{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "VS Code: start as \n", "code . \n", "in directory dmc_project \n", "\n", "Timings:\n", "17s\n", "\n", "added free-slots\\\n", "18s\n", "\n", "passing eta\\\n", "17.6s\n", "\n", "F=Drift(r) -> Drift!(F,R)\\\n", "11.6s\n", "\n", "Rold to MMatrix\\\n", "12s\n", "\n", "major StaticArray use\\\n", "6.8s;\n", "\n", "internal timing:\\\n", "@time main 4.0s\n", "\n", "replacing global Ntry ... with ::VMCStats\\\n", "3.5s\n", "\n", "empty!(free_slots)\\\n", "3.3s\n", "\n", "Original: QMC course 2018\n", "Improved: FYSS5120 2026\n", "\n", "\n", " JULIA\n", "DMC on a helium atom with trial wave function\n", "$$\n", " \\varphi_T = e^{-\\alpha r_1}e^{-\\alpha r_2} \n", " $$ \n", " $r_1,r_2$ distance of els from proton\\\n", " electrons in spin-up and in spin-down states \n", "\n", "Measures only total energy\n", "Hamiltonian in a.u. (Z=2) :\n", "\n", "$$H = -\\frac{1}{2} (\\nabla_1^2 + \\nabla_2^2) - \\frac{2}{r_1} - \\frac{2}{r_2}+ \\frac{1}{r_{12}}$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "# Activate dmc_project and set project paths" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Executable: /home/vap/.julia/juliaup/julia-1.12.7+0.x64.linux.gnu/bin\n", "Active Environment: /home/vap/texst/opetus/FYSS5120_Efficient_Numerical_Programming/juliacodes/dmc_project/Project.toml\n", "Julia Version: 1.12.7\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "\u001b[32m\u001b[1m Activating\u001b[22m\u001b[39m project at `~/texst/opetus/FYSS5120_Efficient_Numerical_Programming/juliacodes/dmc_project`\n", "\u001b[32m\u001b[1m Activating\u001b[22m\u001b[39m project at `~/texst/opetus/FYSS5120_Efficient_Numerical_Programming/juliacodes/dmc_project`\n" ] } ], "source": [ "using Pkg\n", "# project is one up from this src file\n", "Pkg.activate(joinpath(@__DIR__, \"..\"))\n", "\n", "# just for checking:\n", "# 1. Get the path to the active Julia executable (should be juliaup global executable)\n", "println(\"Executable: \", Sys.BINDIR)\n", "\n", "# 2. Get the active environment/project path\n", "import Pkg\n", "println(\"Active Environment: \", Pkg.project().path)\n", "\n", "# 3. Get the exact Julia version\n", "println(\"Julia Version: \", VERSION)\n", "\n", "# make project runnable from anywhere, saving results to path RUNS_DIR \n", "const PROJECT_ROOT = normpath(joinpath(@__DIR__, \"..\"))\n", "const RUNS_DIR = joinpath(PROJECT_ROOT, \"runs\")\n", "\n", "Pkg.activate(PROJECT_ROOT)\n", "# install packages in Project.toml if not done yet\n", "Pkg.instantiate()\n" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [], "source": [ "using Printf\n", "using Distributions\n", "import LinearAlgebra: norm\n", "using Statistics\n", "using Random\n", "using StaticArrays\n", "using BenchmarkTools\n", "#using LoopVectorization # don't use this package, it is not compatible with the current version of Julia" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [], "source": [ "\n", "\n", "using Printf\n", "using Distributions\n", "import LinearAlgebra: norm\n", "using Statistics\n", "using Random\n", "using StaticArrays\n", "using BenchmarkTools\n", "using LoopVectorization\n", "\n", "#Random.seed!(1234)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Parameters" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "He atom DMC using algorithm of order 2\n" ] } ], "source": [ "const order = 2 # 1 or 2\n", "println(\"He atom DMC using algorithm of order \",order)\n", "\n", "const lambda = 0.5 # hbar^2/(2m) in au\n", "const alpha = 27/16 # trial wafe function parameter\n", "const tau = 0.01 # imaginary time step\n", "const blocksize = 1000 # DMC data block size\n", "\n", "\n", "const NVMC = 1000 # number of VMC steps\n", "const NDMC = 10000 # number of DMC steps\n", "const Nwx = 20000 # max number of walkers\n", "const Nw_target = 1000 # on average this many walkers\n", "const kappa = 0.1 # how zealously DMC tries to keep Nw_target walkers\n", "const N = 2 # number of electrons\n", "const D = 3 # dimension\n", "const Eexact = -2.903724377 # accurate ground state energy, for comparison\n", "\n", "const DEAD = 0\n", "const ALIVE = 1\n", "const CHILD = 2\n", "\n", "const minstep = 1e-5\n", "const maxstep = 10.0\n", "\n", "const Coord = MMatrix{D,N,Float64,D*N}\n", "\n", "mutable struct Walker\n", " R ::Coord\n", " alive ::Int\n", " psi ::Float64\n", " E ::Float64\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "analytical energy in the trial wave function, for checking" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Eexpec (generic function with 1 method)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "function Eexpec(alpha)\n", " return alpha^2-27/8*alpha\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "local energy = $\\frac{H\\varphi_T}{\\varphi_T}$" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [], "source": [ "@inline function Elocal(R::Coord)\n", " x1, y1, z1 = R[1,1], R[2,1], R[3,1]\n", " x2, y2, z2 = R[1,2], R[2,2], R[3,2]\n", "\n", " r1 = sqrt(x1*x1 + y1*y1 + z1*z1)\n", " r2 = sqrt(x2*x2 + y2*y2 + z2*z2)\n", "\n", " dx = x1 - x2\n", " dy = y1 - y2\n", " dz = z1 - z2\n", " r12 = sqrt(dx*dx + dy*dy + dz*dz)\n", "\n", " return -alpha^2 - (2-alpha)/r1 - (2-alpha)/r2 + 1/r12\n", "end\n", "\n", "mutable struct VMCStats\n", " step::Float64\n", " Naccept::Int\n", " Ntry::Int\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Walker initialization" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "init (generic function with 1 method)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "\n", "# initialization \n", "function init()\n", " println(\"init\")\n", " # \n", " # Generate Nwx walker\n", " #\n", " walker = [Walker(zero(Coord),0,0.0,0.0) for i in 1:Nwx] # R, alive=0, psi=0, E=0\n", " println(\"generating $Nw_target walkers\")\n", " # set Nw_target walkers\n", " for iw in 1:Nw_target\n", " walker[iw].alive = 1\n", " R = Coord(rand(D, N)) # coordinates\n", " walker[iw].R = copy(R)\n", " walker[iw].psi = ln_psi2(R) # wave function (ln(psi^2))\n", " walker[iw].E = Elocal(R) \n", " end\n", " vmcstat = VMCStats(3.1,0,0)\n", " println(\"init done\")\n", " walker, vmcstat\n", "end\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$ln(\\varphi_T^2) = 2\\ln(\\varphi_T)$" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "metro (generic function with 1 method)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "@inline function ln_psi2(R::Coord)\n", " x1, y1, z1 = R[1,1], R[2,1], R[3,1]\n", " x2, y2, z2 = R[1,2], R[2,2], R[3,2]\n", "\n", " r1 = sqrt(x1*x1 + y1*y1 + z1*z1)\n", " r2 = sqrt(x2*x2 + y2*y2 + z2*z2)\n", "\n", " return -2*alpha*(r1+r2)\n", "end\n", "\n", "@inline function metro(Wold ::Float64, Wnew ::Float64)\n", " if Wnew>Wold\n", " return true\n", " end\n", " if rand()= 0.0 || exp(diff) > rand()\n", "\n", " if accept\n", " vmcstat.Naccept += 1\n", " psi = psi_new\n", " E = Elocal(R)\n", " else\n", " R[1,i] -= d1\n", " R[2,i] -= d2\n", " R[3,i] -= d3\n", " end\n", " end\n", "\n", " return psi, E\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Adjust step to keep VMC acceptance 40-60 %" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "save_r (generic function with 2 methods)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "function adjust_step(vmcstat::VMCStats)\n", " acceptance = vmcstat.Naccept/vmcstat.Ntry*100.0\n", " if acceptance<40.0\n", " vmcstat.step *= 0.9\n", " end\n", " if acceptance>60.0\n", " vmcstat.step *= 1.1\n", " end\n", " vmcstat.step = max(minstep,vmcstat.step)\n", " vmcstat.step = min(maxstep,vmcstat.step)\n", "end\n", "\n", "function save_R(R::Array{Float64,2},mode=\"w\")\n", " open(\"R\",mode) do f\n", " @inbounds for i in 1: size(R,2)\n", " for k in 1:size(R,1)\n", " print(f,R[k,i],\" \")\n", " end\n", " println(f,\" \")\n", " end\n", " end\n", "end\n", "\n", "function save_r(R::Array{Float64,2},mode=\"w\")\n", " open(\"r\",mode) do f\n", " rmin::Float64 = 1.e5\n", " rr::Float64 = 0.0\n", " @inbounds for i in 1: size(R,2)\n", " rr = norm(R[:,i])\n", " rmin = min(rmin,rr)\n", " end\n", " println(f,rmin)\n", " end\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "DMC drift, for ith electron $F_i = 2\\nabla_i \\varphi_T /\\varphi_T$" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Drift! (generic function with 1 method)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "@inline function Drift!(F::Coord, R::Coord)\n", " x1, y1, z1 = R[1,1], R[2,1], R[3,1]\n", " x2, y2, z2 = R[1,2], R[2,2], R[3,2]\n", "\n", " r1 = sqrt(x1*x1 + y1*y1 + z1*z1)\n", " r2 = sqrt(x2*x2 + y2*y2 + z2*z2)\n", "\n", " c = -2*alpha\n", "\n", " F[1,1] = c*x1/r1\n", " F[2,1] = c*y1/r1\n", " F[3,1] = c*z1/r1\n", "\n", " F[1,2] = c*x2/r2\n", " F[2,2] = c*y2/r2\n", " F[3,2] = c*z2/r2\n", "\n", " return nothing\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "one diffusion+drift step in DMC" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "lnG (generic function with 1 method)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "@inline function diffusion_drift_step!(R::Coord, F::Coord, R1::Coord, R2::Coord, eta::Coord)\n", " if order == 1\n", " randn!(eta)\n", " Drift!(F, R)\n", " @inbounds for j in 1:N, i in 1:D\n", " R[i,j] += lambda*tau*F[i,j] + sqrt(2*lambda*tau)*eta[i,j]\n", " end\n", "\n", " elseif order == 2\n", " randn!(eta)\n", " @inbounds for j in 1:N, i in 1:D\n", " R1[i,j] = R[i,j] + sqrt(lambda*tau)*eta[i,j]\n", " end\n", " Drift!(F, R1)\n", " @inbounds for j in 1:N, i in 1:D\n", " R2[i,j] = R1[i,j] + 0.5*lambda*tau*F[i,j]\n", " end\n", "\n", " Drift!(F, R2)\n", " @inbounds for j in 1:N, i in 1:D\n", " R[i,j] = R1[i,j] + lambda*tau*F[i,j]\n", " end\n", "\n", " randn!(eta)\n", " @inbounds for j in 1:N, i in 1:D\n", " R[i,j] += sqrt(lambda*tau)*eta[i,j]\n", " end\n", " end\n", "\n", " return nothing\n", "end\n", "\n", "\n", "function lnG(Rnew::Coord, Rold::Coord, Fold::Coord)\n", " if order != 1\n", " println(\"lnG only for 1st order code\")\n", " exit(1)\n", " end\n", "\n", " Drift!(Fold, Rold)\n", "\n", " s = 0.0\n", " @inbounds for j in 1:N, i in 1:D\n", " x = Rnew[i,j] - Rold[i,j] - lambda*tau*Fold[i,j]\n", " s += x*x\n", " end\n", "\n", " return -s / (4*lambda*tau)\n", "end\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "main (generic function with 1 method)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "\n", "\n", "function main()\n", " #\n", " # Main program \n", " # \n", " # initialize\n", " walker, vmcstat = init() \n", " # thermalization\n", " println(\"thermalizing\")\n", " Nw = Nw_target # for VMC\n", " @inbounds for i in 1:10\n", " @inbounds for iw in 1:Nw \n", " psi,E = vmc_step!(walker[iw].R, vmcstat)\n", " walker[iw].psi = psi\n", " walker[iw].E = E\n", " end\n", " end\n", "\n", " println(\"thermalization done\")\n", "\n", " filename = string(\"../runs/E_heatom_\",order,\"_tau=\",tau)\n", " println(\"output: \",filename)\n", " # init file output\n", " open(filename,\"w\") do f\n", " println(f,\" \")\n", " end\n", " #\n", " # VMC\n", " #\n", " E_ave::Float64 = 0.0\n", " nE::Int = 0\n", " @inbounds for ivmc in 1:NVMC\n", " E = 0.0 \n", " @inbounds for iw in 1:Nw \n", " psiw ,Ew = vmc_step!(walker[iw].R, vmcstat)\n", " walker[iw].psi = psiw\n", " walker[iw].E = Ew\n", " E += Ew \n", " end\n", " E_ave += E/Nw \n", " nE +=1\n", " \n", " if ivmc%10 == 0 \n", " #println(\"VMC Elocal=$(E/Nw) =$(E_ave/nE) _analytical=$(Eexpec(alpha))\")\n", " @printf(\"VMC E = %.10f = %.10f _analytical = %.10f\\n\",E/Nw,E_ave/nE, Eexpec(alpha)) \n", " end\n", " adjust_step(vmcstat)\n", " end\n", " \n", " #\n", " # DMC\n", " #\n", " ET::Float64 = E_ave/nE # trial energy, to be updated\n", " println(\"ET = $ET\")\n", " \n", " Ntherm = floor(Int,1.0/tau) # start DMC measurement after excitations have died out; system specific !\n", " idmc::Int = -Ntherm\n", "\n", " E_ave = 0.0\n", " nE = 0\n", " \n", " Edat::Float64 = 0.0 # single block E \n", " E2dat::Float64 = 0.0 # single block E^2\n", " Ndat::Int = 0 \n", "\n", " # block data\n", " Eb = Array{Float64,1}() # block data E\n", " E2b = Array{Float64,1}() # block data E^2\n", " sizehint!(Eb, 1000)\n", " sizehint!(E2b, 1000)\n", "\n", " \n", " idat::Int = 0 \n", " copies = zeros(Int,Nwx)\n", " Ntry::Int = 0 \n", " Nacc::Int = 0 \n", "\n", " Rold = zero(Coord)\n", " F = zero(Coord)\n", " R1 = zero(Coord)\n", " R2 = zero(Coord)\n", " eta = zero(Coord)\n", " \n", " free_slots = Int[]\n", " sizehint!(free_slots, Nwx)\n", "\n", " while idmcold) + lnpsi2(new)\n", " Wold = lnG(Rnew, Rold, F)+ln_psi2(Rold) # lnG(old->new) + lnpsi2(old)\n", " accept = metro(Wold, Wnew)\n", " else\n", " accept = true\n", " end\n", " Ntry +=1\n", " if accept\n", " Nacc +=1\n", " walker[iw].E = EL\n", " else\n", " EL = ELold\n", " walker[iw].E = ELold\n", " Rnew .= Rold \n", " end\n", "\n", " E += EL\n", " n += 1\n", " if idat>0\n", " Edat += EL\n", " E2dat += EL^2\n", " Ndat +=1 \n", " end\n", " \n", " weight = exp(-tau*(0.5*(ELold+EL)-ET)) # symmetric under R<->R'\n", " copies[iw] = floor(Int,weight + rand()) \n", " end\n", "\n", " empty!(free_slots)\n", " @inbounds for i in 1:Nwx\n", " if walker[i].alive == 0\n", " push!(free_slots, i)\n", " end\n", " end\n", " \n", " next_free = 1\n", " # Branching\n", " @inbounds for iw in 1:Nwx\n", " if walker[iw].alive != 1; continue; end \n", " if copies[iw] == 1\n", " continue\n", " elseif copies[iw] == 0\n", " walker[iw].alive = 0\n", " continue\n", " end\n", " srcR = walker[iw].R\n", " srcE = walker[iw].E\n", " srcpsi = walker[iw].psi\n", "\n", " for _ in 1:copies[iw]-1\n", " if next_free > length(free_slots)\n", " break\n", " end\n", " \n", " \n", " inew = free_slots[next_free]\n", " next_free += 1\n", "\n", " walker[inew].R = copy(srcR)\n", " walker[inew].E = srcE\n", " walker[inew].psi = srcpsi\n", " walker[inew].alive = 2\n", " end\n", " end\n", " \n", " @inbounds for iw in 1:Nwx\n", " if walker[iw].alive == 2\n", " walker[iw].alive =1\n", " end\n", " end\n", "\n", " Nw = sum([walker[i].alive for i in 1:Nwx])\n", " E_ave += E/n # don't use Nw as energy counter, it's not correct\n", " nE += 1\n", "\n", " idmc +=1\n", " \n", " \n", " \n", " if Nw==Nwx\n", " println(\"hit max number of walkers\")\n", " exit(1)\n", " end\n", "\n", " # update local energy; on average Nw_target walkers\n", " # A too large factor kappa will cause a bad feedback, a too small will let number of walkers get too large\n", " ET = E_ave/nE + kappa* log(Nw_target/Nw)\n", "\n", " if idmc<=0\n", " @printf(\"DMC %10d E = %.10f = %.10f ET = %.10f _exact = %.10f %6d Walkers \\n\",\n", " idmc, E/Nw, E_ave/nE, ET, Eexact, Nw)\n", " if order==1;@printf(\"DMC acceptance = %.5f %%\\n\",Nacc*100.0/Ntry);end\n", " end\n", "\n", " # block data\n", " # ==========\n", " # screen and file output\n", " if idat== blocksize \n", " push!(Eb,Edat/Ndat)\n", " push!(E2b,E2dat/Ndat)\n", " Eave::Float64 = mean(Eb)\n", " EStdError::Float64 = 0.0 \n", " if size(Eb,1)>1\n", " EStdError = std(Eb)/sqrt(size(Eb,1)) # sigma/sqrt(N) for N data points\n", " end\n", " @printf(\"DMC %10d E %.10f = %.10f +/- %.10f ET = %.10f _exact = %.10f %6d Walkers \\n\",\n", " idmc, Edat/Ndat ,Eave, EStdError, ET, Eexact, Nw)\n", " open(filename,\"a\") do f\n", " println(f,tau,\" \",Eave,\" \",EStdError,\" \",Eexact)\n", " end\n", " \n", " Edat = 0\n", " E2dat = 0\n", " Ndat = 0\n", " idat = 0\n", " \n", " end\n", " \n", " # \n", " if idmc==0\n", " println(\"THERMALIZATION ENDS\")\n", " E_ave = 0\n", " nE = 0\n", " idat = 1\n", " end\n", " if idmc>0\n", " idat += 1\n", " end\n", " \n", " end\n", "end\n", "\n" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "init\n", "generating 1000 walkers\n", "init done\n", "thermalizing\n", "thermalization done\n", "output: ../runs/E_heatom_2_tau=0.01\n", "VMC E = -2.8271461737 = -2.7989350800 _analytical = -2.8476562500\n", "VMC E = -2.8383850895 = -2.8077780476 _analytical = -2.8476562500\n", "VMC E = -2.8525705009 = -2.8166238477 _analytical = -2.8476562500\n", "VMC E = -2.8598080681 = -2.8225661382 _analytical = -2.8476562500\n", "VMC E = -2.8538934399 = -2.8284213535 _analytical = -2.8476562500\n", "VMC E = -2.8395614220 = -2.8338045684 _analytical = -2.8476562500\n", "VMC E = -2.8544775688 = -2.8361222928 _analytical = -2.8476562500\n", "VMC E = -2.8782582433 = -2.8397932024 _analytical = -2.8476562500\n", "VMC E = -2.9044423892 = -2.8403822288 _analytical = -2.8476562500\n", "VMC E = -2.8829587528 = -2.8439995022 _analytical = -2.8476562500\n", "VMC E = -2.8645690730 = -2.8463886465 _analytical = -2.8476562500\n", "VMC E = -2.8284365856 = -2.8474983281 _analytical = -2.8476562500\n", "VMC E = -2.8763233929 = -2.8476280974 _analytical = -2.8476562500\n", "VMC E = -2.8302850914 = -2.8473575695 _analytical = -2.8476562500\n", "VMC E = -2.8275990885 = -2.8437219945 _analytical = -2.8476562500\n", "VMC E = -2.8085688292 = -2.8418002832 _analytical = -2.8476562500\n", "VMC E = -2.9007119878 = -2.8432603110 _analytical = -2.8476562500\n", "VMC E = -2.9236205975 = -2.8462088166 _analytical = -2.8476562500\n", "VMC E = -2.8074872120 = -2.8467971589 _analytical = -2.8476562500\n", "VMC E = -2.8520219887 = -2.8460261694 _analytical = -2.8476562500\n", "VMC E = -2.8545693551 = -2.8462918782 _analytical = -2.8476562500\n", "VMC E = -2.8619960652 = -2.8465323938 _analytical = -2.8476562500\n", "VMC E = -2.8585935318 = -2.8465641525 _analytical = -2.8476562500\n", "VMC E = -2.8745142669 = -2.8471700179 _analytical = -2.8476562500\n", "VMC E = -2.8541669406 = -2.8470063433 _analytical = -2.8476562500\n", "VMC E = -2.8402672385 = -2.8459561508 _analytical = -2.8476562500\n", "VMC E = -2.8348512562 = -2.8458674630 _analytical = -2.8476562500\n", "VMC E = -2.8682512198 = -2.8448457902 _analytical = -2.8476562500\n", "VMC E = -2.8138553519 = -2.8445679826 _analytical = -2.8476562500\n", "VMC E = -2.8519567280 = -2.8453231994 _analytical = -2.8476562500\n", "VMC E = -2.8358563018 = -2.8460325317 _analytical = -2.8476562500\n", "VMC E = -2.8550765207 = -2.8449551426 _analytical = -2.8476562500\n", "VMC E = -2.8525076852 = -2.8445896707 _analytical = -2.8476562500\n", "VMC E = -2.8327446753 = -2.8452806105 _analytical = -2.8476562500\n", "VMC E = -2.7798754793 = -2.8444825578 _analytical = -2.8476562500\n", "VMC E = -2.8717510067 = -2.8448342186 _analytical = -2.8476562500\n", "VMC E = -2.8371197889 = -2.8447998134 _analytical = -2.8476562500\n", "VMC E = -2.8098321296 = -2.8446420217 _analytical = -2.8476562500\n", "VMC E = -2.8606193660 = -2.8444391907 _analytical = -2.8476562500\n", "VMC E = -2.8537797350 = -2.8442329739 _analytical = -2.8476562500\n", "VMC E = -2.8401811534 = -2.8444631410 _analytical = -2.8476562500\n", "VMC E = -2.9000848086 = -2.8449354306 _analytical = -2.8476562500\n", "VMC E = -2.8459843592 = -2.8454700884 _analytical = -2.8476562500\n", "VMC E = -2.8906630270 = -2.8463930854 _analytical = -2.8476562500\n", "VMC E = -2.7968509228 = -2.8465605299 _analytical = -2.8476562500\n", "VMC E = -2.7942528465 = -2.8462725993 _analytical = -2.8476562500\n", "VMC E = -2.8607013438 = -2.8466567412 _analytical = -2.8476562500\n", "VMC E = -2.8823526898 = -2.8471476968 _analytical = -2.8476562500\n", "VMC E = -2.8112747551 = -2.8470969255 _analytical = -2.8476562500\n", "VMC E = -2.8359567590 = -2.8465557584 _analytical = -2.8476562500\n", "VMC E = -2.8407789092 = -2.8464112194 _analytical = -2.8476562500\n", "VMC E = -2.8730470264 = -2.8463671354 _analytical = -2.8476562500\n", "VMC E = -2.8362858320 = -2.8465636359 _analytical = -2.8476562500\n", "VMC E = -2.8068979054 = -2.8464470638 _analytical = -2.8476562500\n", "VMC E = -2.8434821096 = -2.8461637607 _analytical = -2.8476562500\n", "VMC E = -2.8433920063 = -2.8457513354 _analytical = -2.8476562500\n", "VMC E = -2.8131876208 = -2.8456085543 _analytical = -2.8476562500\n", "VMC E = -2.8500609642 = -2.8454646080 _analytical = -2.8476562500\n", "VMC E = -2.8263536360 = -2.8454341566 _analytical = -2.8476562500\n", "VMC E = -2.8259626961 = -2.8449001772 _analytical = -2.8476562500\n", "VMC E = -2.8269611642 = -2.8439522524 _analytical = -2.8476562500\n", "VMC E = -2.8523777588 = -2.8440733072 _analytical = -2.8476562500\n", "VMC E = -2.8382226800 = -2.8441896517 _analytical = -2.8476562500\n", "VMC E = -2.8372433839 = -2.8439554897 _analytical = -2.8476562500\n", "VMC E = -2.8696764600 = -2.8440137052 _analytical = -2.8476562500\n", "VMC E = -2.8724319943 = -2.8445086425 _analytical = -2.8476562500\n", "VMC E = -2.8527859536 = -2.8448492311 _analytical = -2.8476562500\n", "VMC E = -2.8830340148 = -2.8446080513 _analytical = -2.8476562500\n", "VMC E = -2.8356574205 = -2.8445176206 _analytical = -2.8476562500\n", "VMC E = -2.8459247467 = -2.8446728311 _analytical = -2.8476562500\n", "VMC E = -2.8905162450 = -2.8444983895 _analytical = -2.8476562500\n", "VMC E = -2.8860247974 = -2.8446308956 _analytical = -2.8476562500\n", "VMC E = -2.8534636173 = -2.8448380443 _analytical = -2.8476562500\n", "VMC E = -2.8186491981 = -2.8445991567 _analytical = -2.8476562500\n", "VMC E = -2.9011420295 = -2.8441992382 _analytical = -2.8476562500\n", "VMC E = -2.8114946047 = -2.8441648272 _analytical = -2.8476562500\n", "VMC E = -2.8556242812 = -2.8442804832 _analytical = -2.8476562500\n", "VMC E = -2.8541169353 = -2.8443064082 _analytical = -2.8476562500\n", "VMC E = -2.8651648182 = -2.8442514779 _analytical = -2.8476562500\n", "VMC E = -2.8157358915 = -2.8441562512 _analytical = -2.8476562500\n", "VMC E = -2.8466200726 = -2.8440540084 _analytical = -2.8476562500\n", "VMC E = -2.8249586205 = -2.8439676933 _analytical = -2.8476562500\n", "VMC E = -2.8307088296 = -2.8440047021 _analytical = -2.8476562500\n", "VMC E = -2.8295121779 = -2.8440189205 _analytical = -2.8476562500\n", "VMC E = -2.8279937441 = -2.8441700894 _analytical = -2.8476562500\n", "VMC E = -2.8600991859 = -2.8440116573 _analytical = -2.8476562500\n", "VMC E = -2.8562944997 = -2.8441934900 _analytical = -2.8476562500\n", "VMC E = -2.8570683782 = -2.8441737078 _analytical = -2.8476562500\n", "VMC E = -2.8353658893 = -2.8444490239 _analytical = -2.8476562500\n", "VMC E = -2.8430467982 = -2.8444777906 _analytical = -2.8476562500\n", "VMC E = -2.8466304851 = -2.8444355567 _analytical = -2.8476562500\n", "VMC E = -2.7965263275 = -2.8443478344 _analytical = -2.8476562500\n", "VMC E = -2.8278013069 = -2.8440473823 _analytical = -2.8476562500\n", "VMC E = -2.8167102740 = -2.8439442576 _analytical = -2.8476562500\n", "VMC E = -2.8190484169 = -2.8437254497 _analytical = -2.8476562500\n", "VMC E = -2.8552143998 = -2.8437819565 _analytical = -2.8476562500\n", "VMC E = -2.9107176846 = -2.8440288487 _analytical = -2.8476562500\n", "VMC E = -2.8393354681 = -2.8444142374 _analytical = -2.8476562500\n", "VMC E = -2.8466594057 = -2.8441969817 _analytical = -2.8476562500\n", "VMC E = -2.7921117695 = -2.8441927376 _analytical = -2.8476562500\n", "ET = -2.8441927376249696\n", "DMC -99 E = -2.8407078647 = -2.8350264489 ET = -2.8348262487 _exact = -2.9037243770 998 Walkers \n", "DMC -98 E = -2.8727375790 = -2.8538820140 ET = -2.8536818137 _exact = -2.9037243770 998 Walkers \n", "DMC -97 E = -2.8610732067 = -2.8524566691 ET = -2.8518548619 _exact = -2.9037243770 994 Walkers \n", "DMC -96 E = -2.8360773275 = -2.8490751329 ET = -2.8485738787 _exact = -2.9037243770 995 Walkers \n", "DMC -95 E = -2.8448088588 = -2.8499373407 ET = -2.8497371404 _exact = -2.9037243770 998 Walkers \n", "DMC -94 E = -2.8666104491 = -2.8536736438 ET = -2.8536736438 _exact = -2.9037243770 1000 Walkers \n", "DMC -93 E = -2.9078523534 = -2.8601672370 ET = -2.8598667861 _exact = -2.9037243770 997 Walkers \n", "DMC -92 E = -2.8869310855 = -2.8635127181 ET = -2.8632122672 _exact = -2.9037243770 997 Walkers \n", "DMC -91 E = -2.8194969730 = -2.8595647400 ET = -2.8595647400 _exact = -2.9037243770 1000 Walkers \n", "DMC -90 E = -2.8527417197 = -2.8585971638 ET = -2.8584971138 _exact = -2.9037243770 999 Walkers \n", "DMC -89 E = -2.8252775959 = -2.8547968106 ET = -2.8543960085 _exact = -2.9037243770 996 Walkers \n", "DMC -88 E = -2.8302640151 = -2.8541732263 ET = -2.8543730266 _exact = -2.9037243770 1002 Walkers \n", "DMC -87 E = -2.8382579486 = -2.8527310824 ET = -2.8528310324 _exact = -2.9037243770 1001 Walkers \n", "DMC -86 E = -2.8668131200 = -2.8529186710 ET = -2.8526182201 _exact = -2.9037243770 997 Walkers \n", "DMC -85 E = -2.8712498233 = -2.8537567626 ET = -2.8532555084 _exact = -2.9037243770 995 Walkers \n", "DMC -84 E = -2.8838255391 = -2.8558172060 ET = -2.8554164038 _exact = -2.9037243770 996 Walkers \n", "DMC -83 E = -2.9032387685 = -2.8582637804 ET = -2.8576619732 _exact = -2.9037243770 994 Walkers \n", "DMC -82 E = -2.8873519531 = -2.8595570366 ET = -2.8587538194 _exact = -2.9037243770 992 Walkers \n", "DMC -81 E = -2.8734456195 = -2.8605929219 ET = -2.8599911146 _exact = -2.9037243770 994 Walkers \n", "DMC -80 E = -2.8902179177 = -2.8619287885 ET = -2.8612263270 _exact = -2.9037243770 993 Walkers \n", "DMC -79 E = -2.9018985356 = -2.8639712695 ET = -2.8633694623 _exact = -2.9037243770 994 Walkers \n", "DMC -78 E = -2.9151012138 = -2.8664286623 ET = -2.8659274081 _exact = -2.9037243770 995 Walkers \n", "DMC -77 E = -2.9243344078 = -2.8688185195 ET = -2.8682167123 _exact = -2.9037243770 994 Walkers \n", "DMC -76 E = -2.9569661713 = -2.8727392396 ET = -2.8723384375 _exact = -2.9037243770 996 Walkers \n", "DMC -75 E = -2.9170620471 = -2.8747464541 ET = -2.8745462538 _exact = -2.9037243770 998 Walkers \n", "DMC -74 E = -2.9214333957 = -2.8768798697 ET = -2.8769798197 _exact = -2.9037243770 1001 Walkers \n", "DMC -73 E = -2.9076992327 = -2.8781289125 ET = -2.8783287128 _exact = -2.9037243770 1002 Walkers \n", "DMC -72 E = -2.8896335220 = -2.8787457819 ET = -2.8791449840 _exact = -2.9037243770 1004 Walkers \n", "DMC -71 E = -2.8380382833 = -2.8773420750 ET = -2.8777412772 _exact = -2.9037243770 1004 Walkers \n", "DMC -70 E = -2.9235614510 = -2.8788827209 ET = -2.8792819230 _exact = -2.9037243770 1004 Walkers \n", "DMC -69 E = -2.9323953127 = -2.8807031501 ET = -2.8812019042 _exact = -2.9037243770 1005 Walkers \n", "DMC -68 E = -2.9502815097 = -2.8827857362 ET = -2.8831849383 _exact = -2.9037243770 1004 Walkers \n", "DMC -67 E = -2.9264063955 = -2.8841959000 ET = -2.8846946542 _exact = -2.9037243770 1005 Walkers \n", "DMC -66 E = -2.9076057617 = -2.8849695177 ET = -2.8855677249 _exact = -2.9037243770 1006 Walkers \n", "DMC -65 E = -2.9033774088 = -2.8854954575 ET = -2.8860936646 _exact = -2.9037243770 1006 Walkers \n", "DMC -64 E = -2.8618763581 = -2.8851554611 ET = -2.8861504942 _exact = -2.9037243770 1010 Walkers \n", "DMC -63 E = -2.8580809099 = -2.8848061196 ET = -2.8862949808 _exact = -2.9037243770 1015 Walkers \n", "DMC -62 E = -2.8843878387 = -2.8846455458 ET = -2.8859371683 _exact = -2.9037243770 1013 Walkers \n", "DMC -61 E = -2.8863274137 = -2.8846886706 ET = -2.8859802932 _exact = -2.9037243770 1013 Walkers \n", "DMC -60 E = -2.9012104888 = -2.8851733156 ET = -2.8865636061 _exact = -2.9037243770 1014 Walkers \n", "DMC -59 E = -2.8902294741 = -2.8855051969 ET = -2.8871909086 _exact = -2.9037243770 1017 Walkers \n", "DMC -58 E = -2.8923064386 = -2.8855994179 ET = -2.8871867528 _exact = -2.9037243770 1016 Walkers \n", "DMC -57 E = -2.8960444608 = -2.8858423258 ET = -2.8874296608 _exact = -2.9037243770 1016 Walkers \n", "DMC -56 E = -2.8322005758 = -2.8848766133 ET = -2.8868568760 _exact = -2.9037243770 1020 Walkers \n", "DMC -55 E = -2.8897342286 = -2.8849845603 ET = -2.8869648230 _exact = -2.9037243770 1020 Walkers \n", "DMC -54 E = -2.8595138234 = -2.8843699041 ET = -2.8862520796 _exact = -2.9037243770 1019 Walkers \n", "DMC -53 E = -2.9081093050 = -2.8848749978 ET = -2.8867571732 _exact = -2.9037243770 1019 Walkers \n", "DMC -52 E = -2.8611978746 = -2.8843232275 ET = -2.8861072193 _exact = -2.9037243770 1018 Walkers \n", "DMC -51 E = -2.8795351800 = -2.8844564200 ET = -2.8866325692 _exact = -2.9037243770 1022 Walkers \n", "DMC -50 E = -2.8848938420 = -2.8843522568 ET = -2.8863325195 _exact = -2.9037243770 1020 Walkers \n", "DMC -49 E = -2.8827616600 = -2.8842656522 ET = -2.8861478276 _exact = -2.9037243770 1019 Walkers \n", "DMC -48 E = -2.8651696466 = -2.8838443493 ET = -2.8856283411 _exact = -2.9037243770 1018 Walkers \n", "DMC -47 E = -2.8645622528 = -2.8834274434 ET = -2.8851131551 _exact = -2.9037243770 1017 Walkers \n", "DMC -46 E = -2.8419984298 = -2.8826602395 ET = -2.8843459512 _exact = -2.9037243770 1017 Walkers \n", "DMC -45 E = -2.8957624022 = -2.8827431502 ET = -2.8841334407 _exact = -2.9037243770 1014 Walkers \n", "DMC -44 E = -2.8980407556 = -2.8831183942 ET = -2.8847057291 _exact = -2.9037243770 1016 Walkers \n", "DMC -43 E = -2.9065654779 = -2.8833791785 ET = -2.8846708010 _exact = -2.9037243770 1013 Walkers \n", "DMC -42 E = -2.8815091995 = -2.8833959811 ET = -2.8847862716 _exact = -2.9037243770 1014 Walkers \n", "DMC -41 E = -2.8788436945 = -2.8834631845 ET = -2.8851488962 _exact = -2.9037243770 1017 Walkers \n", "DMC -40 E = -2.8682050736 = -2.8833969001 ET = -2.8854751540 _exact = -2.9037243770 1021 Walkers \n", "DMC -39 E = -2.8584302303 = -2.8828958190 ET = -2.8847779944 _exact = -2.9037243770 1019 Walkers \n", "DMC -38 E = -2.8650553144 = -2.8826080689 ET = -2.8844902443 _exact = -2.9037243770 1019 Walkers \n", "DMC -37 E = -2.8497595090 = -2.8820866632 ET = -2.8839688386 _exact = -2.9037243770 1019 Walkers \n", "DMC -36 E = -2.8633361984 = -2.8817936872 ET = -2.8836758626 _exact = -2.9037243770 1019 Walkers \n", "DMC -35 E = -2.8393442239 = -2.8812263540 ET = -2.8833046079 _exact = -2.9037243770 1021 Walkers \n", "DMC -34 E = -2.8702544058 = -2.8810601123 ET = -2.8831383663 _exact = -2.9037243770 1021 Walkers \n", "DMC -33 E = -2.8688907392 = -2.8808784799 ET = -2.8829567338 _exact = -2.9037243770 1021 Walkers \n", "DMC -32 E = -2.8739098740 = -2.8807346063 ET = -2.8827148690 _exact = -2.9037243770 1020 Walkers \n", "DMC -31 E = -2.8709205408 = -2.8805515818 ET = -2.8824337572 _exact = -2.9037243770 1019 Walkers \n", "DMC -30 E = -2.8587173477 = -2.8801995868 ET = -2.8819835786 _exact = -2.9037243770 1018 Walkers \n", "DMC -29 E = -2.8650295829 = -2.8799859248 ET = -2.8817699166 _exact = -2.9037243770 1018 Walkers \n", "DMC -28 E = -2.8770915299 = -2.8798279657 ET = -2.8813168269 _exact = -2.9037243770 1015 Walkers \n", "DMC -27 E = -2.8887652669 = -2.8798724198 ET = -2.8811640424 _exact = -2.9037243770 1013 Walkers \n", "DMC -26 E = -2.8937088067 = -2.8799821933 ET = -2.8810761874 _exact = -2.9037243770 1011 Walkers \n", "DMC -25 E = -2.8952173946 = -2.8803380609 ET = -2.8818269222 _exact = -2.9037243770 1015 Walkers \n", "DMC -24 E = -2.8626110205 = -2.8800677011 ET = -2.8814579916 _exact = -2.9037243770 1014 Walkers \n", "DMC -23 E = -2.8939734454 = -2.8802853603 ET = -2.8817742216 _exact = -2.9037243770 1015 Walkers \n", "DMC -22 E = -2.8849496432 = -2.8804180387 ET = -2.8821037504 _exact = -2.9037243770 1017 Walkers \n", "DMC -21 E = -2.9049918533 = -2.8806567849 ET = -2.8821456461 _exact = -2.9037243770 1015 Walkers \n", "DMC -20 E = -2.9230231149 = -2.8811503662 ET = -2.8825406567 _exact = -2.9037243770 1014 Walkers \n", "DMC -19 E = -2.8908277510 = -2.8813050366 ET = -2.8827938978 _exact = -2.9037243770 1015 Walkers \n", "DMC -18 E = -2.9118030914 = -2.8816769641 ET = -2.8831658253 _exact = -2.9037243770 1015 Walkers \n", "DMC -17 E = -2.9411776420 = -2.8824287519 ET = -2.8840160868 _exact = -2.9037243770 1016 Walkers \n", "DMC -16 E = -2.9266318825 = -2.8830921484 ET = -2.8850724111 _exact = -2.9037243770 1020 Walkers \n", "DMC -15 E = -2.9405953711 = -2.8837347400 ET = -2.8856169154 _exact = -2.9037243770 1019 Walkers \n", "DMC -14 E = -2.9028998748 = -2.8839575904 ET = -2.8858397658 _exact = -2.9037243770 1019 Walkers \n", "DMC -13 E = -2.9432295919 = -2.8847052766 ET = -2.8867835305 _exact = -2.9037243770 1021 Walkers \n", "DMC -12 E = -2.8957357671 = -2.8847983938 ET = -2.8867786565 _exact = -2.9037243770 1020 Walkers \n", "DMC -11 E = -2.9121085785 = -2.8851373286 ET = -2.8872155825 _exact = -2.9037243770 1021 Walkers \n", "DMC -10 E = -2.9593989972 = -2.8858980464 ET = -2.8877802219 _exact = -2.9037243770 1019 Walkers \n", "DMC -9 E = -2.8923637759 = -2.8859690984 ET = -2.8878512738 _exact = -2.9037243770 1019 Walkers \n", "DMC -8 E = -2.8711682168 = -2.8857163399 ET = -2.8873036748 _exact = -2.9037243770 1016 Walkers \n", "DMC -7 E = -2.9184505084 = -2.8860683202 ET = -2.8876556552 _exact = -2.9037243770 1016 Walkers \n", "DMC -6 E = -2.9261952415 = -2.8864952024 ET = -2.8880825373 _exact = -2.9037243770 1016 Walkers \n", "DMC -5 E = -2.9050083854 = -2.8867502729 ET = -2.8885342648 _exact = -2.9037243770 1018 Walkers \n", "DMC -4 E = -2.9009683748 = -2.8867796417 ET = -2.8881699323 _exact = -2.9037243770 1014 Walkers \n", "DMC -3 E = -2.8843194034 = -2.8866663043 ET = -2.8877602983 _exact = -2.9037243770 1011 Walkers \n", "DMC -2 E = -2.8836922080 = -2.8866359564 ET = -2.8877299504 _exact = -2.9037243770 1011 Walkers \n", "DMC -1 E = -2.8949998424 = -2.8866336674 ET = -2.8874304843 _exact = -2.9037243770 1008 Walkers \n", "DMC 0 E = -2.8539971255 = -2.8863073019 ET = -2.8871041189 _exact = -2.9037243770 1008 Walkers \n", "THERMALIZATION ENDS\n", "DMC 1000 E -2.8956161693 = -2.8956161693 +/- 0.0000000000 ET = -2.8910236480 _exact = -2.9037243770 956 Walkers \n", "DMC 2000 E -2.8971913616 = -2.8964037654 +/- 0.0007875962 ET = -2.9016003061 _exact = -2.9037243770 1054 Walkers \n", "DMC 3000 E -2.8969027180 = -2.8965700830 +/- 0.0004841805 ET = -2.8936907188 _exact = -2.9037243770 972 Walkers \n", "DMC 4000 E -2.9001893349 = -2.8974748959 +/- 0.0009674202 ET = -2.9006960107 _exact = -2.9037243770 1033 Walkers \n", "DMC 5000 E -2.8921053402 = -2.8964009848 +/- 0.0013095138 ET = -2.8997352725 _exact = -2.9037243770 1034 Walkers \n", "DMC 6000 E -2.8957611892 = -2.8962943522 +/- 0.0010745176 ET = -2.9052710795 _exact = -2.9037243770 1094 Walkers \n", "DMC 7000 E -2.8938412421 = -2.8959439079 +/- 0.0009734048 ET = -2.8958261608 _exact = -2.9037243770 999 Walkers \n", "DMC 8000 E -2.8991620698 = -2.8963461781 +/- 0.0009340552 ET = -2.9040995109 _exact = -2.9037243770 1081 Walkers \n", "DMC 9000 E -2.8995416412 = -2.8967012296 +/- 0.0008970177 ET = -2.9031510590 _exact = -2.9037243770 1067 Walkers \n", "DMC 10000 E -2.8998603123 = -2.8970171379 +/- 0.0008622707 ET = -2.9060570882 _exact = -2.9037243770 1095 Walkers \n", " 3.311359 seconds (109.67 k allocations: 1.510 GiB, 4.56% gc time)\n" ] } ], "source": [ "@time main()" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.12.7", "language": "julia", "name": "julia-1.12" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.12.7" } }, "nbformat": 4, "nbformat_minor": 3 }