply(dF2,2,overlap)
F2
#+end_src
#+RESULTS:
#+begin_example
FCI mu_0 mu_025 mu_05 mu_1 mu_2 mu_5 mu_inf
0.9973656 0.9684951 0.9719259 0.9821996 0.9934597 0.9972475 0.9978171 0.9973656
jast
1.0000000
#+end_example
**** F2 optimal
Parameters of the Jastrow:
a r_{12} / (1 + b r_{12})
p r_{12}^2 / (1 + p r_{12}^2)
a = 1/2 b = 1.3557
p_F = 0.8081 (energy minimization)
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
fun.0 <- function(x) ((0.8081 * x) / (1 + 0.8081* x))**2
fun.2 <- function(x) (0.5 * x) / (1 + 1.3557 * x)
p <- p + stat_function(fun = fun.0) + xlim(0,5)
p <- p + stat_function(fun = fun.2) + xlim(0,5)
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-RBQjE9/figure8Oip7n.png]]
#+begin_src R :results output :session *R* :exports both
normalize <- function(x) {x / sqrt(sum(x^2))}
FCI <- normalize(scan("F2_fci.dat"))
dF2 = data.frame(FCI)
dF2$mu_0 <- normalize(scan("F2_mu00.dat"))
dF2$mu_025 <- normalize(scan("F2_mu02.dat"))
dF2$mu_05 <- normalize(scan("F2_mu05.dat"))
dF2$mu_1 <- normalize(scan("F2_mu10.dat"))
dF2$mu_2 <- normalize(scan("F2_mu20.dat"))
dF2$mu_5 <- normalize(scan("F2_mu50.dat"))
dF2$mu_inf<- normalize(scan("F2_diag.dat"))
dF2$jast <- normalize(scan("F2_jastopt.dat"))
overlap <- function(x) {x <- abs(crossprod(x,dF2$jast)[1]) }
F2opt <- apply(dF2,2,overlap)
F2opt
#+end_src
#+RESULTS:
#+begin_example
FCI mu_0 mu_025 mu_05 mu_1 mu_2 mu_5 mu_inf
0.9985079 0.9684980 0.9722671 0.9834621 0.9953808 0.9987209 0.9989324 0.9985079
jast
1.0000000
#+end_example
**** Plot
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
Mu <- c(0., 0.25, 0.5, 1., 2., 5., Inf)
X <- Mu/(Mu+1.)
X[7] <- 1.
H2O_FCI <- H2O["FCI"]
F2_FCI <- F2["FCI"]
H2O <- subset(H2O, names(H2O) != "jast" & names(H2O) != "FCI")
F2 <- subset(F2 , names(F2 ) != "jast" & names(F2 ) != "FCI")
F2opt <- subset(F2opt , names(F2opt) != "jast" & names(F2opt) != "FCI")
ov = data.frame(Mu, X, H2O, F2, F2opt)
print(ov)
Mu <- as.character(Mu)
Mu[7] <- TeX("$\\infty$")
H2O.spline_int <- as.data.frame(spline(ov$"X", ov$"H2O", n=101))
F2.spline_int <- as.data.frame(spline(ov$"X", ov$"F2", n=101))
F2opt.spline_int <- as.data.frame(spline(ov$"X", ov$"F2opt", n=101))
p <- ggplot()
p <- p + geom_point(aes(x=X,y=F2), color='red')
p <- p + geom_point(aes(x=X,y=H2O), color='blue')
#p <- p + geom_point(aes(x=X,y=F2opt), color='green')
p <- p + geom_line(data = F2.spline_int, aes(x=x, y=y, colour='a'))
p <- p + geom_line(data = H2O.spline_int, aes(x=x, y=y, colour='b'))
#p <- p + geom_line(data = F2opt.spline_int, aes(x=x, y=y, colour='c'))
p <- p + scale_x_continuous(name=TeX("$\\mu$ (bohr$^{-1}$)"), breaks=X, labels=mu)
p <- p + scale_y_continuous(name = TeX("Overlap") )
p <- p + scale_colour_manual(name = 'Molecule', guide='legend',
values =c('c'='green', 'b'='blue','a'='red'),
labels = c(expression(F[2]), expression(H[2]*O), expression(F[2]*opt) ))
p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.85, .15))
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-RBQjE9/figureTA79X4.png]]
#+begin_src R :results output :session *R* :exports both
pdf("../Manuscript/overlap.pdf", family="Times", width=8, height=5)
p
dev.off()
#+end_src
#+RESULTS:
:
: png
: 2
*** Optimal mu
**** Table
Find the optimal $\mu$ in cc-pVDZ for each molecule and atom, making
a spline interpolation of the total energy wrt $\mu$.
#+begin_src R :results output :exports both :session *R*
rows <- list()
j <- 0
for (mol in levels(ref$"Molecule") )
{
data <- subset(raw_data, (Method == "DMC" & Basis == "cc-pVDZ" & Molecule == mol & Mu != "opt"))
data$"Mu" <- as.numeric(levels(factor(data$"Mu")))
vdz <- data["Mu"]
vdz$"X" <- data$"Mu"/(data$"Mu"+1.)
vdz$"E" <- data$"TotalEnergy"
vdz$"E.lo" <- data$"TotalEnergy" - data$"TotalEnergyErr"
vdz$"E.hi" <- data$"TotalEnergy" + data$"TotalEnergyErr"
vdz$"Basis" <- "VDZ-BFD"
vdz$"X"[7] <- 1.
vdz$"Mu"[7] <- TeX("$\\infty$")
vdz.spline_int <- as.data.frame(spline(vdz$"X", vdz$"E", n=101))
vdz.spline_int_lo <- as.data.frame(spline(vdz$"X", vdz$"E.lo", n=101))
vdz.spline_int_hi <- as.data.frame(spline(vdz$"X", vdz$"E.hi", n=101))
Emin = min(vdz.spline_int)
i <- which(vdz.spline_int$y == Emin)
x <- vdz.spline_int$x[i]
Eerr <- (vdz.spline_int_hi$y - vdz.spline_int_lo$y)[i]
mu <- x / (1-x)
dat <- data.frame("Molecule"=mol, "Mu"=mu, "TotalEnergy"=Emin, "TotalEnergyErr"=Eerr)
j <- j + 1
rows[[j]] <- dat
}
mu_opt = do.call(rbind, rows)
mu_opt
#+end_src
#+RESULTS:
#+begin_example
Molecule Mu TotalEnergy TotalEnergyErr
1 Be 0.9607843 -1.0079615 2.171751e-05
2 BeH 0.5625000 -1.5805129 4.067433e-04
3 C 2.8461538 -5.4320134 2.060275e-04
4 C2H2 2.1250000 -12.4978013 1.693933e-03
5 C2H4 2.0303030 -13.7496646 3.523327e-03
6 C2H6 2.0303030 -14.9862616 3.724381e-03
7 CH 2.1250000 -6.0641416 4.574267e-04
8 CH2_1A1 3.3478261 -6.7168950 1.009825e-03
9 CH2_3B1 3.1666667 -6.7315434 8.764920e-04
10 CH3 1.2727273 -7.4162327 5.305483e-04
11 CH3Cl 3.5454545 -22.5123924 4.245374e-03
12 CH4 1.7777778 -8.0950231 7.121452e-04
13 Cl Inf -14.9623070 5.120000e-04
14 Cl2 2.5714286 -30.0152826 4.435378e-03
15 ClF Inf -39.2471030 2.942000e-03
16 ClO 2.7037037 -30.9557213 3.402424e-03
17 CN 2.3333333 -15.5136342 1.488321e-03
18 CO 2.4482759 -21.7422903 1.696224e-03
19 CO2 Inf -37.8388280 3.554000e-03
20 CS 3.7619048 -15.8228210 2.003188e-03
21 F 3.1666667 -24.1949312 7.391838e-04
22 F2 4.8823529 -48.4453257 4.050079e-03
23 H 0.0000000 -0.5000220 3.200000e-05
24 H2CO 2.7037037 -22.9171510 3.311087e-03
25 H2O 2.2258065 -17.2591426 1.461246e-03
26 H2O2 13.2857143 -33.2120183 4.669473e-03
27 H2S 2.4482759 -11.4106147 3.903350e-04
28 H3COH Inf -24.1332880 2.934000e-03
29 H3CSH 2.4482759 -18.2993308 4.557326e-03
30 HCl 3.3478261 -15.6303714 1.893189e-03
31 HCN 2.3333333 -16.2207533 1.825787e-03
32 HCO 2.0303030 -22.2673034 3.309921e-03
33 HF 2.3333333 -24.9104643 1.191760e-03
34 HOCl 4.5555556 -31.6166964 4.824703e-03
35 Li 1.0000000 -0.1963480 2.000000e-05
36 Li2 0.7241379 -0.4314662 3.585501e-05
37 LiF 2.5714286 -24.6088303 1.550810e-03
38 LiH 2.4482759 -0.7887656 3.052112e-05
39 N 3.0000000 -9.7986421 3.121059e-04
40 N2 2.1250000 -19.9524868 1.048942e-03
41 N2H4 2.1250000 -22.2699022 4.249663e-03
42 Na 0.6129032 -0.1821443 2.754961e-05
43 Na2 0.0000000 -0.3910930 1.660000e-04
44 NaCl 1.1276596 -15.3111605 5.178434e-03
45 NH 2.3333333 -10.4264840 7.976368e-04
46 NH2 2.1250000 -11.0809890 1.362137e-03
47 NH3 1.8571429 -11.7605192 3.873223e-04
48 NO 2.3333333 -25.9345356 2.074537e-03
49 O 2.7037037 -15.9017110 5.380787e-04
50 O2 4.2631579 -31.9858778 2.794643e-03
51 OH 2.8461538 -16.5657615 1.069651e-03
52 P 2.5714286 -6.4727490 3.693525e-04
53 P2 2.8461538 -13.1215885 2.097094e-03
54 PH2 Inf -7.7124550 8.060000e-04
55 PH3 Inf -8.3503240 1.756000e-03
56 S Inf -10.1249220 3.820000e-04
57 S2 6.6923077 -20.4034043 3.266119e-03
58 Si Inf -3.7651370 1.260000e-04
59 Si2 4.0000000 -7.6457238 9.502980e-04
60 Si2H6 3.5454545 -11.3730818 3.867320e-03
61 SiH2_1A1 6.1428571 -5.0079101 8.651101e-04
62 SiH2_3B1 7.3333333 -4.9746405 7.553165e-04
63 SiH3 Inf -5.6243160 1.250000e-03
64 SiH4 2.5714286 -6.2776913 9.957806e-04
65 SiO 3.7619048 -19.9634155 1.961388e-03
66 SO Inf -26.2122290 4.822000e-03
67 SO2 3.0000000 -42.2998957 6.613564e-03
#+end_example
**** Atoms
Average percentage of the difference E_{DMC}(KS) - E_{DMC}(FCI) for all
atoms.
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
atoms = c("Be", "C", "Cl", "F", "N", "O", "P", "S", "Si")
mid <- rep(0.,7)
hi <- rep(0.,7)
lo <- rep(0.,7)
x <- rep(0.,7)
for (atom in atoms) {
data <- subset(raw_data, (Method == "DMC" & Basis == "cc-pVDZ" & Molecule == atom))
vdz = data["Mu"]
data$"Mu" <- as.numeric(levels(factor(data$"Mu")))
vdz$"X" = data$"Mu"/(data$"Mu"+1.)
vdz$"E" = data$"TotalEnergy"
vdz$"E.lo" = data$"TotalEnergy" - data$"TotalEnergyErr"
vdz$"E.hi" = data$"TotalEnergy" + data$"TotalEnergyErr"
vdz$"Basis" = "VDZ-BFD"
vdz$"X"[7] <- 1.
vdz$"Mu"[7] <- TeX("$\\infty$")
# vdz.spline_int <- as.data.frame(spline(vdz$"X", vdz$"E", n=101))
# vdz.spline_int_lo <- as.data.frame(spline(vdz$"X", vdz$"E.lo", n=101))
# vdz.spline_int_hi <- as.data.frame(spline(vdz$"X", vdz$"E.hi", n=101))
# vdz.spline_int_lo$y <- (vdz.spline_int_lo$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
# vdz.spline_int_hi$y <- (vdz.spline_int_hi$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
# vdz.spline_int$y <- (vdz.spline_int$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
# mid <- mid + vdz.spline_int$y
# hi <- hi + vdz.spline_int_hi$y
# lo <- lo + vdz.spline_int_lo$y
# x <- vdz.spline_int$x
mid <- mid + (vdz$"E" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
hi <- hi + (vdz$"E.hi" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
lo <- lo + (vdz$"E.lo" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
x <- vdz$"X"
}
out = data.frame(mid)
out$mid <- mid / length(atoms)
out$hi <- hi / length(atoms)
out$lo <- lo / length(atoms)
out$x <- x
breaks <- vdz$"X"
labels <- vdz$"Mu"
labels[7] <- TeX("$\\infty$")
p <- ggplot(out, aes(x=x, y=mid, ymin=lo, ymax=hi) )
p <- p + geom_point()
p <- p + geom_ribbon(alpha=0.5)
p <- p + scale_x_continuous(name=TeX("$\\mu$"), breaks=breaks, labels=labels)
p <- p + scale_y_continuous(name = TeX("\\frac{E_{DMC} - E_{DMC}(FCI)}{E_{DMC}(KS) - E_{DMC}(FCI)}"), breaks=seq(0.,1.,0.2))
p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.15, .15))
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-RBQjE9/figuremq7fNJ.png]]
**** Molecules
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
molecules <- c("BeH", "C2H2", "C2H4", "C2H6", "CH", "CH2_1A1", "CH2_3B1", "CH3", "CH3Cl", "CH4", "CN", "CO", "CO2", "CS", "Cl2", "ClF", "ClO", "F2", "H2CO", "H2O", "H2O2", "H2S", "H3COH", "H3CSH", "HCN", "HCO", "HCl", "HF", "HOCl", "Li2", "LiF", "LiH", "N2", "N2H4", "NH", "NH2", "NH3", "NO", "Na2", "NaCl", "O2", "OH", "P2", "PH2", "PH3", "S2", "SO", "SO2", "Si2", "Si2H6", "SiH2_1A1", "SiH2_3B1", "SiH3", "SiH4", "SiO")
molecules <- c("H2O")
molecules <- c("CH2_1A1")
molecules <- c("F2")
mid <- rep(0.,101)
hi <- rep(0.,101)
lo <- rep(0.,101)
x <- rep(0.,101)
#mid <- rep(0.,7)
#hi <- rep(0.,7)
#lo <- rep(0.,7)
#x <- rep(0.,7)
for (molecule in molecules) {
data <- subset(raw_data, (Method == "DMC" & Basis == "cc-pVDZ" & Molecule == molecule & Mu != "opt"))
vdz = data["Mu"]
vdz$"Mu" <- as.numeric(levels(factor(data$"Mu")))
vdz$"X" = vdz$"Mu"/(vdz$"Mu"+1.)
vdz$"E" = data$"TotalEnergy"
vdz$"E.lo" = data$"TotalEnergy" - data$"TotalEnergyErr"
vdz$"E.hi" = data$"TotalEnergy" + data$"TotalEnergyErr"
vdz$"Basis" = "VDZ-BFD"
vdz$"X"[7] <- 1.
vdz$"Mu"[7] <- TeX("$\\infty$")
vdz.spline_int <- as.data.frame(spline(vdz$"X", vdz$"E", n=101))
vdz.spline_int_lo <- as.data.frame(spline(vdz$"X", vdz$"E.lo", n=101))
vdz.spline_int_hi <- as.data.frame(spline(vdz$"X", vdz$"E.hi", n=101))
vdz.spline_int_lo$y <- (vdz.spline_int_lo$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
vdz.spline_int_hi$y <- (vdz.spline_int_hi$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
vdz.spline_int$y <- (vdz.spline_int$y - vdz.spline_int$y[101])/(vdz.spline_int$y[1]-vdz.spline_int$y[101])
mid <- mid + vdz.spline_int$y
hi <- hi + vdz.spline_int_hi$y
lo <- lo + vdz.spline_int_lo$y
x <- vdz.spline_int$x
# mid <- mid + (vdz$"E" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
# hi <- hi + (vdz$"E.hi" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
# lo <- lo + (vdz$"E.lo" - vdz$"E"[7])/(vdz$"E"[1] - vdz$"E"[7])
# x <- vdz$"X"
}
out = data.frame(mid)
out$mid <- mid / length(molecules)
out$hi <- hi / length(molecules)
out$lo <- lo / length(molecules)
out$x <- x
breaks <- vdz$"X"
labels <- vdz$"Mu"
labels[7] <- TeX("$\\infty$")
p <- ggplot(out, aes(x=x, y=mid, ymin=lo, ymax=hi) )
p <- p + geom_line()
p <- p + geom_ribbon(alpha=0.5)
p <- p + scale_x_continuous(name=TeX("$\\mu$"), breaks=breaks, labels=labels)
p <- p + scale_y_continuous(name = TeX("\\frac{E_{DMC} - E_{DMC}(FCI)}{E_{DMC}(KS) - E_{DMC}(FCI)}"))
p <- p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.15, .15))
p
#+end_src
#+RESULTS:
[[file:/tmp/babel-RBQjE9/figureD9kKIs.png]]
*** Figure MAD R DZ
#+begin_src R :results output :session *R* :exports both
data <- data.frame(ref["Molecule"])

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24
Manuscript/rsdft-cipsi-qmc.tex
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@ -142,14 +142,14 @@ factor.\cite{Giner_2016,Dash_2018,Dash_2019}
\begin{enumerate}
\item Total energies and nodal quality:
\begin{itemize}
\item Factual stuffs: KS occupied orbitals closer to NOs than HF
\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 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 ?
\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
@ -166,6 +166,9 @@ factor.\cite{Giner_2016,Dash_2018,Dash_2019}
\item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$
\item large wave functions
\end{itemize}
\begin{itemize}
\item plot $N_{det}$ en fonction de $\mu$
\end{itemize}
\end{itemize}
\end{enumerate}
@ -398,10 +401,18 @@ implemented.\cite{GinPraFerAssSavTou-JCP-18}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
\caption{Fixed-node energies of the water molecule with variable
\caption{Fixed-node energies of the water molecule for different
values of $\mu$.}
\label{fig:h2o-dmc}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{f2-dmc.pdf}
\caption{Fixed-node energies of difluorine for different
values of $\mu$.}
\label{fig:f2-dmc}
\end{figure}
The water molecule was taken at the equilibrium
geometry,\cite{Caffarel_2016} and RSDFT-CIPSI wave functions were
generated with BFD pseudopotentials and the corresponding double-zeta
@ -433,6 +444,13 @@ in the CBS limit we expect the minimum of the FN-DMC energy to be
obtained for the FCI wave function, at $\mu=\infty$.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{overlap.pdf}
\caption{Overlap of the RSDFT-CIPSI wave functions with the
wave function reoptimized in the presence of a Jastrow factor.}
\label{fig:overlap}
\end{figure}
\section{Computational details}
\label{sec:comp-details}