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\"Jastrows/H2O/H2O-cc-pcvTz.wfj-1det_J2.ud_exp.dat\"\n", + "os = 1.-7.92535494e-01\n", + "\n", + "multidet = \"Jastrows/H2O/H2O-cc-pcvTz-multidet.wfj_J2.ud_exp.dat\"\n", + "ms = 1.-8.59001137e-01\n", + "\n", + "onedet_H = \"Jastrows/H2O/H2O-cc-pcvTz.wfj-1det_J1.H_exp.dat\"\n", + "ohs = 1.-1.03221118e+00\n", + "onedet_O = \"Jastrows/H2O/H2O-cc-pcvTz.wfj-1det_J1.O_exp.dat\"\n", + "oos = 1.-1.43887500e+00\n", + "\n", + "#onedet = \"Jastrows/N2H4/N2H4-tr6.wfj_J2.ud_exp.dat\"\n", + "#os = 1.-8.03464242e-01\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "set yrange [1:1.4]\n", + "set xrange [0:3]\n", + "set grid\n", + "set xlabel \"r_{12} (a.u.)\"\n", + "set format y \"%.2f\"\n", + "set format x \"%.1f\"\n", + "set key bottom right\n", + "\n", + "plot onedet u 1:($2+os) w l title \"HF\"\\\n", + ", multidet u 1:($2+ms) w l title \"FCI\"\n", + "\n", + "set yrange [0:1]\n", + "plot onedet_O u 1:($2+oos) w l title \"O\"\\\n", + ", onedet_H u 1:($2+ohs) w l title \"H\"\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "f1(x) = exp(a_1 * x/(1. + b_1*x))\n", + "f2(x) = exp(a_2 * x/(1. + b_2*x))\n", + "f3(x) = exp(a_3 * x**2+b_3)\n", + "f4(x) = exp(a_4 * x**2+b_4)\n", + "a_1 = 0.5\n", + "a_2 = 0.5\n", + "a_3 = 0.5\n", + "a_4 = 0.5\n", + "b_1 = 1.\n", + "b_2 = 1.\n", + "b_3 = 1.\n", + "b_4 = 1.\n", + "\n", + "fit f1(x) onedet u 1:($2+os) via b_1\n", + "fit f2(x) multidet u 1:($2+ms) via b_2\n", + "fit [0:1] f3(x) onedet_O u 1:($2+oos) via a_3, b_3\n", + "fit [0:1] f4(x) onedet_H u 1:($2+ohs) via a_4, b_4\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "set xrange [0:3]\n", + "set yrange [1.:1.3]\n", + "set grid\n", + "set xlabel \"r_{12} (a.u.)\"\n", + "set format y \"%.2f\"\n", + "set format x \"%.1f\"\n", + "set key bottom right\n", + "\n", + "plot onedet u 1:($2+os) w l title \"HF\"\\\n", + ", multidet u 1:($2+ms) w l title \"FCI\"\\\n", + ", f1(x) title \"exp(- 0.5 r_{12} / (1. + 1.93 r_{12}))\"\\\n", + ", f2(x) title \"exp(- 0.5 r_{12} / (1. + 3.99 r_{12}))\"\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "set xrange [0:1]\n", + "set yrange [0.6:1.]\n", + "set grid\n", + "set xlabel \"r_{12} (a.u.)\"\n", + "set format y \"%.2f\"\n", + "set format x \"%.1f\"\n", + "set key bottom right\n", + "\n", + "plot onedet_O u 1:($2+oos) w l title \"O\"\\\n", + ", onedet_H u 1:($2+ohs) w l title \"H\"\\\n", + ", f3(x) title \"exp(- 0.5 r_{12} / (1. + 1.93 r_{12}))\"\\\n", + ", f4(x) title \"exp(- 0.5 r_{12} / (1. + 3.99 r_{12}))\"\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "#set term pdf font \"Times,15pt\"\n", + "#set output \"jastrow_h2o.pdf\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Range-separated Coulomb operator" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "w(x) = 1./x\n", + "w_lr(mu,x) = erf(mu*x)/x\n", + "w_sr(mu,x) = w(x) - w_lr(mu,x)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "set xrange [0:4]\n", + "set yrange [0:3]\n", + "set key top right\n", + "plot w(x) title '1/r_{12}',\\\n", + " w_lr(0.5,x) title '{/Symbol m}=0.5' ls 2,\\\n", + " w_sr(0.5,x) notitle ls 2, \\\n", + " w_lr(1.0,x) title '{/Symbol m}=1.0' ls 4,\\\n", + " w_sr(1.0,x) notitle ls 4" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## DMC Energies" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Single-determinant" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "data = \"Jastrows/data_dmc\"\n", + "set key top right\n", + "set xrange [0:7.5]\n", + "set xlabel \"{/Symbol m} (a.u.)\"\n", + "set ylabel \"Energy (a.u.)\"\n", + "set format y \"%.4f\"\n", + "set yrange [-76.4115:-76.4095]\n", + "plot data index 0 u 1:5:6 w errorlines title \"1 det / cc-pVDZ\"\n", + "set yrange [-76.4215:-76.4195]\n", + "plot data index 1 u 1:5:6 w errorlines title \"1 det / cc-pVTZ\"\n", + "set yrange [-76.4232:-76.4212]\n", + "plot data index 2 u 1:5:6 w errorlines title \"1 det / cc-pVQZ\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Multi-determinant" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "data = \"Jastrows/data_dmc\"\n", + "set key top right\n", + "set xrange [0:7.5]\n", + "set xlabel \"{/Symbol m} (a.u.)\"\n", + "set ylabel \"Energy (a.u.)\"\n", + "set format y \"%.3f\"\n", + "set yrange [-76.44:-76.405]\n", + "plot data index 0 u 1:7:8 w errorlines title \"CIPSI / cc-pVDZ\", \\\n", + "data index 1 u 1:7:8 w errorlines title \"CIPSI / cc-pVTZ\", \\\n", + "data index 2 u 1:7:8 w errorlines title \"CIPSI / cc-pVQZ\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Fit $\\mu$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "-----\n", + "$\\Psi(r_1,\\dots,r_N)$ is a CI trial wave function:\n", + "$$\n", + "|\\Psi \\rangle = \\sum_{I \\in \\mathcal{B}} c_I |I\\rangle\n", + "$$\n", + "When running a FN-DMC calculation, the fixed-node wave function can be written as $\\Phi(r_1,\\dots,r_N) = \\Psi(r_1,\\dots,r_N)\\times w(r_1,\\dots,r_N)$, where $w$ is a positive function, such that\n", + "$$\n", + "E = \\min_w \\frac{ \\langle w \\Psi | H | \\Psi \\rangle } {\\langle w \\Psi | \\Psi \\rangle}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We want to find the change in the potential that would model the effect of the FN-DMC. This corresponds to removing some short-range potential:\n", + "\n", + "$$\n", + "\\frac{ \\langle w \\Psi | V_{ee} | w \\Psi \\rangle } {\\langle w \\Psi | w \\Psi \\rangle} = \n", + "\\frac{ \\langle \\Psi | V_{ee} - \\delta V_{ee} | \\Psi \\rangle } {\\langle \\Psi | \\Psi \\rangle} \n", + "$$\n", + "where\n", + "\n", + "$$\n", + "V_{ee} = \\frac{1}{r_{12}} \n", + "$$\n", + "and\n", + "$$\n", + "\\delta V_{ee} = \n", + "\\alpha \\left( \\frac{1}{r_{12}} - \\frac{\\text{erf}( \\mu r_{12})}{r_{12}} \\right).\n", + "$$\n", + "\n", + "\n", + "$$\n", + "\\frac{ \\langle w \\Psi | V_{ee} | w \\Psi \\rangle } {\\langle w \\Psi | w \\Psi \\rangle} = \n", + "\\frac{ \\langle \\Psi | V_{ee} | \\Psi \\rangle } {\\langle \\Psi | \\Psi \\rangle} -\n", + "\\alpha \\frac{ \\langle \\Psi | V_{ee} | \\Psi \\rangle } {\\langle \\Psi | \\Psi \\rangle} +\n", + "\\alpha \\frac{ \\langle \\Psi | V_{ee}^{lr} | \\Psi \\rangle } {\\langle \\Psi | \\Psi \\rangle} \n", + "$$\n", + "\n", + "$$\n", + "\\frac{\\langle \\Psi | w^2 V_{ee} | \\Psi \\rangle}\n", + "{\\langle w \\Psi | w \\Psi \\rangle}\n", + "=\n", + "(1-\\alpha) \\langle \\Psi | V_{ee} | \\Psi \\rangle +\n", + "\\alpha \\langle \\Psi | V_{ee}^{lr} | \\Psi \\rangle \n", + "$$\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "w^2 = \\left[ (1-\\alpha) + \\alpha\\, \\text{erf}(\\mu\\, r_{12}) \\right]\\langle w \\Psi | w \\Psi \\rangle\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To find the parameter $\\mu$, we minimize\n", + "$$\n", + "\\int \\left|a\\, \\text{erf}(\\mu\\,r_{12}) + b - w(r_{12})^2 \\right|^2 \\text{d}r_{12}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "set yrange [1:1.3]\n", + "set xrange [0:3]\n", + "set grid\n", + "set xlabel \"r_{12} (a.u.)\"\n", + "set format y \"%.2f\"\n", + "set format x \"%.1f\"\n", + "set key bottom right\n", + "\n", + "f(x) = (a * (erf(mu*x)) +b)\n", + "mu = 2.\n", + "b=1.\n", + "a = 1.\n", + "fit f(x) multidet u 1:( ($2+ms)**2 ) via a, b, mu\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plot f(x), multidet u 1:( ($2+ms)**2) title \"{/Symbol m}=3.05\" w l" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "set yrange [1:1.7]\n", + "set xrange [0:2.5]\n", + "set grid\n", + "set xlabel \"r_{12} (a.u.)\"\n", + "set format y \"%.2f\"\n", + "set format x \"%.1f\"\n", + "set key bottom right\n", + "\n", + "f(x) = (a * (erf(mu*x)) +b)\n", + "mu = 2.\n", + "b=1.\n", + "a = 1.\n", + "fit f(x) onedet u 1:( ($2+os)**2) via a, b, mu\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plot f(x), onedet u 1:( ($2+os)**2 ) title \"{/Symbol m}=0.94\" w l" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "gnuplot", + "language": "gnuplot", + "name": "gnuplot" + }, + "language_info": { + "codemirror_mode": "Octave", + "file_extension": ".gp", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://metakernel.readthedocs.io/en/latest/source/README.html" + } + ], + "mimetype": "text/x-gnuplot", + "name": "gnuplot" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Data/Jastrows/H2O/H2O-cc-pcvTz-multidet.wfj_J1.H.dat b/Data/Jastrows/H2O/H2O-cc-pcvTz-multidet.wfj_J1.H.dat new file mode 100644 index 0000000..57f2f84 --- /dev/null +++ b/Data/Jastrows/H2O/H2O-cc-pcvTz-multidet.wfj_J1.H.dat @@ -0,0 +1,10000 @@ + 0.00000000e+00 -3.22698400e-02 0.00000000e+00 1.58733692e-01 + 5.00050005e-04 -3.22698202e-02 7.93175797e-05 1.58504900e-01 + 1.00010001e-03 -3.22697607e-02 1.58520752e-04 1.58276109e-01 + 1.50015002e-03 -3.22696617e-02 2.37609518e-04 1.58047318e-01 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