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<!-- 2021-02-02 Tue 13:25 -->
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<!-- 2021-02-02 Tue 15:42 -->
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<title>Quantum Monte Carlo</title>
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<title>Quantum Monte Carlo</title>
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<h2>Table of Contents</h2>
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<div id="text-table-of-contents">
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<ul>
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<ul>
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<li><a href="#org30f9b00">1. Introduction</a>
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<li><a href="#org4806db2">1. Introduction</a>
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<ul>
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<ul>
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<li><a href="#org2a0c339">1.1. Energy and local energy</a></li>
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<li><a href="#org39e8734">1.1. Energy and local energy</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgd046170">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<li><a href="#org3ff5f5c">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<ul>
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<ul>
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<li><a href="#org04c419e">2.1. Local energy</a>
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<li><a href="#org2f969ef">2.1. Local energy</a>
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<ul>
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<ul>
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<li><a href="#org9662ca2">2.1.1. Exercise 1</a>
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<li><a href="#org1a09b0b">2.1.1. Exercise 1</a>
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<ul>
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<ul>
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<li><a href="#org39a8341">2.1.1.1. Solution</a></li>
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<li><a href="#org6526a1e">2.1.1.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org5310947">2.1.2. Exercise 2</a>
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<li><a href="#orge1801ce">2.1.2. Exercise 2</a>
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||||||
<ul>
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<ul>
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<li><a href="#orgad57cd6">2.1.2.1. Solution</a></li>
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<li><a href="#orgce6e25c">2.1.2.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgf590195">2.1.3. Exercise 3</a>
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<li><a href="#org79375a8">2.1.3. Exercise 3</a>
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<ul>
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<ul>
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<li><a href="#org5552ad3">2.1.3.1. Solution</a></li>
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<li><a href="#org31a08a8">2.1.3.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org6b87999">2.1.4. Exercise 4</a>
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<li><a href="#org8fa6bcf">2.1.4. Exercise 4</a>
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<ul>
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<ul>
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<li><a href="#org10518c2">2.1.4.1. Solution</a></li>
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<li><a href="#orga7c5b22">2.1.4.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgc9278ef">2.1.5. Exercise 5</a>
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<li><a href="#org391c941">2.1.5. Exercise 5</a>
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<ul>
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<ul>
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<li><a href="#orga3e7c1d">2.1.5.1. Solution</a></li>
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<li><a href="#orgefe27af">2.1.5.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org9c14c10">2.2. Plot of the local energy along the \(x\) axis</a>
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<li><a href="#org8ae2dd0">2.2. Plot of the local energy along the \(x\) axis</a>
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<ul>
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<ul>
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<li><a href="#orgd7332c2">2.2.1. Exercise</a>
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<li><a href="#org1a03f83">2.2.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#org4012fb9">2.2.1.1. Solution</a></li>
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<li><a href="#org70196f5">2.2.1.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgcdc18e1">2.3. Numerical estimation of the energy</a>
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<li><a href="#orged0d97d">2.3. Numerical estimation of the energy</a>
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<ul>
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<ul>
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<li><a href="#orgdc606a2">2.3.1. Exercise</a>
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<li><a href="#orgc31228d">2.3.1. Exercise</a>
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||||||
<ul>
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<ul>
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<li><a href="#org210143f">2.3.1.1. Solution</a></li>
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<li><a href="#org6128c99">2.3.1.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orga42d797">2.4. Variance of the local energy</a>
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<li><a href="#org7c02a1e">2.4. Variance of the local energy</a>
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<ul>
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<ul>
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<li><a href="#orgdf66630">2.4.1. Exercise (optional)</a>
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<li><a href="#org436af40">2.4.1. Exercise (optional)</a>
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<ul>
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<ul>
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<li><a href="#org5bea86c">2.4.1.1. Solution</a></li>
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<li><a href="#org6c201e4">2.4.1.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org6ded880">2.4.2. Exercise</a>
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<li><a href="#org761059b">2.4.2. Exercise</a>
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<ul>
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<ul>
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<li><a href="#orgb68ab31">2.4.2.1. Solution</a></li>
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<li><a href="#org66c834d">2.4.2.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org31de66e">3. Variational Monte Carlo</a>
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<li><a href="#orgdaf0632">3. Variational Monte Carlo</a>
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<ul>
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<ul>
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<li><a href="#org6984c54">3.1. Computation of the statistical error</a>
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<li><a href="#org06324d0">3.1. Computation of the statistical error</a>
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<ul>
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<ul>
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<li><a href="#orgdab3180">3.1.1. Exercise</a>
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<li><a href="#org6141752">3.1.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#orge2f6676">3.1.1.1. Solution</a></li>
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<li><a href="#org0d61080">3.1.1.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orga9a234d">3.2. Uniform sampling in the box</a>
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<li><a href="#org223470d">3.2. Uniform sampling in the box</a>
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<ul>
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<ul>
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<li><a href="#org032b417">3.2.1. Exercise</a>
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<li><a href="#orgce01843">3.2.1. Exercise</a>
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||||||
<ul>
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<ul>
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||||||
<li><a href="#org64d30ff">3.2.1.1. Solution</a></li>
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<li><a href="#orgad5e6d9">3.2.1.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgb67c4b6">3.3. Metropolis sampling with \(\Psi^2\)</a>
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<li><a href="#org31ec1ae">3.3. Metropolis sampling with \(\Psi^2\)</a>
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<ul>
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<ul>
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<li><a href="#orgf5ad4c8">3.3.1. Optimal step size</a></li>
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<li><a href="#orgb881ef4">3.3.1. Optimal step size</a></li>
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<li><a href="#org520886f">3.3.2. Exercise</a>
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<li><a href="#org4349247">3.3.2. Exercise</a>
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||||||
<ul>
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<ul>
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<li><a href="#org1ffb571">3.3.2.1. Solution</a></li>
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<li><a href="#org8e41d72">3.3.2.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgdebaed0">3.4. Generalized Metropolis algorithm</a>
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<li><a href="#org76e82d1">3.4. Generalized Metropolis algorithm</a>
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<ul>
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<ul>
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<li><a href="#org29273ad">3.4.1. Gaussian random number generator</a></li>
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<li><a href="#org1a612c2">3.4.1. Gaussian random number generator</a></li>
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<li><a href="#orgd0fe2c3">3.4.2. Exercise 1</a>
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<li><a href="#orgec93d83">3.4.2. Exercise 1</a>
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||||||
<ul>
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<ul>
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||||||
<li><a href="#orgd84d00c">3.4.2.1. Solution</a></li>
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<li><a href="#orgc6b904e">3.4.2.1. Solution</a></li>
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</ul>
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</ul>
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||||||
</li>
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</li>
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<li><a href="#orgc5fbb56">3.4.3. Exercise 2</a>
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<li><a href="#orgcd77b87">3.4.3. Exercise 2</a>
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||||||
<ul>
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<ul>
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<li><a href="#org129879f">3.4.3.1. Solution</a></li>
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<li><a href="#org153ffd5">3.4.3.1. Solution</a></li>
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||||||
</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org439e68a">4. Diffusion Monte Carlo</a>
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<li><a href="#orgc82bf53">4. Diffusion Monte Carlo</a>
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<ul>
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<ul>
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<li><a href="#orgd5b3203">4.1. Schrödinger equation in imaginary time</a></li>
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<li><a href="#org464134c">4.1. Schrödinger equation in imaginary time</a></li>
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<li><a href="#orgc474c5a">4.2. Diffusion and branching</a></li>
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<li><a href="#org70f8d8f">4.2. Diffusion and branching</a></li>
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<li><a href="#org44a65e4">4.3. Importance sampling</a>
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<li><a href="#orgea2a477">4.3. Importance sampling</a>
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<ul>
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<ul>
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<li><a href="#orgcc53a23">4.3.1. Appendix : Details of the Derivation</a></li>
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<li><a href="#orgc70f2e3">4.3.1. Appendix : Details of the Derivation</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgb23685a">4.4. Pure Diffusion Monte Carlo (PDMC)</a></li>
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<li><a href="#orgd228f4a">4.4. Pure Diffusion Monte Carlo (PDMC)</a></li>
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<li><a href="#org1aef98c">4.5. Hydrogen atom</a>
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<li><a href="#org82e6bf1">4.5. Hydrogen atom</a>
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<ul>
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<ul>
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<li><a href="#org30ac455">4.5.1. Exercise</a>
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<li><a href="#org7a36346">4.5.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#orgccb4486">4.5.1.1. Solution</a></li>
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<li><a href="#org6e72432">4.5.1.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgea27c27">4.6. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
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<li><a href="#org3844040">4.6. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org87e0f32">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
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<li><a href="#org90ecd5b">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
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<li><a href="#org54efc49">6. Schedule</a></li>
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<li><a href="#orga52a4c0">6. Schedule</a></li>
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</ul>
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</ul>
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</div>
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</div>
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</div>
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<div id="outline-container-org30f9b00" class="outline-2">
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<div id="outline-container-org4806db2" class="outline-2">
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<h2 id="org30f9b00"><span class="section-number-2">1</span> Introduction</h2>
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<h2 id="org4806db2"><span class="section-number-2">1</span> Introduction</h2>
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||||||
<div class="outline-text-2" id="text-1">
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<div class="outline-text-2" id="text-1">
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||||||
<p>
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<p>
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||||||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
This website contains the QMC tutorial of the 2021 LTTC winter school
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@ -515,8 +515,8 @@ coordinates, etc).
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</p>
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</p>
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</div>
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</div>
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<div id="outline-container-org2a0c339" class="outline-3">
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<div id="outline-container-org39e8734" class="outline-3">
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||||||
<h3 id="org2a0c339"><span class="section-number-3">1.1</span> Energy and local energy</h3>
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<h3 id="org39e8734"><span class="section-number-3">1.1</span> Energy and local energy</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="outline-text-3" id="text-1-1">
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||||||
<p>
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<p>
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For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
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For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
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@ -599,8 +599,8 @@ energy computed over these configurations:
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-orgd046170" class="outline-2">
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<div id="outline-container-org3ff5f5c" class="outline-2">
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<h2 id="orgd046170"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
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<h2 id="org3ff5f5c"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
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<div class="outline-text-2" id="text-2">
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<div class="outline-text-2" id="text-2">
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<p>
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<p>
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In this section, we consider the hydrogen atom with the following
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In this section, we consider the hydrogen atom with the following
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@ -629,8 +629,8 @@ To do that, we will compute the local energy and check whether it is constant.
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</p>
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</p>
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</div>
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</div>
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<div id="outline-container-org04c419e" class="outline-3">
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<div id="outline-container-org2f969ef" class="outline-3">
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<h3 id="org04c419e"><span class="section-number-3">2.1</span> Local energy</h3>
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<h3 id="org2f969ef"><span class="section-number-3">2.1</span> Local energy</h3>
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<div class="outline-text-3" id="text-2-1">
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<div class="outline-text-3" id="text-2-1">
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<p>
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<p>
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You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
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You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
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@ -657,8 +657,8 @@ to catch the error.
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-org9662ca2" class="outline-4">
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<div id="outline-container-org1a09b0b" class="outline-4">
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<h4 id="org9662ca2"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
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<h4 id="org1a09b0b"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
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<div class="outline-text-4" id="text-2-1-1">
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<div class="outline-text-4" id="text-2-1-1">
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<div class="exercise">
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<div class="exercise">
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<p>
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<p>
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@ -703,8 +703,8 @@ and returns the potential.
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</div>
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</div>
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</div>
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<div id="outline-container-org39a8341" class="outline-5">
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<div id="outline-container-org6526a1e" class="outline-5">
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<h5 id="org39a8341"><span class="section-number-5">2.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org6526a1e"><span class="section-number-5">2.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
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||||||
<div class="outline-text-5" id="text-2-1-1-1">
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<div class="outline-text-5" id="text-2-1-1-1">
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<p>
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<p>
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<b>Python</b>
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<b>Python</b>
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@ -745,8 +745,8 @@ and returns the potential.
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|||||||
</div>
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</div>
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</div>
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</div>
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|
||||||
<div id="outline-container-org5310947" class="outline-4">
|
<div id="outline-container-orge1801ce" class="outline-4">
|
||||||
<h4 id="org5310947"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
<h4 id="orge1801ce"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-2">
|
<div class="outline-text-4" id="text-2-1-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -781,8 +781,8 @@ input arguments, and returns a scalar.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgad57cd6" class="outline-5">
|
<div id="outline-container-orgce6e25c" class="outline-5">
|
||||||
<h5 id="orgad57cd6"><span class="section-number-5">2.1.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orgce6e25c"><span class="section-number-5">2.1.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-2-1-2-1">
|
<div class="outline-text-5" id="text-2-1-2-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -809,8 +809,8 @@ input arguments, and returns a scalar.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgf590195" class="outline-4">
|
<div id="outline-container-org79375a8" class="outline-4">
|
||||||
<h4 id="orgf590195"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
<h4 id="org79375a8"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-3">
|
<div class="outline-text-4" id="text-2-1-3">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -891,8 +891,8 @@ Therefore, the local kinetic energy is
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org5552ad3" class="outline-5">
|
<div id="outline-container-org31a08a8" class="outline-5">
|
||||||
<h5 id="org5552ad3"><span class="section-number-5">2.1.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org31a08a8"><span class="section-number-5">2.1.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-2-1-3-1">
|
<div class="outline-text-5" id="text-2-1-3-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -933,8 +933,8 @@ Therefore, the local kinetic energy is
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6b87999" class="outline-4">
|
<div id="outline-container-org8fa6bcf" class="outline-4">
|
||||||
<h4 id="org6b87999"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
<h4 id="org8fa6bcf"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-4">
|
<div class="outline-text-4" id="text-2-1-4">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -993,8 +993,8 @@ are calling is yours.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org10518c2" class="outline-5">
|
<div id="outline-container-orga7c5b22" class="outline-5">
|
||||||
<h5 id="org10518c2"><span class="section-number-5">2.1.4.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orga7c5b22"><span class="section-number-5">2.1.4.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-2-1-4-1">
|
<div class="outline-text-5" id="text-2-1-4-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -1025,8 +1025,8 @@ are calling is yours.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgc9278ef" class="outline-4">
|
<div id="outline-container-org391c941" class="outline-4">
|
||||||
<h4 id="orgc9278ef"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
<h4 id="org391c941"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||||||
<div class="outline-text-4" id="text-2-1-5">
|
<div class="outline-text-4" id="text-2-1-5">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1036,8 +1036,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orga3e7c1d" class="outline-5">
|
<div id="outline-container-orgefe27af" class="outline-5">
|
||||||
<h5 id="orga3e7c1d"><span class="section-number-5">2.1.5.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orgefe27af"><span class="section-number-5">2.1.5.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-2-1-5-1">
|
<div class="outline-text-5" id="text-2-1-5-1">
|
||||||
\begin{eqnarray*}
|
\begin{eqnarray*}
|
||||||
E &=& \frac{\hat{H} \Psi}{\Psi} = - \frac{1}{2} \frac{\Delta \Psi}{\Psi} -
|
E &=& \frac{\hat{H} \Psi}{\Psi} = - \frac{1}{2} \frac{\Delta \Psi}{\Psi} -
|
||||||
@ -1057,8 +1057,8 @@ equal to -0.5 atomic units.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org9c14c10" class="outline-3">
|
<div id="outline-container-org8ae2dd0" class="outline-3">
|
||||||
<h3 id="org9c14c10"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
<h3 id="org8ae2dd0"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
||||||
<div class="outline-text-3" id="text-2-2">
|
<div class="outline-text-3" id="text-2-2">
|
||||||
<p>
|
<p>
|
||||||
The program you will write in this section will be written in
|
The program you will write in this section will be written in
|
||||||
@ -1089,8 +1089,8 @@ In Fortran, you will need to compile all the source files together:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgd7332c2" class="outline-4">
|
<div id="outline-container-org1a03f83" class="outline-4">
|
||||||
<h4 id="orgd7332c2"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
<h4 id="org1a03f83"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-2-2-1">
|
<div class="outline-text-4" id="text-2-2-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1184,8 +1184,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org4012fb9" class="outline-5">
|
<div id="outline-container-org70196f5" class="outline-5">
|
||||||
<h5 id="org4012fb9"><span class="section-number-5">2.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org70196f5"><span class="section-number-5">2.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-2-2-1-1">
|
<div class="outline-text-5" id="text-2-2-1-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -1262,8 +1262,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgcdc18e1" class="outline-3">
|
<div id="outline-container-orged0d97d" class="outline-3">
|
||||||
<h3 id="orgcdc18e1"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
<h3 id="orged0d97d"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||||||
<div class="outline-text-3" id="text-2-3">
|
<div class="outline-text-3" id="text-2-3">
|
||||||
<p>
|
<p>
|
||||||
If the space is discretized in small volume elements \(\mathbf{r}_i\)
|
If the space is discretized in small volume elements \(\mathbf{r}_i\)
|
||||||
@ -1293,8 +1293,8 @@ The energy is biased because:
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-orgdc606a2" class="outline-4">
|
<div id="outline-container-orgc31228d" class="outline-4">
|
||||||
<h4 id="orgdc606a2"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
<h4 id="orgc31228d"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-2-3-1">
|
<div class="outline-text-4" id="text-2-3-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1365,8 +1365,8 @@ To compile the Fortran and run it:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org210143f" class="outline-5">
|
<div id="outline-container-org6128c99" class="outline-5">
|
||||||
<h5 id="org210143f"><span class="section-number-5">2.3.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org6128c99"><span class="section-number-5">2.3.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-2-3-1-1">
|
<div class="outline-text-5" id="text-2-3-1-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -1483,8 +1483,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orga42d797" class="outline-3">
|
<div id="outline-container-org7c02a1e" class="outline-3">
|
||||||
<h3 id="orga42d797"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
<h3 id="org7c02a1e"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||||||
<div class="outline-text-3" id="text-2-4">
|
<div class="outline-text-3" id="text-2-4">
|
||||||
<p>
|
<p>
|
||||||
The variance of the local energy is a functional of \(\Psi\)
|
The variance of the local energy is a functional of \(\Psi\)
|
||||||
@ -1511,8 +1511,8 @@ energy can be used as a measure of the quality of a wave function.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgdf66630" class="outline-4">
|
<div id="outline-container-org436af40" class="outline-4">
|
||||||
<h4 id="orgdf66630"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
<h4 id="org436af40"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||||||
<div class="outline-text-4" id="text-2-4-1">
|
<div class="outline-text-4" id="text-2-4-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1523,8 +1523,8 @@ Prove that :
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org5bea86c" class="outline-5">
|
<div id="outline-container-org6c201e4" class="outline-5">
|
||||||
<h5 id="org5bea86c"><span class="section-number-5">2.4.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org6c201e4"><span class="section-number-5">2.4.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-2-4-1-1">
|
<div class="outline-text-5" id="text-2-4-1-1">
|
||||||
<p>
|
<p>
|
||||||
\(\bar{E} = \langle E \rangle\) is a constant, so \(\langle \bar{E}
|
\(\bar{E} = \langle E \rangle\) is a constant, so \(\langle \bar{E}
|
||||||
@ -1543,8 +1543,8 @@ Prove that :
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
<div id="outline-container-org6ded880" class="outline-4">
|
<div id="outline-container-org761059b" class="outline-4">
|
||||||
<h4 id="org6ded880"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
<h4 id="org761059b"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-2-4-2">
|
<div class="outline-text-4" id="text-2-4-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1620,8 +1620,8 @@ To compile and run:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgb68ab31" class="outline-5">
|
<div id="outline-container-org66c834d" class="outline-5">
|
||||||
<h5 id="orgb68ab31"><span class="section-number-5">2.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org66c834d"><span class="section-number-5">2.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-2-4-2-1">
|
<div class="outline-text-5" id="text-2-4-2-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -1760,8 +1760,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org31de66e" class="outline-2">
|
<div id="outline-container-orgdaf0632" class="outline-2">
|
||||||
<h2 id="org31de66e"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
<h2 id="orgdaf0632"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||||||
<div class="outline-text-2" id="text-3">
|
<div class="outline-text-2" id="text-3">
|
||||||
<p>
|
<p>
|
||||||
Numerical integration with deterministic methods is very efficient
|
Numerical integration with deterministic methods is very efficient
|
||||||
@ -1777,8 +1777,8 @@ interval.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6984c54" class="outline-3">
|
<div id="outline-container-org06324d0" class="outline-3">
|
||||||
<h3 id="org6984c54"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
<h3 id="org06324d0"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||||||
<div class="outline-text-3" id="text-3-1">
|
<div class="outline-text-3" id="text-3-1">
|
||||||
<p>
|
<p>
|
||||||
To compute the statistical error, you need to perform \(M\)
|
To compute the statistical error, you need to perform \(M\)
|
||||||
@ -1818,8 +1818,8 @@ And the confidence interval is given by
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgdab3180" class="outline-4">
|
<div id="outline-container-org6141752" class="outline-4">
|
||||||
<h4 id="orgdab3180"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
<h4 id="org6141752"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-3-1-1">
|
<div class="outline-text-4" id="text-3-1-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -1859,8 +1859,8 @@ input array.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orge2f6676" class="outline-5">
|
<div id="outline-container-org0d61080" class="outline-5">
|
||||||
<h5 id="orge2f6676"><span class="section-number-5">3.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org0d61080"><span class="section-number-5">3.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-3-1-1-1">
|
<div class="outline-text-5" id="text-3-1-1-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -1921,8 +1921,8 @@ input array.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orga9a234d" class="outline-3">
|
<div id="outline-container-org223470d" class="outline-3">
|
||||||
<h3 id="orga9a234d"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
<h3 id="org223470d"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||||||
<div class="outline-text-3" id="text-3-2">
|
<div class="outline-text-3" id="text-3-2">
|
||||||
<p>
|
<p>
|
||||||
We will now perform our first Monte Carlo calculation to compute the
|
We will now perform our first Monte Carlo calculation to compute the
|
||||||
@ -1983,8 +1983,8 @@ compute the statistical error.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org032b417" class="outline-4">
|
<div id="outline-container-orgce01843" class="outline-4">
|
||||||
<h4 id="org032b417"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
<h4 id="orgce01843"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-3-2-1">
|
<div class="outline-text-4" id="text-3-2-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -2086,8 +2086,8 @@ well as the index of the current step.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org64d30ff" class="outline-5">
|
<div id="outline-container-orgad5e6d9" class="outline-5">
|
||||||
<h5 id="org64d30ff"><span class="section-number-5">3.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orgad5e6d9"><span class="section-number-5">3.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-3-2-1-1">
|
<div class="outline-text-5" id="text-3-2-1-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -2193,8 +2193,8 @@ E = -0.48084122147238995 +/- 2.4983775878329355E-003
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgb67c4b6" class="outline-3">
|
<div id="outline-container-org31ec1ae" class="outline-3">
|
||||||
<h3 id="orgb67c4b6"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
<h3 id="org31ec1ae"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||||||
<div class="outline-text-3" id="text-3-3">
|
<div class="outline-text-3" id="text-3-3">
|
||||||
<p>
|
<p>
|
||||||
We will now use the square of the wave function to sample random
|
We will now use the square of the wave function to sample random
|
||||||
@ -2313,8 +2313,8 @@ All samples should be kept, from both accepted <i>and</i> rejected moves.
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-orgf5ad4c8" class="outline-4">
|
<div id="outline-container-orgb881ef4" class="outline-4">
|
||||||
<h4 id="orgf5ad4c8"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
|
<h4 id="orgb881ef4"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
|
||||||
<div class="outline-text-4" id="text-3-3-1">
|
<div class="outline-text-4" id="text-3-3-1">
|
||||||
<p>
|
<p>
|
||||||
If the box is infinitely small, the ratio will be very close
|
If the box is infinitely small, the ratio will be very close
|
||||||
@ -2349,8 +2349,8 @@ the same variable later on to store a time step.
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org520886f" class="outline-4">
|
<div id="outline-container-org4349247" class="outline-4">
|
||||||
<h4 id="org520886f"><span class="section-number-4">3.3.2</span> Exercise</h4>
|
<h4 id="org4349247"><span class="section-number-4">3.3.2</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-3-3-2">
|
<div class="outline-text-4" id="text-3-3-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -2459,8 +2459,8 @@ Can you observe a reduction in the statistical error?
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org1ffb571" class="outline-5">
|
<div id="outline-container-org8e41d72" class="outline-5">
|
||||||
<h5 id="org1ffb571"><span class="section-number-5">3.3.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org8e41d72"><span class="section-number-5">3.3.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-3-3-2-1">
|
<div class="outline-text-5" id="text-3-3-2-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -2607,8 +2607,8 @@ A = 0.50762633333333318 +/- 3.4601756760043725E-004
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgdebaed0" class="outline-3">
|
<div id="outline-container-org76e82d1" class="outline-3">
|
||||||
<h3 id="orgdebaed0"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
|
<h3 id="org76e82d1"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
|
||||||
<div class="outline-text-3" id="text-3-4">
|
<div class="outline-text-3" id="text-3-4">
|
||||||
<p>
|
<p>
|
||||||
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
|
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
|
||||||
@ -2729,8 +2729,8 @@ The algorithm of the previous exercise is only slighlty modified as:
|
|||||||
</ol>
|
</ol>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org29273ad" class="outline-4">
|
<div id="outline-container-org1a612c2" class="outline-4">
|
||||||
<h4 id="org29273ad"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
|
<h4 id="org1a612c2"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
|
||||||
<div class="outline-text-4" id="text-3-4-1">
|
<div class="outline-text-4" id="text-3-4-1">
|
||||||
<p>
|
<p>
|
||||||
To obtain Gaussian-distributed random numbers, you can apply the
|
To obtain Gaussian-distributed random numbers, you can apply the
|
||||||
@ -2794,8 +2794,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-orgd0fe2c3" class="outline-4">
|
<div id="outline-container-orgec93d83" class="outline-4">
|
||||||
<h4 id="orgd0fe2c3"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
|
<h4 id="orgec93d83"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
|
||||||
<div class="outline-text-4" id="text-3-4-2">
|
<div class="outline-text-4" id="text-3-4-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -2837,8 +2837,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgd84d00c" class="outline-5">
|
<div id="outline-container-orgc6b904e" class="outline-5">
|
||||||
<h5 id="orgd84d00c"><span class="section-number-5">3.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orgc6b904e"><span class="section-number-5">3.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-3-4-2-1">
|
<div class="outline-text-5" id="text-3-4-2-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -2871,8 +2871,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgc5fbb56" class="outline-4">
|
<div id="outline-container-orgcd77b87" class="outline-4">
|
||||||
<h4 id="orgc5fbb56"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
|
<h4 id="orgcd77b87"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
|
||||||
<div class="outline-text-4" id="text-3-4-3">
|
<div class="outline-text-4" id="text-3-4-3">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -2968,8 +2968,8 @@ Modify the previous program to introduce the drift-diffusion scheme.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org129879f" class="outline-5">
|
<div id="outline-container-org153ffd5" class="outline-5">
|
||||||
<h5 id="org129879f"><span class="section-number-5">3.4.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org153ffd5"><span class="section-number-5">3.4.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-3-4-3-1">
|
<div class="outline-text-5" id="text-3-4-3-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -3151,12 +3151,12 @@ A = 0.62037333333333333 +/- 4.8970160591451110E-004
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org439e68a" class="outline-2">
|
<div id="outline-container-orgc82bf53" class="outline-2">
|
||||||
<h2 id="org439e68a"><span class="section-number-2">4</span> Diffusion Monte Carlo   <span class="tag"><span class="solution">solution</span></span></h2>
|
<h2 id="orgc82bf53"><span class="section-number-2">4</span> Diffusion Monte Carlo   <span class="tag"><span class="solution">solution</span></span></h2>
|
||||||
<div class="outline-text-2" id="text-4">
|
<div class="outline-text-2" id="text-4">
|
||||||
</div>
|
</div>
|
||||||
<div id="outline-container-orgd5b3203" class="outline-3">
|
<div id="outline-container-org464134c" class="outline-3">
|
||||||
<h3 id="orgd5b3203"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
<h3 id="org464134c"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
||||||
<div class="outline-text-3" id="text-4-1">
|
<div class="outline-text-3" id="text-4-1">
|
||||||
<p>
|
<p>
|
||||||
Consider the time-dependent Schrödinger equation:
|
Consider the time-dependent Schrödinger equation:
|
||||||
@ -3224,8 +3224,8 @@ system.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgc474c5a" class="outline-3">
|
<div id="outline-container-org70f8d8f" class="outline-3">
|
||||||
<h3 id="orgc474c5a"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
<h3 id="org70f8d8f"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
||||||
<div class="outline-text-3" id="text-4-2">
|
<div class="outline-text-3" id="text-4-2">
|
||||||
<p>
|
<p>
|
||||||
The imaginary-time Schrödinger equation can be explicitly written in terms of the kinetic and
|
The imaginary-time Schrödinger equation can be explicitly written in terms of the kinetic and
|
||||||
@ -3322,8 +3322,8 @@ Therefore, in both cases, you are dealing with a "Bosonic" ground state.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org44a65e4" class="outline-3">
|
<div id="outline-container-orgea2a477" class="outline-3">
|
||||||
<h3 id="org44a65e4"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
<h3 id="orgea2a477"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
||||||
<div class="outline-text-3" id="text-4-3">
|
<div class="outline-text-3" id="text-4-3">
|
||||||
<p>
|
<p>
|
||||||
In a molecular system, the potential is far from being constant
|
In a molecular system, the potential is far from being constant
|
||||||
@ -3419,8 +3419,8 @@ energies computed with the trial wave function.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgcc53a23" class="outline-4">
|
<div id="outline-container-orgc70f2e3" class="outline-4">
|
||||||
<h4 id="orgcc53a23"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4>
|
<h4 id="orgc70f2e3"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4>
|
||||||
<div class="outline-text-4" id="text-4-3-1">
|
<div class="outline-text-4" id="text-4-3-1">
|
||||||
<p>
|
<p>
|
||||||
\[
|
\[
|
||||||
@ -3481,8 +3481,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgb23685a" class="outline-3">
|
<div id="outline-container-orgd228f4a" class="outline-3">
|
||||||
<h3 id="orgb23685a"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
<h3 id="orgd228f4a"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
||||||
<div class="outline-text-3" id="text-4-4">
|
<div class="outline-text-3" id="text-4-4">
|
||||||
<p>
|
<p>
|
||||||
Instead of having a variable number of particles to simulate the
|
Instead of having a variable number of particles to simulate the
|
||||||
@ -3563,13 +3563,13 @@ code, so this is what we will do in the next section.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org1aef98c" class="outline-3">
|
<div id="outline-container-org82e6bf1" class="outline-3">
|
||||||
<h3 id="org1aef98c"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
|
<h3 id="org82e6bf1"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
|
||||||
<div class="outline-text-3" id="text-4-5">
|
<div class="outline-text-3" id="text-4-5">
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org30ac455" class="outline-4">
|
<div id="outline-container-org7a36346" class="outline-4">
|
||||||
<h4 id="org30ac455"><span class="section-number-4">4.5.1</span> Exercise</h4>
|
<h4 id="org7a36346"><span class="section-number-4">4.5.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-4-5-1">
|
<div class="outline-text-4" id="text-4-5-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -3668,8 +3668,8 @@ energy of H for any value of \(a\).
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgccb4486" class="outline-5">
|
<div id="outline-container-org6e72432" class="outline-5">
|
||||||
<h5 id="orgccb4486"><span class="section-number-5">4.5.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org6e72432"><span class="section-number-5">4.5.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-4-5-1-1">
|
<div class="outline-text-5" id="text-4-5-1-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -3887,8 +3887,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-orgea27c27" class="outline-3">
|
<div id="outline-container-org3844040" class="outline-3">
|
||||||
<h3 id="orgea27c27"><span class="section-number-3">4.6</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
|
<h3 id="org3844040"><span class="section-number-3">4.6</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
|
||||||
<div class="outline-text-3" id="text-4-6">
|
<div class="outline-text-3" id="text-4-6">
|
||||||
<p>
|
<p>
|
||||||
We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the
|
We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the
|
||||||
@ -3909,8 +3909,8 @@ the nuclei.
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org87e0f32" class="outline-2">
|
<div id="outline-container-org90ecd5b" class="outline-2">
|
||||||
<h2 id="org87e0f32"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2>
|
<h2 id="org90ecd5b"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2>
|
||||||
<div class="outline-text-2" id="text-5">
|
<div class="outline-text-2" id="text-5">
|
||||||
<ul class="org-ul">
|
<ul class="org-ul">
|
||||||
<li class="off"><code>[ ]</code> Give some hints of how much time is required for each section</li>
|
<li class="off"><code>[ ]</code> Give some hints of how much time is required for each section</li>
|
||||||
@ -3924,8 +3924,8 @@ the H\(_2\) molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.</li>
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org54efc49" class="outline-2">
|
<div id="outline-container-orga52a4c0" class="outline-2">
|
||||||
<h2 id="org54efc49"><span class="section-number-2">6</span> Schedule</h2>
|
<h2 id="orga52a4c0"><span class="section-number-2">6</span> Schedule</h2>
|
||||||
<div class="outline-text-2" id="text-6">
|
<div class="outline-text-2" id="text-6">
|
||||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||||
|
|
||||||
@ -3989,7 +3989,7 @@ the H\(_2\) molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.</li>
|
|||||||
</div>
|
</div>
|
||||||
<div id="postamble" class="status">
|
<div id="postamble" class="status">
|
||||||
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
||||||
<p class="date">Created: 2021-02-02 Tue 13:25</p>
|
<p class="date">Created: 2021-02-02 Tue 15:42</p>
|
||||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||||
</div>
|
</div>
|
||||||
</body>
|
</body>
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user