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    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Approximate-Methods-for-Solving-One-Particle-Schrödinger-Equations\" data-toc-modified-id=\"Approximate-Methods-for-Solving-One-Particle-Schrödinger-Equations-7\"><span class=\"toc-item-num\">7&nbsp;&nbsp;</span>Approximate Methods for Solving One-Particle Schrödinger Equations</a></span><ul class=\"toc-item\"><li><span><a href=\"#Expansion-in-a-Basis\" data-toc-modified-id=\"Expansion-in-a-Basis-7.1\"><span class=\"toc-item-num\">7.1&nbsp;&nbsp;</span>Expansion in a Basis</a></span><ul class=\"toc-item\"><li><span><a href=\"#Solving-the-Secular-Equation\" data-toc-modified-id=\"Solving-the-Secular-Equation-7.1.1\"><span class=\"toc-item-num\">7.1.1&nbsp;&nbsp;</span>Solving the Secular Equation</a></span></li><li><span><a href=\"#Example-for-the-Particle-in-a-Box\" data-toc-modified-id=\"Example-for-the-Particle-in-a-Box-7.1.2\"><span class=\"toc-item-num\">7.1.2&nbsp;&nbsp;</span>Example for the Particle-in-a-Box</a></span></li><li><span><a href=\"#Particle-in-a-Box-with-Jacobi-polynomials\" data-toc-modified-id=\"Particle-in-a-Box-with-Jacobi-polynomials-7.1.3\"><span class=\"toc-item-num\">7.1.3&nbsp;&nbsp;</span>Particle-in-a-Box with Jacobi polynomials</a></span><ul class=\"toc-item\"><li><span><a href=\"#🤔-Thought-Provoking-Question:-Why-does-adding-odd-order-polynomials-to-the-basis-set-not-increase-the-accuracy-for-the-ground-state-wavefunction.\" data-toc-modified-id=\"🤔-Thought-Provoking-Question:-Why-does-adding-odd-order-polynomials-to-the-basis-set-not-increase-the-accuracy-for-the-ground-state-wavefunction.-7.1.3.1\"><span class=\"toc-item-num\">7.1.3.1&nbsp;&nbsp;</span>🤔 Thought-Provoking Question: Why does adding odd-order polynomials to the basis set not increase the accuracy for the ground state wavefunction.</a></span></li><li><span><a href=\"#🤔-Thought-Provoking-Question:-Why-does-one-get-exactly-the-same-results-for-the-Jacobi-polynomials-and-the-simpler-$(1-x)(1+x)x^k$-polynomials?\" data-toc-modified-id=\"🤔-Thought-Provoking-Question:-Why-does-one-get-exactly-the-same-results-for-the-Jacobi-polynomials-and-the-simpler-$(1-x)(1+x)x^k$-polynomials?-7.1.3.2\"><span class=\"toc-item-num\">7.1.3.2&nbsp;&nbsp;</span>🤔 Thought-Provoking Question: Why does one get exactly the same results for the Jacobi polynomials and the simpler $(1-x)(1+x)x^k$ polynomials?</a></span></li></ul></li></ul></li><li><span><a href=\"#Perturbation-Theory\" data-toc-modified-id=\"Perturbation-Theory-7.2\"><span class=\"toc-item-num\">7.2&nbsp;&nbsp;</span>Perturbation Theory</a></span><ul class=\"toc-item\"><li><span><a href=\"#The-Perturbed-Hamiltonian\" data-toc-modified-id=\"The-Perturbed-Hamiltonian-7.2.1\"><span class=\"toc-item-num\">7.2.1&nbsp;&nbsp;</span>The Perturbed Hamiltonian</a></span></li><li><span><a href=\"#Hellmann-Feynman-Theorem\" data-toc-modified-id=\"Hellmann-Feynman-Theorem-7.2.2\"><span class=\"toc-item-num\">7.2.2&nbsp;&nbsp;</span>Hellmann-Feynman Theorem</a></span><ul class=\"toc-item\"><li><span><a href=\"#Derivation-of-the-Hellmann-Feynman-Theorem-by-Differentiation-Under-the-Integral-Sign\" data-toc-modified-id=\"Derivation-of-the-Hellmann-Feynman-Theorem-by-Differentiation-Under-the-Integral-Sign-7.2.2.1\"><span class=\"toc-item-num\">7.2.2.1&nbsp;&nbsp;</span>Derivation of the Hellmann-Feynman Theorem by Differentiation Under the Integral Sign</a></span></li><li><span><a href=\"#Derivation-of-the-Hellmann-Feynman-Theorem-from-First-Order-Perturbation-Theory\" data-toc-modified-id=\"Derivation-of-the-Hellmann-Feynman-Theorem-from-First-Order-Perturbation-Theory-7.2.2.2\"><span class=\"toc-item-num\">7.2.2.2&nbsp;&nbsp;</span>Derivation of the Hellmann-Feynman Theorem from First-Order Perturbation Theory</a></span></li></ul></li><li><span><a href=\"#Perturbed-Wavefunctions\" data-toc-modified-id=\"Perturbed-Wavefunctions-7.2.3\"><span class=\"toc-item-num\">7.2.3&nbsp;&nbsp;</span>Perturbed Wavefunctions</a></span></li><li><span><a href=\"#The-Law-of-Diminishing-Returns-and-Accelerating-Losses\" data-toc-modified-id=\"The-Law-of-Diminishing-Returns-and-Accelerating-Losses-7.2.4\"><span class=\"toc-item-num\">7.2.4&nbsp;&nbsp;</span>The Law of Diminishing Returns and Accelerating Losses</a></span></li><li><span><a href=\"#Example:-Particle-in-a-Box-with-a-Sloped-Bottom\" data-toc-modified-id=\"Example:-Particle-in-a-Box-with-a-Sloped-Bottom-7.2.5\"><span class=\"toc-item-num\">7.2.5&nbsp;&nbsp;</span>Example: Particle in a Box with a Sloped Bottom</a></span><ul class=\"toc-item\"><li><span><a href=\"#The-Hamiltonian-for-an-Applied-Uniform-Electric-Field\" data-toc-modified-id=\"The-Hamiltonian-for-an-Applied-Uniform-Electric-Field-7.2.5.1\"><span class=\"toc-item-num\">7.2.5.1&nbsp;&nbsp;</span>The Hamiltonian for an Applied Uniform Electric Field</a></span></li><li><span><a href=\"#Perturbation-Theory-for-the-Particle-in-a-Box-in-a-Uniform-Electric-Field\" data-toc-modified-id=\"Perturbation-Theory-for-the-Particle-in-a-Box-in-a-Uniform-Electric-Field-7.2.5.2\"><span class=\"toc-item-num\">7.2.5.2&nbsp;&nbsp;</span>Perturbation Theory for the Particle-in-a-Box in a Uniform Electric Field</a></span></li><li><span><a href=\"#Variational-Approach-to-the-Particle-in-a-Box-in-a-Uniform-Electric-Field\" data-toc-modified-id=\"Variational-Approach-to-the-Particle-in-a-Box-in-a-Uniform-Electric-Field-7.2.5.3\"><span class=\"toc-item-num\">7.2.5.3&nbsp;&nbsp;</span>Variational Approach to the Particle-in-a-Box in a Uniform Electric Field</a></span></li><li><span><a href=\"#Basis-Set-Expansion-for-the-Particle-in-a-Box-in-a-Uniform-Electric-Field\" data-toc-modified-id=\"Basis-Set-Expansion-for-the-Particle-in-a-Box-in-a-Uniform-Electric-Field-7.2.5.4\"><span class=\"toc-item-num\">7.2.5.4&nbsp;&nbsp;</span>Basis-Set Expansion for the Particle-in-a-Box in a Uniform Electric Field</a></span></li><li><span><a href=\"#Demonstration\" data-toc-modified-id=\"Demonstration-7.2.5.5\"><span class=\"toc-item-num\">7.2.5.5&nbsp;&nbsp;</span>Demonstration</a></span></li></ul></li></ul></li><li><span><a href=\"#🪞-Self-Reflection\" data-toc-modified-id=\"🪞-Self-Reflection-7.3\"><span class=\"toc-item-num\">7.3&nbsp;&nbsp;</span>🪞 Self-Reflection</a></span></li><li><span><a href=\"#🤔-Thought-Provoking-Questions\" data-toc-modified-id=\"🤔-Thought-Provoking-Questions-7.4\"><span class=\"toc-item-num\">7.4&nbsp;&nbsp;</span>🤔 Thought-Provoking Questions</a></span></li><li><span><a href=\"#🔁-Recapitulation\" data-toc-modified-id=\"🔁-Recapitulation-7.5\"><span class=\"toc-item-num\">7.5&nbsp;&nbsp;</span>🔁 Recapitulation</a></span></li><li><span><a href=\"#🔮-Next-Up...\" data-toc-modified-id=\"🔮-Next-Up...-7.6\"><span class=\"toc-item-num\">7.6&nbsp;&nbsp;</span>🔮 Next Up...</a></span></li><li><span><a href=\"#📚-References\" data-toc-modified-id=\"📚-References-7.7\"><span class=\"toc-item-num\">7.7&nbsp;&nbsp;</span>📚 References</a></span></li></ul></li></ul></div>"
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    "![Train Wreck](https://github.com/PaulWAyers/IntroQChem/blob/main/linkedFiles/TrainWreck.jpg?raw=true \"Train wreck at Montparnasse Station; from 1895 and in the public domain\")\n",
    "\n",
    "# Approximate Methods for Solving One-Particle Schr&ouml;dinger Equations\n",
    "Up to this point, we've focused on systems for which we can solve the Schr&ouml;dinger equation. Unfortunately, there are very few such systems, and their relevance for real chemical systems is very limited. This motivates the approximate methods for solving the Schr&ouml;dinger equation. One must be careful, however, if one makes poor assumptions, the results of approximate methods can be very poor. Conversely, with appropriate insight, approximation techniques can be extremely useful. "
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    "## Expansion in a Basis\n",
    "We have seen the eigenvectors of a Hermitian operator are a complete basis, and can be chosen to be orthonormal. We have also seen how a wavefunction can be expanded in a basis,\n",
    "$$\n",
    "\\Psi(x) = \\sum_{k=0}^{\\infty} c_k \\phi_k(x)\n",
    "$$\n",
    "Note that there is no requirement that the basis set, $\\{\\phi_k(x) \\}$ be eigenvectors of a Hermitian operator: all that matters is that the basis set is complete. For real problems, of course, one can choose only a finite number of basis functions, \n",
    "$$\n",
    "\\Psi(x) \\approx \\sum_{k=0}^{N_{\\text{basis}}} c_k \\phi_k(x)\n",
    "$$\n",
    "but as the number of basis functions, $N_{\\text{basis}}$, increases, results should become increasingly accurate. \n",
    "\n",
    "Substituting this expression for the wavefunction into the time-independent Schr&ouml;dinger equation, \n",
    "$$\n",
    "\\hat{H} \\Psi(x) = \\hat{H} \\sum_{k=0}^{\\infty} c_k \\phi_k(x) = E \\sum_{k=0}^{\\infty} c_k \\phi_k(x)\n",
    "$$\n",
    "Multiplying on the left by $\\left(\\phi_j(x) \\right)^*$ and integrating over all space,  \n",
    "$$\n",
    " \\sum_{k=0}^{\\infty} \\left[\\int \\left(\\phi_j(x) \\right)^* \\hat{H} \\phi_k(x) dx \\right] c_k \n",
    " = E \\sum_{k=0}^{\\infty}\\left[ \\int \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx\\right] c_k\n",
    "$$\n",
    "At this stage we usually define the Hamiltonian matrix, $\\mathbf{H}$, as the matrix with elements\n",
    "$$\n",
    "h_{jk} = \\int \\left(\\phi_j(x) \\right)^* \\hat{H} \\phi_k(x) dx \n",
    "$$\n",
    "and the overlap matrix, $\\mathbf{S}$ as the matrix with elements\n",
    "$$\n",
    "s_{jk} = \\int \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx\n",
    "$$\n",
    "If the basis is orthonormal, then the overlap matrix is equal to the identity matrix, $\\mathbf{S} = \\mathbf{I}$ and its elements are therefore given by the Kronecker delta, $s_{jk} = \\delta_{jk}$.\n",
    "\n",
    "The Schr&ouml;dinger equation therefore can be written as a [generalized matrix eigenvalue problem](https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem):\n",
    "$$\n",
    "\\mathbf{Hc}=E\\mathbf{Sc}\n",
    "$$\n",
    "or, in element-wise notation, as:\n",
    "$$\n",
    " \\sum_{k=0}^{\\infty} h_{jk} c_k \n",
    " = E \\sum_{k=0}^{\\infty} s_{jk} c_k\n",
    "$$\n",
    "In the special case where the basis functions are orthogonormal, $\\mathbf{S} = \\mathbf{I}$ and this is an ordinary [matrix eigenvalue problem](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Eigenvalues_and_eigenvectors_of_matrices),\n",
    "$$\n",
    "\\mathbf{Hc}=E\\mathbf{c}\n",
    "$$\n",
    "or, in element-wise notation, as:\n",
    "$$\n",
    " \\sum_{k=0}^{\\infty} h_{jk} c_k \n",
    " = E c_j\n",
    "$$"
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    "### Solving the Secular Equation\n",
    "In the context of quantum chemistry, the generalized eigenvalue problem \n",
    "$$\n",
    "\\mathbf{Hc}=E\\mathbf{Sc}\n",
    "$$\n",
    "is called the *secular equation*. To solve the secular equation:\n",
    "1. Choose a basis, $\\{|\\phi_k\\rangle \\}$ and a basis-set size, $N_{\\text{basis}}$\n",
    "1. Evaluate the matrix elements of the Hamiltonian and the overlap matrix\n",
    "\\begin{align}\n",
    "h_{jk} &= \\int \\left(\\phi_j(x) \\right)^* \\hat{H} \\phi_k(x) dx  \\qquad \\qquad 0 \\le j,k \\le N_{\\text{basis}} \\\\\n",
    "s_{jk} &= \\int \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx \n",
    "\\end{align}\n",
    "1. Solve the generalized eigenvalue problem\n",
    "$$\n",
    " \\sum_{k=0}^{\\infty} h_{jk} c_k \n",
    " = E \\sum_{k=0}^{\\infty} s_{jk} c_k\n",
    "$$\n",
    "\n",
    "Because of the variational principle, the lowest eigenvalue will always be greater than or equal to the true ground-state energy. "
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    "### Example for the Particle-in-a-Box\n",
    "As an example, consider an electron confined to a box with length 2 Bohr, stretching from $x=-1$ to $x=1$. We know that the exact energy of this system is \n",
    "$$E=\\tfrac{(\\pi n)^2}{8}$$ \n",
    "The exact wavefunctions are easily seen to be \n",
    "$$\\psi_n(x) = \n",
    "\\begin{cases}\n",
    "\\cos\\left(\\tfrac{n \\pi x}{2}\\right) & n=1,3,5,\\ldots \\\\\n",
    "\\sin\\left(\\tfrac{n \\pi x}{2}\\right) & n=2,4,6,\\ldots \n",
    "\\end{cases}\n",
    "$$ \n",
    "\n",
    "However, for pedagogical purposes, suppose we did not know these answers. We know that the wavefunction will be zero at $x= \\pm1$, so we might hypothesize a basis like:\n",
    "$$\n",
    "\\phi_k(x) = (x-1)(x+1)x^k = x^{k+2} - x^{k} \\qquad \\qquad k=0,1,2,\\ldots\n",
    "$$\n",
    "The overlap matrix elements are\n",
    "\n",
    "\\begin{align}\n",
    "s_{jk} &= \\int_{-1}^{1} \\left(\\phi_j(x) \\right)^* \\phi_k(x) dx \\\\\n",
    "&= \\int_{-1}^{1} \\left(x^{j+2}-x^{j}\\right) \\left(x^{k+2} - x^{k}\\right) dx \\\\\n",
    "&= \\int_{-1}^{1} \\left(x^{j+k+4}+x^{j+k} - 2 x^{j+k+2}\\right) dx \\\\\n",
    "&= \\left[\\frac{x^{k+j+5}}{k+j+5} + \\frac{x^{k+j+1}}{k+j+1} \n",
    "- 2\\frac{x^{k+j+3}}{k+j+3} \\right]_{-1}^{+1}\n",
    "\\end{align}\n",
    "\n",
    "This integral is zero when $k+j$ is odd. Specifically,\n",
    "$$\n",
    "s_{jk} = \n",
    "\\begin{cases}\n",
    "    0 & j+k \\text{ is odd}\\\\\n",
    "    2\\left(\\frac{1}{k+j+5} - \\frac{2}{k+j+3} + \\frac{1}{k+j+1}  \\right)       & j+k \\text{ is even}\n",
    "\\end{cases}\n",
    "$$\n",
    "and the Hamiltonian matrix elements are\n",
    "\n",
    "\\begin{align}\n",
    "h_{jk} &= \\int_{-1}^{1} \\left(\\phi_j(x) \\right)^* \\hat{H} \\phi_k(x) dx \\\\\n",
    "&= \\int_{-1}^{1} \\left(x^{j+2}-x^{j}\\right) \\left(-\\tfrac{1}{2}\\tfrac{d^2}{dx^2}\\right) \\left(x^{k+2} - x^{k}\\right) dx \\\\\n",
    "&= -\\tfrac{1}{2}\\int_{-1}^{1} \\left(x^{j+2}-x^{j}\\right) \\left((k+2)(k+1)x^{k} - (k)(k-1)x^{k-2}\\right) dx \\\\\n",
    "&= -\\tfrac{1}{2}\\int_{-1}^{1} \\left((k+2)(k+1)x^{k+j+2} + (k)(k-1)x^{k+j-2} -\\left[(k+2)(k+1) + k(k-1) \\right]x^{k+j} \\right) dx \\\\\n",
    "&= -\\tfrac{1}{2}\\left[\\left(\\frac{(k+2)(k+1)}{k+j+3}x^{k+j+3} + \\frac{(k)(k-1)}{k+j-1}x^{k+j-1} \n",
    "- \\frac{(k+2)(k+1) + k(k-1)}{k+j+1}x^{k+j+1} \\right) \\right]_{-1}^{+1}\n",
    "\\end{align}\n",
    "\n",
    "This integral is also zero when $k+j$ is odd. Specifically,\n",
    "$$\n",
    "h_{jk} = \n",
    "\\begin{cases}\n",
    "    0 & j+k \\text{ is odd}\\\\\n",
    "    2\\left(\\frac{(k+2)(k+1)}{k+j+3} - \\frac{(k+2)(k+1) + k(k-1)}{k+j+1} + \\frac{(k)(k-1)}{k+j-1}  \\right)       & j+k \\text{ is even}\n",
    "\\end{cases}\n",
    "$$"
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gW0qFTadfJxiN+dsE/ZLbsnkT9XjJPL9Nkeg3QzLfG1D/e6BR3xvu/jmb31BpBXyL4ObMgwSvnSVmdrOZxSf6IklRgihZzd0r3f1jgn43sXf/i4kZuMPMdiPor9KYBGuzfd39aoJaxudI3M+lDUEn+XfMLOnkKKyB2Z2gH2RdNTe7EvT5+b9kz5vgOhvd/Xl3/7W7709wR/IYgtrRWO1IzZyB9f1A2hqJfjiNaOQ5mpRoh+r7YZpo8vF6+6S6+xo2r2FoTdAXcTBBn5Ja78TV7DX1MdQ16EStefWUxV9zixq3JDQ17vpeT41+3psoHbGn6hppT5gy6DNkax5rohYAW3sz6x4adxMgGfW9D5uiyt1n1LPETgif6PlNyesrQQuMhp77K2jkQFJs3ed7us6b0u8N4IcELRXGEozqHa8jwUi4Y+Nu1ogkJWObgIk0hrsvM7OlbD5xeuxokt9m8xsi6wiaQY0hGAACgr5TxzRwnWeAZ8IvuW3DawwiGIChtharjKDPzA2NiH8xwTQcl5pZR4LEoD/BiH6xzQwvMrM/J2oK11juvgF41syWEdQGxIodXXIBwR3mWpcSzCXZkGQ7/TfWKwT95GLv5P7MzO5JUItUl/kJtvUlmC4jVkXceiWJv8y31r0Er5lapxDMbRUrvrnrErZ8Hs5gy//LxqrvtbWA4HVZqyLBPts2cP745/5JkhvdckkS+yQr0Y/vAWw54EW8RK+b4TRck1LVQHl919jGzCxBjVlF3Hp8U9+0a8bPkPj/rwF1PCeJxD/3m42cG3YN2CaJ8yQUNseMH5X0IYJRr+cR/NAfRhKj+8bZ6s/4rZCoKXEFwXyssRp63hK97lsTTNVTK36gqHjxg0t9QJA0TiPo9lHElp+V6VI7/2LsZ26iAagqEmxL6fszfO3fB9wXvoa3I3guhxDM59wh3HVnggGOnkjl9SX3qQZRckLYvr9r3ObYO3DxzTtedPcb3X2iu88gaLa4Wz3nzzezTecP+wNOc/fn3P3PwHVxh/QnSWbWPbbpqLsvc/d33f1ed/8OweiMtTqwZTO/+s79AzP7vzDprEuiIbhjR0x9I67sWGBuXXeiCZq79I67Ix2fuDW5dtHdl7Nlf6UBwC1Wz5D7ZnaqmQ0MzzELmB23y8i4/Y1gBNlY49w9HTVTr8XFcySbj5i6gbgf1GEc78Sd57j6agkI+hV12Io4349bPzJBbfkPGjhH/OvpQOqp3SD4MTk4xX1pxrNlE66rEjXhNrNCM2sNm1438cngUQ0859vSuKbAb8atdyauf3TYImJwA8elRIZ8hrwed3wFwaA2ieItjdu0PG49/rP5NIIaz6ZK1HTwPHd/w92/DB9Pnd8tmSiMOf6GzGb9asOkJL6PdLzlCbZtev7DG60NfV7EP79/cPdn3H1qGOeODRwPKfr+CV+P8aNtn5Xgc+PsuPXZ7h7/fdNkZtbavhm1uLZmf4q7P+3uV7DlzYikf4+I1FKCKNkodh7E7c3sWBIPxR47VH78l92IsDN3fzM7gqAPY32dyEuAOWb2lJldYGbDzGzH8PjjCKbOiLUmwTnqcjEwzcyuN7MTLRhIp5+Z7W5m17LlD/rGnLsrwdDr88NO7BeY2X5mtkN4nXPYskP9XHePHSQkftju/YBXwlh3tWAutEPM7Ndm9hYwhS3niIvvN9gt7FO5ffj/2KsRjwmCvhZz47adA7xvZmeb2W7hYzzAzC42s0kEAyp0itn/X3HHf8vMbjOzvcxsf4I51+JrBuKPSYmwRjh2aPNigr6otZ4OE+N48f83p4Sv0aPC/5ddzOzIcPCCDwkSvEFbEeoDceutCeYpPCK83mUEzZrq8yibDyzRCXjdzH5kwWBPO4Sv0fPN7EmCWqSfb0XMW3D39Wz5uj8QeDF87rY3s8Fm9mPgIzZv6nt73HGXmNnd4XugvwUDRJxgZteY2ecENd59SN5Ytmxqfm+YqA2wYA7JZ+LKl7HlQCipkgmfIY+x5TQ/t5rZH81sj/D/a7iZXQ88G7dffK3XUeF1djaz77HlFAiNlahm+49mNtDMhprZX0jx67eJCmK+NxMt8f08H4xbP8XM/h4+puEEc4E21Pd7Gls2fxwVfi4PJZhnsa55Y2vFP7+/MLN9w9fNBSRXMxv//TMofI/2Cx97Y5oHx38H7A48amYHmtmeZnYLwfQbsUY14vzJ2JZgbIIHw+/R2vfjThYM0Bc/L25jfjOIBDwDhlLVoqW+hcZNc1G7zAPax5xjnySOWRC3Pjrm+LaNvP4hMccOjy+Pe3zxUwfUt7zZyOfu0iY8dz9JcJ5EU0vUt1wZd3xngru4de0/pwmvi0EEPxobE9fwmOPbEPRnTPbYF4mbBoEtp4gYmSDOBvcJ99uxnmsfU8cxeQTJQmOeg5FNiS9m/4caOH/89BGbPe/hOU4laEKXbMxj4o6/sr7yZPYheE1+keT1B8UcV8KW058k/bpL8rV9EPW/X+KX7yXxuXllY2LIwM+QbyX5mvko7rhdkjguvjz+2g29lj5r4Pzx3y3x3wEVCY6paMr/V8w5Rzfy+Y5/3noQ1ADWd0z8e31GgjgebuRzH//c3N7Y55YtP28GNnCON2P2HVnfYyLow/pSI57XSUCbxr4369uHYEyCZK9fDfTfmteSlpa5qAZRctF8gqZ2K2s3uPs7BD9Q6nID8HyKrv8Pd385ReeKNZ9gdNPGWE7wJZEMB/7m7v9MUPYz4G9JnquauP4WHkz3kHBY+qZy948Imtk9l+QhG4np++LB4DBHsGUzzUSeIZifMG39gtz9M7ZswglBEpzwtRnGcyrBCH/JqCTxKLCNcS5BE826zh8/xUPt9k3c/SGCPpfJNhuNry3eauFr8mC2nFC+oePWEbxuXkrykNV808852Wu8SjCC7coGdq0Eznf3+KlSUmk5mfEZ8gTBa72h52TzgNw/of7+4Hey9c1zz6XuEV0/Jxg5O6u4+zyCvtB1Pa5P2HIKnUR+Td39h5cQtKCpz+UkHikUgs/zExsKwN0nkaLvdw+a9p9I3NQzdRgHHBF+10TBgYvdvbn6aEoO0SA1kgvWE3zRfELQ7OVuj5mLrpa7X2BmHxA0B92F4MfVZOCf7v6gmY2u5xprgL0ImqHtT9AsphtB87j1BM3g3iWodXytkfFfEx57IEEH8/Lw3IUEP86mECRBt3niZoZ1cvfbzOxpgh+0+xHcedyGoNlqAcEX7HSCycrvdvcJdZynCrjQzO4kGD1tGMFd7/YEfbnmENwpfQ14yt0TDebxE4L57U4naLLXOsE+jeLucwmajA0mGMn2QIJBAzoS3JleRPB//BLwUHxc7j4vbE56Ynj8ngT99JxgoIZ3COa5emFrY03SPWw5398Dvnl/zs2EPz7OMLObCPpl7UfQXLoNQXIyi6CZ5CvAf919qwbZcfcVZnYAcBHB/2VfggTodYKmiImSvvjmgbj7fWb2AsHr6TCCwRQ6EAxssYhgZMw3w5jjBw9KCXefaWb7AscRJB97EQzxnxfGMIWg5vjLuOMWA4eZ2WEEz8E+BO/bYr55T31A8Lp7tik/EN39STPrB5xH0Cd1R4L325ownpeBmz3oF5k2mfQZ4u6PmNmrBM3JDyf4HO9IUJM1j+Bz9NEEl7+IoLnjjwiex0qC/5+b3P1xMxvTmOckQVyvm9leBIOnDCd4780BHieYHim+qXpWcPcXzWx34HcEN1PaE/TBfQT4M1tOw5PoHNPNbG/g98ChQCnB/9XTBJ8XOzdw/CIz25OgFvc4gkHglgCvEvRHnGrJzf50IsG0VN8meN01NJpzfTGtBI4L3/9nEbz/ywg+N5YQ3Oh7EHg0TTcVPyN4/xxI8J6sIPjNUErwXphJ8H683d0/SMP1pQUw92RvDIqIiNTPzC4l+PFYaynQLZ21ryLS/MxsJJv3P53p7hXRRCMiqaQmpiIikrRwMIYHwoFp2sZsb2dm5xHUosT6t5JDERGR7KEmpiIi0hhFBKPknQa4mX1N0Jy3C1tOJP0VQRM7ERERyRKqQRQRkaYygtFAu7JlcjgOODgcDEZERESyhGoQRUSkMcYTDJwyAhhAMKhLKcHgKfMIBmh4CHheTUtFRESyjwapEREREREREUBNTEVERERERCSkBFFEREREREQAJYgiIiIiIiISUoIoIiIiIiIigBJEERERERERCSlBFBEREREREUAJooiIiIiIiISUIIqIiIiIiAigBFFERERERERCShBFREREREQEUIIoIiIiIiIiISWIIiIiIiIiAihBFBERERERkZASRBEREREREQGUIIqIiIiIiEhICaKIiIiIiIgAShBFREREREQkpARRREREREREACWIIiIiIiIiElKCKCIiIiIiIoASRBEREREREQkpQRQRERERERFACaKIiIiIiIiElCCKiIiIiIgIoARRREREREREQkoQRUREREREBFCCKCIiIiIiIiEliCIiIiIiIgIoQRQREREREZGQEkQREREREREBlCCKiIiIiIhISAmiiIiIiIiIAFAQdQDNrUuXLl5RURF1GCIikmYTJkxY4u5do44jW+j7UUSk5ajvO7LFJYgVFRWMHz8+6jBERCTNzGxm1DFkE30/ioi0HPV9R6qJqYiIiIiIiABKEEVERERERCSkBFFEREREREQAJYgiIiIiIiISUoIoIiIiIiIigBJEERERERERCSlBFBEREREREUAJooiIiIiIiISUIIqIiIiIiAigBFFERERERERCShBFREREREQEUIIoIiLA6NGjadu2bdRhiIiISMSUIIqIpNCHH35Ifn4+++23X9ShNMqpp57KV199lfbrjBkzBjNjyZIlab+WiIiINJ4SRBGRFPrXv/7Fj3/8YyZPnsyUKVPSfr2amhqqq6u3+jwlJSV069YtBRGJiIhINlOCKCI56ckP57LfNa/S99Jn2e+aV3nyw7lpv+a6det44IEH+OEPf8hJJ53EnXfeualsxowZmBkPPPAA+++/P8XFxey44468+OKLm/aprV175plnGDRoEMXFxQwZMoQJEyZs2qe2Kej//vc/dt11V4qKipgyZQrLli3jrLPOomPHjpSUlHDIIYfwySefALB48WLKy8v5/e9/v+k8kyZNori4mEcffXSz89a68sor2XXXXbnnnnuoqKigbdu2nH322VRWVnLLLbfQu3dvOnfuzK9+9Stqamo2HXffffexxx570K5dO7p168bJJ5/M3LlzNz0HI0aMAKBr166YGSNHjgTA3fnLX/5Cv379KCkpYcCAAdx3332p+q8RERGRJDVLgmhmd5nZIjObXEf56WY2KVzeNrPdYspmmNnHZvaRmY2P2d7JzF4ys2nhvx3T/Tii+MEpIo335Idzuezxj5m7fB0OzF2+jsse/zjt79lHH32UbbbZhoEDB3LmmWdy7733snHjxs32ueSSS/jZz37GRx99xKGHHsrxxx+/KYGqddFFF3Httdcyfvx4tt12W44++mjWrl27qXz9+vVcffXV3H777Xz66adss802jBw5knfffZennnqK9957j9atW3PEEUewbt06unbtyujRo7n66qt55513WLduHaeddhqnnXYaJ510Up2PZ8aMGTz11FM888wzPPbYYzzyyCMcf/zxvP/++7z44ovccccd3HTTTTzxxBObjqmsrOSqq65i4sSJPPPMMyxZsoTTTjsNgN69e/PYY48B8MknnzB//nxuvPFGAH77299y5513cvPNN/Ppp59y2WWX8aMf/Yhnn3126/5TpNnoO1JEJDcUNNN1RgP/BO6to3w6MMzdl5nZkcAoYK+Y8hHuHt9h5VLgFXe/xswuDdd/ndqwv1H7g3PdxqApV+0PToATBvdM12VFJHTq7e9sse2YgeWcuU8F6yqrGXn3e5u2fzhrOZXVNZvtu25jNZc8OokH35u1adsZe2/Dsbv1YN7ydfzyoY822/+hH+3T6BjvuOMOzjzzTACGDRtG69atefrppznxxBM37XP++edzyimnAHDjjTfywgsvcOutt3L11Vdv2ufyyy/n8MMPB+Duu++mV69ePPDAA5xzzjkAVFdXc9NNNzFkyBAApk2bxtNPP83YsWM58MADAfj3v/9Nnz59uP/++znnnHM4/PDD+fGPf8zpp5/OsGHD2LBhAzfddFO9j6e6upq7776b0tJSdt11V4444gjGjh3L3LlzKSoqYqeddmK//fbjtdde2/QYv//97286ftttt+XWW29lp512Ys6cOfTq1YtOnToB0K1bN7p06QLAmjVr+Nvf/saLL77IAQccAEDfvn157733uPnmmzn66KMb/X8hzUvfkSIiuaNZahDd/XXg63rK33b3ZeHqOKBXEqc9Hrgn/Pse4IStibEh170wddMXX611G6u57oWp6bysiDRBfHLY0PZU+OKLL3jrrbf47ne/C4CZcfrpp3PHHXdstt8++3yTeObl5bHXXnvx6aef1rlP27ZtGTBgwGb7FBQUMGjQoE3rU6ZMIS8vb7PjSktLtzju2muvpaioiHvvvZf777+/wVFL+/TpQ2lp6ab17t27s8MOO1BUVLTZtkWLFm1a/+CDDzj++OPZZpttaNeuHUOHDgVg1qxvEvN4n376KevXr+eII46gbdu2m5Zbb72VL7/8st4YJTPoO1JEJHc0Vw1iY/wAeC5m3YEXzcyB2919VLi9u7vPB3D3+WZW5+gKZnYucC4EP3iaYt7ydY3aLiKpVV+NXklR/mbl+13zKnMTvDd7dihJeJ4edWxvjDvuuIPq6urNPmPcHYDZs2dv1bnjtWrVivz8/C2uk4iZbfp7xowZzJ49GzPjq6++Yq+99qrzOIDCwsItzpVoW+0gOWvWrOHwww/nkEMO4d///jfdunVjyZIlHHDAAVRWVtZ5ndo+jP/973+3+IyOv55kJn1HiojkjowapMbMRhAkiLFNRfdz992BI4ELzOzAxp7X3Ue5+1B3H9q1a9cmxdajQ0mjtotIdC4+vD8lhfmbbSspzOfiw/un5XpVVVXcc889/PnPf+ajjz7atEycOJGBAwdy9913b9p33Lhxm/52d9577z122mmnzc4Xu8+aNWuYPHnyFvvE2nnnnampqeGdd75phrty5Uo+/vhjdt55ZwA2btzI6aefznHHHcf111/P+eefX2+tXlN89tlnLFmyhD/96U8ceOCB7LjjjpvVLgKbah9jR17deeedadWqFTNnzmS77bbbbNlmm21SGqOkh74jRURyR8bUIJrZQOAO4Eh3X1q73d3nhf8uMrMngD2B14GFZlYe1h6WA4sSnTdVLj68/2b9KyC9PzhFpOlq+zxd98JU5i1fR48OJVx8eP+09YV69tlnWbJkCT/84Q/p3LnzZmXf+c53uPXWWznjjDMAuPXWW9lhhx0YMGAAt9xyCzNnzuT888/f7Jirr76arl270qNHD37/+99TVFS0qelqIttvvz3HH388P/rRjxg1ahQdOnTgN7/5De3bt9903OWXX86iRYt4+eWXKS0t5fnnn+fMM8/ktddeIy8vNfcK+/TpQ6tWrfjnP//JBRdcwJQpU7j88ss322ebbbbBzHj22Wc59thjKSkpoV27dlx00UVcdNFFuDsHHnggq1evZty4ceTl5XHuueemJD5Jn0Tfka0K8vQdKSKShTKiBtHM+gCPA2e6++cx29uYWbvav4HDgNqRUJ8Gzgr/Pgt4Kp0xnjC4J3/+9gDK2hcDUFpSyJ+/PUCd70Uy1AmDe/LWpQcx/ZqjeevSg9L6Xr3zzjsZMWLEFskhwMknn8zMmTN5+eWXAbjmmmv429/+xm677cbzzz/PE088Qa9em3e7vuaaa7jwwgvZfffdmTZtGs888wxt2rSpN4a7776bPffck+OOO44999yTtWvX8vzzz1NSUsLYsWP561//yr333kuHDh0wM0aPHs2UKVO49tprU/Y8dO3alXvuuYcnn3ySnXfemauuuoq//e1vm+3Ts2dPrrrqKn7zm9/QvXt3fvKTnwDwhz/8gSuvvJLrr7+eXXbZhUMPPZTHHnuMvn37piw+SZ/a78ieHUowwIBdytvrO1JEJAtZfX1XUnYRsweB4UAXYCFwBVAI4O63mdkdwInAzPCQKncfambbArXjpxcAD7j7H8NzdgYeBvoAs4CT3b3OgXBqDR061MePH9/QbnWqrnH6//Y5zj1wWy45Yscmn0dEWpYZM2bQt29f3n///U0Dt8QbM2YMI0aMYPHixZtG+JSmM7MJ7p74yc5xZnYdcCxQCXwJnO3uy+s7Zmu/H2Pd9Mo01m2s5uLD+2/WD1ZERDJDfd+RzdLE1N1Pa6D8HOCcBNu/Anbb8ggIm6EenJIAGyE/z3j4vH3o3bF1c19aREQkWS8Bl7l7lZldC1xGGqeCivfTg7dvrkuJiEiKZUQT02yze5+OdG3XKuowREREEnL3F929KlxNdvqoVMfAa1MXMWnO8ua+tIiIbIWMGaQmm0yY+TWfL1zNaXs2bcoMEWl5Kioq6p2OAmD48OEN7iPSBN8HHmrui26oquHXj05im86tefhH+6ipqYhIllANYhM8P3kBV/33E/2QExGRyJjZy2Y2OcFyfMw+vwGqgPvrOMe5ZjbezMYvXrw4pfEVF+bzs4O35/0ZyxgzNbXnFhGR9FGC2ARlpSWs31jDinUbow5FRERaKHc/xN13TbA8BWBmZwHHAKd7HXc0UzFPcH1O3aM323RuzV9emEpNjW6qiohkAyWITVBeGkx1MX/F+ogjERER2ZKZHUEwKM1x7r42qjgK8/P41aE7MGX+Sv47aV5UYYiISCMoQWyCsjBBXKAEUUREMtM/gXbAS2b2kZndFlUgxw7swfD+XclTH0QRkaygQWqaQDWIIiKSydx9u6hjqJWXZ4w+e8+owxARkSSpBrEJurUr5o1LRnDSkGYfNVxERCQrbayu4T/vzWJdZXXUoYiISD2UIDZBfp7Ru1Nrigr09ImIiCRj0pzlXPr4x9zzzoyoQxERkXoow2miJz+cy736khORGCNHjsTMtlj23nvvZrn+lVdeya677tos1xJprCHbdGJE/67cOuZLjQIuIpLBlCA20fOTF3DvOzOjDkNEMswhhxzC/PnzN1v+97//RR2WSEa46PD+rFi3kVGvfxl1KCIiUgcliE1U3qFYo5iKyBZatWpFWVnZZkunTp0YO3YshYWFjBkzZtO+t912G+3bt+err74C4Pnnn+eAAw6gY8eOdOrUicMPP5wpU6Zsdv558+Zx+umn07lzZ1q3bs2gQYN47bXXGD16NFdddRWffPLJpprL0aNHN+MjF2nYLj1KOXa3Htz15gwWrdJ3qIhIJtIopk1UXlrM6g1VrFq/kXbFhVGHIyIZbtiwYVx88cWceeaZTJw4kUWLFnHhhRdyyy23sO222wKwZs0afvGLXzBw4EDWrVvH1VdfzbHHHsunn35KUVERa9asYdiwYXTr1o0nnniCnj17MnHiRABOPfVUJk+ezDPPPLMpCS0tLY3q4YrU6VeH7sDMpWtYurqSbu2Kow5HRETiKEFsorLSEiCYC1EJokiaPXcpLPi4ea9ZNgCOvKbRhz3//PO0bdt2s20XXHAB1157LVdddRUvvfQS55xzDjNmzOCYY47hrLPO2rTfiSeeuNlxd999N+3bt+e9995j//3354EHHmDBggW88847dOnSBYB+/fpt2r9t27YUFBRQVlbW6LhFmkvfLm146oL9MM2LKCKSkZQgNlHtXIiLV21g++7tIo5GRDLFgQceyKhRozbb1qFDBwAKCwt54IEH2GWXXejWrRuvvvrqZvt9+eWXXH755bz77rssXryYmpoaampqmDVrFgAffvghAwcO3JQcimQrM2PF2o288cVijhnYI+pwREQkhhLEJhrcuwNTrz6CVgX5UYcikvuaUJMXldatW7PddnXPUT5u3DhqampYvnw5ixcv3pQ8Ahx77LH07NmT22+/nZ49e1JQUMDOO+9MZWUlAO6e7vBFms2oN77kljFfsn23dvQv041WEZFMoUFqmqggP0/JoYg0yowZM/jJT37CzTffzKGHHsrpp59OVVUVAEuXLmXKlCn83//9H4cccgg77bQTq1at2lQOsPvuuzNp0iSWLFmS8PxFRUVUV2sScskO5+y/LW2LCrj+xalRhyIiIjGUIG6Fv704lX+P01QXIvKNDRs2sGDBgs2WxYsXU11dzRlnnMGwYcP40Y9+xB133MGcOXO46qqrAOjYsSNdunThX//6F1988QVjx47lvPPOo6Dgm4Ye3/3ud+nWrRsnnHACb7zxBtOnT+fpp5/mtddeA6CiooKZM2fywQcfsGTJEjZs2BDJcyCSjI5tivjRsG156dOFfDBrWdThiIhISAniVnht6mJe/nRh1GGISAZ5+eWXKS8v32wZPHgwf/rTn/jiiy+48847AejcuTP33HMP11xzDW+++SZ5eXk89NBDTJo0iV133ZULLriAP/zhD7Rq1WrTudu0acPYsWPp2bMnxx57LLvssgtXXHHFpsE+TjzxRI466igOPvhgunbtyoMPPhjJcyCSrLP360uXtkX85fnP1IRaRCRDWEv7QB46dKiPHz8+Jef64b3jmbV0LS/88sCUnE9ERFLHzCa4+9Co48gWqfx+bIzRb03nzS+W8o/TBtG6SEMjiIg0h/q+I/VJvBXKS4t5b/rXUYchIiKStc7at4KR+/WNOgwREQmpielWKCstZsW6jaytrGp4ZxEREdlCbRPpGUvWMGGm+iKKiERNCeJW6NmhhC5tW/H1msqoQxEREcla7s6P7/+Aix+dSFV1TdThiIi0aEoQt8Lxg3oy/reH0Ktj66hDERERyVpmxs8P2Z6vFq/hsQ/mRB2OiEiLpgRRREREInfYzt0Z1LsDf395Gus3aj5PEZGoKEHcCtU1zrn3jufh8bOjDkVERCSrmRmXHNGf+SvWc5/mGBYRiYwSxK2Qn2e8P+NrJs5eHnUoIiIiWW/ffl04aMdurFyvwd9ERKKiaS62UllpCQtWrI86DBERkZxwx/eGkpdnUYchItJiqQZxK5WXFjNfCaKIiEhK1CaH475aytLVGyKORkSk5VGCuJXKSotZsFIJooiISKrMW76O7/5rHLeM+TLqUEREWhwliFtp+25t6dWxRPM2iYiIpEiPDiWcNKQX/35nJnOXr4s6HBGRFkUJ4lY6e7++PP2T/SnI11MpIiKSKj8/ZAcAbnz584gjERFpWZTViIiISMbp2aGEM/behkcnzOGLRaujDkdEpMVQgriVFq5cz3H/fJPnJy+IOhQREZGccsGIfvToUMKMJWuiDkVEpMXQNBdbqW2rAibNWcF0fXmJiIikVOe2rRh78QjyNe2FiEizUQ3iVmrTqoD2xQUsWKFO9CIiIqmWn2dU1zhjpi6KOhQRkRZBCWIKlJeWaC5EERGRNHlk/GxG3v0+b3+xJOpQRERynhLEFNBciCIiIulzwuCelJcWc+0LU3H3qMMREclpShBTYM++ndiprH3UYYiIiOSk4sJ8fn7w9kycvZwXP10YdTgiIjmtWRJEM7vLzBaZ2eQ6yk83s0nh8raZ7RZXnm9mH5rZMzHbBpnZODP7yMzGm9me6X4cdblgxHZce9LAqC4vIiKS804a0ottu7Th+hemUl2jWkQRkXRprhrE0cAR9ZRPB4a5+0DgD8CouPKfA1Pitv0FuMrdBwG/C9dFREQkBxXk53HhYf2pdme+BoYTEUmbZkkQ3f114Ot6yt9292Xh6jigV22ZmfUCjgbuiD8MqG3XWQrMS1nAjTRh5tfs8ceXmTCzzocoIiIiW+nIXct48RcH0qtj66hDERHJWZnYB/EHwHMx638HLgFq4vb7BXCdmc0Grgcuq+uEZnZu2Ax1/OLFi1MbLdC2VSGLV21g3nINVCMiIpIueXlGQX4eqzdU8dHs5VGHIyKSkzIqQTSzEQQJ4q/D9WOARe4+IcHu5wO/dPfewC+BO+s6r7uPcveh7j60a9euKY+7rLQYgAWa6kJERCTtLnp4Iufc8z5rNlRFHYqISM7JmATRzAYSNCM93t2Xhpv3A44zsxnAf4CDzOy+sOws4PHw70eAyAapaV9cQJuifM2FKCIi0gzOHbYtS1ZXcvdb06MORUQk52REgmhmfQiSvTPd/fPa7e5+mbv3cvcK4DvAq+5+Rlg8DxgW/n0QMK0ZQ96MmYVzIarTvIiISLrt3qcjh+7cndvHfsWyNZVRhyMiklOaa5qLB4F3gP5mNsfMfmBm55nZeeEuvwM6A7fUTluRxGl/CPzVzCYCfwLOTUvwSTpmYA9279MxyhBERERajIsO68/qyipuG/tl1KGIiOSUgua4iLuf1kD5OcA5DewzBhgTs/4mMCQF4aXELw/dIeoQREREWoz+Ze341qCefLl4Ne6OmUUdkohITmiWBLGlqKyqoTDf9CUlIiLSDP584gBaFeRHHYaISE7JiD6IueCR8bPZ4bfPsWClBqoRERFpDrXJ4dzl65i3XOMAiIikghLEFOnctgjQVBciIiLNaf3Gao7+xxv8+bnPog5FRCQnKEFMkbL2JYASRBERkeZUXJjP6Xv14b8T5/HJvBVRhyMikvWUIKZIeWkxgOZCFBERaWbnHtiP0pJCrn9hatShiIhkPSWIKdKhdSGtCvLUB1FERKSZlZYUct6wfrw2dTHvTf866nBERLKaEsQUMTN+PHw79qzoFHUoIiIiLc7IfSvoUVrMR7OXRR2KiEhW0zQXKfTzQ7aPOgQREckQZtYROBQYDHQElgMfAi+5u6q5UqykKJ+XLxxG6yL9tBER2Rr6FE2hDVXVLFldSc8OJVGHIiIiETGzfsCVwElAERA7Oa4DlWb2CHClu3/V/BHmrtrk8JN5K9iprD15eZqXWESksdTENIX+/vI0hl/3GjU1HnUoIiISnSnA6cAs4E7gQuCH4b93htvPAD6NKsBcNu6rpRz9jzf576R5UYciIpKVVIOYQuWlxWysdpas2UC3dsVRhyMiItF4GPibu39Q1w5mNgT4RbNF1ILsWdGJHcva8beXPueoAeUU5uteuIhIY+hTM4XK2gdJoeZCFBFpudz9jPqSw3CfCe5+ZnPF1JLk5RmXHNGfmUvX8tD7s6MOR0Qk66gGMYXKS4O+h/NXrGdgr4iDERGRSJhZn0bsvtLdl6crlpZqRP9uDN2mI/94ZRon7t6LkqL8qEMSEckaqkFMobJS1SCKiAgzgOlJLr+LJsTcZmZccsSOrK2s5pN5K6IOR0Qkq6gGMYU6tyni/47akaEVHaMORUREorUaaGgqi8bUNEoj7dm3E+9cdhDtigujDkVEJKsoQUyhvDzj3AP7RR2GiIhE7053/1V9O5hZTXMF01K1Ky7E3Zm2aDU7dG8XdTgiIllBTUxTbMGK9Xw6b2XUYYiISHT+CLyYwv1kK1z/4lSO/+dbLFql7h8iIslQgphif/zfFM6/f0LUYYiISETc/XJ3fz5V+8nWOWlIbyqra/jnq19EHYqISFZQgphi5aXFzF+xHnePOhQREckwZtbBzH5mZj+LOpaWom+XNpy6R28efG8Ws79eG3U4IiIZTwliipW1L6ayqoZlazdGHYqIiGSe7sDfgRsijqNF+dlB25Nnxg0vfR51KCIiGU8JYoqVh1NdzF+xLuJIREQkA60FXg8XaSZlpcWM3LeC8TOXsa6yOupwREQymkYxTbHYuRB36VEacTQiIpJJ3H02MDzqOFqinx28Pb86bAdaFeRHHYqISEZTDWKK9evWlptOG8yAnkoORUREMkWbVgW0Kshn/cZqtfIREamHahBTrH1xIcfu1iPqMEREJAOYWV3tGd3d9R3czNydk297hzat8nnwh3tjZlGHJCKScVSDmAYfzV7OhJnLog5DRESiZ/Us6b+42UVm5mbWpTmul+nMjJOG9GLcV1/zxrQlUYcjIpKRlCCmwVX//YS/vTQ16jBERCR6fWOW7YBvAfOAi9N9YTPrDRwKzEr3tbLJaXv2oVfHEq57YSo1NZqSSkQknhLENKidC1FERFo2d58Zs3zl7k8B9wPfb4bL3wBcAigLilFUkMcvD9mBj+eu4LnJC6IOR0Qk46j/QxqUtS9hzNTFuLv6N4iItGBm9ru4Te2Bs4DCNF/3OGCuu0+s73vIzM4FzgXo06dPOkPKKCcM7sntr3/JK1MWcvTA8qjDERHJKEoQ06BHh2LWVlazcn0VpSVp/Q0gIiKZ7Uo2r8Grzdbu39oTm9nLQFmCot8A/wcc1tA53H0UMApg6NChLaamMT/PeOCHe9O5TVHUoYiIZBwliGkQOxeiEkQRkRbtXjZPENcAHwL/3toTu/shibab2QCCPo+1tYe9gA/MbE93V5vKUJe2rQBYvGoD7YoLKC7U/IgiIqAEMS322bYzj563D306tY46FBERiZC7j4zgmh8D3WrXzWwGMNTdNWxnnLnL13Ho38byq0N34JwDto06HBGRjKBBatKgc9tWDK3oREmR7kaKiMg3zKyNmR0X9hGUiPXsUMKQbTpy82tfsGr9xqjDERHJCEoQ0+Spj+by/oyvow5DREQySy/gSeDx5rqgu1eo9rBuFx/en2VrN3LHG9OjDkVEJCMoQUyTPz47hUfHz4k6DBERySwbCeYlnB11IBIY2KsDR+5axh1vfMXS1RuiDkdEJHJKENOkvLSY+Ss1F6KIiHwjnAuxwt37Rh2LfOPCw3ZgQ1UNY6YujjoUEZHIaZCaNCkrLWb6kjVRhyEiIiIN2K5bO9749QjKS0uiDkVEJHKqQUyT8tIS5q9QDaKISEtmZu3N7BYz+8zMlprZ1+GyNOrYZHO1yeEitf4RkRZOCWKalJUWs2p9Fas3VEUdioiIROdm4DxgB6Aj0CFmkQzz5Idz2e/aV/li0aqoQxERiYwSxDQ5ZWhv3vz1CFpr4l0RkZbscOBzgpFLHTgVWAD8LMKYpA4HbN+Fovw8/vri51GHIiISGSWIadKpTRG9OrYmL8+iDkVERKLTAXgZ+DJcfwL4H/CDqAKSunVu24pzDtiW5yYvYNKc5VGHIyISiWZJEM3sLjNbZGaT6yg/3cwmhcvbZrZbXHm+mX1oZs/Ebf+pmU01s0/M7C/pfAyNtbayilvGfMEHs5ZFHYqIiERnBVACLArXbyaoVewfWURSr3MO6EvH1oVc98LUqEMREYlEc9UgjgaOqKd8OjDM3QcCfwBGxZX/HJgSu8HMRgDHAwPdfRfg+pRFmwJ5Zvzl+am8NU1zE4uItGBTgF2BlwADzgF6Au9EGZTUrV1xIReM2I73Z3zN7K/XRh2OiEiza5ZpLtz9dTOrqKf87ZjVcUCv2hUz6wUcDfwR+FXMfucD17j7hvAci8ggxYX5dG5TpLkQRURatlOAdu4+zcxOD9cXAldFG5bU54y9t+G43XrQrX1x1KGIiDS7TOyD+APguZj1vwOXADVx++0AHGBm75rZWDPbo64Tmtm5ZjbezMYvXtx8k+CWlRYzf/m6ZrueiIhkFndf4O7Twr8fdPdvuft57j4/6tikbsWF+XRrX4y78/WayqjDERFpVhmVIIbNRn8A/DpcPwZY5O4TEuxeQDBk+N7AxcDDZpZwRBh3H+XuQ919aNeuXdMTfALlpcWaC1FEpIUxs1lmdnmq9pPoXPLoJE69/R2qazzqUEREmk2zNDFNhpkNBO4AjnT32gmE9wOOM7OjgGKgvZnd5+5nAHOAx93dgffMrAboAjRfFWEDykqL+WDW8qjDEBGR5tULGGBmByaxX8dmiEeaaHj/bjwyYQ5PfjiXE4f0avgAEZEckBEJopn1AR4HznT3TZMPuftlwGXhPsOBi8LkEII5pQ4CxpjZDkARkFEjwlx65E5ccewuUYchIiLN78RwqY+qpTLckbuWMaBnKTe8/DnH7FZOqwLNbSwiua9ZEkQzexAYDnQxsznAFUAhgLvfBvwO6AzcErYSrXL3oQ2c9i7grnDqjErgrLA2MWO0bZUR+beIiDSv10k++fsinYHI1snLMy4+vD/fu+s9Hnx3FiP36xt1SCIiaddco5ie1kD5OQRDf9e3zxhgTMx6JXBGXftngrnL13H72C/57l592LGsfdThiIhIM3D34VHHIKlzwPZd2HvbTjw8fg5n7VtBHcMdiIjkDFVxpdGGjdXc+85MBvfpoARRREQkC5kZfz1lEJ1aFyk5FJEWQQliGpWVBvMnaSRTERGR7NWzQwkAG6tr2FhdQ+si/XwSkdyVUdNc5JrWRQWUlhSyQAmiiIhIVlu/sZrDb3idG1+eFnUoIiJppVtgaaa5EEVERLJfcWE+ndoUcvvrXzHq9a/o0aGEiw/vzwmDe0YdWso9+eFcrnthKvOWr8vpxykiiakGMc16dChhQ1VN1GGIiEgzM7N8M9toZv+JOhbZek9+OJeP564EgiFq5y5fx2WPf8yTH86NNrAUe/LDuVz2+MfMXb4upx+niNStzhpEM7urEed50d31BZjAnWcNVad2EZEWyN2rzexzQHcJc8B1L0zd4obvuo3VXPfCVB6dMIcPZy3brGxgrw48eO7eAJxw81tMW7hqs/J9+nXmjrP2AOCwG8Yyd9m6zcoP3qk7/zhtMAD7XfMqy9dWblZ+3KCe/PnbAwAY/PsXqYyL7Tt79uHyY3ZmY3UNg656cYvH84P9+/Krw/qzfG0l+13z6qbtayurt5ijpfZxqhZRpGWor4npSIKbZA1lNw4sB5QgJqDkUESkRbsfuNLMpgJvAxtqC9z99ciikkabt3xdndt/eEBfdixrt9n2HuHANgBHDShj0TYdNyvv27XNpr+P260Hy9du3Ky8f8z5Tty9J2srqzcrH9CrdNPfp+zRm+rqzdO6IeH18sw4bc8+W8Q9qE8HAFoV5G9Wfseb07d8kNT9+EUk9zTUB/El4MF6yo1gwnqpw4SZy7h1zJf84YRdKC8tafgAERHJJVcT3Ej9Xdx2R+MAZJUeHUqYmyBJ6tGhhJH79a332HMP7Fdv+U8O2r7e8l8d1r/e8suO3KnOsvw847fH7FxneUlR/mblz01eUOfjFJGWoaEvp0/d/Z76dmhkU9QWZ82GKl6espBzD9xWCaKISMszC7ZosSdZ6OLD+3PZ4x+zbuM3NXklhflcfHj9yVu2SfQ4iwvzcu5xikjd6ksQC0mu30Sy+7VI5ZvmQlTTDBGRlsbdK6KOQVKjtv9dro/uGf84HTiof7ece5wiUrc6E0R3r66rrCn7tVRlYYKouRBFRFomMysFTga2AWYAj7n78ihjkqY5YXDPFpEoxT7O7931HhNmLWNjdQ2F+Rr8XqQlaNI73cy2M7OvzWxpqgPKNe2KC2nbqkBzIYqItEBmtgPwKXA78H/AKOATM1N7PckKZ+9bwcKVG3h+8oKoQxGRZtLUW0H5QIdwkQYM6FlKq0LddRMRaYGuB8qBicBDwEfh+l8ijEkkacN26Mrxg3rQuW1R1KGISDNp6ghqs4ARqQwkl9XOgyQiIi3O3gRzBR9Ru8HMngP2iS4kkeTl5Rk3fmdw1GGISDNqUrWWu69z97HuPjbVAYmIiOQQAyrjtlXS8BzDIhll8aoNvPCJmpmKtARJ1SCaWfz8TbXc3f+Qwnhy0lMfzeWON6bzxI/3pUAdvEVEWpIJwNFm9gowBdiRoAXOi5FGJdJIN7/2BQ+8O4u3Lj2Iru1aRR2OiKRRsk1Mr6TueZyUIDZgbWU1H89dwaJVGzTRrIhIy3IJMJYgKRxOUHO4Arg0wphEGu3MfbZh9NszePC9Wfzs4O2jDkdE0ijZBPFevkkQ84FdgUHAk6kPKfeUbZoLcb0SRBGRFsTdJ5nZTsCZQAXBNBf3ufv8KOMSaax+XdsybIeu/HvcTM4b1o+iArWIEslVSSWI7j4yfpuZPQRoDsQklLXXXIgiIi2NmRUALwBvu/vlUccjsrVG7lfB2Xe/z3OT53P8oNyfD1KkpWrqPIhtgWLg8NSGk5vKN9Ugros4EhERaS7uXkXQ57As6lhEUmHY9l3p17UNny1YFXUoIpJGyQ5SU1dN4VcpjCVnlZYUsmdFJ0pLCqMORUREmtfvgavN7D6CmsSNUQck0lR5ecazPzuA4sL8qEMRkTRKtg9iouG45wDnpzCWnGVmPHyeprwSEWmBbiXow/8qBN8HIXf3ps5FLBKZ2uRw2ZpKOrYpijgaEUmHZJuY9o1burl7H3d/OW2RiYiI5AZLsGiED8laD7w7i73//AqLVmlsBZFclNQXlLvPjFuWpDuwXPPXF6dyzE1vRB2GiIg0r/gbrLGLSFbap19nNlTV8MC7s6IORUTSoKmD1GxjZh+Y2YRUB5Sratz5bP4qqmvqmk5SRERySTiK6V3AOQlutM6MOj6RpurbpQ0j+nflvnGzqKyqiTocEUmxpjZxKSaYB3FQyiLJcWWlJVTVOEtXb4g6FBERaQYaxVRy2cj9+rJk9Qb+97Gm9BTJNU1NEOcDZwPfT2EsOa28fe1UF2qvLyLSgvweOMHMhpmZhrKWnHHAdl3Ytmsb7hunynCRXNOkEdTcfSVwT4pjyWllMXMh7ta7Q7TBiIhIc9EoppKT8vKMG04ZRI8OJVGHIiIplvSXk5n1AfYFuhIz7YW7/yMNceWcXh1LOGpAGR1ba0hoEZEWJtFUUYm2iWQV3fAWyU1JJYhmdiLwb6BVgmIliEno0LqIW04fEnUYIiLSvDRaqeS0yXNXcPWzn/KP7wymW9idRkSyW7J9EC8HCoGFBHc9pwBVwNg0xZWzNIqpiEjLkWj0Uo1iKrmkbasC3p3+NfdryguRnJFsgrgj8BjwH4J+E7sCrwPvpiuwXHTOPe9z2r/GRR2GiIikWTgV1K/Cv08zs7tiyv5kZkuji04kdSq6tOGg/t24/91ZbKiqjjocEUmBZBPEKmAxsBrAzPoCa4Hz0hRXTmrTqoAFGsVURKQlGAT0Cv/eEzgrpqwE6NDM8Yikzcj9KliyegPPTtKUFyK5INkEcR5QDkwjaGI6FTgGWJemuHJSWWkxC1asx13NTEVERCQ37L9dF/p1bcPdb83QbxyRHJBsgng7wdyHjwGfEAxu4wTzO0mSytsXU1ldw9drKqMORURE0k+/lKVFMDN+eegOnDSkFxpqQST7JTWKqbv/tfZvM9sd2BlY4u5z0xVYLiorDeYKmr9iPZ3bJhoQVkREcsiPzexsgialmNnX4XZNHCc555iBPaIOQURSpM4aRDP7vpntGb/d3Te6+8Ta5LCu/WRL/cva8YP9+9KuWHMji4i0AK0I+hq2Iuie0SFmXSTnrKus5oF3Z2m8BZEsV1+mcgdwA/BeA+e4A/h7Evu1eH27tOHyY3aOOgwREUm/EVEHINLcFq/awG+e/JifjNiOCw/rH3U4ItJEDVVlnWxmgxvYR63NG2H9xmo2bKyhtHVh1KGIiEiauLvmCZYWp0/n1hy8Y3ceeHcWF4zYjuLC/KhDEpEmaGiQml7A8AYWa+giZnaXmS0ys8l1lJ9uZpPC5W0z2y2uPN/MPjSzZxIce5GZuZl1aSiOTHDwX8dy1X8/iToMERERkZQ7e78Klq6p5BlNeSGSteqrQWxM85g5DZSPBv4J3FtH+XRgmLsvM7MjgVHAXjHlPwemAO1jDzKz3sChwKxGxBqpstJi5qttvoiIiOSgfft1Zvtubbn7remcuHtPzBqsRxCRDFNngpjK5jHu/rqZVdRT/nbM6ji+mVwYM+sFHA38EfhV3KE3AJcAT6Uq1nQrKy3m03krow5DREREJOXMjJH7VfDw+7NZsW4jHVoXRR2SiDRSsvMgNqcfAM/FrP+dIAmsid3JzI4D5rr7xIZOaGbnmtl4Mxu/ePHiVMbaaOXti5m/Yp0mkhURaQHMTGP/S4tz2h59eOon+ys5FMlSGZUgmtkIggTx1+H6McAid58Qt19r4DfA75I5r7uPcveh7j60a9euKY66ccpKi1m/sYYV6zZGGoeIiDSLWWb2ipn9wMw6RB2MSHPIywualS5bU8nytZURRyMijZUxCaKZDSSYMuN4d18abt4POM7MZgD/AQ4ys/uAfkBfYGJY1gv4wMzKmj3wRtqnX2d+e/ROmz48RUQkp60h6NM/CphvZo+b2clmprkQJaetWLeR/a59lX+98VXUoYhIIyWVIJrZ3ukMwsz6AI8DZ7r757Xb3f0yd+/l7hXAd4BX3f0Md//Y3bu5e0VYNgfY3d0XpDPOVNilRynnHLAt7Ys1zYWISAvQGTgCuBVYCJxAcMNzkZndbWZ71XOsSNYqLSlkv+268MC7s1i/sTrqcESkEZKtQXzbzL4wsz+YWaNnejezB4F3gP5mNidsanOemZ0X7vI7gi/RW8zsIzMb39hrZIuaGmfGkjUsWqmRTEVEcp27V7n7i+7+E+BY4CWC6aHaAWcRfL9eHGWMIuly9n4VLFu7kacnzos6FBFphGQTxM+BbQn6/X0czkl4UTjCaIPc/TR3L3f3wrBG8E53v83dbwvLz3H3ju4+KFyGJjjHGHc/po7zV7j7kiQfS6Rq3Dnor2O4792smZlDRESayMx6hN+XHwEfEUzNtB64n6BlzETg0sgCFEmjfbbtTP/u7Rj91gwNzieSRZJKEN19R2AHgtFE3wR2Ba4FZpjZa2Z2cvpCzC0F+Xl0a1fMghXrog5FRETSbybB9+VAgmTwp0C5u5/p7g8D9wEdI4xPJG1qp7z4dP5KPluwKupwRCRJdc6DGM/dvzCzG4CpBBPXH0TQTGYYcKCZ7efuv0hLlDmme2kx81eoiamISAuwBngAuMPdP0hQ/i/gsXRc2Mx+CvwEqAKedfdL0nEdkfqcMKgne/btRL+ubaMORUSSlFSCGHaiPxM4haCvoAFzgdHAfwnujn4f+EU6gsw15e2L+WLx6qjDEBGR9Ctz9zrvCLr7KiDlVSvhtFHHAwPdfYOZdUv1NUSSUVKUvyk5dHfMNIq7SKZLtgbxnfDfKuAp4E7gOXevATCzpwlqEiUJZaXFvPlFVnSZFBGRrfOnOn4QbyDo3/+Yu69Mw3XPB65x9w0A7r4oDdcQSUpNjfOz/3xI3y5tuPCw/lGHIyINSDZBnEaQFI5O9CXj7jcAN6QysFz27d17skdFJ2pqXPMhiojktl8A9Y3OcZWZDXf3VE8WtwNwgJn9kWBQnIvc/f34nczsXOBcgD59+qQ4BJFAXp5RVe3cN24mF4zYjuLC/KhDEpF6JDtITX93/4vuQKbGwF4dOHpguZJDEZHc9xBBgjgp/HtiuP4UQZ/+XsAfmnJiM3vZzCYnWI4nuAHcEdgbuBh42BJUZbr7KHcf6u5Du3bt2pQwRJIysnbKi4805YVIpku2D+LjdRTVNpG5y91npiyqHLeuspoPZy+jX9e2dG9fHHU4IiKSPhuBl9z9yNoNZvYcQb/DXYCXgf2bcmJ3P6SuMjM7H3jcg7kF3jOzGqALsLgp1xLZWnv17cSOZe24++0ZnDy0l/oiimSwZOdBPIGgs/sJMcvxwKnAb4FJZjYo1cHlqiWrN/Ddf73L2Kn6nhYRyXHfBorithUA3w6TtzFAOgaQeZJgtHHMbIcwBnV+l8iYGSP3rWDK/JW8N/3rqMMRkXok2wfxnwQd3p8FPgV2Ao4F7gAqCCb+/T1wXOpDzD21tYaa6kJEJOdNB4ab2TSCJqXbAdsDk8PynYB0dN+4C7jLzCYDlcBZrpnKJWLHD+rJgpXr6du1TdShiEg9kk0QtwWecPdTajeY2cNAD3c/PGyCulc6AsxFRQV5dGnbigUr10UdioiIpNf5BNNB9QsXgOXA+WbWnuB7+NZUX9TdK4EzUn1eka1RUpTPLw7ZIeowRKQBySaII4BPzKzY3debWSugD7BrWP4hcGSdR8sWykuLVYMoIpLj3P1NM6sAjgF6Eswh/Iy7rwh3OaWuY0Vy1StTFrJ0dSWn7NE76lBEJIFkE8SPgT2AhWY2m2DUtXbAu2H5UGBO6sPLXWWlxcz+em3UYYiISJqYWQHwGfCCu18QdTwimeLRCXN4+8ulHLtbD0qKNOWFSKZJdpCa7xP0o2gH7Ay0B74Evh82kfkSuDotEeaonx+8PX85aWDUYYiISJq4exXB92zrqGMRySQj961gxbqNPPnR3KhDEZEEkqpBdPdPzaw/wXxKtU1kxrl7dbjLr9IUX87atWdp1CGIiEj63Qj80czGAG8TTA8FgLvPiiookSjt2bcTO5W3Z/RbM/jOHr015YVIhmmwBtHM8s1sI3C/u7/l7g+H/1Y3dKzUbdGq9Tw2YQ5fr6mMOhQREUmfG4ASglFFPyNojTMd+CrKoESiZGacvV8FUxeu4p2vlkYdjojEaTBBDBPBz4Ga9IfTcny5aA0XPjKRz+avjDoUERFJL0uwJNvFQyQnHbdbD3br3YE1G1TfIJJpkh2k5n7gSjObypZNZF5PR2C5rrxUcyGKiOQ6d1ciKJJAcWE+T12wX9RhiEgCySaIVwMO/C5uuzfiHBKjLEwQF6xUgigiksvMrJBguqh+BE1NuwNL3F1DWUuLt35jNdMWrmZAL43NIJIpkr2zOauOZXaa4sp5xYX5dGxdyPwV66IORURE0sTMegMfAc8BNwEdgC+AP0QXlUjmuPzJyZx+xzjWVlZFHYqIhJJKEN29wt37JlrSHWAuKystYYGamIqI5LIbgJ2AJYC5+0LgdeCISKMSyRCn7NGbleurePLDeVGHIiKhpPtGmFknMzvXzK41s1IzO9DMuqczuFx302mD+dO3B0QdhoiIpM+BwDPAAzHbvgD6RBOOSGYZuk1HdunRntFvT8fdow5HREgyQTSzXYApwK3AReHm54Ar0xNWy7Bdt7Z0a1ccdRgiIpI+DsTPZ1QBrG7+UEQyj5kxct8KPl+4mne+1JQXIpkg2RrEvxH0m/gKwN1XAGOBQ9ITVsswZf5Kbn7tCzZUaYhnEZEcNR44BjgUwMweCf9+L8qgRDLJsbv1oFObIl6asjDqUESE5BPEPYEnCZrJ1JoJ9Eh1QC3J5LkruO6FqSxcsaHhnUVEJBtdAqwFdiaY//BEYAXwmyiDEskkxYX5PPPT/fndMTtHHYqIkPwUFWuAtnHbBgJqC7AVyktLAJi/Yh19OreOOBoREUk1d//EzHYCvgdsA8wA7nf3+ZEGJpJhenQIfhNVVddQkK/pQ0WilGyC+CZwEjAAwMzeA4YAD6YprhZBcyGKiOS+cOTS66KOQyTTPTJ+Nn976XNe/tUw2rTSNNsiUUn2Fs3FBHc9exE0kRkKTAf+Lz1htQy1CeJ8TXUhIpKTzKy3md1jZpPM7KuY5cuoYxPJNNt2bcP8Fet54sO5UYci0qIldXvG3Web2QCCjva1TWT+5+5r0xhbzmvbqoB2xQWaC1FEJHc9COxDcHM1lsbzF4mze5+ODOhZyui3Z3D6Xn0wi3/biEhzSLr+3t3XAY+kMZYW6dULh9OxdWHUYYiISHoMBmYTTAu1HCWGInWqnfLiwkcm8tYXS9l/+y5RhyTSIiU7D+JAMxtrZivNrDpmqUp3gLmua7tW6owtIpK7xgNvuPtod3/S3Z+qXaIOTCQTHbNbOV3aFjH67elRhyLSYiVbg/hvwgFq4qjufyu9+tlC3v3qay47aqeoQxERkdR7C7jYzJYBHwGbbqy6+71RBSWSqVoV5PPnbw+kV8eSqEMRabGSTRC3Az4BfoqayKTUxNkrGPXGV1x0eH8KVZMoIpJrLiX4zrwgQZkSRJEEDt25e9QhiLRoySaIrwAb3X1MGmNpkcpLi3GHRas20LOD7paJiOSY19FNVZFG+2LRam4f+yVXHreLprwQaWbJvuPmAuea2dNs2UTm92mIq8XYNBfiinVKEEVEcoy7D486BpFstHL9Rh6ZMIeBvUo5c5+KqMMRaVGSbdP4I4L+hscAvwGuIBiR7Yr0hNVylJcGSeG85ZrqQkQkV4SDu5XXUbaLmR3X3DGJZJPBvTuwW69gyouaGlXCizSnZGsQ70VNZNKirLSY/DxjxbqNUYciIiKp8yHwd+BCM7sB+Km7137nngP8DMiPKDaRjGdmjNyvgl8+NJE3v1jCgTt0jTokkRYjqQTR3UemOY4Wq31xAZ9ffST5eRoQVkQkhxibj/StD3mRRjpqQDl/fPYzRr89QwmiSDOqt4mpmR1nZruEf3cwsz4xZQeY2c/SHWCuMzMlhyIiIiJxWhXkc96wbdm+e1vc1ZBNpLk01AfxSeD74d9XALGzln4buCENMbU4d781nT8++2nUYYiISGr1C/sa9gMws2Nj10WkYeccsC2XHbkTZrqZLtJckhmkRu/INJs8dyXPTJofdRgiIpJaxwBPAEeH60/GrYtIEmpqnDFTF7F6Q1XDO4vIVmuWmdnN7C4zW2Rmk+soP93MJoXL22a2W1x5vpl9aGbPxGy7zsw+C495wsw6pPlhpE15aTGLVm2gqrom6lBERCQ1ZiWxiEgSPpm3kpF3v89jE+ZEHYpIi5BMgjg47Gs4GMDMfhq7nqTRwBH1lE8Hhrn7QOAPwKi48p8DU+K2vQTsGh7zOXBZI+LJKOUdiqmucZasrow6FBERSQF3r3D3vvUtUccoki0G9Cplt94duEdTXog0i2QSxAMJ+hoeQNDc9O/h+oHJXsTdXwe+rqf8bXdfFq6OA3rVlplZL4LmOHfEHfOiu1clOibblJcWAzB/xbqIIxERERHJPGfvW8FXS9bw+rTFUYcikvMamubidZp//sMfAM/FrP8duARoV88x3wceqqvQzM4FzgXo06dPXbtFpry0hB6lxazbWB11KCIiIiIZ56gB5fzxf1MY/fYMhvfvFnU4Ijmt3gTR3Yc3UxwAmNkIggRx/3D9GGCRu08ws4SxmNlvgCrg/rrO6+6jCJutDh06NOPaJuxU3p63Lzs46jBEREREMlJRQR5n7LUND70/i9UbqmjbKqmpvEWkCTLm3WVmAwmakR7p7kvDzfsBx5nZUUAx0N7M7nP3M8JjziIYJe5g1wQ5IiIiIjnr3AO35ccj+lGY3yxjLIq0WBnxDjOzPsDjwJnu/nntdne/zN17uXsF8B3g1Zjk8Ajg18Bx7r42grBT6rLHP+ba5z+LOgwREUkDM9shvNkpIk1UUpRPYX4elVU1rFe3HJG0aa5pLh4E3gH6m9kcM/uBmZ1nZueFu/wO6AzcYmYfmdn4JE77T4J+iS+Fx9yWnuibx/Qlqxk/o85xfEREJAuZWScze5lgJO6nzazCzKrM7A9RxyaSjZau3sD+177Kf97TTDEi6dIsTUzd/bQGys8BzmlgnzHAmJj17VIRW6YoLy3hfSWIIiK55jrgIGADUOTuM8xsHEH3iMsjjUwkC3Vu24peHUu4552ZfG+fCvLyLOqQRHJO0jWIZlZoZoeZ2Y/NrJWZ9TGz1ukMriUpKy1m4cr1mt9HRCS3HEFwczO2lcunwLaRRCOSA87at4LpS9YwVlNeiKRFUgmimfUGPiKYfuIfQAfgC4JJ7SUFykuL2VjtLF1TGXUoIiKSOiXA/LhtXQB92Is00ZG7ltOtXStGvzUj6lBEclKyNYg3ADsBSwBz94UEcyQeka7AWpq+XdowZJuOrKtUp2sRkRwyiaA56V4AZnYdcCwwMcqgRLJZUUEeZ+y9DWM/X8z0JWuiDkck5yTbB/FA4BngS+Bn4bYvCL/wZOsdsH1XDti+a9RhiIhIav0WeBnYO1y/ENgIXBlVQCK54PS9+rB7n45UdFZvJ5FUSzZBdLZsDlMBrE5pNCIiIjnE3d80syHA+QTfmzOAUe4+Kcq4RLJd57at2H/7VlGHIZKTkm1iOp6gicyhAGb2SPj3e2mKq8Vxd47755v889VpUYciIiIpYma/A3q6+0/c/Rh3/wngZvbtqGMTyXYbq2u4+plPeVBTXoikVLIJ4q+BtcDOgAEnAiuA36QprhbHzPh6TSVfLlZbehGRHHIlW/bX/z7wSPOHIpJbCvPz+HD2cm4b+6VGgRdJoaSamLr7ZDPbCTiTb5rI3O/u8SOzyVYoLy1m/op1UYchIiJbycy+F7O6c8x6HjACqGr+qERyz8h9K/jpgx8y5vNFHLRj96jDEckJSSWIZnYX8KK7Xx+zbZiZnebuf0tbdC1MWWkJH89ZHnUYIiKy9UYT9N93gi4Zh8aUGRrFVCQljti1jO7tW3H3WzOUIIqkSLJNTEcCe8Zt+xZwXUqjaeGCGsT1uKuZhIhIlns9XAyYF7P+GnAPQYscEdlKhfl5nLn3NrwxbQlfLNLYiSKpUG8NYti5vtbeMet5BPM4aaLfFBrUuwPzdu7Ohqoaigvzow5HRESayN2HA5jZa8Cj7n5ztBGJ5K7T9uzDjKVrKcy3qEMRyQkNNTG9km+ayOzF5vMeGvBWesJqmY4aUM5RA8qjDkNERFLE3UcAmFkZ0JXgu7O2TFNdiKRA57atuP7k3aIOQyRnNJQg3kuQHJ4FTOGbaS2qgdnAv9IXWsvl7pjpLpiISLYzs12Ah4Ed44qc5OciFpEkTJ67gq/XVHLgDl2jDkUkq9X75eTuI4HaZOVFd3+wGWJqsZas3sDBfx3LJUf05/S9tok6HBER2Xr/AHaKOgiRluCq/37CwpUbeO2i4eTn6Ua7SFMlNUiNu58NPG1me5vZsWZ2XO2S5vhalI6ti1i9oYoFK9ZHHYqIiKTGHsBY4J8EtYbdCUYw/U6UQYnkopH79mXW12t57bNFUYciktWSneZiGEETmS5xRWoik0L5eUa3dq2YrwRRRCRXFAGTgTXh+nrgI+B64JGIYhLJSYft0p3y0mJGvz2DQ3bWlBciTZXsNBfXEHSuX0fQwX4FQT/EWWmKq8UqKy1WDaKISO5YDHQA5hJ8f74BnAKURhiTSE4qzM/jjL234c0vljBt4aqowxHJWskmiAOBZwkGpaltIjMBuCFNcbVY5aXFzFuxLuowREQkNd4AKoDHCWoRBwIlqPZQJC1O27MPXdoW8flCzYko0lTJNg/NA2YAa8P1UuAr4LfATakPq+U6eMfubNe1bdRhiIhICrj7d2v/NrPBwNHAQuChyIISyWGd2hTxzmUHU5ifbB2IiMRL9t2zgKD/4QyCJjKTCTrYazb3FDtxSC9+dVj/qMMQEZEUMLNXzex8AHf/wt1vBCqBG6ONTCR3Febn4e4sXKkuOyJNkWyC+DhQCDxKkCx2I0gUb01TXC3a2soqKqtqog5DRES23nBg+wTbLmj2SERakEsencSJt75NdY1HHYpI1kmqiam7X1j7t5ntChwALHT3cekKrKWaMHMZJ976NqPP3oPh/btFHY6IiDSBmd0Vs3pYzHoecBTBoG8ikiYjduzGIxPm8MqUhRy2S1nU4YhklaRqEM3sKzP7DYC7f+3uTwG7mtlTaY2uBerevhWARjIVEcluI4GzCAZ22zlcHwl8j6DLxusRxSXSIhy28zdTXohI4yQ7SE0F0Dlu22DgmJRGI3RrV4wZmgtRRCS7XRX+ewXwLvB8uF4NzEajmIqkVUF+Hmfusw1/eX4qUxeson9Zu6hDEska9SaIZvZqzOpJZjYo/DsPGAJokpkUKyrIo0vbVupYLSKSxdz9KgAzAxjn7i9EG5FIy3PaHn248eVpPPT+bH537M5RhyOSNRqqQRwe/utAr3CJdX+qAxLoUVqsGkQRkRwQkyjmA8cC7YA33H1GlHGJtAQd2xTx4Ll7s2uP0qhDEckqDSWIZxOMVnoX8BLwQLi9tomM+lCkwZn7VFCQZ1GHISIiTWRmfweOAPZ29+XAM8BhYfF6MzvO3V9J4/UHAbcBxUAV8GN3fy9d1xPJVLv36Rh1CCJZp94E0d3vATAzBz519/ebJaoW7qQh8RW1IiKSZQ4Dlrr7cjPbHTg83D4R2I2gb2LaEkTgL8BV7v6cmR0Vrg9P4/VEMtYzk+Zx7zszeeCcvSjIT3aGN5GWK6l3ibvf4+7vm1l7M/uFmV1uZvumO7iWav3Gar5cvJqN1ZoLUUQkS/UEPg3/rq05fMDdBwNPA7um+foOtA//LgXmpfl6IhmrIC+P96Z/zctTFkUdikhWqDdBNLP/mNkaM+tgQU/7V4G/AlcCr5vZqc0QY4vzwicLOPivY5m5dG3UoYiISNPk800rnX0IErbnwvWFQFGar/8L4Dozmw1cD1yWaCczO9fMxpvZ+MWLF6c5JJFoHLJTN3p2KGH029OjDkUkKzRUgzgEeDfsP7EfsDuwkeDuZw1wcVqja6HK2hcDmgtRRCSLTQNONLMbCJqXOsFNVoDtCJLErWJmL5vZ5ATL8cD5wC/dvTfwS+DOROdw91HuPtTdh3bt2nVrQxLJSAX5eXxvn20Y99XXTJm/MupwRDJeQwliGfB5+Pch4b93u/u3gMeA7dMVWEtWXloCwPwV6yKOREREmujPQGvg5wS1hQ+4+3wzKwP2J5gbcau4+yHuvmuC5SngLODxcNdHgD239noi2ezUPXpTXJjHve/MiDoUkYzX0CimNQRDcgPsS3AHtLZT/RqCEU4lxbq1bwWoBlFEJFu5+8Nm9gUwDJgPPBwWlQK/At5OcwjzwmuPAQ4iqNEUabE6tC7isiN3om+XNlGHIpLxGkoQJwMnmVlrgi+Yar5JEHck+NKTFCsuzKdzmyLmr1SCKCKSrdz9A+CDuG1TganNcPkfAjeaWQGwHji3Ga4pktHO2rci6hBEskJDTUx/R9Dn8Phw3xvcfZmZbQvsDbyZ5vharN8duzPfHtwz6jBERCQFzOwGM6tqruu5+5vuPsTdd3P3vdx9QnNdWySTzV+xjptemUaVRooXqVND8yC+YmY7AHsB8919XFhUmzR+kub4WqzjByk5FBHJMeqWIRKxj+es4K8vfc523dpy5IDyqMMRyUgNzoPo7vPc/Ql3HxdOd9HH3We7+7PuPqMZYmyRFq1cz9tfLok6DBEREZGccfBO3enVsYS7354RdSgiGavBBDHOFYAmkWkGj0yYw3f/9S7rN1ZHHYqIiGy994B7ow5CpKXLzzPO2qeC96Z/zSfzVkQdjkhGamyCKM1EcyGKiOQOd3/Q3c+OOg4RgVOG9qakMJ97VIsoklBDo5imhJndBRwDLHL3XROUnw78OlxdDZzv7hNjyvOB8cBcdz8m3NYJeAioAGYAp7j7sjQ+jGZVXhokiPNXrKdCQzKLiGQNM3u84b1wdz8x7cGIyBZKWxdyytBerKmsxt0xU/dgkViNTRCfB5Y34TqjgX9Sd/Oa6cCwcITUI4FRBAPj1Po5MAVoH7PtUuAVd7/GzC4N139NjigLE8QFK9dFHImIiDTSCXHrzpYD1HjzhCIiiVx53C489dE89r/2NeYtX0ePDiVcfHh/TsixEeSf/HAu170wNacfo6ReoxJEd38BeKGxF3H3182sop7y2AmDxwG9alfMrBdwNPBHgsmFax0PDA//vodgMuCcSxDnq4mpiEi2uSrm787A+cBTwOdAf4IE8rbmD0tEaj310Twue/xj1oVjPcxdvo7LHv8YIGcSqCc/nJvzj1HSo84E0cw+qKsshrv7kBTGA/AD4LmY9b8DlwDt4vbr7u7zwyDmm1m3uk5oZucSThLcp0+flAabLq2LChh15hB2Km/f8M4iIpIx3H1TgmhmTwOPuft3YrY9BGwbRWwiErjuhambEqda6zZWc8mjk9h7286UlRbzxIdzuH/crC2OvfOsPShtXcgD787i8Q/mbFF+/w/3olVBPne+OZ3nPp6/WVlenvHwj/YB4ObXvuC1zxZtVt6mVQH3fH9PAP764lTe+XLpZuWd2xZx+5lDAbj6mU/5aPbyzcp7dSzh798ZDMD/PfFxwsd43QtTlSBKveqrQRwUt572JjJmNoIgQdw/XK/ttzjBzIY39bzuPoqg2SpDhw7NmmY9h+1SFnUIIiKydQ4GJptZK3ffYGbFBH3nt+iPLyLNZ97yxF14KqtryAt/7eaZ0aowwXiOYXlBfuJyC3coTFCeF9PfsSBvy/KigryY8rwtylsV5H9Tnr9leezxaysTj4Rf12MXqVVfghg72lpfguabN/NNE5kLgL+mKhAzGwjcARzp7rW3S/YDjjOzo4BioL2Z3efuZwALzaw8rD0sBxYlPnP2mjx3BfNXrOfQnbtHHYqIiDTNx8AeBN9Zcwi6ULQjmPZCRCLSo0MJcxMkSj07lNAtHEn++EE9OX5Q3TVtpwztzSlDe9dZ/r19KvjePhV1lv9oWD9+NKxfneU/P2R7YPs6yy89csc6yyB4LIkeY48OJfUeJ1LnNBfufk/tAhwEPOTuF7n7KHe/kGAE0RGpCMLM+gCPA2e6++cxMVzm7r3cvQL4DvBqmBwCPA2cFf59FkH/jpxy/7szN7UVFxGRrHQ2wUBs7YGdw3+nE7SWEZGIXHx4f0oK8zfbVlKYz8WH948ootRrCY9R0iPZQWp2BzqbWQ93n2dmPQnuiCbVoc/MHiQYUKZLeAf1CqAQwN1vA35H0JH/lnCo4Sp3H9rAaa8BHjazHwCzgJOTfCxZo6x9CUtWb6CyqmazJgMiIpId3H2KmfUH9gZ6AnOBce6euO2XiDSL2j54uTzCZ0t4jJIeySaIbwOHALPNbB1QWzf9cjIHu/tpDZSfA5zTwD5jCEYqrV1fStC3I2fVzoW4cOV6endqHXE0IiLSFO5ebWZTgNnhpp5mhrtvOfqFiDSbEwb3zPlkKfYxfjBrGY+Mn81xu/UgL09zP0rdkq2WOgt4naBbbuvw3zeA76cpLiF2LkRNdSEiko3M7GAzmwEsJmhaWrt8FWVcItLyzP56LQ++N5t3vlra8M7SoiWVILr7fHcfDvQmaCbT292Hu/vcdAbX0pVrLkQRkWx3G0F3DItb1G9ARJrV4buU0a64gEfGz254Z2nRkv6CMrO2QHeC0UT7mdmBZnZg2iIT+nRuzVMX7MeI/l2jDkVERJqmDBhHMO9hJ6BjzCIi0myKC/M5brcePDd5ASvXb4w6HMlgSSWIZvZdYB7wPvBazPJq+kKTVgX57Na7A+2KC6MORUREmuZxYLW7z3D35e6+onaJOjARaXlOHtqbDVU1PDNxftShSAZLdpCavwBtgWpgDZA1k81nuxc+WUBVtXP0wPKoQxERkcZrDRxsZpOAj4CqcLu7u6a6EJFmtVuvUkb070pBvgapkbolmyC2AV4EjnH3qoZ2ltS5b9xMVq6vUoIoIpKdTgz/3TVcajmaC1FEmpmZcffZe0YdhmS4ZBPEu4F9CTrWSzMqa1/MtIVLog5DRESa5veo1Y2IZJjKqhpmLl3D9t3bRR2KZKBkE8Q9wmVeOJdTbBOZnJ6LMGrlpcUsWrWequoaCvI16J2ISDZx9yujjkFEJN4lj07krS+X8s6lB+n3pWwh2QRxv/DfzsD+Mdt1VzTNykpLqHFYvHoD5aUlUYcjIiKNYGa/q6PI3f0PzRqMiEjoyAHlPPnRPF6ftpiDduwedTiSYZJNEM9OaxRSp9i5EJUgiohknSup+2aqEkQRicRBO3ajc5siHhk/RwmibCGpBNHd70l3IJLY3tt25r3/O5gubVtFHYqIiDTevXyTIOYTDFQzCHgyonhERCjMz+OEwT25950ZfL2mkk5tiqIOSTJIUgmimd1VR5GG6U6zkqJ8Soryow5DRESawN1Hxm8zs4cIpo0SEYnMyUN7ceeb03l+8gK+u1efqMORDJJsE9ORBHdAa0cxrf1bw3Q3g9vHfknvTq05aoCmuhARyWZm1hYoZvP+/CIizW7HsvY8ecF+DOxZGnUokmGSTRCvivk7HxgAHAfUVbMoKfTQ+7PZqUd7JYgiIlnGzOqqKfyqWQMREUlgUO8OUYcgGSjZPohXxW8zs1FAz5RHJFsoKy1mwYr1UYchIiKNl2j+4DnA+c0diIhIIn/+3xRaFebzq0N3iDoUyRDJ9kGMb5jcHugP7JbyiGQLZe2LeXf611GHISIijdc3bn2Nuy+JJBIRkQTmLFvHW18u4YIR/WhVoHEvBJKdGXN63DKRoP/EjPSEJbHKSotZuHI91TWadlJEJJu4+0yCGsPtgUOAQWaWbPcOEZG0O3loL5av3cgrUxZFHYpkiGQTRItb1gFvEwxeI2lWOxfi12sqI45EREQaw8zKgQ+AF4BR4b8TzKxHpIGJiIQO2L4rZe2LeWT87KhDkQyRbB/EZBNJSYNT9+jD6XttQ15eoq4sIiKSwa4jGNhtEcHANNuG638BzogwLhERAPLzjG/v3pPbxn7JwpXr6d6+OOqQJGJJN3Mxs1LgZGAbgqalj7n78vSEJbGKCpSfi4hkqRHAO8Awd68ys0JgTLhdRCQjnDy0N0tXV1Kl7kxC8oPU7AC8BpTFbP69mR3k7lPTEplssn5jNVf99xMO2rE7h+7cPepwREQkea2Bme5eBeDuG81sJrBztGGJiHyjb5c2XHvSwKjDkAyRbA3i9UA58BHwGcEIpoMJmsgcn5bIZJNWBXk8NmEu7UsKlSCKiGSXj4FTzGwtwffnjgStcd6JNCoRkTjuzqQ5K2hfUkjfLm2iDkcilGyCuDfworsfUbvBzJ4D9klLVLIZM9NciCIi2em3wMvA2eG6AVXA5ZFFJCKSwNrKar4zahwnDO7Bn7+t2sSWrDGjmMYPoVlJ4gmAJQ3KSouZrwRRRCSruPvrwBDgNuB54FZgD3cfG2lgIiJx2rQq4KgB5fx34nzWVVZHHY5EKNkaxAnA0Wb2CjCFoInMCODFdAUmmysvLebDWcujDkNERJIUznd4O/Cxu18QdTwiIg05eWgvHvtgDs9/Mp9vDe4VdTgSkWRrEC8BVhIkhecDB4Xrl6YpLonTp1NrCvINd40uJSKSDcKBaY4nuKkqIpLx9urbiT6dWvPI+DlRhyIRSipBdPdJwE7Arwmayfwa2NndJ6YxNolx4WH9efXC4ZipVa+ISBa5CTjKzPpGHYiISEPMjJOG9GLy3BWsWLcx6nAkIg02MQ2byLwAvO3u6lQvIiKSvJFAT2CamS0ENoTb3d37RRaViEgdzt6vgh8esC0lRflRhyIRabAGMWwisyObz4EozWz212s56673GPfV0qhDERGR5G1DMKBbHsF0URUxi4hIxmlXXEhJUT7urq5NLVSyfRB/D5xgZsPMrDCdAUliRQV5jP18MdMWrY46FBERSd6IOpaDogxKRKQ+XyxaxSF/G8s7qphokZIdxfRWwIFXgdh+cO7uyZ5DtkKXtq3IzzMWrFgXdSgiIpIkTWchItmoZ4fWLFq5gUfHz2Hffl2iDkeaWbI1iBA0kYlfGnO8bIX8PKN7u1aaC1FEJAuY2YFm9hcz2y5c/ypuOSHiEEVE6lRSlM+xg3rwv8nzWbVeg9W0NMkmeH3rWaSZlHcoYYESRBGRbHAhcCYwM1yviFvOjCAmEZGknTykF+s31vDspPlRhyLNLNlpLmbWtaQ7QPnGgJ6ldGvXKuowRESkYbsTjP4de+v9v8C3gLeAwZFEJSKSpEG9O7Bdt7Y8MkFzIrY09fYfNLNvAacBv3P3z8ysOm6Xc9z97rRFJ5u58rhdog5BRESS0xVYErP+EPCsuz9lZocDQ6IJS0QkOWbGRYftQFVNMJqp5uJuORoaYOZ8YHtgarge/8o4GVCCKCIisrnVBLWIALj7aTFlA/lmPkQRkYx1xK7lUYcgEWioiemuwLu++SQo9xM0jXk+LJdm8u5XSznshrFMW7gq6lBERKR+44HdzezPZlYMYGatzOwKYF9gQqTRiYgkadHK9dzxxldUVddEHYo0k4YSxM5AbDZyLfCgu08EZgHd0hWYbKkgP4/PF65mznJNdSEikuFuIGh1cwmwyszmEXyf/o5g2qgbIoxNRCRpH8xaxtXPTuGNaUsa3llyQkMJ4jJgXzPLA3D3y9z9f2HZXmyePEqalZcWA2gkUxGRDOfuLwAXAVVAPlBG0K2jCvi1uz8bYXgiIkk7aMfudGpTxCMTZkcdijSThhLEt4EdgXvNbAczKzKzfmZ2B0EfirfTHqFs0rVdK/IMzYUoIpIF3P1vQD/gXOC34b/93P36SAMTEWmEooI8ThjUk5c/XcSyNZVRhyPNoKEE8VqghmAk0ynAOuBz4Oxw+7XJXMTM7jKzRWY2uY7y081sUri8bWa7hduLzew9M5toZp+Y2VUxxwwys3Fm9pGZjTezPZOJJZsV5ufRtV0rFqxQE1MRkWzg7nPc/Q53/1P4r8aLF5Gsc/LQXlRW1/DUR3OjDkWaQb0Joru/C3yXoKmpxSzLgDPcPdkaxNHAEfWUTweGuftA4A/AqHD7BuAgd98NGAQcYWZ7h2V/Aa5y90EEfTr+kmQsWe3gnbpT0aVN1GGIiEgdzOye2hud9ewzyMzuba6YRES2xk7l7dmtdwfmLFMlRUvQ0DQXuPsjZvYMsB/BoDSLgLfcPelXiLu/bmYV9ZTHJprjgF7hdicYKhygMFxqR1R1oH34dykwL9l4stmfvjUg6hBERKR+3wXOMLNpwBiCFjirgHbATsBwgimkqoDvRROiiEjjPHrePhTmN9T4UHJBgwkiQJgMvpzmWGr9AHiudsXM8gmGA98OuDms1QT4BfCCmV1PUBO6b10nNLNzCfp+0KdPn/RELSIiEtgFuAr4NrAD39zYhKAVzkbgIeCK5g9NRKRpapPDNRuqaNMqqRRCslRG3QYwsxEECeKva7e5e3XYjLQXsKeZ1c69eD7wS3fvDfwSuLOu87r7KHcf6u5Du3btmrb4m8ND789iwJUvsGr9xqhDERGRBNz9c3c/jeB760zgeoLvqL8S1Bj2dvfvuvu0CMMUEWm0m16ZxrDrXqOySnMi5rKMSf/NbCBwB3Ckuy+NL3f35WY2hqAv42TgLODnYfEj4bE5r7gwn1Xrq1i4cj3tigujDkdEROrg7ouB+8NFRCTr7dqzlCWrK3n1s4UcsWt51OFImmREDaKZ9QEeB850989jtnc1sw7h3yXAIcBnYfE8YFj490FAi7gTW15aAsC85ZrqQkRERESazwHbd6Fbu1Y8Ml4DMueyZqlBNLMHCTrldzGzOQT9LgoB3P02glFIOwO3mBlAlbsPBcqBe8J+iHnAw+7+THjaHwI3mlkBsJ6wj2GuKy8tBmCB5kIUERERkWZUkJ/HiUN6Mer1r1i0aj3d2hVHHZKkQbMkiGFfjPrKzwHOSbB9EjC4jmPeBIakJMAs0q19KwDmK0EUERERkWZ28pBe3DrmS574YC4/GtYv6nAkDTKmD6Ikp1VBPt/dqw87dG8bdSgiIiIi0sJs27UtN5y6G/tvl90DP0rdlCBmIc2FKCKS+cIuEN8FBgFlwAJgIvCAu2soahHJWt8a3CvqECSNlCBmqXWV1ZQU5UcdhoiIJGBmOwDPA9sQzH1Yy4Erzezw2EHZRESyzTOT5rFo5Qa+v3/fqEORFMuIUUylcf7wzKfs/edXog5DRETqdiNQAbwF/B34Y7htHEHSeOPWXsDMTjazT8ysxsyGxpVdZmZfmNlUMzt8a68lIhLv1SmLuOGlz1lXWR11KJJiqkHMQp3bFrFi3UbWVlbRukj/hSIiGegA4Gl3PyG+wMyeJpieaWtNBr4N3B53/p2B7wC7AD2Al81sB3fXrzgRSZmThvbi8Q/n8sInCzhhcM+ow5EUUg1iFtJUFyIiGa8S6BpO07SJmRUCXcLyreLuU9x9aoKi44H/uPsGd58OfAHsubXXExGJtXffzvTqWMIjE2ZHHYqkmBLELFTWvgRQgigiksGeBfYBFpjZK2b2lJm9AswH9grL06UnEPuLbU64bQtmdq6ZjTez8YsXL05jSCKSa/LyjJOG9OLtL5cyZ9naqMORFFL7xCxUW4OouRBFRDLWz4GuwGHAiLiyl8LyBpnZywQjoMb7jbs/VddhCbZ5oh3dfRQwCmDo0KEJ9xERqcuJu/fixU8WsnjVBnp1bB11OJIiShCzUFlpMecN68f2mgtRRCQjufvXwBFmthswGOgOLAQ+cvePGnGeQ5pw+TlA75j1XsC8JpxHRKRevTu15n8/PyDqMCTFlCBmoeLCfC49cseowxARkQa4+0SCuQ+b09PAA2b2N4JBarYH3mvmGESkBVm9oYq1G6ro1r446lAkBdQHMUut3lDFvOXrog5DREQayczuNrOqFJznW2Y2h6Cv47Nm9gKAu38CPAx8SjAX4wUawVRE0qW6xjn4r2O49vlEY2ZJNlKCmKV++sAH/PDe8VGHISIiTZOon2CjuPsT7t7L3Vu5e3d3Pzym7I/u3s/d+7v7c1t7LRGRuuTnGQft2I3/fTyf1Ru2+t6XZAAliFmqrLREo5iKiGQoM6uuawG+F3V8IiKpdNKQ3qzbWM2zk9TdORcoQcxS5aXFLF1TyfqNajUkIpKBrIFFRCRn7N6nA9t2bcMj4+dEHYqkgBLELFUWTnWxaOWGiCMREZEElgEvEoxgGr/8N8K4RERSzsw4ZWhvxs9cxuyvNSdittMoplnqm7kQ19Gns+adERHJMO8C/cJRTDdjZstQLaKI5JhThvZm2A5d6d1Jv0uznRLELLVjWXuuOm4XvQlFRDLTOUCXOsouBK5oxlhERNKuU5siOrUpijoMSQE1Mc1SXdu14qx9K+jRoSTqUEREJI67z3P3SXWULXX3mc0dk4hIui1dvYFfPvQRb05bEnUoshVUg5jFvly8GnfYrlvbqEMREZEEzOxndRRtAD4HXtcchSKSK9oVFzJm6iIqq2rYf/u6GlFIplOCmMXOv28Cfbu04fYzh0YdioiIJPZ3wOsp/8jMjnL3hc0Uj4hI2hQV5HH8oJ488O4slq+tpENrNTnNRmpimsU0F6KISMZ7gyBBXEIwcM3icP09YDkwCLgqothERFLu5KG9qKyu4amPNCditlKCmMXK2xczXwmiiEgmG0+QGPZy932B3sA44O3w7w+AQ6MLT0QktXbpUcrO5e15ZMLsqEORJlKCmMXKSotZvHoDG6trog5FREQSOwdY5O4bAcJ/FwE/cPe1BHMi9ogwPhGRlPv+/n3Zt18X/UbNUuqDmMXKS4txh0WrNtBTo5mKiGSixcBxZvYKMAXYHjgE+DIs7wMsiyg2EZG0OGlIr6hDkK2gGsQstv/2XRh15hA6lBRGHYqIiCR2CVANjAB+TNCctBq4xMzaAzsBD0cXnohIelTXOG9MW0xllWoRs41qELNYr46t6dWxddRhiIhIHdz9cTPbCfgO0BOYCzzo7l+Fu+wXWXAiImn0xrTFjLz7fW4/cwiH71IWdTjSCKpBzGI1Nc7YzxczdcGqqEMREZE6uPuXwEPAY8BDMcmhiEjO2n+7LnRr14pHxmuwmmyjBDGLmQVzIT6sN56ISEYys1Iz+x8wFXgRmGpmz5lZacShiYikVUF+Ht/avSevTV3MolUadT+bKEHMYmZGWWmx5kIUEclcfwSOIOh3uDD89zDgT1EGJSLSHE4e0pvqGufJD+dGHYo0ghLELFdeWsz8FeuiDkNERBI7BpgMlLt7D6Ac+AQ4OtKoRESawXbd2rJ7nw6M/Xxx1KFII2iQmixX1r6Et79cEnUYIiKSWCfgNXdfCuDuS81sAnBitGGJiDSPW04fQpe2RVGHIY2gBDHLlZcWs2jVBqqqayjIV4WwiEiGmQp8x8y+BD4DdgROJahFFBHJeWWlxQC4O2YWcTSSDCWIWe7UPXpz+C5lesOJiGSmPxGMXnpVzDYDrokmHBGR5vfspPnc+MrnPP2T/SkuzI86HGmAqpyyXO9OrRnQq5T8PCWIIiKZxt2fIOhv+DxBDeLzwDHu/likgYmINKOOrQv5fOFqXvhkQdShSBKUIGa51RuqeOj9WUxbqLkQRUQykbs/5+5Hu/su7n40sNLMjos6LhGR5rL3tp3p1bGERyfMiToUSYISxCxXWVXDrx/7mDemaaAaEZEscR3wRNRBiIg0l7w848Tde/HmF0uYu1yj72c6JYhZrmPrQooK8liwUnMhioiIiEhmOmlIL9zhcdUiZjwliFnOzMK5EJUgioiIiEhm6t2pNRceugP79OscdSjSAI1imgPKS4tZsELV9SIimaKBPoYdmy0QEZEM8tODt486BElCsySIZnYXcAywyN13TVB+OvDrcHU1cL67TzSzYuB1oFUY66PufkXMcT8FfgJUAc+6+yXpfSSZqby0hPEzv446DBER+caTgNdRZvWUiYjktC8WrWLK/FUcu1uPqEOROjRXDeJo4J/AvXWUTweGufsyMzsSGAXsBWwADnL31WZWCLxpZs+5+zgzGwEcDwx09w1m1i39DyMzXXrkjprmQkQks8xCSaCIyBbueGM6T0+cx0E7dqNNKzVmzETN8r/i7q+bWUU95W/HrI4DeoXbnaBGEaAwXGq/cM8HrnH3DeG+i1Icdtbo3r446hBERCSGu1dEHYOISCY6eWgv/vP+bJ79eD6nDO0ddTiSQCYOUvMD4LnaFTPLN7OPgEXAS+7+bli0A3CAmb1rZmPNbI+6Tmhm55rZeDMbv3jx4nTGHok5y9Zyw0ufM/vrtVGHIiIiIiJSp937dGTbrm14ZPzsqEOROmRUghg2G/0B3/RHxN2r3X0QQa3inmZW24exgKCj/97AxcDDZpawnaW7j3L3oe4+tGvXrul8CJH4ek0lN74yjSnzV0YdioiIiIhIncyMk4b04v0Zy5i+ZE3U4UgCGZMgmtlA4A7geHdfGl/u7suBMcAR4aY5wOMeeA+oAbo0T7SZpaw0aGKquRBFREREJNOduHsv2hTl88m8FVGHIglkRIJoZn2Ax4Ez3f3zmO1dzaxD+HcJcAjwWVj8JHBQWLYDUAQsab6oM0eXNq0oyDPNhSgiIiIiGa97+2ImXH4oxwzUSKaZqLmmuXgQGA50MbM5wBUEA87g7rcBvwM6A7eErUSr3H0oUA7cY2b5BMnsw+7+THjau4C7zGwyUAmcFQ5q0+Lk5Rnd2xezQAmiiIiIiGSB4sJ8ANZvrN70t2SG5hrF9LQGys8BzkmwfRIwuI5jKoEzUhJgDigvLWbRKiWIIiIiIpIdzrzzXUpLCvnnd3ePOhSJoclHcsQ939+T1kW6+yIiIiIi2aFf17Y88N4sVqzdSGnrwqjDkVBG9EGUrdemVQF1DOIqIiIiIpJxThrSi8qqGp6eODfqUCSGEsQc8cGsZVzy6ERWrN0YdSgiIiIiIg3atWcpO5W355EJc6IORWIoQcwRC1es5+Hxc5i7fF3UoYiIiIiIJOWkIb2YNGcFUxesijoUCakPYo74Zi7Edezco33E0YiIiIiINOxbg3vSrriAXh1Log5FQkoQc0Rtgqi5EEVEREQkW3RqU8QpQ3tHHYb8f3t3HiVHfZ57/Pv27DMajfZ1JA0yRkCEkUAGHGIuq7Ed22wmNiexHR8THF/sgJNgG+c4xluC7+ViJ86JczDGwhsOiwBfbBbZgDE4QLQBAiQ2CbRrhJZZpFl65s0fVS1aM9OzdPdMddU8n3PqtLqqu/qp6lG/9av6VVUWdTFNiOkTqkgZuheiiIiIiMRKR3cPtzy+if969c2oowhqICZGeVmKeVNq6Uz3Rh1FRERERGTYylPG93/3Krc8sSnqKIK6mCbKo39/pm51ISIiIiKxUl6W4uKlc/nh45tobu1ken1V1JHGNR1BTBA1DkVEREQkji5d1ki617lnre6JGDU1EBPk3nXb+NTy/8bdo44iIiIiIjJsR8+oZ8m8Sdyxeou2ZSOmBmKC7G7p5LcbdtPamY46ioiIiIjIiHzknfOYWldFS4e2ZaOkcxAT5PC9EA90MLG6IuI0IiIiIiLD99F3zuOyU+ZHHWPc0xHEBJmteyGKiIiISExlrqfR3NpJR3dPxGnGLzUQEyRzBHHH/kMRJxERERERGbkXtrfwrn/+LQ+9sCvqKOOWGogJMqO+mqaptaRSupqpiIiIiMTPsbPqmTmxmjtWbYk6yrilcxATpLI8xaPXnBV1DBERERGRvKRSxiUnzeV7j7zC9v2HmDOpJupI446OIIqIiIiISMn48MnzcIcVa7ZGHWVcUgMxYb77m5e4/NZVUccQEREREcnL/Km1nHrUFFas2aZ7IkZAXUwTZl97F09tejPqGCIiIiIiefvqB/+IhtqKw1c2lbGjBmLCzGqoobUjTXtnmroqfb0iIiIiEj/Hz5kYdYRxS11MEyZzL8SdLboXooiIiIjE17Nb9/PXP1lNe2c66ijjihqICZO5F+LOA2ogioiIiEh8daV7eeD5nfz6uR1RRxlX1EBMmMbJNZy2cAoVZfpqRURERCS+Tl4wmYXT6rhjta5mOpbUikiYxsm1/OKKd3HKUVOijiIiIiIikjcz45KTG3l6014272mPOs64oQaiiIiIiIiUpEtOaiRlcKeOIo4ZNRAT6NM/WcWVP18TdQwRERERkYLMaqjmslPmM3tSddRRxg3dByGB0j3OG3t1GF5ERERE4u9bF50QdYRxRUcQE2hWQzU7DxyKOoaIiIiISFF0dPew+vV9UccYF9RATKDZDdXsO9hNR3dP1FFERGSUmNmlZva8mfWa2bKs8eeZ2Wozey58PDvKnCIixXDDgxu57AdPcuBgd9RREk8NxASa1VAD6F6IIiIJtx64GHisz/g9wAfd/QTgE8BPxjqYiEixXbBkLl3pXn757PaooySeGogJdOysei5aOpeUWdRRRERklLj7i+6+cYDxa909swX1PFBtZlVjm05EpLgWz53IsbPquXPVlqijJJ4aiAm0eG4D3/nIEuZPrY06ioiIROsSYK27dw400cyuMLNVZraqubl5jKOJiAyfmfHhkxt5ZusBXtrVGnWcRFMDMaHcne6e3qhjiIhIAczsN2a2foDhgmG894+AbwOfzvUad7/J3Ze5+7Lp06cXM7qISNFdtHQu5Slj5Qu7oo6SaLrNRUKdfv3DnHv8TL5+weKoo4iISJ7c/dx83mdmjcDdwMfd/dXiphIRicbUCVU89PkzOGpaXdRREk0NxISqr65ghy5SIyIy7pjZJOBXwLXu/kTEcUREimrh9AlRR0g8dTFNqOBeiGogiogklZldZGZbgXcBvzKzB8NJnwWOBr5iZuvCYUZkQUVEiuyGBzfyd7c/E3WMxFIDMaFmN1TrCKKISIK5+93u3ujuVe4+093PD8d/093r3H1J1rA76rwiIsXSme7h3nXb2NM24PW3pEBqICbUrIZq9rR10pXWhWpEREREJDkuXTaPdK9zz9ptUUdJJDUQE+q0hVP532e+TVcyFREREZFEOWZmPSfOm8Sdq7fi7lHHSZwxaSCa2S1mttvM1ueY/udm9mw4/MHMTgzHV5vZ02b2jJk9b2ZfG+C9f29mbmbTRns54uS0hVP5wnuPpa5K1yESERERkWS59ORGNuxsZf22lqijJM5YtR6WA/8G/DjH9E3A/3L3fWb2PuAm4FSgEzjb3dvMrAJ43Mzud/cnAcxsHnAe8MZoL0DcuDsth9KkUsEVTUVEREREkuKDJ85h485WJtboYEixjckRRHd/DNg7yPQ/uPu+8OmTQGM43t29LRxfEQ7Zx5G/A3yhzzgBWjvTnPj1h7jtabWdRURERCRZGmoq+MaFi1kwVfdELLZSPAfxU8D9mSdmVmZm64DdwEp3fyoc/yFgm7sPeY1bM7vCzFaZ2arm5uZRil1a6qvKqass05VMRURERCSR3J1Vm/eybsv+qKMkSkk1EM3sLIIG4hcz49y9x92XEBxVPMXMFptZLfAPwD8OZ77ufpO7L3P3ZdOnTx+F5KXHzJg9qUb3QhQRERGRROp1+Nxta7lx5UtRR0mUkmkgmtk7gJuBC9z9zb7T3X0/8CjwXuBtwFHAM2a2maDxuMbMZo1V3jjQvRBFREREJKnKUsYlJzXy+5eb2XHgUNRxEqMkGohmNh9YAXzM3V/KGj/dzCaF/64BzgU2uPtz7j7D3ZvcvQnYCpzk7jvHPn3pmjWxWkcQRURERCSxPnxyI+6wYo3uiVgsY3LZHzO7DTgTmGZmW4GvElxwBnf/D4KuolOBfzczgLS7LwNmA7eaWRlBY/Z2d79vLDLH3T1rt/Hwht282d7F6dc/zDXnL+LCpXOjjpXTPWu38X8f3Mj2/YeYM6mmpPMq6+iIU1aIV15lFRGRpGqaVsfCaXXcuPIlbnhwY6Jrx1jVyDFpILr7ZUNMvxy4fIDxzwJLhzH/przDJdA9a7dx7YrnONTdA8C2/Yf44l3Psqulg3OOmwnAnEnV1FaW09rRza6Wzn7zaJxcQ3VFGQcOddPc2n/6vCk1VJWXsf9gF3vauvpNXzC1loqyFHvbu9jb3n/6UdPqKEsZe9o6uWv1Vm5c+RKd6d7Dea9d8SwAZy2aQUe6BzMoM6MsZaRSxsTw1h3dPcF7yswwC869HE0DrdtrVzwHUHI/RMo6euKUV1lFRCTJ7lm7jS37DtLTG9zUIKm1YyxrpLmPrztELFu2zFetWpX/DO7/Eux8rniBRsGaN/bRFTaccjlu1kQaaip4s72Tl3e39Zu+eE4DE6rK2d3awWt72vtNf0djA7UV5ew4cIjX9x7sN33pvElUlZexdf9Btu7r3yd82YLJlKdSvL63Ped5kpVlKSbVVrC7TwM1ZXBK01QAXmluY0/bkdMrylKcPH8yAC/vbmX/wW4wMIKhqqKMxXMagvfvbqW1M40RNDABairLOGZGPQCvNrdxqLsHI3hzW2eagf7LmHG40Tqhqpx5k2sPz7+798g3TKyuYO6kGgA27mqlt88MJ9VUMLshmL5hZ0u/e7hMqatkZn01ve5s3NXaL8u0CVVMn1CV8++gPGUsWzCFrp5eXm3u/93PmljN5NpKOrp72PRm/+9+TkMNDTUVHOxKD/jdN06uob6qgrbONFv29Z8+f0otdZXltHR0s21/8LfR0tE95HrNWDitjqryMva2d7JrgJ0XR0+fQEVZij1tnTS39Z9+zMx6yszY3drBmwPsvDh2Vj2GsbPlEPsOdh8xLWXGopn1Oddt9t/mln0HaetMHzG9sizF26ZPAOCNvQdp7zpyenV5GUdNCy7XvfnN9sNFIKO2ouzw5bxf29N2eKdKxkB/e7nWbcr63yO1WH976d7eAX9XZtRXMbWuKuffXntnmnT4/+WF3gV8Pf1xAOZOquGJL53dfyGGYGarw94oMgwF10cRkQicfv3Dh7cnslWVpzh1YVCT3z5jAl/5wPEAXLviWbbtP3Lb84S5E7nm/GMB+Px/ruu3fXBK02Q+e/bbAfjMT1fT3nVkfT7j7dO4/N0LAfjkj56mp08BPe/4mXzstAV0pXu5/Mf9f2c/+I7ZXLpsHgcOdfO529b2m37pyY1cf/+GAZdzNGqk7iyZQIM1Do+eEWyc1lSWAcEGZWZctqqK4PTUiTUVA06vLAumT6qtpKK8/6ms5eH0KXWVVFeU9ZueCltj0yZU5WwgdvX0Mm1CFXVV5Yc3cB3HeOsoYTD/FPhbN8NMZR1FbKipoKIsFUzz4P0VZW/lraksP/y+zGsqs6anUsFRSw8n5tqf4s7hPVc9WQ3CdK8f8RygJ2smPb3er4GY/fJ0b/8P7O098v39pofzy/V3kJmne/9s2e/3HPN3hpie9V0NOj1rnQ1nvb71+ZmcA39+9nIMPn2o9/ef3muDr9vslw/0+T2W9d0NNL3P38aIpw/wt5dr3Q64fEX628s9PXjM9bc30GcCbB+gIIqIiEDuGtGZ7qXlULCj92DWDtm2zp7D4zMOZjX4WjvSQ07vuwP4UNb0lo50vxrXGe7wdbzfvDNZwxcMOL0r3ZtzOUejRuoIYgLl2pOS7x6G0RanvMo6OuKUFeKVdzxn1RHEkRkP9VFEkidOda4QY1kjS+IqplJc15y/iJo+R+1qKsq45vxFESUaXJzyKuvoiFNWiFdeZRURkSQbL7VjLJdTXUwTKHOialyuBBinvMo6OuKUFeKVV1lFRCTJxkvtGMvlVBdTERFJJHUxHRnVRxGR8UNdTEVERERERGRIaiCKiIiIiIgIoAaiiIiIiIiIhNRAFBEREREREUANRBEREREREQmpgSgiIiIiIiKAGogiIiIiIiISUgNRREREREREADUQRUREREREJKQGooiIiIiIiABqIIqIiIiIiEhIDUQREREREREB1EAUERERERGRkBqIIiIiIiIiAqiBKCIiIiIiIiFz96gzjCkzawZeL3A204A9RYgzFuKUFeKVV1lHR5yyQrzyjresC9x9ejHCjAdFqo8Qr7+zfI2HZQQtZ5KMh2UELedI5KyR466BWAxmtsrdl0WdYzjilBXilVdZR0ecskK88iqrjIXx8N2Nh2UELWeSjIdlBC1nsaiLqYiIiIiIiABqIIqIiIiIiEhIDcT83BR1gBGIU1aIV15lHR1xygrxyqusMhbGw3c3HpYRtJxJMh6WEbScRaFzEEVERERERATQEUQREREREREJqYEoIiIiIiIigBqII2Jm88zsETN70cyeN7Oros6Ui5lVm9nTZvZMmPVrUWcaipmVmdlaM7sv6ixDMbPNZvacma0zs1VR5xmMmU0yszvNbEP4t/uuqDMNxMwWheszM7SY2dVR58rFzD4f/t9ab2a3mVl11JlyMbOrwpzPl+I6NbNbzGy3ma3PGjfFzFaa2cvh4+QoM8rg4lQfCxHH2pqvONXkfMWplhciLtsBhYjbNkS+xmrbQ+cgjoCZzQZmu/saM6sHVgMXuvsLEUfrx8wMqHP3NjOrAB4HrnL3JyOOlpOZ/S2wDJjo7h+IOs9gzGwzsMzdS/5mrGZ2K/B7d7/ZzCqBWnffH3GsQZlZGbANONXdi3Hj7qIys7kE/6eOd/dDZnY78Gt3Xx5tsv7MbDHwC+AUoAt4APiMu78cabAsZnYG0Ab82N0Xh+P+D7DX3a83sy8Bk939i1HmlNziVB8LEcfamq841eR8xamWFyKO2wGFKPVtiHyN5baHjiCOgLvvcPc14b9bgReBudGmGpgH2sKnFeFQsnsDzKwR+FPg5qizJImZTQTOAH4I4O5dMSkK5wCvlvgPezlQY2blQC2wPeI8uRwHPOnuB909DfwOuCjiTEdw98eAvX1GXwDcGv77VuDCscwkIxOn+liIuNXWfKkmJ0eMtwMKEYdtiHyNybaHGoh5MrMmYCnwVMRRcgq7h6wDdgMr3b1kswLfBb4A9EacY7gceMjMVpvZFVGHGcRCoBn4UdhV6GYzq4s61DB8FLgt6hC5uPs24AbgDWAHcMDdH4o2VU7rgTPMbKqZ1QLvB+ZFnGk4Zrr7DggaH8CMiPPIMMWhPhYiZrU1X98lXjU5X3Gp5YWI63ZAIUp6GyJfY7ntoQZiHsxsAnAXcLW7t0SdJxd373H3JUAjcErY1azkmNkHgN3uvjrqLCNwurufBLwPuDLsIleKyoGTgO+7+1KgHfhStJEGF3Z/+RBwR9RZcgnPh7sAOAqYA9SZ2V9Em2pg7v4i8G1gJUH30meAdKShJLHiUh8LEZfamq+Y1uR8xaWWFyJ22wGFiMM2RL7GcttDDcQRCs85uAv4mbuviDrPcIRdCR4F3httkpxOBz4UngvwC+BsM/tptJEG5+7bw8fdwN0E53eVoq3A1qw93HcSFIpS9j5gjbvvijrIIM4FNrl7s7t3AyuAP444U07u/kN3P8ndzyDoylky5x8OYld4Xlvm/LbdEeeRIcSxPhYiBrU1X7GryfmKUS0vRBy3AwoRh22IfI3ZtocaiCMQnpz+Q+BFd78x6jyDMbPpZjYp/HcNwR/VhkhD5eDu17p7o7s3EXQLeNjdS/JoDICZ1YUXYSDspvEegm58JcfddwJbzGxROOocoNQvGnEZpd815A3gNDOrDX8XziE456okmdmM8HE+cDGlv34Bfgl8Ivz3J4B7I8wiQ4hTfSxEnGprvuJWk/MVp1peiJhuBxQiDtsQ+RqzbY/y0Zhpgp0OfAx4Ljz/AODL7v7r6CLlNBu4NbySUwq43d0Te6nqMTYTuDv4v0k58HN3fyDaSIP6HPCzsNvFa8AnI86TU3iO3HnAp6POMhh3f8rM7gTWEHTXXAvcFG2qQd1lZlOBbuBKd98XdaBsZnYbcCYwzcy2Al8FrgduN7NPERTFS6NLKMMQp/pYCNXW5IhbLS9EbLYDChGXbYh8jeW2h25zISIiIiIiIoC6mIqIiIiIiEhIDUQREREREREB1EAUERERERGRkBqIIiIiIiIiAqiBKCIiIiIiIiE1EGVcM7PNZuZmdmYJZCkzs+Vmtj/MdENEOTwcmqL4/GIzs6bMMkWdRUQkLlQfB8yh+ijjghqIErmsIrQ9vPEwZrZkHP5oXUJwQ/A08D3gsb4vMLMzswqUm9lBM3vFzL4e3jS1GP4lHFpG+kYzu65Pvszw3SJlG+rzl4efd13W6BbeWiYRkdhQfTxM9bFAqo8yEuVRBxDJMhv4DHBj1EEKYWYV7t6dx1uPCR9/7e5/M8Rru4DvA/XAnwNfAV4BfpzH5x7B3a8udB4EN+L9/1nPf1eEeebF3fcCV0f1+SIiRaD6GFB9LCLVR8nJ3TVoiHQANgMO9AK7gFpgSTjOs17n4dAUPr8ufL48fP6X4fNnCIpoG/ACsBT4BnCA4If5PQN89t8C64BW4G5gatZr/gR4FNgHbAduyUwHmrJy/XU4/ZEcy9kE3AHsCOf1CHBqn2XJHv5ygHmcGU7bnzXurnDcP4XPZxPsXd0DdAPNwE+BSeH0SuAHwE6gE9gC/HKQ9Xw18CrQEc7rUWBRjmXMLMc9OZa/73e6PBx3XZ/330lQzNsICvu5We+ZAvxrVqbXgA9kzSt7WJ7jc3N+F+H0R8P3/HO4Lg8CTwALhrMONWjQoKEYA6qP2cui+qj6qGGMBh1BlFJyB/BnwJXAygLmcwLBD+eLwDKCH7cdwJPAewgKWGOf91xH8KPbAFxIUIwvMbPFwG8J9kjeR1BcPgk0mdk5febxLeAegh/FI5hZHfAwcBRvFaeLgYfN7B1htqeAU8PcDxEU71xqwm4pE4D3h/kyeyTrgZrw+SHgXIK9qG0ERfrjwOXA88AKYA7w7oE+xMyOBr4T5l0OTAROC9fDxkHyndCn28zPgd2DvL6vSwjW+3qCdXILMN/MUgTr+N3ANoLC3gQsJFhnpwDHEazLJ4GnB1imQb8Ld3816+VfAH4RfsYfA98EPsYI1qGISBGoPqo+Zqg+yqhTA1FKyX8Ci4FrgP8qYD7tBD/6pxIUvwaCH68tBP3t55rZdHdvznrPV9z9X8zsRII9pReb2QSCLj2VwFqCvbe7CArAWcAigj10GZe6+8M5Mv0pwQ/ua8CZ7u5mdjdBsf2Uu3/ZzE4LMz/tQ3djqQSuynq+nmDvJe7+kpldAZwHzCD4kX47cHb42orw8TngZwSFNtf5FJnXbif4oX/B3beaWdkQ+Rb2ybeOkRXA58P8TQTrbJ6ZTQufv5tgvb/T3XfAW92WzOw9BAXwAXe/LpzW1Gfeg34XwJezXvsf7n6lmX2SoAgvDcePZB2KiBRK9VH1MUP1UUadLlIjpaQX+BowHfjsMF6f60d4s7sfAvZnjdvo7q1Zz+v6vOfF8HFD1ri5BD+4EBSmq8KhKhx3dJ95PDFI1sx8NroHfTCyPmvBIO/L5YC7G1AN/D+CDYebAMzsMmAN8G3g7wh+2CFYrxB0TbkduAB4HHgTuD/cc3gEd38R+CrBungQ2GJmGwiKzGDudXfLGpbneF2u73BduJ72Z42bQFC4AN7IFL8w50jOaWkKH4fzXawNHzM5JoSPw16HIiJFoPo4fKqPqo9SIDUQpdTcQbDX6c8GmHYwfJwYPi7OMY+eviPcvd+4PjI/6MdmjdtGcA4GwI3ZP+jAQne/r89ndA4y/8x8jsm6mtqi8PH1IbLlFH5mZm9y5iT+j4SPNxMU68zzzOem3f0jBOvxOOA3BHsjL+47/3BP6LfcfRpBcfh2mPvzecRtz5rvUN9hOnz0PuM3hY/zzWxW1vwyvSEy3/Ngv22bw8fhfBe5cgx7HYqIFInq4wioPh6en+qjjJi6mEpJCbszfI3gfIe+1gKnA/9mZhsJ9k4VyzfC7jNnhc/vdvc2M7sJ+CvgKjNbSNAf/ziCLjkj2cHyK4If3rcBj5jZHuAignMgbskjb+Yciyre2gOaKYS7wsf3EVzJ7f193nuZmX0RWEVw3sUJ4fj9A3zOPOApM3uMoAvM6YO8dlDu3mxmWwnOb/mpmXUQXGxhJNYAvyfoRvPfZvYAwd7b+wkufb4lfN1fmFkDwfkYm/rMoxjfxUjWoYhIwVQfh031UfVRCqQjiFKKVhD0ye/rcwR7T5cQ/Ij+qIifeR1wEkE3k18CVwC4+zME52s8BpwBfJTgJPfrRzJzd28HziG4otqx4Tx/B5zj7q/kkTdzjsVfEXQ9+hHB+SAQdEN6BJgKnAz8U5/3biQo5O8nOKegi+Dk8vvor4XgRPbTw8+aQ3BS+jfzyEz4ea8RFLBe4N6RvNndewkK/vcIrkD3cYL1uTl8yQ+APxAUxb8hWP6+8yjGdzGSdSgiUiyqj0NTfVR9lALZW12MRUREREREZDzTEUQREREREREB1EAUERERERGRkBqIIiIiIiIiAqiBKCIiIiIiIiE1EEVERERERARQA1FERERERERCaiCKiIiIiIgIoAaiiIiIiIiIhP4HnBaxIL7yBZ8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 1080x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.linalg import eigh\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\n",
    "def compute_energy_ground_state(n_basis):\n",
    "    \"\"\"Compute ground state energy by solving the Secular equations.\"\"\"\n",
    "    # assign S & H to zero matrices\n",
    "    s = np.zeros((n_basis, n_basis))\n",
    "    h = np.zeros((n_basis, n_basis))\n",
    "\n",
    "    # loop over upper-triangular elements & compute S & H elements\n",
    "    for j in range(0, n_basis):\n",
    "        for k in range(j, n_basis):\n",
    "            if (j + k) % 2 == 0:\n",
    "                s[j, k] = s[k, j] = 2 * (1 / (k + j + 5) - 2 / (k + j + 3) + 1 / (k + j + 1))\n",
    "                h[j, k] = h[k, j] = -1 * (((k + 2) * (k + 1)) / (k + j + 3) -  ((k + 2) * (k + 1) + k * (k - 1)) / (k + j + 1) + (k**2 - k) / (k + j - 1))\n",
    "    \n",
    "    # solve Hc = ESc to get eigenvalues E\n",
    "    e_vals = eigh(h, s, eigvals_only=True)\n",
    "    return e_vals[0]\n",
    "\n",
    "\n",
    "# plot basis set convergence of Secular equations\n",
    "# -----------------------------------------------\n",
    "\n",
    "# evaluate energy for a range of basis functions\n",
    "n_values = np.arange(2, 11, 1)\n",
    "e_values = np.array([compute_energy_ground_state(n) for n in n_values])\n",
    "expected_energy = (1 * np.pi)**2 / 8.\n",
    "\n",
    "plt.rcParams['figure.figsize'] = [15, 8]\n",
    "fig, axes = plt.subplots(1, 2)\n",
    "fig.suptitle(\"Basis Set Convergence of Secular Equations\", fontsize=24, fontweight='bold')\n",
    "\n",
    "for index, axis in enumerate(axes.ravel()):\n",
    "    if index == 0:\n",
    "        # plot approximate & exact energy\n",
    "        axis.plot(n_values, e_values, marker='o', linestyle='--', label='Approximate')\n",
    "        axis.plot(n_values, np.repeat(expected_energy, len(n_values)), marker='', linestyle='-', label='Exact')\n",
    "        # set axes labels\n",
    "        axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n",
    "        axis.set_ylabel(\"Ground-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n",
    "        axis.legend(frameon=False, fontsize=14)\n",
    "    else:\n",
    "        # plot log of approximate energy error (skip the last two values because they are zero)\n",
    "        axis.plot(n_values[:-2], np.log10(e_values[:-2] - expected_energy), marker='o', linestyle='--')\n",
    "        # set axes labels\n",
    "        axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n",
    "        axis.set_ylabel(\"Log10 (Ground-State Energy Error [a.u.])\", fontsize=12, fontweight='bold')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "czech-nigeria",
   "metadata": {},
   "source": [
    "### Particle-in-a-Box with Jacobi polynomials\n",
    "Similar results can be obtained with different basis functions. It is often convenient to use an orthonormal basis, where $s_{jk} = \\delta_{jk}$. For the particle-in-a-box with $-1 \\le x \\le 1$, one such set of basis functions can be constructed from the (normalized) [Jacobi polynomials](https://en.wikipedia.org/wiki/Jacobi_polynomials),\n",
    "$$\n",
    "\\phi_j(x) = N_j(1-x)(1+x)P_j^{(2,2)}(x)\n",
    "$$\n",
    "where $N_j$ is the normalization constant\n",
    "$$\n",
    "N_j = \\sqrt{\\frac{(2j+5)(j+4)(j+3)}{32(j+2)(j+1)}}\n",
    "$$\n",
    "To evaluate the Hamiltonian it is useful to know that:\n",
    "\\begin{align}\n",
    "\\frac{d^2\\phi_j(x)}{dx^2} &= N_j\n",
    "\\left(-2 P_j^{(2,2)}(x) - 4x \\frac{d P_j^{(2,2)}(x)}{dx} + (1-x)(1+x)\\frac{d^2 P_j^{(2,2)}(x)}{dx^2} \\right) \\\\\n",
    "&= N_j\n",
    "\\left(-2 P_j^{(2,2)}(x) - 4x \\frac{j+5}{2} P_{j-1}^{(3,3)}(x) + (1-x^2)\\frac{(j+5)(j+6)}{4}P_{j-2}^{(4,4)}(x) \\right) \n",
    "\\end{align}\n",
    "The Hamiltonian matrix elements could be evaluated analytically, but the expression is pretty complicated. It's easier to merely evaluate them numerically as:\n",
    "$$\n",
    "h_{jk} = -\\frac{1}{2}N_j N_k \\int_{-1}^1 (1-x)(1+x) P_k^{(2, 2)}(x) \\left(-2 P_j^{(2,2)}(x) - 4x \\frac{j+5}{2} P_{j-1}^{(3,3)}(x) + (1-x^2)\\frac{(j+5)(j+6)}{4}P_{j-2}^{(4,4)}(x) \\right) dx \n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "bizarre-failing",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.linalg import eigh\n",
    "from scipy.special import eval_jacobi\n",
    "from scipy.integrate import quad\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\n",
    "def compute_energy_ground_state(n_basis):\n",
    "    \"\"\"Compute ground state energy for a particle-in-a-Box with Jacobi basis.\"\"\"\n",
    "    \n",
    "    def normalization(i):\n",
    "        return np.sqrt((2 * i + 5) * (i + 4) * (i + 3) / (32 * (i + 2) * (i + 1)))\n",
    "    \n",
    "    def phi_squared(x, j):\n",
    "        return (normalization(j) * (1 - x) * (1 + x) * eval_jacobi(j, 2, 2, x))**2\n",
    "    \n",
    "    def integrand(x, j, k):\n",
    "        term = -2 * eval_jacobi(j, 2, 2, x)\n",
    "        if j - 1 >= 0:\n",
    "            term -= 2 * x * (j + 5) * eval_jacobi(j - 1, 3, 3, x)\n",
    "        if j - 2 >= 0:\n",
    "            term += 0.25 * (1 - x**2) * (j + 5) * (j + 6) * eval_jacobi(j - 2, 4, 4, x)\n",
    "        return (1 - x) * (1 + x) * eval_jacobi(k, 2, 2, x) * term\n",
    "    \n",
    "    # assign H to a zero matrix\n",
    "    h = np.zeros((n_basis, n_basis))\n",
    "\n",
    "    # compute H elements\n",
    "    for j in range(n_basis):\n",
    "        for k in range(n_basis):\n",
    "            integral = quad(integrand, -1.0, 1.0, args=(j, k))[0]\n",
    "            h[j, k] = -0.5 * normalization(j) * normalization(k) * integral\n",
    "\n",
    "    # solve Hc = Ec to get eigenvalues E\n",
    "    e_vals = eigh(h, None, eigvals_only=True)\n",
    "    return e_vals[0]\n",
    "\n",
    "\n",
    "# plot basis set convergence of particle-in-a-Box with Jacobi basis\n",
    "# -----------------------------------------------------------------\n",
    "\n",
    "# evaluate energy for a range of basis functions\n",
    "n_values = np.arange(2, 11, 1)\n",
    "e_values = np.array([compute_energy_ground_state(n) for n in n_values])\n",
    "expected_energy = (1 * np.pi)**2 / 8.\n",
    "\n",
    "plt.rcParams['figure.figsize'] = [15, 8]\n",
    "fig, axes = plt.subplots(1, 2)\n",
    "fig.suptitle(\"Basis Set Convergence of Particle-in-a-Box with Jacobi Basis\", fontsize=24, fontweight='bold')\n",
    "\n",
    "for index, axis in enumerate(axes.ravel()):\n",
    "    if index == 0:\n",
    "        # plot approximate & exact energy\n",
    "        axis.plot(n_values, e_values, marker='o', linestyle='--', label='Approximate')\n",
    "        axis.plot(n_values, np.repeat(expected_energy, len(n_values)), marker='', linestyle='-', label='Exact')\n",
    "        # set axes labels\n",
    "        axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n",
    "        axis.set_ylabel(\"Ground-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n",
    "        axis.legend(frameon=False, fontsize=14)\n",
    "    else:\n",
    "        # plot log of approximate energy error (skip the last two values because they are zero)\n",
    "        axis.plot(n_values[:-2], np.log10(e_values[:-2] - expected_energy), marker='o', linestyle='--')\n",
    "        # set axes labels\n",
    "        axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n",
    "        axis.set_ylabel(\"Log10 (Ground-State Energy Error [a.u.])\", fontsize=12, fontweight='bold')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "latter-shopper",
   "metadata": {},
   "source": [
    "#### &#x1f914; Thought-Provoking Question: Why does adding odd-order polynomials to the basis set not increase the accuracy for the ground state wavefunction. \n",
    "Hint: The ground state wavefunction is an even function.\n",
    "A function is said to be even if it is symmetric about the origin, $f(x) = f(-x)$. A function is said to be odd if it is antisymmetric around the origin, $f(x) = - f(-x)$. Even-degree polynomials (e.g., $1, x^2, x^4, \\ldots$) are even functions; odd-degree polynomials (e.g.; $x, x^3, x^5, \\ldots$) are odd functions. $\\cos(ax)$ is an even function and $\\sin(ax)$ is an odd function. $\\cosh(ax)$ is an even function and $\\sinh(ax)$ is an odd function. In addition,\n",
    "- A linear combination of odd functions is also odd.\n",
    "- A linear combination of even functions is also even.\n",
    "- The product of two odd functions is even.\n",
    "- The product of two even functions is even.\n",
    "- The product of an odd and an even function is odd.\n",
    "- The integral of an odd function from $-a$ to $a$ is always zero.\n",
    "- The integral of an even function from $-a$ to $a$ is always twice the value of its integral from $0$ to $a$; it is also twice its integral from $-a$ to $0$.\n",
    "- The first derivative of an even function is odd.\n",
    "- The first derivative of an odd function is even.\n",
    "- The k-th derivative of an even function is odd if k is odd, and even if k is even.\n",
    "- The k-th derivative of an odd function is even if k is odd, and odd if k is even.\n",
    "\n",
    "These properties of odd and even functions are often very useful. In particular, the first and second properties indicate that if you know that the exact wavefunction you are looking for is odd (or even), it will be a linear combination of basis functions that are odd (or even). E.g., odd basis functions are useless for approximating even eigenfunctions.\n",
    "\n",
    "#### &#x1f914; Thought-Provoking Question: Why does one get exactly the same results for the Jacobi polynomials and the simpler $(1-x)(1+x)x^k$ polynomials? \n",
    "Hint: Can you rewrite one set of polynomials as a linear combination of the others?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bound-diabetes",
   "metadata": {},
   "source": [
    "![Balancing Rock](https://github.com/PaulWAyers/IntroQChem/blob/main/linkedFiles/Perturbation.jpg?raw=true \"A precariously balanced rock. CC BY-NC-SA 2.0 license by Tatiana Gerus\")\n",
    "\n",
    "## Perturbation Theory\n",
    "It is not uncommon that a Hamiltonian for which the Schr&ouml;dinger equation is difficult to solve is \"close\" to another Hamiltonian that is easier to solve. In such cases, one can attempt to solve the easier problem, then *perturb* the system towards the actual, more difficult to solve, system of interest. The idea of leveraging easy problems to solve difficult problems is the essence of [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fancy-tampa",
   "metadata": {},
   "source": [
    "### The Perturbed Hamiltonian\n",
    "Suppose that for some Hamiltonian, $\\hat{H}$, we know the eigenfunctions and eigenvalues,\n",
    "$$\n",
    "\\hat{H} |\\psi_k \\rangle = E_k |\\psi_k \\rangle\n",
    "$$\n",
    "However, we are not interested in this Hamiltonian, but a different Hamiltonian, $\\tilde{H}$, which we can write as:\n",
    "$$\n",
    "\\tilde{H} = \\hat{H} + \\hat{V}\n",
    "$$\n",
    "where obviously\n",
    "$$\n",
    "\\hat{V} = \\tilde{H} - \\hat{H}\n",
    "$$\n",
    "\n",
    "Let us now define a family of perturbed Hamiltonians, \n",
    "$$\n",
    "\\hat{H}(\\lambda) = \\hat{H} + \\lambda \\hat{V}\n",
    "$$\n",
    "where obviously:\n",
    "$$\n",
    "\\hat{H}(\\lambda) = \n",
    "\\begin{cases}\n",
    "    \\hat{H} & \\lambda = 0\\\\\n",
    "    \\tilde{H} & \\lambda = 1\n",
    "\\end{cases}\n",
    "$$\n",
    "Writing the Schr&ouml;dinger equation for $\\hat{H}_\\lambda$, we have:\n",
    "$$\n",
    "\\hat{H}(\\lambda) |\\psi_k(\\lambda) \\rangle = E_k(\\lambda) |\\psi_k(\\lambda) \\rangle \n",
    "$$\n",
    "This equation holds true for all values of $\\lambda$. Since we know the answer for $\\lambda = 0$, and we *assume* that the perturbed system described by $\\tilde{H}$ is close enough to $\\hat{H}$ for the solution at $\\lambda =0$ to be useful, we will write the expand the energy and wavefunction as [Taylor-MacLaurin series](https://en.wikipedia.org/wiki/Taylor_series)\n",
    "\n",
    "\\begin{align}\n",
    "E_k(\\lambda) &= E_k(\\lambda=0) + \\lambda \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0} \n",
    "+ \\frac{\\lambda^2}{2!} \\left[\\frac{d^2E_k}{d \\lambda^2} \\right]_{\\lambda=0} \n",
    "+ \\frac{\\lambda^3}{3!} \\left[\\frac{d^3E_k}{d \\lambda^3} \\right]_{\\lambda=0} + \\cdots \\\\\n",
    "|\\psi_k(\\lambda) \\rangle &= |\\psi_k(\\lambda=0) \\rangle + \\lambda \\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0} \n",
    "+ \\frac{\\lambda^2}{2!} \\left[\\frac{d^2|\\psi_k \\rangle}{d \\lambda^2} \\right]_{\\lambda=0} \n",
    "+ \\frac{\\lambda^3}{3!} \\left[\\frac{d^3|\\psi_k \\rangle}{d \\lambda^3} \\right]_{\\lambda=0} + \\cdots \n",
    "\\end{align}\n",
    "\n",
    "When we write this, we are implicitly assuming that the derivatives all exist, which is not true if the zeroth-order state is degenerate (unless the perturbation does not break the degeneracy).\n",
    "\n",
    "If we insert these expressions into the Schr&ouml;dinger equation for $\\hat{H}(\\lambda)$, we obtain a polynomial of the form:\n",
    "$$\n",
    "0=p(\\lambda)= a_0 + a_1 \\lambda + a_2 \\lambda^2 + a_3 \\lambda^3 + \\cdots\n",
    "$$\n",
    "This equation can only be satisfied for *all* $\\lambda$ if all its terms are zero, so \n",
    "$$\n",
    "0 = a_0 = a_1 = a_2 = \\cdots\n",
    "$$ \n",
    "The key equations that need to be solved are listed below. First there is the zeroth-order equation, which is automatically satisfied:\n",
    "$$\n",
    "0 = a_0 = \\left( \\hat{H}(0) - E_k(0) \\right) | \\psi_k(0) \\rangle \n",
    "$$\n",
    "The first-order equation is:\n",
    "$$\n",
    "0 = a_1 = \\left( \\hat{H}(0) - E_k(0) \\right) \\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0} +\\left(\\hat{V} -  \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(\\lambda=0) \\rangle \n",
    "$$\n",
    "The second-order equation is:\n",
    "$$\n",
    "0 = a_2 = \\tfrac{1}{2} \\left( \\hat{H}(0) - E_k(0) \\right) \\left[\\frac{d^2|\\psi_k \\rangle}{d \\lambda^2} \\right]_{\\lambda=0} +\\left(\\hat{V} -  \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)\\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0}\n",
    "-\\tfrac{1}{2} \\left[\\frac{d^2E_k}{d \\lambda^2} \\right]_{\\lambda=0} |\\psi_k(\\lambda=0) \\rangle \n",
    "$$\n",
    "Higher-order equations are increasingly complicated, but still tractable in some cases. One usually applies perturbation theory only when the perturbation is relatively small, which usually suffices to ensure that the Taylor series expansion converges rapidly and higher-order terms are relatively insignificant."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cathedral-jamaica",
   "metadata": {},
   "source": [
    "### Hellmann-Feynman Theorem\n",
    "The [Hellmann-Feynman theorem](https://en.wikipedia.org/wiki/Hellmann%E2%80%93Feynman_theorem) has been discovered many times, most impressively by [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman), who included it in his undergraduate senior thesis. In simple terms:\n",
    "> **Hellmann-Feynman Theorem**: Suppose that the Hamiltonian, $\\hat{H}(\\lambda)$ depends on a parameter. Then the first-order change in the energy with respect to the parameter is given by the equation,\n",
    "$$\n",
    "\\left[\\frac{dE}{d\\lambda}\\right]_{\\lambda = \\lambda_0} = \\int \\left( \\psi(\\lambda_0;x)\\right)^* \\left[\\frac{d\\hat{H}}{d \\lambda} \\right]_{\\lambda = \\lambda_0}\\psi(\\lambda_0;x) \\; dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "encouraging-rouge",
   "metadata": {},
   "source": [
    "#### Derivation of the Hellmann-Feynman Theorem by Differentiation Under the Integral Sign\n",
    "The usual way to derive the Hellmann-Feynman theorem uses the technique of [differentiation under the integral sign](https://en.wikipedia.org/wiki/Leibniz_integral_rule). Therefore, \n",
    "$$\n",
    "\\frac{dE}{d\\lambda} = \\frac{d}{d\\lambda}\\int \\left( \\psi(\\lambda;x)\\right)^* \\hat{H}\\psi(\\lambda;x) \\; dx = \\int \\frac{d\\left( \\psi(\\lambda;x)\\right)^* \\hat{H}\\psi(\\lambda;x) }{d\\lambda}\\; dx\n",
    "$$\n",
    "While such an operation is not always mathematically permissible, it is usually permissible, as should be clear from the definition of the derivative as a limit of a difference,\n",
    "$$\n",
    "\\left[\\frac{dE}{d\\lambda}\\right]_{\\lambda = \\lambda_0} = \\lim_{h\\rightarrow0} \\frac{E(\\lambda_0 + h) - E(\\lambda_0)}{h}\n",
    "$$\n",
    "and the fact that the integral of a sum is the sum of the integrals. Using the product rule for derivatives, one obtains:\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{dE}{d\\lambda} &= \\int \\frac{d\\left( \\psi(\\lambda;x)\\right)^* \\hat{H}\\psi(\\lambda;x) }{d\\lambda}\\; dx \\\\\n",
    "&=\\int \n",
    "  \\frac{\\left(\\psi(\\lambda;x)\\right)^*}{d\\lambda} \\hat{H} \\psi(\\lambda;x) \n",
    "+ \\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\hat{H}}{d \\lambda}\\psi(\\lambda;x) \n",
    "+ \\left( \\psi(\\lambda;x)\\right)^* \\hat{H} \\frac{d\\psi(\\lambda;x)}{d\\lambda}\n",
    "\\; dx \\\\\n",
    "&=\\int \n",
    "  \\frac{\\left(\\psi(\\lambda;x)\\right)^*}{d\\lambda} E(\\lambda) \\psi(\\lambda;x) \n",
    "+ \\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\hat{H}}{d \\lambda}\\psi(\\lambda;x) \n",
    "+ \\left( \\psi(\\lambda;x)\\right)^* E(\\lambda) \\frac{d\\psi(\\lambda;x)}{d\\lambda}\n",
    "\\; dx \\\\\n",
    "&=E(\\lambda) \\int \n",
    "+  \\frac{\\left(\\psi(\\lambda;x)\\right)^*}{d\\lambda} \\psi(\\lambda;x) \n",
    "+ \\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\psi(\\lambda;x)}{d\\lambda}\n",
    "\\; dx\n",
    "+\\int  \n",
    "\\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\hat{H}}{d \\lambda}\\psi(\\lambda;x) \n",
    "\\; dx \\\\\n",
    "&=\\int  \n",
    "\\left( \\psi(\\lambda;x)\\right)^* \\frac{d\\hat{H}}{d \\lambda}\\psi(\\lambda;x) \n",
    "\\; dx \n",
    "\\end{align}\n",
    "\n",
    "In the third-from-last line we used the eigenvalue relation and the Hermitian property of the Hamiltonian; in the last step we have used the fact that the wavefunctions are normalized and the fact that the derivative of a constant is zero to infer that the terms involving the wavefunction derivatives vanish. Specifically, we used:\n",
    "\\begin{align}\n",
    "\\int &\\left(\\left[\\frac{d\\left( \\psi(\\lambda_0;x)\\right)^*}{d \\lambda}\\right]_{\\lambda = \\lambda_0} \\psi(\\lambda_0;x) \n",
    "+ \\left( \\psi(\\lambda_0;x)\\right)^* \\left[\\frac{d \\psi(\\lambda_0;x)}{d \\lambda}\\right]_{\\lambda = \\lambda_0}\\right) \\; dx \\\\\n",
    "&= \\left[\\frac{d}{d \\lambda} \\int \\left( \\psi(\\lambda_0;x)\\right)^* \\psi(\\lambda_0;x) \\; dx \\right]_{\\lambda = \\lambda_0}\\\\\n",
    "&= \\frac{d 1}{d\\lambda} \\\\\n",
    "&= 0\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "immediate-stopping",
   "metadata": {},
   "source": [
    "#### Derivation of the Hellmann-Feynman Theorem from First-Order Perturbation Theory"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "unauthorized-audit",
   "metadata": {},
   "source": [
    "Starting with the equation from first-order perturbation theory,\n",
    "$$\n",
    "0 = a_1 = \\left( \\hat{H}(0) - E_k(0) \\right) \\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0} +\\left(\\hat{V} -  \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(0) \\rangle \n",
    "$$\n",
    "multiply on the left-hand-side by $\\langle \\psi_k(0) |$. (I.e., multiply by $\\psi_k(0;x)^*$ and integrate.) Then: \n",
    "$$\n",
    "0 = \\langle \\psi_k(0) |\\left( \\hat{H}(0) - E_k(0) \\right) \\left[\\frac{d|\\psi_k \\rangle}{d \\lambda} \\right]_{\\lambda=0} +\\langle \\psi_k(0) |\\left(\\hat{V} -  \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(0) \\rangle \n",
    "$$\n",
    "Because the Hamiltonian is Hermitian, the first term is zero. The second term can be rearranged to give the Hellmann-Feynman theorem, \n",
    "$$\n",
    "\\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0} \\langle \\psi_k(0) |\\psi_k(0) \\rangle = \\langle \\psi_k(0) |\\hat{V}|\\psi_k(0) \\rangle = \\langle \\psi_k(0) |\\left[\\frac{d\\hat{H}}{d\\lambda} \\right]_{\\lambda=0}|\\psi_k(0) \\rangle\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "innocent-nylon",
   "metadata": {},
   "source": [
    "### Perturbed Wavefunctions\n",
    "To determine the change in the wavefunction, \n",
    "$$\\psi_k'(\\lambda) = \\frac{d |\\psi_k\\rangle}{d\\lambda}$$\n",
    "it is helpful to adopt the convention of intermediate normalization, whereby\n",
    "$$\n",
    "\\langle \\psi_k(0) | \\psi_k(\\lambda) \\rangle = 1\n",
    "$$\n",
    "for all $\\lambda$. Inserting the series expansion for $|\\psi(\\lambda) \\rangle$ one finds that \n",
    "\\begin{align}\n",
    "1 &= \\langle \\psi_k(0) | \\psi_k(0) \\rangle + \\lambda \\langle \\psi_k(0) | \\psi_k'(0) \\rangle + \\tfrac{\\lambda^2}{2!} \\langle \\psi_k(0) | \\psi_k''(0) \\rangle + \\cdots \\\\\n",
    "1 &= 1 + \\lambda \\langle \\psi_k(0) | \\psi_k'(0) \\rangle + \\tfrac{\\lambda^2}{2!} \\langle \\psi_k(0) | \\psi_k''(0) \\rangle + \\cdots \n",
    "\\end{align} \n",
    "where in the second line we have used the normalization of the zeroth-order wavefunction, $\\langle \\psi_k(0) | \\psi_k(0) \\rangle = 1$. Since this equation holds for all $\\lambda$, it must be that\n",
    "$$\n",
    "0=\\langle \\psi_k(0) | \\psi_k'(0) \\rangle\\\\\n",
    "0=\\langle \\psi_k(0) | \\psi_k''(0) \\rangle\\\\\n",
    "\\vdots\n",
    "$$\n",
    "Because the eigenfunctions of $\\hat{H}(0)$ are a complete basis, we can expand $ | \\psi_k'(0) \\rangle$ as:\n",
    "$$\n",
    " | \\psi_k'(0) \\rangle = \\sum_{j=0}^{\\infty} c_j | \\psi_j(0) \\rangle\n",
    "$$\n",
    "but because $\\langle \\psi_k(0) | \\psi_k'(0) \\rangle=0$, it must be that $c_k = 0$. So:\n",
    "$$\n",
    " | \\psi_k'(0) \\rangle = \\sum_{j=0\\\\\n",
    " j \\ne k}^{\\infty} c_j | \\psi_j(0) \\rangle\n",
    "$$\n",
    "We insert this expansion into the expression from first-order perturbation theory:\n",
    "$$\n",
    "0 = \\left( \\hat{H}(0) - E_k(0) \\right) \\sum_{j=0\\\\\n",
    " j \\ne k}^{\\infty} c_j | \\psi_j(0) \\rangle +\\left(\\hat{V} -  \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(0) \\rangle \n",
    "$$\n",
    "and multiply on the left by $\\langle \\psi_l(0) |$, with $l \\ne k$. \n",
    "\\begin{align}\n",
    "0 &= \\langle \\psi_l(0) |\\left( \\hat{H}(0) - E_k(0) \\right) \\sum_{j=0\\\\\n",
    " j \\ne k}^{\\infty} c_j | \\psi_j(0) \\rangle +\\langle \\psi_l(0) |\\left(\\hat{V} -  \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\right)|\\psi_k(0) \\rangle \\\\\n",
    "&= \\sum_{j=0\\\\ j \\ne k}^{\\infty} c_j\\langle \\psi_l(0) |\\left( E_l(0) - E_k(0) \\right)  | \\psi_j(0) \\rangle +\\langle \\psi_l(0) |\\hat{V} |\\psi_k(0) \\rangle - \\left[\\frac{dE_k}{d \\lambda} \\right]_{\\lambda=0}\\langle \\psi_l(0) |\\psi_k(0) \\rangle \\\\\n",
    "&= \\sum_{j=0\\\\ j \\ne k}^{\\infty} c_j \\left( E_l(0) - E_k(0) \\right) \\delta_{lj} +\\langle \\psi_l(0) |\\hat{V} |\\psi_k(0) \\rangle \\\\\n",
    "&=c_l \\left( E_l(0) - E_k(0) \\right)+\\langle \\psi_l(0) |\\hat{V} |\\psi_k(0) \\rangle \n",
    "\\end{align}\n",
    "\n",
    "Assuming that the k-th state is nondegenerate (so that we can safely divide by $ E_l(0) - E_k(0)$), \n",
    "$$\n",
    "c_l = \\frac{\\langle \\psi_l(0) |\\hat{V} |\\psi_k(0) \\rangle }{E_k(0) - E_l(0)}\n",
    "$$\n",
    "and so:\n",
    "$$\n",
    " | \\psi_k'(0) \\rangle = \\sum_{j=0\\\\\n",
    " j \\ne k}^{\\infty} \\frac{\\langle \\psi_j(0) |\\hat{V} |\\psi_k(0) \\rangle }{E_k(0) - E_j(0)} | \\psi_j(0) \\rangle\n",
    "$$\n",
    "\n",
    "Higher-order terms can be determined in a similar way, but we will only deduce the expression for the second-order energy change. Using the second-order terms from the perturbation expansion,\n",
    "$$\n",
    "0 = a_2 = \\tfrac{1}{2} \\left( \\hat{H}(0) - E_k(0) \\right) |\\psi_k''(0) \\rangle +\\left(\\hat{V} -  E_k'(0)\\right)|\\psi_k'(0) \\rangle-\\tfrac{1}{2} E_k''(0) |\\psi_k(0) \\rangle \n",
    "$$\n",
    "Projecting this expression against $\\langle \\psi_k(0) |$, one has:\n",
    "\\begin{align}\n",
    "0 &= \\tfrac{1}{2} \\langle \\psi_k(0) |\\left( \\hat{H}(0) - E_k(0) \\right) |\\psi_k''(0) \\rangle +\\langle \\psi_k(0) |\\left(\\hat{V} -  E_k'(0)\\right)|\\psi_k'(0) \\rangle-\\tfrac{1}{2} \\langle \\psi_k(0) |E_k''(0) |\\psi_k(0) \\rangle \\\\\n",
    " &= \\tfrac{1}{2} \\langle \\psi_k(0) |\\left(E_k(0) - E_k(0) \\right) |\\psi_k''(0) \\rangle \n",
    "+\\langle \\psi_k(0) |\\hat{V} |\\psi_k'(0) \\rangle\n",
    "-E_k'(0)\\langle \\psi_k(0) |\\psi_k'(0) \\rangle\n",
    "-\\tfrac{1}{2} E_k''(0) \\\\\n",
    " &= \\langle \\psi_k(0) |\\hat{V} |\\psi_k'(0) \\rangle\n",
    "-\\tfrac{1}{2} E_k''(0) \n",
    "\\end{align}\n",
    "To obtain the last line we used the intermediate normalization of the perturbed wavefunction, $\\langle \\psi_k(0) | \\psi_k'(0) \\rangle = 0$. Rewriting the expression for the second-order change in the energy, and then inserting the expression for the first-order wavefunction, gives\n",
    "\\begin{align}\n",
    " E_k''(0) &= 2\\langle \\psi_k(0) |\\hat{V} |\\psi_k'(0) \\rangle \\\\\n",
    " &= 2\\langle \\psi_k(0) |\\hat{V} \\sum_{j=0\\\\\n",
    " j \\ne k}^{\\infty} \\frac{\\langle \\psi_j(0) |\\hat{V} |\\psi_k(0) \\rangle }{E_k(0) - E_j(0)} | \\psi_j(0) \\rangle \\\\\n",
    " &= 2 \\sum_{j=0\\\\j \\ne k}^{\\infty}\\frac{\\langle \\psi_j(0) |\\hat{V} |\\psi_k(0) \\rangle \\langle \\psi_k(0) |\\hat{V} | \\psi_j(0) \\rangle}{E_k(0) - E_j(0)}  \\\\\n",
    " &= 2 \\sum_{j=0\\\\j \\ne k}^{\\infty}\\frac{ \\left|\\langle \\psi_j(0) |\\hat{V} |\\psi_k(0) \\rangle \\right|^2}{E_k(0) - E_j(0)} \n",
    "\\end{align}\n",
    "Notice that for the ground state ($k=0$), where $E_0 - E_{j>0} < 0$, the second-order energy change is never positive, $ E_0''(0) \\le 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "commercial-milan",
   "metadata": {},
   "source": [
    "### The Law of Diminishing Returns and Accelerating Losses\n",
    "Suppose one is given a Hamiltonian that is parameterized in the general form used in perturbation theory, \n",
    "$$\n",
    "\\hat{H}(\\lambda) = \\hat{H}(0) + \\lambda \\hat{V}\n",
    "$$\n",
    "According to the Hellmann-Feynman theorem, I have:\n",
    "$$\n",
    "\\frac{dE_0}{d\\lambda} = E_0'(\\lambda) = \\langle \\psi(\\lambda) | \\hat{V} |\\psi(\\lambda) \\rangle\n",
    "$$\n",
    "Consider two distinct values for the perturbation parameter, $\\lambda_1 < \\lambda_2$. According to the variational principle, if one evaluates the expectation value of $\\hat{H}(\\lambda_1)$ with $\\psi(\\lambda_2)$ one will obtain an energy above the true ground-state energy. I.e., \n",
    "$$\n",
    "E_0(\\lambda_1) = \\langle \\psi(\\lambda_1) | \\hat{H}(\\lambda_1) |\\psi(\\lambda_1) \\rangle < \\langle \\psi(\\lambda_2) | \\hat{H}(\\lambda_1) |\\psi(\\lambda_2) \\rangle\n",
    "$$\n",
    "Or, more explicitly, \n",
    "$$\n",
    "\\langle \\psi(\\lambda_1) | \\hat{H}(0) +\\lambda_1\\hat{V} |\\psi(\\lambda_1) \\rangle < \\langle \\psi(\\lambda_2) | \\hat{H}(0) +\\lambda_1\\hat{V}  |\\psi(\\lambda_2) \\rangle\n",
    "$$\n",
    "Similarly, the energy expectation value $\\hat{H}(\\lambda_2)$ evaluated with $\\psi(\\lambda_1)$ is above the true ground-state energy, so\n",
    "$$\n",
    "\\langle \\psi(\\lambda_2) | \\hat{H}(0) +\\lambda_2\\hat{V} |\\psi(\\lambda_2) \\rangle < \\langle \\psi(\\lambda_1) | \\hat{H}(0) +\\lambda_2\\hat{V}  |\\psi(\\lambda_1) \\rangle\n",
    "$$\n",
    "Adding these two inequalities and cancelling out the factors of $\\langle \\psi(\\lambda_2) | \\hat{H}(0) |\\psi(\\lambda_2) \\rangle $ that appear on both sides of the inequality, one finds that:\n",
    "$$\n",
    "\\left(\\lambda_2 - \\lambda_1 \\right) \\left(\\langle \\psi(\\lambda_2) | \\hat{V} |\\psi(\\lambda_2) \\rangle - \\langle \\psi(\\lambda_1) | \\hat{V}  |\\psi(\\lambda_1) \\rangle \\right) < 0\n",
    "$$\n",
    "or, using the Hellmann-Feynman theorem (in reverse),\n",
    "$$\n",
    "\\left(\\lambda_2 - \\lambda_1 \\right) \\left( E_0'(\\lambda_2) - E_0'(\\lambda_1)\\right) < 0\n",
    "$$\n",
    "\n",
    "Recall that $\\lambda_2 > \\lambda_1$. Thus $E_0'(\\lambda_2) < E_0'(\\lambda_1)$. If the system is losing energy at $\\lambda_1$ (i.e., $E'(\\lambda_1) < 0$), then at $\\lambda_2$ the system is losing energy even faster ($E_0'(\\lambda_2)$ is more negative than $E_0'(\\lambda_1)$. This is the law of accelerating losses. If the system is gaining energy a $\\lambda_1$ (i.e., $E_0'(\\lambda_1) > 0$), then at $\\lambda_2$ the system is gaining energy more slowly (or even losing energy) ($E_0'(\\lambda_2)$ is smaller than $E_0'(\\lambda_1)$). This is the law of diminishing returns. \n",
    "\n",
    "If the energy is a twice-differentiable function of $\\lambda$, then one can infer that the second derivative of the energy is always negative\n",
    "$$\n",
    "\\lim_{\\lambda_2 \\rightarrow \\lambda_1} \\frac{E_0'(\\lambda_2) - E_0'(\\lambda_1)}{\\lambda_2 - \\lambda_1} = \\left[\\frac{d^2E_0}{d\\lambda^2}\\right]_{\\lambda = \\lambda_1}= E_0''(\\lambda_1)< 0 \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "authentic-starter",
   "metadata": {},
   "source": [
    "### Example: Particle in a Box with a Sloped Bottom\n",
    "#### The Hamiltonian for an Applied Uniform Electric Field\n",
    "When a system cannot be solved exactly, one can solve it approximately using\n",
    "- perturbation theory. \n",
    "- variational methods using either an explicit wavefunction form or basis-set expansion.\n",
    "\n",
    "To exemplify these approaches, we will use the particle-in-a-box with a sloped bottom. This is obtained when an external electric field is applied to a charged particle in the box. The force on the charged particle due to the field is\n",
    "$$\n",
    "\\text{force} = \\text{charge} \\cdot \\text{electric field}                                                                      \n",
    "$$\n",
    "so for an electron in a box on which an electric field of magnitude $F$ is applied in the $+x$ direction, the force is\n",
    "$$\n",
    "\\text{force} = -e F\n",
    "$$\n",
    "where $e$ is the magnitude of the charge on the electron. The potential is\n",
    "$$\n",
    "\\text{potential} = - \\nabla \\text{force}\n",
    "$$\n",
    "Assuming that the potential is zero at the origin for convenience, $V(0) = 0$, the potential is thus:\n",
    "$$\n",
    "V(x) = eFx\n",
    "$$\n",
    "\n",
    "The particle in a box with an applied field has the Hamiltonian\n",
    "$$\n",
    "\\hat{H} = -\\frac{\\hbar^2}{2m} \\frac{d^2}{dx^2} + V(x) + eFx\n",
    "$$\n",
    "or, in atomic units,\n",
    "$$\n",
    "\\hat{H} = -\\tfrac{1}{2} \\tfrac{d^2}{dx^2} + V(x) + Fx\n",
    "$$\n",
    "For simplicity, we assume the case where the box has length 2 and is centered at the origin,\n",
    "$$\n",
    "V(x) =\n",
    "\\begin{cases}\n",
    "\\infty & x \\le -1 \\\\\n",
    "0 & -1 < x < 1 \\\\\n",
    "\\infty & 1 \\le x \n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "For small electric fields, we can envision solving this system by perturbation theory. We also expect that variational approaches can work well. We'll explore how these strategies can work. It turns out, however, that this system can be [solved exactly](https://www.utupub.fi/bitstream/handle/10024/117904/progradu-tuomas-riihim%C3%A4ki.pdf), though the treatment is far beyond the scope of this course. There are a few useful equations, however: for a field strength of $F=\\tfrac{1}{16}$ the ground-state energy is 1.23356 a.u. and for a field strength of $F=\\tfrac{25}{8}$ the ground-state energy is 0.9063 a.u.; these can be compared to the unperturbed result of 1.23370 a.u.. Some approximate formulas for the higher eigenvalues are available:\n",
    "\n",
    "\\begin{align}\n",
    "E\\left(F=\\tfrac{1}{16};n\\right) \n",
    "&= \\frac{10.3685}{8}  \n",
    "\\left( 0.048 + 5.758 \\cdot 10^{-5} n + 0.952 n^2 + 3.054 \\cdot 10^{-7} n^3\\right)  - \\frac{1}{16} \\\\\n",
    "E\\left(F=\\tfrac{25}{8};n\\right)\n",
    "&= \\frac{32.2505}{8}  \n",
    "\\left( 0.688 + 0.045 n + 0.300 n^2 + 2.365 \\cdot 10^{-4} n^3\\right) - \\frac{25}{8}\n",
    "\\end{align}\n",
    "\n",
    "> Note, to obtain these numbers from the reference data in [solved exactly](https://www.utupub.fi/bitstream/handle/10024/117904/progradu-tuomas-riihim%C3%A4ki.pdf), you need to keep in mind that the reference data assumes the mass is $1/2$ instead of $1$, and that the reference data is for a box from $0 \\le x \\le 1$ instead of $-1 \\le x \\le 1$. This requires dividing the reference field by 16, and shifting the energies by the field, and dividing the energy by 8 (because both the length of the box and the mass of the particle has doubled). In the end, $F = \\tfrac{1}{16} F_{\\text{ref}}$ and $E = \\tfrac{1}{8}E_{\\text{ref}}-\\tfrac{1}{16}F_{\\text{ref}}= \\tfrac{1}{8}E_{\\text{ref}}-F$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "endless-delaware",
   "metadata": {},
   "source": [
    "#### Perturbation Theory for the Particle-in-a-Box in a Uniform Electric Field\n",
    "##### The First-Order Energy Correction is Always Zero\n",
    "The corrections due to the perturbation are all zero to first order. To see this, consider that, from the Hellmann-Feynman theorem,\n",
    "\n",
    "\\begin{align}\n",
    "\\left[\\frac{dE_n}{dF}\\right]_{F=0} &= \\int_{-1}^{1} \\psi_n(x) \\left[ \\frac{d \\hat{H}}{dF} \\right]_{F=0} \\psi_n(x) dx \\\\\n",
    "&= \\int_{-1}^{1} x|\\psi_n(x)|^2 dx  \\\\\n",
    "&= \\int_{-1}^{1} \\text{(even function)} \\text{(odd function) } dx \\\\\n",
    "&= \\int_{-1}^{1}\\text{(odd function) } dx \\\\\n",
    "&= 0\n",
    "\\end{align}\n",
    "\n",
    "This reflects the fact that this system has a vanishing dipole moment. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "conservative-exploration",
   "metadata": {},
   "source": [
    "##### The First-Order Correction to the Wavefunction\n",
    "To determine the first-order correction to the wavefunction, one needs to evaluate integrals that look like:\n",
    "$$\n",
    "V_{mn} = \\int_{-1}^{1} \\psi_m(x) (x) \\psi_n(x) dx $$\n",
    "From the properties of odd and even functions, and the fact that $\\psi_n(x)$ is odd if $n$ is even, and *vice versa*, it's clear that $V_mn = 0$ unless $m+n$ is odd. (That is, either $m$ or $n$, but not both, must be odd.) The integrals we need to evaluate all have the form\n",
    "$$\n",
    "V_{mn} = \\int_{-1}^{1} x \\sin \\left(\\frac{m \\pi x}{2} \\right)\\cos \\left(\\frac{n \\pi x}{2} \\right) dx \n",
    "$$\n",
    "where $m$ is even and $n$ is odd. Using the trigonometric identity\n",
    "$$\n",
    "\\sin(ax) \\cos(bx) = \\tfrac{1}{2} \\sin((a+b)x) + \\tfrac{1}{2} \\sin((a-b)x)\n",
    "$$\n",
    "we can deduce that the integral is\n",
    "$$\n",
    "V_{mn} = \\left[2\\frac{\\sin\\left( \\frac{(m-n)\\pi x}{2} \\right)}{(m-n)^2 \\pi^2} \n",
    "+ 2\\frac{\\sin\\left( \\frac{(m+n)\\pi x}{2} \\right)}{(m+n)^2 \\pi^2} \n",
    "-\\frac{x\\cos\\left( \\frac{(m-n)\\pi x}{2} \\right)}{(m-n) \\pi} \n",
    "-\\frac{x\\cos\\left( \\frac{(m+n)\\pi x}{2} \\right)}{(m+n)^2 \\pi} \\right]_{=1}^{1}\n",
    "$$\n",
    "As mentioned before, this integral is zero unless $m+n$ is odd. The cosine terms therefore vanish. For odd $p$, $\\sin \\tfrac{p \\pi}{2} = -1^{(p-1)/2}$, we have\n",
    "$$\n",
    "V_{mn} = \n",
    "\\begin{cases}\n",
    "0  & m+n \\text{ is even} \\\\\n",
    "\\dfrac{4}{\\pi^2} \\left( \\dfrac{-1^{(m-n-1)/2} }{(m-n)^2} + \\dfrac{-1^{(m+n-1)/2}}{(m+n)^2} \\right) & m+n \\text{ is odd}\n",
    "\\end{cases} \n",
    "$$\n",
    "The first-order corrections to the ground-state wavefunction is then: \n",
    "$$\n",
    " \\left[\\frac{d\\psi_n(x)}{dF} \\right]_{F=0} = | \\psi_m'(0) \\rangle = \\sum_{m=1\\\\\n",
    " m \\ne n}^{\\infty} \\frac{V_{mn}}{E_n(0) - E_m(0)} | \\psi_m(0) \\rangle\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "reduced-metallic",
   "metadata": {},
   "source": [
    "##### The Second-Order Correction to the Energy\n",
    "The second-order correction to the energy is \n",
    "$$\n",
    " E_n''(0) = 2 \\sum_{m=1\\\\m \\ne n}^{\\infty}\\frac{ V_{mn}^2}{E_m(0) - E_n(0)} \n",
    "$$\n",
    "This infinite sum is not trivial to evaluate, but we can investigate the first non-vanishing term for the ground state. (This is the so-called Unsold approximation.) Thus:\n",
    "$$\n",
    "E_0''(0) = 2 \\frac{V_{21}^2}{E_1(0) - E_2(0)} = 2 \\frac{\\left(\\tfrac{4}{\\pi^2}(1-\\tfrac{1}{9})\\right)^2}{\\tfrac{\\pi^2}{8} - \\tfrac{4\\pi^2}{8}} \n",
    "= -\\frac{16384 }{243 \\pi^6} = -0.0701\n",
    "$$\n",
    "Using this, we can estimate the ground-state energy for different field strengths as \n",
    "$$\n",
    "E(F) \\approx E(0) - \\frac{1}{2!}\\frac{16384 }{243 \\pi^6} F^2\n",
    "$$\n",
    "For the field strengths for which we have exact results readily available, this gives \n",
    "$$\n",
    "E(\\tfrac{1}{16}) \\approx 1.23356 \\text{ a.u.} \\\\\n",
    "E(\\tfrac{25}{8}) \\approx 0.8913 \\text{ a.u.} \\\\\n",
    "$$\n",
    "These results are impressively accurate, especially considering all the effects we have neglected."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "trained-douglas",
   "metadata": {},
   "source": [
    "#### Variational Approach to the Particle-in-a-Box in a Uniform Electric Field\n",
    "When the field is applied, it becomes more favorable for the electron to drift to the $x<0$ side of the box. To accomodate this, we can propose a wavefunction ansatz for the ground state,\n",
    "$$\n",
    "\\psi_c(x) = (1 - cx)\\cos\\left(\\frac{\\pi x}{2} \\right)\n",
    "$$\n",
    "Clearly $c = 0$ in the absence of a field, but $c > 0$ is to be expected when the field is applied. We can determine the optimal value of $c$ using the variational principle. First we need to determine the energy as a function of $c$:\n",
    "$$\n",
    "E(c) = \\frac{\\langle \\psi_c | \\hat{H} | \\psi_c \\rangle}{\\langle \\psi_c | \\psi_c \\rangle}\n",
    "$$\n",
    "The denominator of this expression is easily evaluated\n",
    "$$\n",
    "\\langle \\psi_c | \\psi_c \\rangle = 1 + \\gamma c^2 \n",
    "$$\n",
    "where we have defined the constant:\n",
    "$$\n",
    "\\gamma = \\int_{-1}^1 x^2 \\cos^2\\left(\\frac{\\pi x}{2}\\right) dx = \\tfrac{1}{3} - \\tfrac{2}{\\pi^2}\n",
    "$$\n",
    "The numerator is \n",
    "\n",
    "$$\n",
    "\\langle \\psi_c | \\hat{H} | \\psi_c \\rangle = \\frac{\\pi^2}{8} \n",
    "+  c^2 \\frac{\\gamma \\pi^2}{8} - 2cF\\gamma + \\tfrac{1}{2}c^2\n",
    "$$\n",
    "where we have used the integral:\n",
    "\n",
    "$$\n",
    "\\int_{-1}^1 x \\cos\\left(\\frac{\\pi x}{2}\\right) \\sin\\left(\\frac{\\pi x}{2} \\right) dx = \\frac{1}{\\pi}\n",
    "$$\n",
    "\n",
    "This equation can be solved analytically because it is a cubic equation, but it is more convenient to solve it numerically."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "lyric-excitement",
   "metadata": {},
   "source": [
    "#### Basis-Set Expansion for the Particle-in-a-Box in a Uniform Electric Field\n",
    "As a final approach to this problem, we can expand the wavefunction in a basis set. The eigenfunctions of the unperturbed particle-in-a-box are a sensible choice here, though we could use polynomials (as we did earlier in this worksheet) without issue if one wished to do so. The eigenfunctions of the unperturbed problem are orthonormal, so the overlap matrix is the identity matrix\n",
    "$$\n",
    "s_{mn} = \\delta_{mn}\n",
    "$$\n",
    "The Hamiltonian matrix elements are:\n",
    "and the Hamiltonian matrix elements are\n",
    "\n",
    "\\begin{align}\n",
    "h_{mn} &= \\int_{-1}^{1} \\cos\\left(\\frac{m \\pi x}{2} \\right) \\hat{H} \\cos\\left(\\frac{n \\pi x}{2} \\right) dx \\\\\n",
    "&= \\int_{-1}^{1} \\cos\\left(\\frac{m \\pi x}{2} \\right)\\left[-\\frac{1}{2}\\frac{d^2}{dx^2} + Fx \\right] \\cos\\left(\\frac{n \\pi x}{2} \\right) dx \\\\\n",
    "&= \\frac{\\pi^2 n^2}{2} \\delta_{mn} + F V_{mn}\n",
    "\\end{align}\n",
    "Using the results we have already determined for the matrix elements, then, \n",
    "\n",
    "$$\n",
    "h_{mn} = \n",
    "\\begin{cases}\n",
    "    0 & m\\ne n \\text{ and }m+n \\text{ is even}\\\\\n",
    "    \\dfrac{\\pi^2n^2}{8} & m = n \\\\\n",
    "    \\dfrac{4F}{\\pi^2} \\left( \\dfrac{-1^{(m-n-1)/2} }{(m-n)^2} + \\dfrac{-1^{(m+n-1)/2}}{(m+n)^2} \\right) & m+n \\text{ is odd}\n",
    "\\end{cases}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "spectacular-association",
   "metadata": {},
   "source": [
    "#### Demonstration\n",
    "In the following code block, we'll demonstrate how the energy converges as we increase the number of terms in our calculation. For the excited states, it seems the reference data is likely erroneous."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "swiss-liechtenstein",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Energy of these models vs. reference values:\n",
      "Energy of the unperturbed ground state (field = 0): 1.2337005501361697\n",
      " \n",
      "Field value:  0.0625\n",
      "Exact Energy of the ground state: 1.2335625\n",
      "Energy of the ground state estimated with 2nd-order perturbation theory: 1.2335633924534153\n",
      "Energy of the ground state estimated with the variational principle: 1.2335671162852282\n",
      "Energy of the ground state estimated with basis set expansion: 1.2335633952295426\n",
      " \n",
      "Field value:  6.25\n",
      "Exact Energy of the ground state: 0.9063125000000003\n",
      "Energy of the ground state estimated with 2nd-order perturbation theory: 0.8908063432502749\n",
      "Energy of the ground state estimated with the variational principle: 0.9250111494073084\n",
      "Energy of the ground state estimated with basis set expansion: 0.9063121281470087\n",
      " \n",
      " \n",
      "Energy of the unperturbed first excited state (field = 0): 4.934802200544679\n",
      " \n",
      "Field value:  0.0625\n",
      "Exact Energy of the first excited state: 4.9348375\n",
      "Energy of the first excited state estimated with 2nd-order perturbation theory: 4.93484310142371\n",
      "Energy of the first excited state estimated with basis set expansion: 4.934843098735417\n",
      " \n",
      "Field value:  6.25\n",
      "Exact Energy of the first excited state: 5.022125000000001\n",
      "Energy of the first excited state estimated with 2nd-order perturbation theory: 5.343810990856903\n",
      "Energy of the first excited state estimated with basis set expansion: 5.161850795200653\n",
      " \n",
      " \n",
      "Energy of the unperturbed second excited state (field = 0): 11.103304951225528\n",
      " \n",
      "Field value:  0.0625\n",
      "Exact Energy of the second excited state: 11.103325\n",
      "Energy of the second excited state estimated with 2nd-order perturbation theory: 11.103329317896025\n",
      "Energy of the first excited state estimated with basis set expansion: 11.103329317809841\n",
      " \n",
      "Field value:  6.25\n",
      "Exact Energy of the ground state: 11.163625\n",
      "Energy of the second excite state estimated with 2nd-order perturbation theory: 11.34697165619853\n",
      "Energy of the second excited state estimated with basis set expansion: 11.336913933805935\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.linalg import eigh\n",
    "from scipy.optimize import minimize_scalar\n",
    "from scipy.integrate import quad\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\n",
    "def compute_V(n_basis):\n",
    "    \"\"\"Compute the matrix <k|x|l> for an electron in a box from -1 to 1 in a unit external field, in a.u.\"\"\"\n",
    "    \n",
    "    # initialize V to a zero matrix\n",
    "    V = np.zeros((n_basis,n_basis))\n",
    "    \n",
    "    # Because Python is zero-indexed, our V matrix will be shifted by 1. I'll\n",
    "    # make this explicit by making the counters km1 and lm1 (k minus 1 and l minus 1)\n",
    "    for km1 in range(n_basis):\n",
    "        for lm1 in range(n_basis):\n",
    "            if (km1 + lm1) % 2 == 1:\n",
    "                # The matrix element is zero unless (km1 + lm1) is odd, which means that (km1 + lm1) mod 2 = 1.\n",
    "                # Either km1 is even or km1 is odd. If km1 is odd, then the km1 corresponds to a sine and\n",
    "                # lm1 is even, and corresponds to a cosine. If km1 is even and lm1 is odd, then the roles of the\n",
    "                # sine and cosine are reversed, and one needs to multiply the first term below by -1. The\n",
    "                # factor -1**lm1 achieves this switching.\n",
    "                V[km1,lm1] = 4. / np.pi**2 * (-1**((km1 - lm1 - 1)/2) / (km1-lm1)**2 * -1**(lm1)\n",
    "                                   + -1**((km1 + lm1 - 1)/2) / (km1+lm1+2)**2)\n",
    "    return V\n",
    "\n",
    "def energy_pt2(k,F,n_basis):\n",
    "    \"\"\"kth excited state energy in a.u. for an electron in a box of length 2 in the field F estimated with 2nd-order PT.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    k : scalar, int \n",
    "        k = 0 is the ground state and k = 1 is the first excite state.\n",
    "    F : scalar\n",
    "        the external field strength\n",
    "    n_basis : scalar, int\n",
    "        the number of terms to include in the sum over states in the second order pert. th. correction.\n",
    "\n",
    "    Returns\n",
    "    -------\n",
    "    energy_pt2 : scalar\n",
    "         The estimated energy of the kth-excited state of the particle in a box of length 2 in field F.\n",
    "    \"\"\"\n",
    "    # It makes no sense for n_basis to be less than k.\n",
    "    assert(k < n_basis), \"The excitation level of interest should be smaller than n_basis\" \n",
    "    \n",
    "    # Energy of the kth-excited state in a.u.\n",
    "    energy = (np.pi**2 / 8.) * np.array([(k + 1)**2  for k in range(n_basis)])\n",
    "    V = compute_V(n_basis)\n",
    "    \n",
    "    der2 = 0\n",
    "    for j in range(n_basis):\n",
    "        if j != k:\n",
    "            der2 += 2*V[j,k]**2/(energy[k]-energy[j])  \n",
    "            \n",
    "    return energy[k] + der2 * F**2 / 2\n",
    "\n",
    "\n",
    "def energy_variational(F):\n",
    "    \"\"\"ground state energy for a electron0in-a-Box in a box of length 2 in the field F estimated with the var. principle.\n",
    "    The variational wavefunction ansatz is psi(x) = (1+cx)cos(pi*x/2) where c is a variational parameter.\n",
    "    \"\"\"\n",
    "    gamma = 1/3 - 2/np.pi**2\n",
    "    def func(c):\n",
    "        return (np.pi**2/8*(1+gamma*c**2)+c**2/2 - 2*c*gamma*F) / ( 1 + gamma * c**2)\n",
    "\n",
    "    res = minimize_scalar(func,(0,1))\n",
    "    \n",
    "    return res.fun\n",
    "\n",
    "\n",
    "def energy_basis(F,n_basis):\n",
    "    \"\"\"Eigenenergies in a.u. of an electron in a box of length 2 in the field F estimated by basis-set expansion.\n",
    "    n_basis basis functions from the F=0 case are used.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    F : scalar\n",
    "        the external field strength\n",
    "    n_basis : scalar, int\n",
    "        the number of terms to include in the sum over states in the second order pert. th. correction.\n",
    "\n",
    "    Returns\n",
    "    -------\n",
    "    energy_basis_exp : array_like\n",
    "        list of n_basis eigenenergies\n",
    "    \"\"\"\n",
    "    energy = (np.pi**2 / 8.) * np.array([(k + 1)**2  for k in range(n_basis)])\n",
    "    V = compute_V(n_basis)\n",
    "    \n",
    "    # assign Hamiltonian to the potential matrix, times the field strength:\n",
    "    h = F*V \n",
    "    np.fill_diagonal(h,energy)\n",
    "    # solve Hc = Ec to get eigenvalues E\n",
    "    e_vals = eigh(h, None, eigvals_only=True)\n",
    "    return e_vals\n",
    "\n",
    "print(\"Energy of these models vs. reference values:\")\n",
    "print(\"Energy of the unperturbed ground state (field = 0):\", np.pi**2 / 8.) \n",
    "print(\" \")\n",
    "print(\"Field value: \", 1./16)\n",
    "print(\"Exact Energy of the ground state:\", 10.3685/8 - 1./16)\n",
    "print(\"Energy of the ground state estimated with 2nd-order perturbation theory:\", energy_pt2(0,1./16,50)) \n",
    "print(\"Energy of the ground state estimated with the variational principle:\", energy_variational(1./16)) \n",
    "print(\"Energy of the ground state estimated with basis set expansion:\", energy_basis(1./16,50)[0]) \n",
    "print(\" \")\n",
    "print(\"Field value: \", 25./4)\n",
    "print(\"Exact Energy of the ground state:\", 32.2505/8 - 25./8)\n",
    "print(\"Energy of the ground state estimated with 2nd-order perturbation theory:\", energy_pt2(0,25./8,50)) \n",
    "print(\"Energy of the ground state estimated with the variational principle:\", energy_variational(25./8)) \n",
    "print(\"Energy of the ground state estimated with basis set expansion:\", energy_basis(25./8,50)[0]) \n",
    "print(\" \")\n",
    "print(\" \")\n",
    "print(\"Energy of the unperturbed first excited state (field = 0):\", np.pi**2 * 2**2 / 8.) \n",
    "print(\" \")\n",
    "print(\"Field value: \", 1./16)\n",
    "print(\"Exact Energy of the first excited state:\", 39.9787/8 - 1./16)\n",
    "print(\"Energy of the first excited state estimated with 2nd-order perturbation theory:\", energy_pt2(1,1./16,50)) \n",
    "print(\"Energy of the first excited state estimated with basis set expansion:\", energy_basis(1./16,50)[1]) \n",
    "print(\" \")\n",
    "print(\"Field value: \", 25./4)\n",
    "print(\"Exact Energy of the first excited state:\", 65.177/8 - 25./8)\n",
    "print(\"Energy of the first excited state estimated with 2nd-order perturbation theory:\", energy_pt2(1,25./4,50)) \n",
    "print(\"Energy of the first excited state estimated with basis set expansion:\", energy_basis(25./4,50)[1]) \n",
    "print(\" \")\n",
    "print(\" \")\n",
    "print(\"Energy of the unperturbed second excited state (field = 0):\", np.pi**2 * 3**2 / 8.) \n",
    "print(\" \")\n",
    "print(\"Field value: \", 1./16)\n",
    "print(\"Exact Energy of the second excited state:\", 89.3266/8 - 1./16)\n",
    "print(\"Energy of the second excited state estimated with 2nd-order perturbation theory:\", energy_pt2(2,1./16,50)) \n",
    "print(\"Energy of the first excited state estimated with basis set expansion:\", energy_basis(1./16,50)[2]) \n",
    "print(\" \")\n",
    "print(\"Field value: \", 25./4)\n",
    "print(\"Exact Energy of the ground state:\", 114.309/8 - 25./8)\n",
    "print(\"Energy of the second excited state estimated with 2nd-order perturbation theory:\", energy_pt2(2,25./4,50)) \n",
    "print(\"Energy of the second excited state estimated with basis set expansion:\", energy_basis(25./4,50)[2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "hungarian-statement",
   "metadata": {
    "code_folding": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# user-specified parameters\n",
    "F = 25.0 / 8\n",
    "nbasis = 20\n",
    "\n",
    "# plot basis set convergence of energy estimates at a given field\n",
    "# ---------------------------------------------------------------\n",
    "\n",
    "# evaluate energy for a range of basis functions at a given field\n",
    "n_values = np.arange(2, 11, 1)\n",
    "e_pt2_basis = np.array([energy_pt2(0, F, n) for n in n_values])\n",
    "e_var_basis = np.repeat(energy_variational(F), len(n_values))\n",
    "e_exp_basis = np.array([energy_basis(F, n)[0] for n in n_values])\n",
    "\n",
    "# evaluate energy for a range of fields at a given basis\n",
    "f_values = np.arange(0.0, 100., 5.)\n",
    "e_pt2_field = np.array([energy_pt2(0, f, nbasis) for f in f_values])\n",
    "e_var_field = np.array([energy_variational(f) for f in f_values])\n",
    "e_exp_field = np.array([energy_basis(f, nbasis)[0] for f in f_values])\n",
    "\n",
    "\n",
    "plt.rcParams['figure.figsize'] = [15, 8]\n",
    "fig, axes = plt.subplots(1, 2)\n",
    "# fig.suptitle(\"Basis Set Convergence of Particle-in-a-Box with Jacobi Basis\", fontsize=24, fontweight='bold')\n",
    "\n",
    "for index, axis in enumerate(axes.ravel()):\n",
    "    if index == 0:\n",
    "        # plot approximate energy at a fixed field\n",
    "        axis.plot(n_values, e_pt2_basis, marker='o', linestyle='--', label='PT2')\n",
    "        axis.plot(n_values, e_var_basis, marker='', linestyle='-', label='Variational')\n",
    "        axis.plot(n_values, e_exp_basis, marker='x', linestyle='-', label='Basis Expansion')\n",
    "        # set axes labels\n",
    "        axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n",
    "        axis.set_ylabel(\"Ground-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n",
    "        axis.set_title(f\"Field Strength = {F}\", fontsize=24, fontweight='bold')\n",
    "        axis.legend(frameon=False, fontsize=14)\n",
    "    else:\n",
    "        # plot approximate energy at a fixed basis\n",
    "        axis.plot(f_values, e_pt2_field, marker='o', linestyle='--', label='PT2')\n",
    "        axis.plot(f_values, e_var_field, marker='', linestyle='-', label='Variational')\n",
    "        axis.plot(f_values, e_exp_field, marker='x', linestyle='-', label='Basis Expansion')\n",
    "        # set axes labels\n",
    "        axis.set_xlabel(\"Field Strength [a.u.]\", fontsize=12, fontweight='bold')\n",
    "        axis.set_ylabel(\"Ground-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n",
    "        axis.set_title(f\"Number of Basis = {nbasis}\", fontsize=24, fontweight='bold')\n",
    "        axis.legend(frameon=False, fontsize=14)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "manufactured-insulin",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1080x576 with 2 Axes>"
      ]
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   "source": [
    "# user-specified parameters\n",
    "F = 25.0 / 8\n",
    "nbasis = 20\n",
    "\n",
    "# plot basis set convergence of 1st excited state energy at a given field\n",
    "# ---------------------------------------------------------------\n",
    "\n",
    "# evaluate energy for a range of basis functions at a given field\n",
    "n_values = np.arange(2, 11, 1)\n",
    "e_pt2_basis = np.array([energy_pt2(1, F, n) for n in n_values])\n",
    "e_exp_basis = np.array([energy_basis(F, n)[1] for n in n_values])\n",
    "\n",
    "# evaluate energy for a range of fields at a given basis\n",
    "f_values = np.arange(0.0, 100., 5.)\n",
    "e_pt2_field = np.array([energy_pt2(1, f, nbasis) for f in f_values])\n",
    "e_exp_field = np.array([energy_basis(f, nbasis)[1] for f in f_values])\n",
    "\n",
    "\n",
    "plt.rcParams['figure.figsize'] = [15, 8]\n",
    "fig, axes = plt.subplots(1, 2)\n",
    "# fig.suptitle(\"Basis Set Convergence of Particle-in-a-Box with Jacobi Basis\", fontsize=24, fontweight='bold')\n",
    "\n",
    "for index, axis in enumerate(axes.ravel()):\n",
    "    if index == 0:\n",
    "        # plot approximate energy at a fixed field\n",
    "        axis.plot(n_values, e_pt2_basis, marker='o', linestyle='--', label='PT2')\n",
    "        axis.plot(n_values, e_exp_basis, marker='x', linestyle='-', label='Basis Expansion')\n",
    "        # set axes labels\n",
    "        axis.set_xlabel(\"Number of Basis Functions\", fontsize=12, fontweight='bold')\n",
    "        axis.set_ylabel(\"1st Excited-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n",
    "        axis.set_title(f\"Field Strength = {F}\", fontsize=24, fontweight='bold')\n",
    "        axis.legend(frameon=False, fontsize=14)\n",
    "    else:\n",
    "        # plot approximate energy at a fixed basis\n",
    "        axis.plot(f_values, e_pt2_field, marker='o', linestyle='--', label='PT2')\n",
    "        axis.plot(f_values, e_exp_field, marker='x', linestyle='-', label='Basis Expansion')\n",
    "        # set axes labels\n",
    "        axis.set_xlabel(\"Field Strength [a.u.]\", fontsize=12, fontweight='bold')\n",
    "        axis.set_ylabel(\"1st Excited-State Energy [a.u.]\", fontsize=12, fontweight='bold')\n",
    "        axis.set_title(f\"Number of Basis = {nbasis}\", fontsize=24, fontweight='bold')\n",
    "        axis.legend(frameon=False, fontsize=14)\n",
    "\n",
    "plt.show()"
   ]
  },
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   "source": [
    "## &#x1fa9e; Self-Reflection\n",
    "- When is a basis set appropriate? When is perturbation theory more appropriate?\n",
    "- Consider the hydrogen molecule ion, $\\text{H}_2^+$. Is it more sensible to use the secular equation (basis-set-expansion) or perturbation theory? What if the bond length is very small? What if the bond length is very large?\n",
    "\n",
    "## &#x1f914; Thought-Provoking Questions\n",
    "- Show that if you minimize the energy as a function of the basis-set coefficients using the variational principle, then you obtain the secular equation.\n",
    "- If a uniform external electric field of magnitude $F$ in the $\\hat{\\mathbf{u}} = [\\hat{u}_x,\\hat{u}_y,\\hat{u}_z]^T$ direction is applied to a particle with charge $q$, the potential $V(x,y,z) = -qF(u_x x + u_y y + u_z z)$ is added to the Hamiltonian. (This follows from the fact that the force applied to the particles is proportional to the electric field, $\\text{force} = q \\vec{E} = q F \\hat{\\mathbf{u}}$ and the force is $\\text{force} = - \\nabla V(x,y,z)$. If the field is weak, then perturbation theory can be used, and the energy can be written as a Taylor series. The coefficients of the Taylor series give the dipole moment ($\\mu$), dipole polarizability ($\\alpha$), first dipole hyperpolarizability ($\\beta$), second dipole hyperpolarizability ($\\gamma$) in the $\\hat{\\mathbf{u}}$ direction.\n",
    "   - The dipole moment, $\\mu$, of any spherical system is zero. Explain why.\n",
    "   - The polarizability, $\\alpha$, of any system is always positive. Explain why.\n",
    "\n",
    "\\begin{align}\n",
    "E_k(F) &= E_k(0) + F \\left[\\frac{dE_k}{d F} \\right]_{F=0} \n",
    "+ \\frac{F^2}{2!} \\left[\\frac{d^2E_k}{d F^2} \\right]_{F=0} \n",
    "+ \\frac{F^3}{3!} \\left[\\frac{d^3E_k}{d F^3} \\right]_{F=0} \n",
    "+ \\frac{F^4}{4!} \\left[\\frac{d^4E_k}{d F^4} \\right]_{F=0} + \\cdots \\\\\n",
    "&= E_k(0) - F \\mu_{F=0} \n",
    "- \\frac{F^2}{2!} \\alpha_{F=0} \n",
    "- \\frac{F^3}{3!} \\beta_{F=0} \n",
    "- \\frac{F^4}{4!} \\gamma_{F=0} + \\cdots \\\\\n",
    "\\end{align}\n",
    "\n",
    "- The Hellmann-Feynman theorem indicates that given the ground-state wavefunction for a molecule, the force on the nuclei can be obtained. Explain how.\n",
    "- What does it mean that perturbation theory is inaccurate when the perturbation is large?\n",
    "- Can you explain why the energy goes down when the electron-in-a-box is placed in an external field?\n",
    "- For a sufficiently-highly excited state, the effect of an external electric field is negligible. Why is this true intuitively? Can you show it graphically? Can you explain it mathematically?\n",
    "\n",
    "## &#x1f501; Recapitulation\n",
    "- What is the secular equation?\n",
    "- What is the Hellmann-Feynman theorem?\n",
    "- How is the Hellmann-Feynman theorem related to perturbation theory? \n",
    "- What is perturbation theory? What is the expression for the first-order perturbed wavefunction?\n",
    "\n",
    "## &#x1f52e; Next Up...\n",
    "- Multielectron systems\n",
    "- Approximate methods for multielectron systems.\n",
    "\n",
    "## &#x1f4da; References\n",
    "My favorite sources for this material are:\n",
    "- [Randy's book](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/DumontBook.pdf?raw=true)\n",
    "- D. A. MacQuarrie, Quantum Chemistry (University Science Books, Mill Valley California, 1983)\n",
    "\n",
    "There are also some excellent wikipedia articles:\n",
    "- [Perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory)\n",
    "- [Variational method](https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics))"
   ]
  }
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