{ "cells": [ { "cell_type": "markdown", "id": "8f4c0199", "metadata": {}, "source": [ "# M1.D3: Schrodinger's hydrogen atom\n", "\n", "## Learning objectives\n", "\n", "* When given the time-independent Schrodinger equation identify the different parts of the Hamiltonian. Identify the eigenfunctions and eigenvalues as the solutions of the Schrodinger equation.\n", "* Describe how the quantum numbers and their rules are necessary\n", "* Plot with matplotlib the energy levels of the hydrogen atom and identify their degeneracy\n", "* Plot the radial distribution function of the electron in hydrogen. Explain how we can use it to tell how far the electron is.\n", "* Distinguish between most probable value and expectation (average) value of the electron.\n", "* Plot 3D wavefunctions and coming up with an isosurface\n", "\n", "Concepts that need to be learned (remembered?) for the hydrogen atom.\n", "* [principal quantum number](https://en.wikipedia.org/wiki/Principal_quantum_number): $n=1,2,3,4...$\n", "* [angular or azimuthal quantum number](https://en.wikipedia.org/wiki/Azimuthal_quantum_number) : $\\ell=0,1,2..n-1$ \n", "* [magnetic quantum number](https://en.wikipedia.org/wiki/Magnetic_quantum_number) $m_l = -\\ell,-\\ell+1,...,-1,0,1,\\ell-1,\\ell$" ] }, { "cell_type": "markdown", "id": "a11c88fc", "metadata": {}, "source": [ "# The Hamiltonian for the Hydrogen Atom\n", "\n", "The first step to figure out a quantum system is to solve the time-independent Schrodinger Equation\n", "\n", "$$\n", "\\hat{H}\\Psi = E\\Psi\n", "$$\n", "\n", "We are not going to solve step by step this equation for the Hydrogen atom (check the references for the mathematical development), rather, we are looking at the main points so that students can achieve the learning objectives.\n", "\n", "What changes for each system is the Hamiltonian $\\hat{H}$. The Hamiltonian operator has the form\n", "\n", "$$\\hat{H} = -\\frac{\\hbar^2}{2m}\\frac{\\delta^2}{\\delta x^2} + V(x)$$\n", "\n", "Unlike the particle in a box, now in the hydrogen atom there is an electrostatic attractive potential between the electron and the proton. Considering their charges and the distance between them. $$ V(r) = \\frac{1}{4\\pi\\epsilon_o}\\frac{q_{e^{-}} q_{p^{+}}}{r}$$ The above equation is in metric units, but if we define an energy unit called Hartree (not metric) we can make $4\\pi\\epsilon_o = 1$ and then we have $\\frac{q_{e^{-}} q_{p^{+}}}{r}$\n", "\n", "The Hamiltonian is \n", "$$\\hat{H} = -\\frac{\\hbar^2}{2\\mu}\\frac{\\delta^2}{\\delta (x,y,z)^2} + \\frac{q_{e^{-}} q_{p^{+}}}{r} $$\n", "\n", "where $\\mu$ is reduced mass for the proton-electron system $\\mu = \\frac{m_e \\cdot m_p}{m_e + m_p}$. So the Schrodinger equation is\n", "\n", "$$-\\frac{\\hbar^2}{2\\mu}\\frac{\\delta^2\\Psi}{\\delta (x,y,z)^2} + \\frac{q_{e^{-}} q_{p^{+}}}{r}\\Psi = E\\Psi $$\n", "\n", "Commonly, the first derivative with respect to a multidimensional variable is a vector and it is indicated with the symbol $\\nabla$. So for the first derivative of any function we write what is called the gradient\n", "\n", "$$\n", "\\nabla f = \\frac{\\delta f}{\\delta x} + \\frac{\\delta f}{\\delta y} + \\frac{\\delta f}{\\delta z}\n", "$$\n", "\n", "and for the second derivative we actually have more than 3 elements because of all cross-derivatives\n", "\n", "$$\n", "\\nabla^2 f = \\frac{\\delta^2 f}{\\delta x^2} + \\frac{\\delta^2 f}{\\delta y \\delta x} + \\frac{\\delta^2 f}{\\delta z \\delta x} + \\frac{\\delta^2 f}{\\delta y^2}... \n", "$$\n", "\n", "Using this notation the Schrodinger equation is\n", "\n", "$$ -\\frac{\\hbar^2}{2\\mu}\\nabla^2 \\Psi + \\frac{q_{e^{-}} q_{p^{+}}}{r}\\Psi = E\\Psi$$" ] }, { "cell_type": "markdown", "id": "6453fdde", "metadata": {}, "source": [ "We won't be spending any time solving that differential equation, but there are few aspects that need to be kept in mind.\n", "\n", "While the potential depends only on the distance between particles \"r\", the derivatives are over the three coordinates of the space (x,y,z). This creates a coordinate problem that forces us to convert the x,y,z into three other coordinates called [spherical coordiantes](https://en.wikipedia.org/wiki/Spherical_coordinate_system)($r,\\theta,\\phi$)\n", "\n", "\n", "\n", "The Schrodinger equation in these coordinates may look more complicated \n", "\n", "$$ -\\frac{\\hbar^2}{2\\mu}\\Big [\\nabla^2_r +\\frac{1}{r^2}\\nabla_{\\theta,\\phi}^2 \\Big ]\\Psi(r,\\theta,\\phi) -V(r)\\Psi(r,\\theta,\\phi)=E\\Psi(r,\\theta,\\phi) $$\n", "\n", "but it can actually be solved more easily by splitting it into a radial part ($R(r)$) and an angular part ($Y(\\theta,\\phi)$) such that \n", "\n", "$$ \\Psi(r,\\theta,\\phi) =R(r)Y(\\theta,\\phi) $$\n", "\n", "So we can rewrite the overall Schrodinger equation for the Hydrogen atom as:\n", "\n", "$$ Y(\\theta,\\phi)\\Big [-\\frac{\\hbar^2}{2\\mu}\\nabla^2_r -V(r)-E\\Big ]R(r) - R(r) \\frac{\\hbar^2}{2\\mu r^2}\\nabla_{\\theta,\\phi}^2 Y(\\theta,\\phi) = 0 $$\n" ] }, { "cell_type": "markdown", "id": "75a92677", "metadata": {}, "source": [ "# Solutions to the Schrodinger equation for hydrogen: the electronic orbitals\n", "\n", "The separation of coordinates in our previous equation will lead into a wavefunction with a radial and an angular solution.\n", "\n", "$$ \\Psi(r,\\theta,\\phi) =R(r)Y(\\theta,\\phi) $$\n", "\n", "To translate what is above into English, we mean that the hydrogen wavefunctions (orbitals) will have two parts, a radial part that dictates the size of the orbital (1s,2s,3s...) or how far the electron is from the nucleus (r), and an angular part that \n", "dictates the shape of the orbital (s,p,d,f...)\n", "\n", "$$ Orbital = RadialSize(r) \\times AngularShape(\\theta,\\phi) $$\n", "\n", "* The wavefunction for the radial part $R_{nl}(r)$ contained what is called the [Laguerre polynomials](https://en.wikipedia.org/wiki/Laguerre_polynomials) $L^{2l+1}_{n-l-1}$\n", "\n", "$$ R_{nl}(r) = \\sqrt{\\Big(\\frac{2}{n a_0}\\Big)^3 \\frac{(n-l-1)!}{2n (n+l)!}} e^{-r/n a_0} \\Big( \\frac{2r}{na_0}\\Big)^l \\cdot L^{2l+1}_{n-l-1} \\Big(\\frac{2r}{n a_0} \\Big)$$\n", "\n", "* The wavefunction for the angular part $Y_{lm}(\\phi,\\theta)$ are called the [Spherical Harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) and they contain the also famous Legendre Polynomials $P_{lm}(cos \\phi)$\n", "\n", "$$\n", "Y_{lm}(\\phi,\\theta) = \\sqrt{\\frac{2l+1}{4\\pi} \\frac{(l-m)!}{(l+m)!} } P_{lm}(cos \\phi) \\cdot e^{im\\theta}\n", "$$\n", "\n", "the energy of the orbitals (the eigenvalues) unsurprisingly looks like Bohr's hydrogen atom, just that this time we\n", "obtained it by using the Schrodinger equation.\n", "\n", "$$ E_n = -\\frac{\\mu e^4}{8 \\epsilon_0^2 h^2 n^2} $$\n", "\n", "You can see here the actual functions depending on the values of n l and m\n", "\n", "http://chemdata.umr.umn.edu/chem2331/orbitals/h.html\n", "\n", "http://chemdata.r.umn.edu/chem2331/rdf/" ] }, { "cell_type": "markdown", "id": "bccb08e7", "metadata": { "tags": [ "Exercise" ] }, "source": [ "* Looking at the radial function what values of n and l will write a negative value inside the square root and therefore invalid?\n", "* Looking at the angular function, what values of l and m will write a negative value inside the square root and therefore invalid?\n", "* Write the overall rules of n, l, and m based on your previous answers.\n" ] }, { "cell_type": "markdown", "id": "ac4e7aa7", "metadata": {}, "source": [ "# The energy levels in the hydrogen atom\n", "\n", "Given the energy from the Schrodinger equation\n", "$$ E_n = -\\frac{\\mu e^4}{8 \\epsilon_0^2 h^2 n^2} $$\n", "If we use SI units for all those magnitudes \n", "* $\\mu = \\frac{m_e \\cdot m_p}{m_e + m_p}\\approx m_e = 9.11x10^{-31} kg$\n", "* $e^- = 1.6x10^{-19} C$\n", "* $\\epsilon_0 = 8.85x10^{-12} F m^{-1}$\n", "* $h = 6.63x10{-34} J s$\n", "\n", "$$\n", "E_n = -\\frac{2.17x10^{-18} J}{n^2} \\; or \\;\n", "E_n = -\\frac{13.6 eV}{n^2}\n", "$$\n", "\n", "What we also know is the degeneracy" ] }, { "cell_type": "code", "execution_count": 1, "id": "52255b47", "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#Lets plot the energy levels applying the quantum number rules\n", "import matplotlib.pyplot as plt\n", "fig = plt.figure(figsize=(15,10))\n", "ax = plt.subplot()\n", "\n", "for n in range(1,5):\n", " ene = -13.6/n**2\n", " delta = 0\n", " for l in range(n):\n", " for m in range(-l,l+1):\n", " #print(\"n=\",n,\" l=\",l,\"ml\",m)\n", " ax.hlines(ene,xmin=delta,xmax=delta+0.5)\n", " delta = delta + 1\n", "ax.set_title(\"Energy of Hydrogen orbitals including degeneracy\")\n", "ax.set_ylabel(\"Energy (eV)\");" ] }, { "cell_type": "markdown", "id": "2c517414", "metadata": { "tags": [ "Exercise" ] }, "source": [ "* Look at the code above. What is the value \"delta\" used for?\n", "* What is the degeneracy level for n=1, n=2, n=3, and n=4? Can you come up with a rule that gives us the degeneracy at a given level \"n\"?" ] }, { "cell_type": "markdown", "id": "c191f7b7", "metadata": {}, "source": [ "# The radial solution: the size of the atom\n", "\n", "If we go back to the the radial part of the wavefunction ($R_{nl}(r)$), we see that the Laguerre polynomials $L^{2l+1}_{n-l-1}(\\frac{2r}{n a_0})$ are what makes the wavefunction different at different levels. These polynomials are so well known and ubiquitous that are included in the *scipy* python package. \n", "\n", "$$ R_{nl}(r) = \\sqrt{\\Big(\\frac{2}{n a_0}\\Big)^3 \\frac{(n-l-1)!}{2n (n+l)!}} e^{-r/n a_0} \\Big( \\frac{2r}{na_0}\\Big)^l \\cdot L^{2l+1}_{n-l-1} \\Big(\\frac{2r}{n a_0} \\Big)$$\n", "\n", "We will use scipy to plot the radial wavefunction of hydrogen. We will use atomic units: $a_0=1,\\hbar=1,m_{e^-}=1,q_{e^-}=1$" ] }, { "cell_type": "code", "execution_count": 5, "id": "8c8a5f2c", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#Lets plot the radial function\n", "import numpy as np\n", "import scipy.special as spe\n", "def psi_R(r,n,l):\n", " coeff = np.sqrt((2.0/n)**3 * spe.factorial(n-l-1) /(2.0*n*spe.factorial(n+l)))\n", " laguerre = spe.assoc_laguerre(2.0*r/n,n-l-1,2*l+1)\n", " return coeff * np.exp(-r/n) * (2.0*r/n)**l * laguerre\n", "\n", "ax1 = plt.subplot(111) \n", "r = np.linspace(0,10,100)\n", "n=2\n", "l=1 #2p\n", "R = psi_R(r,n,l)\n", "ax1.plot(r,R,lw=2)\n", "ax1.set_title(\"Radial wavefunction for n=\"+str(n)+\" and l=\"+str(l))\n", "ax1.set_xlabel(\"Distance to nucleus in bohrs $a_{0}$\");\n", "ax1.set_ylabel(\"Radial wavefunction $R_{nl}(r)$\");" ] }, { "cell_type": "markdown", "id": "816e3b6a", "metadata": { "tags": [ "Exercise" ] }, "source": [ "* Spend some time understanding every line in the code above. What does the function psi_R do?\n", "* What variables decide the orbital that we want to plot?\n", "* What is the variable \"r\" and what is linspace for?" ] }, { "cell_type": "code", "execution_count": 6, "id": "cbd48a04", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=(12,7))\n", "ax1 = plt.subplot(121) #121 means: the subplots will be 1 row 2 columns and this one is subplot number 1\n", "ax2 = plt.subplot(122, sharey = ax1) #share-y means that it will share the y-axis with the ax1 plot\n", "\n", "r = np.linspace(0,10,100)\n", "n=3\n", "l=2 #3d\n", "R = psi_R(r,n,l)\n", "ax1.plot(r,R,lw=2)\n", "ax1.set_title(\"Radial wavefunction for n=\"+str(n)+\" and l=\"+str(l))\n", "\n", "n=3\n", "l=1 #3p\n", "R = psi_R(r,n=3,l=1)\n", "ax2.plot(r,R,lw=2);\n", "ax2.set_title(\"Radial wavefunction for n=\"+str(n)+\" and l=\"+str(l));" ] }, { "cell_type": "markdown", "id": "0b30e781", "metadata": {}, "source": [ "## Calculate the size of atoms\n", "\n", "Remember that the wavefunction plot above does not carry any physical meaning. The square of the wavefunction however gives the **probability** or **probabiilty density**.\n", "\n", "$$\n", "\\text{Probability density}(r,\\theta,\\phi) = |\\Psi(r)\\cdot\\Psi(\\theta,\\phi)|^2\n", "$$\n", "\n", "Because we are interested in knowing how far from the nucleus the electron is, we could just look at the square of the radial wavefunction.\n", "\n", "$$\n", "\\text{Radial probability density}(r)=|\\Psi(r)|^2\n", "$$\n", "\n", "Another magnitude more useful other than the probability is what's called the **radial distribution function** where we just consider the probability as a function of r and integrate over all values of $\\theta$ and $\\phi$. \n", "\n", "\n", "$$\n", " \\int_0^\\infty\\int_0^\\pi \\int_0^{2\\pi} |\\Psi(r)|^2 r^2 sin(\\theta)dr d\\theta d\\phi \n", "$$\n", "So, integrating for $\\theta$ and $\\phi$ we have the radial distribution function which for the hydrogen ground state is obtained by multiplying the square of the wavefunction by a spherical shell volume element ($4\\pi r^2$).\n", "\n", "$$\n", "\\text{Radial distribution function}(r) = |\\Psi(r)|^2 \\cdot 4\\pi r^2 \n", "$$\n", "We will use atomic units $a_{0}=1,\\hbar=1,me=1,e=1$\n", "\n", "Notice that because of the units the probability is not normalized (the values do not integrate to one)" ] }, { "cell_type": "code", "execution_count": 7, "id": "103412b1", "metadata": {}, "outputs": [ { "data": { "image/png": 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3McYES/skoapvqWqFqs5Q1ctVNe2/Ffc3tnLQna21f0Fu4Es9EUSEyinDAtsvrt+XsHMbY0ywtE8SmWiDp6ppUnlpXCOtQ6mcMjzwvHr93l6ONMaY+FiSSIIeVU3lie9dceaEIRS4XWrf29PA7rqmhF/DGGPAkkRSeEsSkxPYaN2luDCvxwjul6zKyRiTJJYkksCbJBLZaO3Vs13CqpyMMclhSSLBVJUNNY2B7UR2f/Xytku8sukAre3WFdYYk3iWJBJsV10zjS3tAJT1z2dYaWFSrjNhWDGjBzm9phpb2nlje21SrmOMyW6WJBJsw56e7REiie3Z1EVEqPKUJqxdwhiTDJYkEixZI61D8bZLVFuSMMYkgSWJBOtRkkhSo3WXMzxdYdfXNLDX1r42xiSYJYkES2VJon9BHqeMLQtsL920P6nXM8ZkH0sSCaSqbD9wJLA9fljyl744e1J3ldPSjZYkjDGJZUkigQ4daQv0bOqXn8uQ4oKkX/OsiUMDz5du2k86rw9ijMk8liQSaMfB7lLEmMH9ktazyWv6qIEM7JcPwN6Glh5jNIwxJl6WJBJoR233HEpjBvVPyTVzc4R5E4cEtl/eaL2cjDGJY0kigXqWJFKTJADOmuhpl7DGa2NMAlmSSKCdniTRNRo6Fc6e1N0u8dqWA7S0d6Ts2saYvs2SRAL1qG5KYUlizOD+jB3iXK+5rZOVtlqdMSZBLEkkUI/qphS1SXTp0cvJusIaYxLEkkSCdHYqOw92lyRGD05ddRP0rHKydgljTKLkxfImcfp2XgBcCJwDHA8MBZqAvcBbwAvAE6r6QUIiTXP7GlsC03UP7JfPgKL8lF7/jAlDyRHoVHjngzrqjrQxsH9qYzDG9D1RlSREpL+IXA9sAZ4G/gU4DSjDSQ7twHjgn4Dbga0i8oiInBlrgCKyTUTeEZG3ROSNWM+TbDtqe46RSLWB/fI5eXQZAKqwbMuBlMdgjOl7Ik4SInI1sBH4NU6J4SackkSZqvZX1dGqOgSndDIN+CLwCHAx8LKIPCgix8cYZ5WqzlLVihjfn3R+tkd0mTehe7zEK1blZIxJgGhKEncCy4HTVXWaqv5cVRerar33IHW8p6p3q+qVwAjg28BZwBcSFHfa2elTzyYvb+P1K5stSRhj4hdNkqhQ1Y+r6goAETlRIph3QlXrVfU/gQnAQzHEqMCzIrJSRBbE8P6U6FmSSH11E8ApYwdRmOf8SLfsO8zuuqZjvMMYY3onsU4IJyKdwL2q+rnEhnTUdY5T1V0iMhx4Dvimqi4JOmYBsACgvLx8zqJFi2K6VmNjIyUlJTG9999fb2JdrdNw/d05hcwYFlOfgLj9ZkUTaw84cXz55ALOGtV743U895yp7J6zg91zdKqqqlaGqtKP55vsILAjjvdHRFV3uf/uFZHHcBrKlwQdsxBYCFBRUaGVlZUxXau6uppY3/uT5S/gNNXAJefOZeJwf34517KJtU+vB6A2bxiVlbN6PT6ee85Uds/Zwe45MeIZJ/EyMDVRgYQiIsUiUtr1HLgIeDeZ14xFe0cnu+u6V4VL5ZQcweZN6NkuYVOHG2PiEU+S+BVwqYicmqhgQigHlorIauB14B+q+nQSrxeT3XXNdHQ6X8bDSgspys/1LZbpowYyoMgpINbUt7B5n00dboyJXTxJ4hM4A+aeF5HPJyieHlR1i6rOdB8nqeqvknGdePUYI+FjKQKcqcPP6NEV1sZLGGNiF0+S+D7wIaAUuEtEdovIHSJyjYjMFhF/Wm594J2Ow6/ur17zvF1hbbyEMSYO8XyRnwec4nlMxhlAd7X7equIvAO8oapfiyvKNJcOA+m8vEli2ZYDtHd0kpdr03QZY6IXc5JQ1WqgumtbRPoDM+mZOGYCc4C+nSR8npIj2PihxYwYUMSe+mYamtt5d1c9s8aU+R2WMSYDJezPS1U9oqrLVPV2Vf2Sqs7GqYpKZsN2WthxMPXLlvZGRKzKyRiTEEmtg1DVVlVdlcxrpIOdPi1b2hvvuteWJIwxsYpmgr+461EScY5009bRyd6GFgBEYMTAIp8jcnhLEm9sP0hzmy1paoyJXjQlia0icp2IFEZ7ERGZKSKP40wt3qfsa2iha7zakOJC8tOkgbh8QFFg1HdreydvbLMlTY0x0YvmG+1Z4BZgt4j8UUSqeisZiMh4EblWRJYBq3AasV+ML9z0s6e+e6R1+YCo82dSeacOt9XqjDGxiLh3k6p+TkR+j7OeRNeEeh0isg7YjTOXUxEwBJiCs1KdADXAj4H/p6otiQ3ff3s9SWLEgPSoauoyb+JQ7lm2HYBXbepwY0wMouoCq6pvABeJyCTgS8D5wCzg5KBD9wGP4iw69IiqtsUfanra45mzqTxN2iO6nD5+SI8lTQ8daaWsf4HfYRljMkhM4yRUdSNwPQTGR4zCKUE0AXtVdXfCIkxze+q7C0fpVpIY2C+fGaPLeGvHIVThtS0HmD99pN9hGWMySNytrO74iI2q+pqqrs6mBAFQk8ZtEtCzK6y1SxhjopXwrjgi8oyI/LuIXCEiUyNZvS6T9UwS6VWSgOB5nGyyP2NMdJIxCd+bOO0Un8OZ6vuIO4fTW+7jTeBtVW0O8/6M4u3dlC5jJLxOOd5Z0rSlvZOt+w/zwaEmRpX1ueEqxpgkSXhJQlWvV9X5qjoSGIkzpfjjwCDgO8CrQIOIrE30tf1QU5e+vZsAivJzOe2EwYHtpRv3+RiNMSbTJHtajhpVfVpV/01Vr1DVqTjzOZ0D3JbMa6dCQ3Mbh1udkcwFeTkM7Nf7etJ+OctT5fTyRmuXMMZELqIkISJlInKpiJwZ3MbgLjF6Q6QXVNUmdyLAP0UbbLqpCRojka7NL2dPGhZ4/urmA3R22pKmxpjIHDNJiMhJwDqcKqOlwAoRGes5pAS4Mcx7zxWRlSLyvoi8ICK3iMjnRGRGX1iUqCaNu796TR1RytASZ3xE7eFW1u6u9zkiY0ymiKQkcTOwDBiIMx5iC/CKO6DuWO4EdgA/ApYAE4Bf4jRgZ/ziy+k8kM4rJ6fn1OFW5WSMiVQkf83PBapU9TBwGPiUiNwCVItIFVDXy3tHABep6hbvThEZDMyOMea00WPeptL0GyPhddbEoTz+1i4Alm7ax7WVE3yOyBiTCSIpSRQCPSqxVfW7wEPAS8CJvbz3ZWBc8E5VrVXVxZGHmZ5q0rz7q5e3XWLFNps63BgTmUiSxHqgIninqn4HJ1E83st7/wj8VESG9XLMMYlIroi8KSJPxnOeREv3gXReIwb2nDr89a21PkdkjMkEkSSJx4ArQ72gqtcB9+LM9hrK34BzgQ0icq+IfN3tIRXt8m3X4TSep5Ue8zaleUkCenaFtSk6jDGROGaSUNWbVfXiXl7/uqqGO88Y4MPAb4EC4Fs4Ddh1kQ6mE5HRwKXAHZEcn0regXTlpemfJM6eZI3XxpjoiGpq+8y7pYiZwIxIxkqIyMM4PaxKgX9R1ctCHNO1vgXl5eVzFi1aFFNsjY2NlJSURHRspypffvYIXUMOFl7Yn4Lc9Bwn0aWpXfnG4iN0uDHfVtWf3LbDEd9zXxHNz7mvsHvODvHcc1VV1UpVPappIeVjFVT1CE6X2mXHOlZELsOZenyliFT2cs6FwEKAiooKrawMe2ivqqurifS9NfXNdD7jtL0P6p/PRedXxXTNVJuzeVmgPaJj2CQG1m+K+J77imh+zn2F3XN2SMY9p8eCzOHNAz4iItuARcB5InKvvyE5MqnR2uvcyd19CJZssHmcjDG9izpJiMh1IrJFRBpF5HUR+XiY40a4a1w/F2twqvpDVR2tquOAK4AXVPUzsZ4vkXoMpMvUJLFxP50prm40xmSWqKqbROQTwP/z7KoA/ioi16rqQhEZAHwWpzfUXML3esp4wfM2ZYppIwcwtKSA/Y2t1B5uZXt9uhcmjTF+ivYb4ltAO3AtcAJwEfAO8BsRuQjYCvweOBOoBx4APpWIQFW1OlSjtV96jLbOgO6vXXJyhHM8A+ve2W+D6owx4UWbJKYAf1PVP6nqdlV9HjgfZ0T2wzhrRjwMfAgYrqqfUdVHEhpxmsiUyf1COXeKJ0nssyRhjAkv2iQxDHjPu0NVDwBPAMXAd1T106r6nKq2JyjGtNRzSo70nrcp2FkTh9I1q/nmuk7qmtr8DcgYk7ZiqZAO9eW/3f33njhiySjehuvhGTCQzmtISSEnjxoIQKfCqzb62hgTRqJaLTsAVPVQgs6X9tJ9betj8fZyesm6whpjwoglSdwgImtF5C9ud9izcBYeyhpNrR00NDsFqrwcYXD/Ap8jil5wkkj1yHtjTGaIdsT1Ypx1IKa6j3/2vigiC4FVwEpgtaq2JiLIdLO/sbvRelhpITk5mdfTd9aYMkqL8mhobmd3XTMbahqZMqLU77CMMWkmqpKEql6oqkOBiTiD2/4DZ02JepwxEV8GbgdeAxpEZJWbOPqUvQ3dSWJoSWY1WnfJy83p0RX2hff2+hiNMSZdxdQmoapbVPUhVf2Bqp6nqoNwShafAW4DXgFagFnAlxIVbLrY19CzJJGpqqYODzx/4b0aHyMxxqSrhE3wp6obgA3A/QAiIjir1s1J1DXSRY/qpgwtSQBUThmG4AxyWbn9IAcPtzKoOPPaV4wxyZO0ORnUsVZV/5Ksa/ilr5QkhpYUMn6g8yvQqbBko/VyMsb0ZBP3xGBfY99IEgAzh+cGni9eZ+0SxpieLEnEYF8faLjuMnNYd5KoXr+X9o5OH6MxxqQbSxIxCO4Cm8mOL80JzD1V39zOqvcP+RuQMSatWJKIQV9pkwAQEc47sbuX02Lr5WSM8Yg7SYhIoYjMcB99fuS1qvapJAFw3hRPV1hrlzDGeCSiJDEVeAtnpPU5CThfWmtoaael3am3L8rPobgg9xjvSH/zJg6lMM/5Vdi4t5H3DxzxOSJjTLpIRJJoBd4HdgB9/ttlf1ApQiTzpuQI1q8glzMnDAlsP7fOqpyMMY64k4SqrlPVcap6gqpWJyCmtNajqinDezZ5XXTSiMDzZ9bs8TESY0w6sYbrKPWlMRJeF5xYHliI6I1ttRzw3KcxJntZkohSX2u07jKstJA5xw8CnNHXNrDOGANpniREpEhEXheR1SKyRkRu8jsm7xiJTB9IF+xDVuVkjAmS1kkCZybZ81R1Js6MsvNFZK6fAfXVkgTARSeVB56/vGk/jS19eplyY0wE0jpJuJMENrqb+e7D1yXU+mrDNcDYIcVMdRceam3vZIkta2pM1pN0X7ZSRHJxVrqbCNyuqv8a4pgFwAKA8vLyOYsWLYrpWo2NjZSU9D4e8MZXm9he74yT+OncIiaUZfY4ieB7fmxjK49vbgNg7shcvjoz89bvPpZIfs59jd1zdojnnquqqlaqakXw/oStJ5EsqtoBzBKRMuAxEZmuqu8GHbMQWAhQUVGhlZWVMV2rurqaY7235dXFQDMAF51zBmMG94/pWuki+J6HTa7j8d8vBWDNQeHMs86hIC+tC5xRi+Tn3NfYPWeHZNxzRP/3i0iZiFwqImdK0OgxESkWkRsSGlUIqnoIqAbmJ/ta4XR2ap+a3C+UaSMHMKqsHwANze28unm/zxEZY/x0zCQhIicB64DHgaXAChEZ6zmkBLgxGcGJyDC3BIGI9AMuAN5LxrUicaipjfZOp3qutCiPovzMrmoKRUSYP727l9M/3t7tYzTGGL9FUpK4GVgGDARGAVuAV0RkUjIDc40EXhSRt4EVwHOq+mQKrhtSX+7Z5HXpjJGB58+s2UNru60xYUy2iqRNYi5QpaqHgcPAp0TkFqBaRKqAumQFp6pvA7OTdf5o9eUxEl6zx5QxqqwfHxxqor65naWb9nHe1PJjv9EY0+dEUpIoJKjbqap+F3gIeAk4MQlxpaVsKUmICJd5ShNPrrYqJ2OyVSRJYj1wVLcoVf0OTqJ4PNFBpau+PEYi2GUzjgs8f3ZtDc1tHT5GY4zxSyRJ4jHgylAvqOp1wL1A5s+XHYG+OrlfKNNHDWDsEKd7b2NLOy/ZwDpjstIxk4Sq3qyqFwfvF5ETRURU9euq2rc60ocRvJZEX3ZUlZP1cjImK8Xz5b4GuCdRgWSCHiWJPl7dBD2rnBavq6Gp1aqcjMk28SSJgzir0WWNbGm47jJ1RCkThhUDcKS1g8Xv2Yp1xmSbeJLEyzjrW2eNbEsSIsKHZ3aXJh5b9YGP0Rhj/BBPkvgVcKmInJqoYNJZe0cntUdaARCBwcUFPkeUGh+bPSrwvHrDvh5jRYwxfV88SeITwAvA8yLy+QTFk7ZqD7fSNWHuoP4F5OdmRVs9Y4cUc+o4Z8W6jk7libd2+RyRMSaV4vmm+z7wIaAUuEtEdovIHSJyjYjMFpG0n2E2GnuzaIxEsI+fMjrw/JFVO32MxBiTavEkifNwEsX9OAPuhgFfBP4beANocJce/UPcUaaBHlNylGZHVVOXS04eGZgufM2uet7bU+9zRMaYVIk5SahqtareoqqfUdVpwABgHvAt4G6c2VpnAl9JRKB+y6bR1sEG9svnomndczc9ag3YxmSNmJOEiMzwbqvqEVVdpqq3q+qXVHU2TlVUn2jYzqbR1qH8k6fK6bE3P6C9w2aGNSYbxFPd9KaI/Li3A1S1VVVXxXGNtLG/oTXwvC/PABvO2ZOGBu57X0MLSzfZYkTGZIN4koQAF4nI1W5Ddf5RB4iMEpHvxnGNtJHtJYm83Bwun9U9ZuLBFVk1jtKYrBVvP86zgDtwGqobRWS1iNwjIt8RkQuATwI/jzfIdOCdtykbSxIAnz51TOD5c2tr2NvQ7GM0xphUiLeb6oPAm8Ap7mM6cDLwWbrXoDgQ5zXSQraXJAAmlZdy6rhBrNh2kPZO5a9v7OTrVRP9DssYk0TxJon3VPW3XRsiUoyzklxXwhiMs+ZExsu2KTnCuer0sazYdhCAB15/n2vPnUBOTlbMFG9MVkrogDd3idOl7qPPaGnvoK6pDYAccUZcZ6v500cw6O/5HDzSxs6DTSzZuI/KKcP9DssYkyTZMbdEnA40dvdsGlJSSG4W/+VclJ/bozvs/cvf9zEaY0yyxZskPiEiPxeRj4vIhIRE5CEiY0TkRRFZJyJrROS6RF8jEj1GW2dpo7XXlacfH3i++L291NRbA7YxfVW8SWI68BPgr8AGEakTkZdF5L9E5MsicqqIFMVx/nbge6p6IjAX+LqITIsz5qhZe0RPE4aVMHf8YMCZ9M9KE8b0XfEmiT8AHwd+AfwdZyGiecDXgD8BrwExT/Sjqru7BuOpagOwDhjV+7sSb1+P7q/Z2x7h9Zm5YwPP71u+neY2W7XOmL5ItGv+62jfKHIJUKeqrwTtLwNmuY/ZwAx3io64iMg4YAkwXVXrg15bACwAKC8vn7No0aKYrtHY2EhJSclR+5/Y3MqjG52G60tOyOdTU/pOogh3z8fS3qn8YEkTtc3O788XpxdwzuijxlOmpVjvOZPZPWeHeO65qqpqpapWBO+PuXeTqv5fmP2HgGr3kRAiUgI8Anw7OEG411wILASoqKjQysrKmK5TXV1NqPe+WPcubNwOwCnTJlJ59viYzp+Owt1zJL6Su5mbn3oPgFf2FfLTq85GJP0b9eO550xl95wdknHPCe/dJCK3iMgnE3i+fJwEcZ+qPpqo80Zjv6d3k7VJdLvitOPpX5ALwPqaBpvPyZg+KBldYL8NXJiIE4nzZ+mdwDpVvSUR54xFNk8T3puB/fL55Jzu7rB3Lt3qYzTGmGSIqrpJRL4Y4aFTvMeq6l1RRdVtHs4UH++IyFvuvh+Fq+pKln09FhyyJOF19bwT+N/XtqMK1ev3sbGmgUnlpX6HZYxJkGjbJO6ge06mcBRn4r+zcGaKVSCmJKGqS91z+Gq/lSTCGje0mAtOLOe5tTUA/PnlLfzmEzN9jsoYkyixNFw34ixReiTEawLcAKzC6RKb8ZrbOmhoaQcgL0cY2C8zevCk0jVnjw8kiUdXfcA3z5vEmMH9fY7KGJMI0SaJzwO/xxkbcbX7l34PInIDsEpVb0pAfL7bFzRFuE1md7TTThjMaScM5vWttbR3Kn98aTO//tjJfodljEmAqBquVfUvwAxgO1AtIr8TkT5d/2JThEfmuvMnBZ7/9Y0d7DrU5GM0xphEibp3k6ruUNULgO/hjKx+U0ROS3hkacJGW0fmzAlDOOX4MgDaOpQ/vbTZ34CMMQkRcxdYVb0NmIPTNvGKiNwcagnTTLffShIRERG+5SlNPLBiB3tt4j9jMl5c4yRUdR1wOnAz8C84q9TFNs9HmgpukzDhnTt5GDNGDwSgtb2TP1RbacKYTBf3YDpV7VDVG3DGNOSTBl1WE8lKEpETEb51Xndp4r7l29lRG6oTnDEmUyRsxLWqvg5MAwYB30rUef1mJYnonH/i8B5tE799Zr2/ARlj4pLQaTncUkWdqvaZymibtyk6IsKPLjkxsP3E6l28s7POx4iMMfGw5UuPwRYcil7FuMFcNK08sH3zU+uIdUp6Y4y/LEkcg1U3xeYH86cG1gJ/dfMBXtqwz+eIjDGxsCTRi8Mt7TS5K64V5OUwoCjm5TeyzsThJVxx6pjA9i//sY7W9k4fIzLGxMKSRC+CpwjPhAV10sl1F0yipNBJrJv2NtpU4sZkIEsSvdhd193+PnyAVTVFa3hpEd+5cHJg+/eLN7LzoHWJNSaTWJLoxZ767vmHjhvYz8dIMtfnzxjL1BHO+hJNbR3c9Pe1PkdkjImGJYleeEsSIwYW+RhJ5srLzeFXH5se2H5ubQ3Pu9OKG2PSnyWJXuw+1J0kRlqSiNmcsYP5dEV3I/ZPH3+XuqY2HyMyxkTKkkQvvCWJkVbdFJd/vXgqg4udWXR31zVz09/X+ByRMSYSliR6sbuuu01iZJmVJOIxuLiAX13eXe306KoPeGbNHh8jMsZEwpJEL/bUWXVTIl188kgun3VcYPtHj77TYwJFY0z6SfskISJ3icheEXk3lddtbuvgwGFn3qbcHGF4qSWJRLjpI9MZMcD5LA8cbuX6R96xKTuMSWNpnySAu4H5qb5ojWfBnOGlhYEpJkx8BvbP598/MSOw/fy6GhYu2eJjRMaY3qR9klDVJUBtqq+7y3o2Jc25k4dx9bxxge3fPLOe5VsO+BeQMSastE8SfvEOpBtZZj2bEu2HF58YWHeio1P5xgNv2nKnxqQhyYT6YBEZBzypqtPDvL4AWABQXl4+Z9GiRTFdp7GxkZKSEgCe3NzKwxudvvwfGpfHlVP75rQc3ntOtdrmTm58pYkGd8jE5EE5fP/UIvKTXLXn5z37xe45O8Rzz1VVVStVtSJ4f5+Y1lRVFwILASoqKrSysjKm81RXV9P13sWH3oWN2wE4bfpkKs86IRGhph3vPfth+IT9fPau5ajChoOd/L2mjFs/PYucJCYKv+/ZD3bP2SEZ92zVTWH0GCNhbRJJc9akoVw/f2pg+4nVu/jds7bkqTHpIu2ThIg8ACwDpojIThH5Uiquu9vGSKTMgnPGc9Xpxwe2/1C9mfuXv+9jRMaYLmlf3aSqV/px3T02JUfKiAg3feQkdh1q4sX1zgp2P/7bO/QvyOXy2aN8js6Y7Jb2JQk/BA+ks7Wtky8vN4f/+udTmD5qAACq8N2H3uKJ1bt8jsyY7GZJIgTvQLpyG0iXMsWFedxz9WlMKXfWn+hU+M6Db/GPt3f7HJkx2cuSRAg9BtLZGImUGlJSyH3XnM6k4U43vo5O5ZsPrOK+5dt9jsyY7GRJIgTvQDpbbCj1hpYUcv81c5kwrBhwShQ/fuxdbnl2vc3zZEyKWZIIwVuSOM6ShC+GlRby4FfOYMbogYF9v39hE99/+G1a2jt8jMyY7GJJIoQ9PZYtteomvwwtKeSBa+ZSOWVYYN/DK3fyqf9exgeHmnp5pzEmUSxJhOAdSGclCX8VF+bx589V8Mk5owP7Vu+s48P/uZSlG/f7GJkx2cGSRAi7e5QkLEn4LT83h998YgY3fngaeW5Ps9rDrXz2ruX88sm1NLdZ9ZMxyWJJIgRvkjjOejelBRHh6nkn8MCCuYFxK6pwx9KtXPafS3l75yF/AzSmj7IkEaS5rYNadyBdXo4wtMQG0qWTU8cN5h/fPItzJne3U2za28jlt7/Cz55YQ11Tm4/RGdP3WJII4m20Lh9QZAPp0tDwAUXcc/Wp/PLy6fQvyAWcbrJ3v7qN835XzUMrdtDRaV1ljUkESxJBNu1tDDy3if3Sl4jwmbljeeq6s5k3cUhg/4HDrfzgkbeZf+sSnnpnt42rMCZOliSCvOZZRnO2u3KaSV9jhxRz75dO5w9XndKjJ9rGvY1ce98qPvxfS3ny7V20d3T6GKUxmSvtZ4FNtde2dieJMyYM6eVIky5EhEtOHknllGH86aUt3Ll0K40t7QC8+0E937j/TUYP6sfV807gE6eMPsbZjDFeVpLwONymrNlVD0COQMW4wT5HZKLRvyCP71w4mSU/qGLBOeMpzOv+9d55sIlfPLmW0379PH96u5llmw/Qae0WxhyTlSQ81td20FWFffKogQwoyvc3IBOTwcUF/OiSE7nm7PH85bXt/GXZNg4ecXo9tbR3smwXLPvza4wYUMQlJ4/kspkjmTW6LKlLphqTqSxJeLxX2z0oa65VNWW8YaWFfPfCyVx77gQefXMn9y9/P1BSBNhT38xdr2zlrle2MrSkkMopwzhv6nDOGD+EQcUFPkZuTPqwJOGxrra7cXPueEsSfUW/glyuOn0sV50+lnc/qOOWv73GmwckULoA2N/YwsMrd/Lwyp2IwNQRAzhj/BDmjB3EKWPLbHVCk7UsSbgOHm5lR4OTJHJzhFOtPaJPmj5qIJ87qZA/nX0OyzYf4Mm3d7F43d7ASoTgjORet7uedbvrueuVrQCUDyjk5FEDmTZyANOOG8Ck8lLGDu5PXq4165m+zZKEa/nW2sDzk0cNpKTQPpq+LD83h3MmD+OcycPo7FRW7zzEi+/tZcnG/bzzQd1Rg/Fq6luoqd/L8+v2BvYV5OYwbmh/ThhazNghxYwd0p/Rg/ozqqwfo8r60c8d6GdMJkv7b0IRmQ/cBuQCd6jqvyXjOt7xEdb1Nbvk5Aizjx/E7OMH8d2LptDY0s6KbbW8vrWWt94/xNs7D3G49ehJBFs7OtlQ08iGmsYQZ4WB/fIpH1BI+YAihpUUMrS0kCHFBQwuLmBQ/wIGFeczsF8BA/rlMbBfPoV5llRM+knrJCEiucDtwIXATmCFiDyhqmsTfa1lmz1JwtojslpJYR5VU4ZTNWU44CyhunlfI2t31bPWrYbatLexx0SQodQ1tVHX1BY2iQQryMuhtDCPkqI8igvyKC7MpX9BHv0LculXkEu//FyK8nMpys+hKC+XwvwcCnJzKMjLpSAvx3nkCvm5OeTl5pCfI+Tl5pCXK2yr62Dtrnryc4XcnKCHCDld/4qQk4Pzr/tcEHLE2SfijEsx2SOtkwRwGrBJVbcAiMgi4KNAQpPEgcYW1tc0AM6kfnPGDkrk6U2Gy80RJpeXMrm8lMtnjwrsb2huY/O+w2w/cJjtB46w7cBhdh1qYtehZnbXNdHWEd04jNb2Tg60t/ZoH0moZS8n7FQ5brIQnOSB85+TRHCTCd3HAIFjoDvRSNA+b/rpmYsk5P7e0lVrayuFrz5/1H7p9V2hrp0cybhEc0sLL81tozSB3ffTPUmMAnZ4tncCpyf6It72iJljyii29ggTgdKifGaNKWPWmLKjXuvsVGqPtFJT30xNfTP7G1rZf7iF/Q2tHDrSSu2RVg4eaaO+yXnUNbXRnkGD+zoVAoOKSOO4W1r8jiDlEv3TSPdvw1DJ9qjPQEQWAAsAysvLqa6ujuoir2xrIz8H2jrhuNzGqN+fyRobs+t+IfX3nAMMdx+Uuo8e8lHNo60TmtqhqV1pbldaOqC5Q2lph9ZOpbUDWjqgrVNp63D2tXc6v7dtnUpHJ7R3Qrs6VWQd6mx3KrS2d4Dk0KnQoc6+wAPoVA1sqzr/kx31PGWfmInH0peX0j8/ceWUdE8SO4Exnu3RwK7gg1R1IbAQoKKiQisrK6O6SCVwQ3sHdz9Rzfxz5zJ2SHGs8Wac6upqov28Mp3dc+xUFVUnqXQVJjrdEoWTUDSQWNQ9puu1ro2uvepJPN5jA8fT83h3I9TTEHHCsmWvcsYZZ/bcH0GqS8XEwcm6xGvLljH//MqEzh6Q7kliBTBJRE4APgCuAP45GRcqzMtlyuDcrEoQxkRL3MbrnKTUqCfWoKKcrFt+eEi/nIRPL5PWSUJV20XkG8AzOF1g71LVNT6HZYwxWSOtkwSAqv4f8H9+x2GMMdnI5hQwxhgTliUJY4wxYVmSMMYYE5YlCWOMMWFZkjDGGBOWaCpGjqSQiOwDtsf49qHA/gSGkwnsnrOD3XN2iOeex6rqsOCdfS5JxENE3lDVCr/jSCW75+xg95wdknHPVt1kjDEmLEsSxhhjwrIk0dNCvwPwgd1zdrB7zg4Jv2drkzDGGBOWlSSMMcaEZUnCJSLzRWS9iGwSkev9jifZRGSMiLwoIutEZI2IXOd3TKkgIrki8qaIPOl3LKkgImUi8rCIvOf+rM/wO6ZkE5HvuL/T74rIAyLS5+YLF5G7RGSviLzr2TdYRJ4TkY3uvwlZh9mSBM4XB3A7cDEwDbhSRKb5G1XStQPfU9UTgbnA17PgngGuA9b5HUQK3QY8rapTgZn08XsXkVHAt4AKVZ2Os8TAFf5GlRR3A/OD9l0PLFbVScBidztuliQcpwGbVHWLqrYCi4CP+hxTUqnqblVd5T5vwPnyGOVvVMklIqOBS4E7/I4lFURkAHAOcCeAqraq6iFfg0qNPKCfiOQB/QmxmmWmU9UlQG3Q7o8C97jP7wEuT8S1LEk4RgE7PNs76eNfmF4iMg6YDSz3OZRkuxX4Ac6yztlgPLAP+B+3iu0OEenTSy+q6gfA74D3gd1Anao+629UKVOuqrvB+SMQd1n1eFmScIRa7y8run2JSAnwCPBtVa33O55kEZHLgL2qutLvWFIoDzgF+KOqzgYOk6AqiHTl1sN/FDgBOA4oFpHP+BtVZrMk4dgJjPFsj6YPFlGDiUg+ToK4T1Uf9TueJJsHfEREtuFUJ54nIvf6G1LS7QR2qmpXCfFhnKTRl10AbFXVfaraBjwKnOlzTKlSIyIjAdx/9ybipJYkHCuASSJygogU4DR0PeFzTEklIoJTV71OVW/xO55kU9UfqupoVR2H8/N9QVX79F+YqroH2CEiU9xd5wNrfQwpFd4H5opIf/d3/Hz6eGO9xxPA593nnwceT8RJ036N61RQ1XYR+QbwDE5viLtUdY3PYSXbPOCzwDsi8pa770fumuKm7/gmcJ/7x88W4Gqf40kqVV0uIg8Dq3B68L1JHxx5LSIPAJXAUBHZCdwI/BvwkIh8CSdZfjIh17IR18YYY8Kx6iZjjDFhWZIwxhgTliUJY4wxYVmSMMYYE5YlCWOMMWFZkjDGGBOWJQljYiQilSKinsd7Kbjm0KBrWh92k1Q2mM6Y+L0EVAP7U3CtI8BN7vMvAGNTcE2TxSxJGBO/alX9WSoupKpHgJ+BU5LBkoRJMqtuMsYYE5YlCWN6ISJVbt3/b0XkNBF5XERq3baAkyI8xz+LyH0iskFEGkTkoIisEJE+PY+S6RssSRjTu66ptacDL+MsWPQn4H7gmA3VIlKKs0rYePf9/4UzffUJwF0i8q9JiNmYhLE2CWN615UkzgLOVdXXul5w2wSORYEx7rTdASLyE2ADzqys/56QSI1JAksSxvSuK0lc500QkVLVRqAxxP7dIrILGBxnfMYklVU3GROGux70ZJwVvu6O8RyDROSnIvKa2xbR4RnfMBln9Tjv8V8Tka0i0iwiK0Xk7Hjvw5h4WEnCmPBm4vwh9Q9V7Yz2zSIyA3gWKAdex1k2tRZow2mT+Byw2nP8p4HbgK8BS91/nxKRaar6fny3YkxsLEkYE15XVdPyXo8K7y9AGVClqtXeF0Tk5+7TNzy7vwvcrap/dre/KSLzgWuBH8YYgzFxseomY8LrShJv9HpUCCIyBpiBM9CuOui1MrqXEV3p7isA5uCUPLyeBc6M9vrGJIolCWPCOwVoBd6J4b3N7r/jRSS/a6eIDAEeBEbjrMH8lvvSUJz11WuCzlMDjIjh+sYkhFU3GROCiBQC04C3VbU12ver6j4ReQE4D1guIs8DI4GLgRdxxlusVdXm4LcGhxJinzEpYyUJY0KbDuTjVgfF6AqcdokxOO0KE3DaHX6N8/+etxprP9DB0aWG4RxdujAmZawkYUwIqroS56/4eM6xD6cHUygSdGyriKwELgT+6nnpQuCReOIwJh5WkjAmfjcmaD2JW4AviMiXReREEbkNOA74764DvOtJAOfGeT1jjslKEsbEbhvdaztAnOtJqOqDbsP2T3DaL94FLlHV7Z7DvOtJGJN0omptYsYYY0Kz6iZjjDFhWZIwxhgTliUJY4wxYVmSMMYYE5YlCWOMMWFZkjDGGBOWJQljjDFhWZIwxhgTliUJY4wxYf1/MsDkoz3zcGcAAAAASUVORK5CYII=", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#Lets plot the radial distribution functions\n", "\n", "r = np.linspace(0,10,100)\n", "n=1\n", "l=0\n", "R = psi_R(r,n,l)\n", "plt.plot(r, 4*np.pi*(r**2 )* R**2, lw=3)\n", "plt.xlabel('$r [a_0]$',fontsize=20)\n", "plt.ylabel('$4\\pi r^{2} \\cdot R^2_{nl}(r)$', fontsize=20)\n", "plt.title(\"Radial probability distribution for n=\"+str(n)+\" l=\"+str(l))\n", "plt.grid('True')" ] }, { "cell_type": "code", "execution_count": 9, "id": "9916b90c", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#Comparing radial distribution functions\n", "fig = plt.figure(figsize=(12,7))\n", "ax1 = plt.subplot(121) #121 means: the subplots will be 1 row 2 columns and this one is subplot number 1\n", "ax2 = plt.subplot(122, sharey = ax1) #share-y means that it will share the y-axis with the ax1 plot\n", "\n", "r = np.linspace(0,10,100)\n", "n=2\n", "l=0\n", "R = psi_R(r,n,l)\n", "ax1.plot(r, 4*np.pi*(r**2 )* R**2, lw=3)\n", "ax1.set_xlabel('$r [a_0]$',fontsize=20)\n", "ax1.set_ylabel('$4\\pi r^{2} \\cdot R^2_{nl}(r)$', fontsize=20)\n", "ax1.set_title(\"Radial probability distribution for n=\"+str(n)+\" l=\"+str(l))\n", "ax1.grid('True')\n", "n=2\n", "l=1\n", "R = psi_R(r,n,l)\n", "ax2.plot(r, 4*np.pi*(r**2 )* R**2, lw=3)\n", "ax2.set_xlabel('$r [a_0]$',fontsize=20)\n", "ax2.set_ylabel('$4\\pi r^{2} \\cdot R^2_{nl}(r)$', fontsize=20)\n", "ax2.set_title(\"Radial probability distribution for n=\"+str(n)+\" l=\"+str(l))\n", "ax2.grid('True')" ] }, { "cell_type": "markdown", "id": "affea69c", "metadata": {}, "source": [ "or more interactively" ] }, { "cell_type": "code", "execution_count": 10, "id": "76993ec6", "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "f55eb028bdcb4ad2a791bcae7e780526", "version_major": 2, "version_minor": 0 }, "text/plain": [ "interactive(children=(Dropdown(description='n', options=(1, 2, 3, 4, 5, 6, 7, 8, 9), value=1), Dropdown(descri…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "nmax=10\n", "import numpy as np\n", "import ipywidgets as widgets\n", "@widgets.interact(n = np.arange(1,nmax,1), l = np.arange(0,nmax-1,1))\n", "\n", "def plot_radial(n=1,l=0):\n", " r = np.linspace(0,250,10000)\n", " psi2 = psi_R(r,n,l)**2 * (r**2) * 4*np.pi\n", " \n", " plt.plot(r, psi2, lw=2, color='red')\n", " plt.xlabel('$r [a_0]$')\n", " plt.ylabel('$4\\pi r^2 R^2_{nl}(r)$')\n", " rmax = n**2*(1+0.5*(1-l*(l+1)/n**2))\n", " plt.xlim([0, 2*rmax])" ] }, { "cell_type": "markdown", "id": "221e1190", "metadata": { "tags": [ "Exercise" ] }, "source": [ "* What is the distance that an electron in a 1s orbital most probable to be? And in 2s? And in 2p?\n", "* Are the most probable distance the average distance? The average distance is called expectation value. Would you expect the expectation value of the 1s be higher or lower than the most probable distance?\n", "* Predict if the expected value for \"r\" of 2s is lower or higher than for 2p." ] }, { "cell_type": "markdown", "id": "7978bf59", "metadata": {}, "source": [ "# Plotting the entire wavefunction: probability and cutoff\n", "\n", "If we want to plot the entire wavefunction we need to plot that huge function that depends on $r$, $\\theta$ and $\\phi$. Luckily for us, again, all the polynomials (the Spherical harmonics and Laguerre) are available on python. Even though the angular function depends on $\\theta$ and $\\phi$ it will be more visually understandable if we plot the wavefunctions as a function of x,y,z So rather than plotting it as a function of the angles, we will build a mesh of xyz and calculate the value of the wavefunction at that point.\n", "\n", "One problem is that this is a 3D function, so we would need a fourth dimension to plot its value. The only solution is to plot an \"isosurface\", that is, plot a surface not a volume where the function has a specific value. This is why we need to come up with a value. The variable iso is set as 30% of the maximum value of the function.\n", "\n", "First we will calculate the whole wavefunction for a given n,l, and m." ] }, { "cell_type": "code", "execution_count": 11, "id": "56525605", "metadata": {}, "outputs": [], "source": [ "#source: https://github.com/wmfschneider/CHE30324/blob/master/Outline/CHE30324-outline.org\n", "import plotly.graph_objects as go\n", "from sympy.physics.hydrogen import R_nl\n", "from sympy import Ynm\n", "from sympy import var\n", "\n", "var(\"r theta phi\")\n", "\n", "n = 3\n", "l = 2\n", "m = 0\n", "Z = 1 # this is the charge of the nucleus. Make it Z=2 to see He+ or Z=3 for Li(2+)\n", "\n", "xmesh=[]\n", "ymesh=[]\n", "zmesh=[]\n", "psi=[]\n", "points=20\n", "\n", "for z in np.linspace(-10.,10.,points):\n", " for y in np.linspace(-10.,10.,points):\n", " for x in np.linspace(-10.,10.,points):\n", " xmesh.append(x)\n", " ymesh.append(y)\n", " zmesh.append(z)\n", " r = np.sqrt(x**2+y**2+z**2)\n", " theta=np.arccos(z/r)\n", " phi=np.arctan(y/x)\n", " value=R_nl(n,l,r,Z)* Ynm(l,m,theta,phi).evalf() \n", " psi.append(float(value))" ] }, { "cell_type": "markdown", "id": "82eafeb6", "metadata": {}, "source": [ "and now we can plot it deciding on the value of iso" ] }, { "cell_type": "code", "execution_count": 4, "id": "ae367e82", "metadata": {}, "outputs": [ { "data": { "text/html": [ " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "caps": { "x": { "show": false }, "y": { "show": false }, "z": { "show": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#source: https://github.com/wmfschneider/CHE30324/blob/master/Outline/CHE30324-outline.org\n", "iso = 0.2*max(psi)\n", "\n", "fig= go.Figure(data=go.Isosurface(\n", " x=xmesh,\n", " y=ymesh,\n", " z=zmesh,\n", " value=psi,\n", " isomin=-iso,\n", " isomax=iso,\n", " surface_count=2,\n", " opacity=1.0,\n", " caps=dict(x_show=False, y_show=False,z_show=False)\n", "))\n", "\n", "fig.show()" ] }, { "cell_type": "markdown", "id": "16891255", "metadata": {}, "source": [ "# Quantum weirdness\n", "\n", "**Popular (non-academic) Resources:**\n", "\n", "* On Schrodinger's cat and quantum weirdness: https://www.youtube.com/watch?v=uWMTOrux0LM\n", "* The double slit experiment: Quantum superposition, the wave nature of reality and the role of observation https://www.youtube.com/watch?v=NvzSLByrw4Q\n", "\n", "**Critical thinking questions on quantum weirdness**\n", "\n", "It seems that \"reality\" exists as many situations at the same time. In other words, it is not that the electron is a wave, rather, it is that we can only know about quantum objects as a wave (superposition) of possibilities, and it is not until we measure the system that one of those possibilities is defined. In other words, our observing changes the nature of what we observe. Can objective science exist? If the observer cannot observe a system without changing it, is there an independent reality outside of the observer?\n", "\n", "More philosophical implications here: https://plato.stanford.edu/entries/qt-issues/\n", " " ] }, { "cell_type": "markdown", "id": "a3776915", "metadata": {}, "source": [ "# Exercise\n", "\n", "* Plot the energy levels for the atom of hydrogen for each allowed combination of n,l, and m.\n", "* What is the degeneracy level for each n?\n", "* Plot the radial wavefunction for the 2p and 3d.\n", "* Plot the radial distribution functions for 3s and 3d. What is the most probable distance to the nucleus for these two orbitals? Do you think the most probable is the average distance?\n", "* Is the most probable distance the size of the atom?\n", "* Use plotly to plot the entire wavefunction for 2 orbitals of the 3d. What cutoff or iso value did you use? What's a cutoff and why do we need it?" ] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "pchem", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.2" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": { "height": "calc(100% - 180px)", "left": "10px", "top": "150px", "width": "165px" }, "toc_section_display": true, "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 5 }