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-0.962417543053159E-02 + 0.184373918079801E-01 + -0.989491154802612E-02 + 0.774306458315819E-02 + -0.105787596138723E-01 + 0.909433477249537E-02 + 0.118499711575993E-01 + -0.751835852411262E-02 + -0.848519965383221E-02 + -0.909234653785843E-02 + 0.118699201154356E-01 + -0.853741564737365E-02 + 0.853048364121520E-02 + -0.745588510310792E-02 + -0.745562214442801E-02 + -0.745811402415452E-02 + -0.745339227754012E-02 + -0.927969157493385E-02 + -0.167362163413626E-01 + -0.968030677743506E-02 + -0.122370948600498E-01 + 0.122383617904124E-01 diff --git a/Data/RSDFT-CIPSI.org b/Data/RSDFT-CIPSI.org index 79cc01a..b6b357d 100644 --- a/Data/RSDFT-CIPSI.org +++ b/Data/RSDFT-CIPSI.org @@ -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"]) diff --git a/Manuscript/f2-dmc.pdf b/Manuscript/f2-dmc.pdf new file mode 100644 index 0000000..bbe6e69 Binary files /dev/null and b/Manuscript/f2-dmc.pdf differ diff --git a/Manuscript/g2-dmc-dz.pdf b/Manuscript/g2-dmc-dz.pdf index 51b6e3c..213a576 100644 Binary files a/Manuscript/g2-dmc-dz.pdf and b/Manuscript/g2-dmc-dz.pdf differ diff --git a/Manuscript/g2-dmc-qz.pdf b/Manuscript/g2-dmc-qz.pdf index 6dc3f46..e321374 100644 Binary files a/Manuscript/g2-dmc-qz.pdf and b/Manuscript/g2-dmc-qz.pdf differ diff --git a/Manuscript/g2-dmc-tz.pdf b/Manuscript/g2-dmc-tz.pdf index 82dcc14..d1aeb3c 100644 Binary files a/Manuscript/g2-dmc-tz.pdf and b/Manuscript/g2-dmc-tz.pdf differ diff --git a/Manuscript/h2o-dmc.pdf b/Manuscript/h2o-dmc.pdf index 9edce06..d529d2a 100644 Binary files a/Manuscript/h2o-dmc.pdf and b/Manuscript/h2o-dmc.pdf differ diff --git a/Manuscript/overlap.pdf b/Manuscript/overlap.pdf new file mode 100644 index 0000000..22b40b8 Binary files /dev/null and b/Manuscript/overlap.pdf differ diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index 0f3d1b4..0cbb305 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -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}