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<!-- 2021-01-31 Sun 09:26 -->
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<!-- 2021-01-31 Sun 16:02 -->
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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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="#orgda3f50b">1. Introduction</a>
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<li><a href="#orga91139b">1. Introduction</a>
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<ul>
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<ul>
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<li><a href="#org036e966">1.1. Energy and local energy</a></li>
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<li><a href="#org57a402c">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="#orgc590c55">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<li><a href="#org58a42cc">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="#org18ff41d">2.1. Local energy</a>
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<li><a href="#orgf1230c2">2.1. Local energy</a>
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<ul>
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<ul>
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<li><a href="#org683e4e2">2.1.1. Exercise 1</a>
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<li><a href="#org244a69e">2.1.1. Exercise 1</a>
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<ul>
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<ul>
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<li><a href="#org3f5fc9d">2.1.1.1. Solution</a></li>
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<li><a href="#org6763bba">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="#orgc407b41">2.1.2. Exercise 2</a>
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<li><a href="#org61975e4">2.1.2. Exercise 2</a>
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<ul>
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<ul>
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<li><a href="#org9104fc1">2.1.2.1. Solution</a></li>
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<li><a href="#orge084b4d">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="#orgb70d960">2.1.3. Exercise 3</a>
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<li><a href="#orgb158f45">2.1.3. Exercise 3</a>
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<ul>
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<ul>
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<li><a href="#org77aed71">2.1.3.1. Solution</a></li>
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<li><a href="#org2ab2a29">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="#org7ae5fd2">2.1.4. Exercise 4</a>
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<li><a href="#org890847f">2.1.4. Exercise 4</a>
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<ul>
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<ul>
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<li><a href="#orgc9c223c">2.1.4.1. Solution</a></li>
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<li><a href="#org35d7e90">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="#org9fad9d8">2.1.5. Exercise 5</a>
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<li><a href="#org4ce2a48">2.1.5. Exercise 5</a>
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<ul>
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<ul>
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<li><a href="#org6035a20">2.1.5.1. Solution</a></li>
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<li><a href="#org2543a5f">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="#org541ad7b">2.2. Plot of the local energy along the \(x\) axis</a>
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<li><a href="#orgd06a1e7">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="#org6fe4366">2.2.1. Exercise</a>
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<li><a href="#org269a10b">2.2.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#org3f1245c">2.2.1.1. Solution</a></li>
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<li><a href="#org7a72ea6">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="#org887fd3f">2.3. Numerical estimation of the energy</a>
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<li><a href="#org0cc6339">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="#org9274039">2.3.1. Exercise</a>
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<li><a href="#org7d44f1d">2.3.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#org451ad90">2.3.1.1. Solution</a></li>
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<li><a href="#orga0f2716">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="#org276e07e">2.4. Variance of the local energy</a>
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<li><a href="#org360dc66">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="#org43e53cc">2.4.1. Exercise (optional)</a>
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<li><a href="#org7c4cd23">2.4.1. Exercise (optional)</a>
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<ul>
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<ul>
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<li><a href="#org38d9972">2.4.1.1. Solution</a></li>
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<li><a href="#orgf1c30f6">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="#org733ec8a">2.4.2. Exercise</a>
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<li><a href="#orged66d34">2.4.2. Exercise</a>
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<ul>
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<ul>
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<li><a href="#org739fa49">2.4.2.1. Solution</a></li>
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<li><a href="#org08ccc7f">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="#org81347ce">3. Variational Monte Carlo</a>
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<li><a href="#orga7eaa2d">3. Variational Monte Carlo</a>
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<ul>
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<ul>
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<li><a href="#orgd013ab9">3.1. Computation of the statistical error</a>
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<li><a href="#orgbd090a7">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="#org32ca3c3">3.1.1. Exercise</a>
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<li><a href="#orgc3fdd7d">3.1.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#orgebeb146">3.1.1.1. Solution</a></li>
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<li><a href="#org646758c">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="#orgb2a9889">3.2. Uniform sampling in the box</a>
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<li><a href="#org4c8f35b">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="#org4e6a1f7">3.2.1. Exercise</a>
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<li><a href="#org40abc16">3.2.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#orgbffcbab">3.2.1.1. Solution</a></li>
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<li><a href="#orgfc967b4">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="#org0733fed">3.3. Metropolis sampling with \(\Psi^2\)</a>
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<li><a href="#orga44380c">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="#org0c0af3c">3.3.1. Exercise</a>
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<li><a href="#org5859423">3.3.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#org6c72dc3">3.3.1.1. Solution</a></li>
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<li><a href="#org2606981">3.3.1.1. 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="#orga0b0b8d">3.4. Gaussian random number generator</a></li>
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<li><a href="#org10c0223">3.4. Gaussian random number generator</a></li>
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<li><a href="#org57a608a">3.5. Generalized Metropolis algorithm</a>
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<li><a href="#org89c81c5">3.5. Generalized Metropolis algorithm</a>
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<ul>
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<ul>
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<li><a href="#orgaa87988">3.5.1. Exercise 1</a>
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<li><a href="#org2359a6c">3.5.1. Exercise 1</a>
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<ul>
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<ul>
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<li><a href="#orgf391a29">3.5.1.1. Solution</a></li>
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<li><a href="#orga3e9097">3.5.1.1. 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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<li><a href="#org6abfe47">3.5.2. Exercise 2</a>
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<li><a href="#org475617c">3.5.2. Exercise 2</a>
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||||||
<ul>
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<ul>
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||||||
<li><a href="#org53b5631">3.5.2.1. Solution</a></li>
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<li><a href="#orgc22b63a">3.5.2.1. 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="#org5f1d81b">4. Diffusion Monte Carlo</a>
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<li><a href="#orgcf7b82d">4. Diffusion Monte Carlo</a>
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<ul>
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<ul>
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<li><a href="#org95ab9cd">4.1. Schrödinger equation in imaginary time</a></li>
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<li><a href="#orgfb5c875">4.1. Schrödinger equation in imaginary time</a></li>
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<li><a href="#org9fcbab9">4.2. Diffusion and branching</a></li>
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<li><a href="#org8524696">4.2. Diffusion and branching</a></li>
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<li><a href="#org0aba1e8">4.3. Importance sampling</a>
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<li><a href="#orgfb0b533">4.3. Importance sampling</a>
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<ul>
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<ul>
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<li><a href="#orgcc09947">4.3.1. Appendix : Details of the Derivation</a></li>
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<li><a href="#orgb989591">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="#org2f26efe">4.4. Fixed-node DMC energy</a></li>
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<li><a href="#org98fdfce">4.4. Fixed-node DMC energy</a></li>
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<li><a href="#orgfc317d3">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li>
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<li><a href="#org9a74f72">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li>
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<li><a href="#orgf410952">4.6. Hydrogen atom</a>
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<li><a href="#org0e4c39d">4.6. Hydrogen atom</a>
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<ul>
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<ul>
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<li><a href="#org5899d68">4.6.1. Exercise</a>
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<li><a href="#org406db4f">4.6.1. Exercise</a>
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<ul>
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<ul>
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<li><a href="#orge0716bd">4.6.1.1. Solution</a></li>
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<li><a href="#orgb3872d9">4.6.1.1. 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="#org5e287d8">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
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<li><a href="#org8ac323b">4.7. <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="#org17d4d49">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
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<li><a href="#org4c7a4a9">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</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-orgda3f50b" class="outline-2">
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<div id="outline-container-orga91139b" class="outline-2">
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<h2 id="orgda3f50b"><span class="section-number-2">1</span> Introduction</h2>
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<h2 id="orga91139b"><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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@ -514,8 +514,8 @@ 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-org036e966" class="outline-3">
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<div id="outline-container-org57a402c" class="outline-3">
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<h3 id="org036e966"><span class="section-number-3">1.1</span> Energy and local energy</h3>
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<h3 id="org57a402c"><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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@ -593,8 +593,8 @@ $$ E ≈ \frac{1}{N<sub>\rm MC</sub>} ∑<sub>i=1</sub><sup>N<sub>\rm MC</
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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-orgc590c55" class="outline-2">
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<div id="outline-container-org58a42cc" class="outline-2">
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<h2 id="orgc590c55"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
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<h2 id="org58a42cc"><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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@ -623,8 +623,8 @@ 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-org18ff41d" class="outline-3">
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<div id="outline-container-orgf1230c2" class="outline-3">
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<h3 id="org18ff41d"><span class="section-number-3">2.1</span> Local energy</h3>
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<h3 id="orgf1230c2"><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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@ -651,8 +651,8 @@ to catch the error.
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</div>
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</div>
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</div>
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<div id="outline-container-org683e4e2" class="outline-4">
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<div id="outline-container-org244a69e" class="outline-4">
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<h4 id="org683e4e2"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
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<h4 id="org244a69e"><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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@ -696,8 +696,8 @@ 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-org3f5fc9d" class="outline-5">
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<div id="outline-container-org6763bba" class="outline-5">
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<h5 id="org3f5fc9d"><span class="section-number-5">2.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org6763bba"><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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@ -737,8 +737,8 @@ 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-orgc407b41" class="outline-4">
|
<div id="outline-container-org61975e4" class="outline-4">
|
||||||
<h4 id="orgc407b41"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
<h4 id="org61975e4"><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>
|
||||||
@ -773,8 +773,8 @@ input arguments, and returns a scalar.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org9104fc1" class="outline-5">
|
<div id="outline-container-orge084b4d" class="outline-5">
|
||||||
<h5 id="org9104fc1"><span class="section-number-5">2.1.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orge084b4d"><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>
|
||||||
@ -801,8 +801,8 @@ input arguments, and returns a scalar.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgb70d960" class="outline-4">
|
<div id="outline-container-orgb158f45" class="outline-4">
|
||||||
<h4 id="orgb70d960"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
<h4 id="orgb158f45"><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>
|
||||||
@ -883,8 +883,8 @@ Therefore, the local kinetic energy is
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org77aed71" class="outline-5">
|
<div id="outline-container-org2ab2a29" class="outline-5">
|
||||||
<h5 id="org77aed71"><span class="section-number-5">2.1.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org2ab2a29"><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>
|
||||||
@ -925,8 +925,8 @@ Therefore, the local kinetic energy is
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org7ae5fd2" class="outline-4">
|
<div id="outline-container-org890847f" class="outline-4">
|
||||||
<h4 id="org7ae5fd2"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
<h4 id="org890847f"><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>
|
||||||
@ -969,8 +969,8 @@ local kinetic energy.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgc9c223c" class="outline-5">
|
<div id="outline-container-org35d7e90" class="outline-5">
|
||||||
<h5 id="orgc9c223c"><span class="section-number-5">2.1.4.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org35d7e90"><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>
|
||||||
@ -1000,8 +1000,8 @@ local kinetic energy.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org9fad9d8" class="outline-4">
|
<div id="outline-container-org4ce2a48" class="outline-4">
|
||||||
<h4 id="org9fad9d8"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
<h4 id="org4ce2a48"><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>
|
||||||
@ -1011,8 +1011,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6035a20" class="outline-5">
|
<div id="outline-container-org2543a5f" class="outline-5">
|
||||||
<h5 id="org6035a20"><span class="section-number-5">2.1.5.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org2543a5f"><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} -
|
||||||
@ -1032,8 +1032,8 @@ equal to -0.5 atomic units.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org541ad7b" class="outline-3">
|
<div id="outline-container-orgd06a1e7" class="outline-3">
|
||||||
<h3 id="org541ad7b"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
<h3 id="orgd06a1e7"><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">
|
||||||
<div class="note">
|
<div class="note">
|
||||||
<p>
|
<p>
|
||||||
@ -1044,8 +1044,8 @@ choose a grid which does not contain the origin.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6fe4366" class="outline-4">
|
<div id="outline-container-org269a10b" class="outline-4">
|
||||||
<h4 id="org6fe4366"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
<h4 id="org269a10b"><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>
|
||||||
@ -1128,8 +1128,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org3f1245c" class="outline-5">
|
<div id="outline-container-org7a72ea6" class="outline-5">
|
||||||
<h5 id="org3f1245c"><span class="section-number-5">2.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org7a72ea6"><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>
|
||||||
@ -1204,8 +1204,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org887fd3f" class="outline-3">
|
<div id="outline-container-org0cc6339" class="outline-3">
|
||||||
<h3 id="org887fd3f"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
<h3 id="org0cc6339"><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 \(\mathbf{r}_i\)
|
||||||
@ -1235,8 +1235,8 @@ The energy is biased because:
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org9274039" class="outline-4">
|
<div id="outline-container-org7d44f1d" class="outline-4">
|
||||||
<h4 id="org9274039"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
<h4 id="org7d44f1d"><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>
|
||||||
@ -1305,8 +1305,8 @@ To compile the Fortran and run it:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org451ad90" class="outline-5">
|
<div id="outline-container-orga0f2716" class="outline-5">
|
||||||
<h5 id="org451ad90"><span class="section-number-5">2.3.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orga0f2716"><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>
|
||||||
@ -1421,8 +1421,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org276e07e" class="outline-3">
|
<div id="outline-container-org360dc66" class="outline-3">
|
||||||
<h3 id="org276e07e"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
<h3 id="org360dc66"><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 \(\Psi\)
|
||||||
@ -1449,8 +1449,8 @@ energy can be used as a measure of the quality of a wave function.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org43e53cc" class="outline-4">
|
<div id="outline-container-org7c4cd23" class="outline-4">
|
||||||
<h4 id="org43e53cc"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
<h4 id="org7c4cd23"><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>
|
||||||
@ -1461,8 +1461,8 @@ Prove that :
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org38d9972" class="outline-5">
|
<div id="outline-container-orgf1c30f6" class="outline-5">
|
||||||
<h5 id="org38d9972"><span class="section-number-5">2.4.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orgf1c30f6"><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}
|
||||||
@ -1481,8 +1481,8 @@ Prove that :
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
<div id="outline-container-org733ec8a" class="outline-4">
|
<div id="outline-container-orged66d34" class="outline-4">
|
||||||
<h4 id="org733ec8a"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
<h4 id="orged66d34"><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>
|
||||||
@ -1556,8 +1556,8 @@ To compile and run:
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org739fa49" class="outline-5">
|
<div id="outline-container-org08ccc7f" class="outline-5">
|
||||||
<h5 id="org739fa49"><span class="section-number-5">2.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org08ccc7f"><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>
|
||||||
@ -1694,8 +1694,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org81347ce" class="outline-2">
|
<div id="outline-container-orga7eaa2d" class="outline-2">
|
||||||
<h2 id="org81347ce"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
<h2 id="orga7eaa2d"><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
|
||||||
@ -1711,8 +1711,8 @@ interval.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgd013ab9" class="outline-3">
|
<div id="outline-container-orgbd090a7" class="outline-3">
|
||||||
<h3 id="orgd013ab9"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
<h3 id="orgbd090a7"><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 \(M\)
|
||||||
@ -1752,8 +1752,8 @@ And the confidence interval is given by
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org32ca3c3" class="outline-4">
|
<div id="outline-container-orgc3fdd7d" class="outline-4">
|
||||||
<h4 id="org32ca3c3"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
<h4 id="orgc3fdd7d"><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>
|
||||||
@ -1791,8 +1791,8 @@ input array.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgebeb146" class="outline-5">
|
<div id="outline-container-org646758c" class="outline-5">
|
||||||
<h5 id="orgebeb146"><span class="section-number-5">3.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org646758c"><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>
|
||||||
@ -1851,8 +1851,8 @@ input array.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgb2a9889" class="outline-3">
|
<div id="outline-container-org4c8f35b" class="outline-3">
|
||||||
<h3 id="orgb2a9889"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
<h3 id="org4c8f35b"><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
|
||||||
@ -1913,8 +1913,8 @@ compute the statistical error.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org4e6a1f7" class="outline-4">
|
<div id="outline-container-org40abc16" class="outline-4">
|
||||||
<h4 id="org4e6a1f7"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
<h4 id="org40abc16"><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>
|
||||||
@ -2014,8 +2014,8 @@ well as the index of the current step.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgbffcbab" class="outline-5">
|
<div id="outline-container-orgfc967b4" class="outline-5">
|
||||||
<h5 id="orgbffcbab"><span class="section-number-5">3.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orgfc967b4"><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>
|
||||||
@ -2129,8 +2129,8 @@ E = -0.49518773675598715 +/- 5.2391494923686175E-004
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0733fed" class="outline-3">
|
<div id="outline-container-orga44380c" class="outline-3">
|
||||||
<h3 id="org0733fed"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
<h3 id="orga44380c"><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
|
||||||
@ -2163,15 +2163,15 @@ initial position \(\mathbf{r}_0\), we will realize a random walk:
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
according to the following algorithm.
|
following the following algorithm.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
At every step, we propose a new move according to a transition probability \(T(\mathbf{r}_{n+1},\mathbf{r}_n)\) of our choice.
|
At every step, we propose a new move according to a transition probability \(T(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1})\) of our choice.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
For simplicity, let us move the electron in a 3-dimensional box of side \(2\delta L\) centered at the current position
|
For simplicity, we will move the electron in a 3-dimensional box of side \(2\delta L\) centered at the current position
|
||||||
of the electron:
|
of the electron:
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
@ -2188,7 +2188,7 @@ where \(\delta L\) is a fixed constant, and
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
After having moved the electron, add the
|
After having moved the electron, we add the
|
||||||
accept/reject step that guarantees that the distribution of the
|
accept/reject step that guarantees that the distribution of the
|
||||||
\(\mathbf{r}_n\) is \(\Psi^2\). This amounts to accepting the move with
|
\(\mathbf{r}_n\) is \(\Psi^2\). This amounts to accepting the move with
|
||||||
probability
|
probability
|
||||||
@ -2196,7 +2196,7 @@ probability
|
|||||||
|
|
||||||
<p>
|
<p>
|
||||||
\[
|
\[
|
||||||
A{\mathbf{r}_{n+1},\mathbf{r}_n) = \min\left(1,\frac{T(\mathbf{r}_{n},\mathbf{r}_{n+1}) P(\mathbf{r}_{n+1})}{T(\mathbf{r}_{n+1},\mathbf{r}_n)P(\mathbf{r}_{n})}\right)\,,
|
A{\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1}) = \min\left(1,\frac{T(\mathbf{r}_{n},\mathbf{r}_{n+1}) P(\mathbf{r}_{n+1})}{T(\mathbf{r}_{n+1},\mathbf{r}_n)P(\mathbf{r}_{n})}\right)\,,
|
||||||
\]
|
\]
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
@ -2206,7 +2206,7 @@ which, for our choice of transition probability, becomes
|
|||||||
|
|
||||||
<p>
|
<p>
|
||||||
\[
|
\[
|
||||||
A{\mathbf{r}_{n+1},\mathbf{r}_n) = \min\left(1,\frac{P(\mathbf{r}_{n+1})}{P(\mathbf{r}_{n})}\right)= \min\left(1,\frac{\Psi(\mathbf{r}_{n+1})^2}{\Psi(\mathbf{r}_{n})^2}
|
A{\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1}) = \min\left(1,\frac{P(\mathbf{r}_{n+1})}{P(\mathbf{r}_{n})}\right)= \min\left(1,\frac{\Psi(\mathbf{r}_{n+1})^2}{\Psi(\mathbf{r}_{n})^2}
|
||||||
\]
|
\]
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
@ -2258,13 +2258,14 @@ The size of the move should be adjusted so that it is as large as
|
|||||||
possible, keeping the number of accepted steps not too small. To
|
possible, keeping the number of accepted steps not too small. To
|
||||||
achieve that, we define the acceptance rate as the number of
|
achieve that, we define the acceptance rate as the number of
|
||||||
accepted steps over the total number of steps. Adjusting the time
|
accepted steps over the total number of steps. Adjusting the time
|
||||||
step such that the acceptance rate is close to 0.5 is a good compromise for the current problem.
|
step such that the acceptance rate is close to 0.5 is a good
|
||||||
|
compromise for the current problem.
|
||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org0c0af3c" class="outline-4">
|
<div id="outline-container-org5859423" class="outline-4">
|
||||||
<h4 id="org0c0af3c"><span class="section-number-4">3.3.1</span> Exercise</h4>
|
<h4 id="org5859423"><span class="section-number-4">3.3.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-3-3-1">
|
<div class="outline-text-4" id="text-3-3-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -2371,8 +2372,8 @@ Can you observe a reduction in the statistical error?
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6c72dc3" class="outline-5">
|
<div id="outline-container-org2606981" class="outline-5">
|
||||||
<h5 id="org6c72dc3"><span class="section-number-5">3.3.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="org2606981"><span class="section-number-5">3.3.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-3-3-1-1">
|
<div class="outline-text-5" id="text-3-3-1-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -2517,8 +2518,8 @@ A = 0.51695266666666673 +/- 4.0445505648997396E-004
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orga0b0b8d" class="outline-3">
|
<div id="outline-container-org10c0223" class="outline-3">
|
||||||
<h3 id="orga0b0b8d"><span class="section-number-3">3.4</span> Gaussian random number generator</h3>
|
<h3 id="org10c0223"><span class="section-number-3">3.4</span> Gaussian random number generator</h3>
|
||||||
<div class="outline-text-3" id="text-3-4">
|
<div class="outline-text-3" id="text-3-4">
|
||||||
<p>
|
<p>
|
||||||
To obtain Gaussian-distributed random numbers, you can apply the
|
To obtain Gaussian-distributed random numbers, you can apply the
|
||||||
@ -2581,8 +2582,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org57a608a" class="outline-3">
|
<div id="outline-container-org89c81c5" class="outline-3">
|
||||||
<h3 id="org57a608a"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3>
|
<h3 id="org89c81c5"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3>
|
||||||
<div class="outline-text-3" id="text-3-5">
|
<div class="outline-text-3" id="text-3-5">
|
||||||
<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.
|
||||||
@ -2607,13 +2608,13 @@ probability of transition from \(\mathbf{r}_n\) to
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
In the previous example, we were using uniform random
|
In the previous example, we were using uniform sampling in a box centered
|
||||||
numbers. Hence, the transition probability was
|
at the current position. Hence, the transition probability was symmetric
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
\[
|
\[
|
||||||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
|
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n})
|
||||||
\text{constant}\,,
|
\text{constant}\,,
|
||||||
\]
|
\]
|
||||||
</p>
|
</p>
|
||||||
@ -2624,7 +2625,7 @@ wave functions.
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
Now, if instead of drawing uniform random numbers we
|
Now, if instead of drawing uniform random numbers, we
|
||||||
choose to draw Gaussian random numbers with zero mean and variance
|
choose to draw Gaussian random numbers with zero mean and variance
|
||||||
\(\delta t\), the transition probability becomes:
|
\(\delta t\), the transition probability becomes:
|
||||||
</p>
|
</p>
|
||||||
@ -2639,9 +2640,9 @@ choose to draw Gaussian random numbers with zero mean and variance
|
|||||||
|
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
To sample even better the density, we can "push" the electrons
|
Furthermore, to sample the density even better, we can "push" the electrons
|
||||||
into in the regions of high probability, and "pull" them away from
|
into in the regions of high probability, and "pull" them away from
|
||||||
the low-probability regions. This will mechanically increase the
|
the low-probability regions. This will ncrease the
|
||||||
acceptance ratios and improve the sampling.
|
acceptance ratios and improve the sampling.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
@ -2656,20 +2657,8 @@ To do this, we can use the gradient of the probability density
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
and add the so-called drift vector, so that the numerical scheme becomes a drifted diffusion:
|
and add the so-called drift vector, so that the numerical scheme becomes a
|
||||||
</p>
|
drifted diffusion with transition probability:
|
||||||
|
|
||||||
<p>
|
|
||||||
\[
|
|
||||||
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta t\, \frac{\nabla
|
|
||||||
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi \,,
|
|
||||||
\]
|
|
||||||
</p>
|
|
||||||
|
|
||||||
<p>
|
|
||||||
where \(\chi\) is a Gaussian random variable with zero mean and
|
|
||||||
variance \(\delta t\).
|
|
||||||
The transition probability becomes:
|
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -2682,11 +2671,28 @@ The transition probability becomes:
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The algorithm of the previous exercise is only slighlty modified summarized:
|
and the corrsponding move is proposed as
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
\[
|
||||||
|
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta t\, \frac{\nabla
|
||||||
|
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi \,,
|
||||||
|
\]
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
where \(\chi\) is a Gaussian random variable with zero mean and
|
||||||
|
variance \(\delta t\).
|
||||||
|
</p>
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The algorithm of the previous exercise is only slighlty modified as:
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<ol class="org-ol">
|
<ol class="org-ol">
|
||||||
<li>For the starting position compute \(\Psi\) and the drif-vector \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\)</li>
|
|
||||||
<li><p>
|
<li><p>
|
||||||
Compute a new position \(\mathbf{r'} = \mathbf{r}_n +
|
Compute a new position \(\mathbf{r'} = \mathbf{r}_n +
|
||||||
\delta t\, \frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi\)
|
\delta t\, \frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi\)
|
||||||
@ -2709,8 +2715,8 @@ Evaluate \(\Psi\) and \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\) at th
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-orgaa87988" class="outline-4">
|
<div id="outline-container-org2359a6c" class="outline-4">
|
||||||
<h4 id="orgaa87988"><span class="section-number-4">3.5.1</span> Exercise 1</h4>
|
<h4 id="org2359a6c"><span class="section-number-4">3.5.1</span> Exercise 1</h4>
|
||||||
<div class="outline-text-4" id="text-3-5-1">
|
<div class="outline-text-4" id="text-3-5-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -2744,8 +2750,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgf391a29" class="outline-5">
|
<div id="outline-container-orga3e9097" class="outline-5">
|
||||||
<h5 id="orgf391a29"><span class="section-number-5">3.5.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orga3e9097"><span class="section-number-5">3.5.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-3-5-1-1">
|
<div class="outline-text-5" id="text-3-5-1-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -2778,13 +2784,13 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6abfe47" class="outline-4">
|
<div id="outline-container-org475617c" class="outline-4">
|
||||||
<h4 id="org6abfe47"><span class="section-number-4">3.5.2</span> Exercise 2</h4>
|
<h4 id="org475617c"><span class="section-number-4">3.5.2</span> Exercise 2</h4>
|
||||||
<div class="outline-text-4" id="text-3-5-2">
|
<div class="outline-text-4" id="text-3-5-2">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
Modify the previous program to introduce the drifted diffusion scheme.
|
Modify the previous program to introduce the drift-diffusion scheme.
|
||||||
(This is a necessary step for the next section).
|
(This is a necessary step for the next section on diffusion Monte Carlo).
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
</div>
|
</div>
|
||||||
@ -2873,8 +2879,8 @@ Modify the previous program to introduce the drifted diffusion scheme.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org53b5631" class="outline-5">
|
<div id="outline-container-orgc22b63a" class="outline-5">
|
||||||
<h5 id="org53b5631"><span class="section-number-5">3.5.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orgc22b63a"><span class="section-number-5">3.5.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-3-5-2-1">
|
<div class="outline-text-5" id="text-3-5-2-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -3060,12 +3066,12 @@ A = 0.78839866666666658 +/- 3.2503783452043152E-004
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org5f1d81b" class="outline-2">
|
<div id="outline-container-orgcf7b82d" class="outline-2">
|
||||||
<h2 id="org5f1d81b"><span class="section-number-2">4</span> Diffusion Monte Carlo   <span class="tag"><span class="solution">solution</span></span></h2>
|
<h2 id="orgcf7b82d"><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-org95ab9cd" class="outline-3">
|
<div id="outline-container-orgfb5c875" class="outline-3">
|
||||||
<h3 id="org95ab9cd"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
<h3 id="orgfb5c875"><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:
|
||||||
@ -3073,12 +3079,12 @@ Consider the time-dependent Schrödinger equation:
|
|||||||
|
|
||||||
<p>
|
<p>
|
||||||
\[
|
\[
|
||||||
i\frac{\partial \Psi(\mathbf{r},t)}{\partial t} = \hat{H} \Psi(\mathbf{r},t)
|
i\frac{\partial \Psi(\mathbf{r},t)}{\partial t} = \hat{H} \Psi(\mathbf{r},t)\,.
|
||||||
\]
|
\]
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
We can expand \(\Psi(\mathbf{r},0)\), in the basis of the eigenstates
|
We can expand a given starting wave function, \(\Psi(\mathbf{r},0)\), in the basis of the eigenstates
|
||||||
of the time-independent Hamiltonian:
|
of the time-independent Hamiltonian:
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
@ -3099,7 +3105,7 @@ The solution of the Schrödinger equation at time \(t\) is
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
Now, let's replace the time variable \(t\) by an imaginary time variable
|
Now, if we replace the time variable \(t\) by an imaginary time variable
|
||||||
\(\tau=i\,t\), we obtain
|
\(\tau=i\,t\), we obtain
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
@ -3110,10 +3116,10 @@ Now, let's replace the time variable \(t\) by an imaginary time variable
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
where \(\psi(\mathbf{r},\tau) = \Psi(\mathbf{r},-i\tau) = \Psi(\mathbf{r},t)\)
|
where \(\psi(\mathbf{r},\tau) = \Psi(\mathbf{r},-i\,)\)
|
||||||
and
|
and
|
||||||
\[
|
\[
|
||||||
\psi(\mathbf{r},\tau) = \sum_k a_k \exp( -E_k\, \tau) \phi_k(\mathbf{r}).
|
\psi(\mathbf{r},\tau) = \sum_k a_k \exp( -E_k\, \tau) \phi_k(\mathbf{r}).
|
||||||
\]
|
\]
|
||||||
For large positive values of \(\tau\), \(\psi\) is dominated by the
|
For large positive values of \(\tau\), \(\psi\) is dominated by the
|
||||||
\(k=0\) term, namely the lowest eigenstate.
|
\(k=0\) term, namely the lowest eigenstate.
|
||||||
@ -3124,8 +3130,8 @@ system.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org9fcbab9" class="outline-3">
|
<div id="outline-container-org8524696" class="outline-3">
|
||||||
<h3 id="org9fcbab9"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
<h3 id="org8524696"><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 <a href="https://en.wikipedia.org/wiki/Diffusion_equation">diffusion equation</a> of particles is given by
|
The <a href="https://en.wikipedia.org/wiki/Diffusion_equation">diffusion equation</a> of particles is given by
|
||||||
@ -3179,8 +3185,8 @@ the combination of a diffusion process and a branching process.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0aba1e8" class="outline-3">
|
<div id="outline-container-orgfb0b533" class="outline-3">
|
||||||
<h3 id="org0aba1e8"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
<h3 id="orgfb0b533"><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,
|
||||||
@ -3237,8 +3243,8 @@ error known as the <i>fixed node error</i>.
|
|||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgcc09947" class="outline-4">
|
<div id="outline-container-orgb989591" class="outline-4">
|
||||||
<h4 id="orgcc09947"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4>
|
<h4 id="orgb989591"><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>
|
||||||
\[
|
\[
|
||||||
@ -3300,8 +3306,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org2f26efe" class="outline-3">
|
<div id="outline-container-org98fdfce" class="outline-3">
|
||||||
<h3 id="org2f26efe"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3>
|
<h3 id="org98fdfce"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3>
|
||||||
<div class="outline-text-3" id="text-4-4">
|
<div class="outline-text-3" id="text-4-4">
|
||||||
<p>
|
<p>
|
||||||
Now that we have a process to sample \(\Pi(\mathbf{r},\tau) =
|
Now that we have a process to sample \(\Pi(\mathbf{r},\tau) =
|
||||||
@ -3353,8 +3359,8 @@ energies computed with the trial wave function.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgfc317d3" class="outline-3">
|
<div id="outline-container-org9a74f72" class="outline-3">
|
||||||
<h3 id="orgfc317d3"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
<h3 id="org9a74f72"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
||||||
<div class="outline-text-3" id="text-4-5">
|
<div class="outline-text-3" id="text-4-5">
|
||||||
<p>
|
<p>
|
||||||
Instead of having a variable number of particles to simulate the
|
Instead of having a variable number of particles to simulate the
|
||||||
@ -3406,13 +3412,13 @@ code, so this is what we will do in the next section.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgf410952" class="outline-3">
|
<div id="outline-container-org0e4c39d" class="outline-3">
|
||||||
<h3 id="orgf410952"><span class="section-number-3">4.6</span> Hydrogen atom</h3>
|
<h3 id="org0e4c39d"><span class="section-number-3">4.6</span> Hydrogen atom</h3>
|
||||||
<div class="outline-text-3" id="text-4-6">
|
<div class="outline-text-3" id="text-4-6">
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org5899d68" class="outline-4">
|
<div id="outline-container-org406db4f" class="outline-4">
|
||||||
<h4 id="org5899d68"><span class="section-number-4">4.6.1</span> Exercise</h4>
|
<h4 id="org406db4f"><span class="section-number-4">4.6.1</span> Exercise</h4>
|
||||||
<div class="outline-text-4" id="text-4-6-1">
|
<div class="outline-text-4" id="text-4-6-1">
|
||||||
<div class="exercise">
|
<div class="exercise">
|
||||||
<p>
|
<p>
|
||||||
@ -3511,8 +3517,8 @@ energy of H for any value of \(a\).
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orge0716bd" class="outline-5">
|
<div id="outline-container-orgb3872d9" class="outline-5">
|
||||||
<h5 id="orge0716bd"><span class="section-number-5">4.6.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
<h5 id="orgb3872d9"><span class="section-number-5">4.6.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||||
<div class="outline-text-5" id="text-4-6-1-1">
|
<div class="outline-text-5" id="text-4-6-1-1">
|
||||||
<p>
|
<p>
|
||||||
<b>Python</b>
|
<b>Python</b>
|
||||||
@ -3728,8 +3734,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org5e287d8" class="outline-3">
|
<div id="outline-container-org8ac323b" class="outline-3">
|
||||||
<h3 id="org5e287d8"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
|
<h3 id="org8ac323b"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
|
||||||
<div class="outline-text-3" id="text-4-7">
|
<div class="outline-text-3" id="text-4-7">
|
||||||
<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
|
||||||
@ -3750,8 +3756,8 @@ the nuclei.
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="outline-container-org17d4d49" class="outline-2">
|
<div id="outline-container-org4c7a4a9" class="outline-2">
|
||||||
<h2 id="org17d4d49"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2>
|
<h2 id="org4c7a4a9"><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>
|
||||||
@ -3767,7 +3773,7 @@ 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: 2021-01-31 Sun 09:26</p>
|
<p class="date">Created: 2021-01-31 Sun 16:02</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