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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2021-02-01 Mon 21:10 -->
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<!-- 2021-02-02 Tue 10:29 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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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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@ -329,151 +329,152 @@ for the JavaScript code in this tag.
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org0752ab6">1. Introduction</a>
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<li><a href="#org6cc7181">1. Introduction</a>
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<ul>
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<li><a href="#orga8b4aa7">1.1. Energy and local energy</a></li>
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<li><a href="#org5afd847">1.1. Energy and local energy</a></li>
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</ul>
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</li>
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<li><a href="#orgb06cb17">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<li><a href="#org2ec7e1b">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<ul>
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<li><a href="#org9feb61d">2.1. Local energy</a>
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<li><a href="#orgfcc8aee">2.1. Local energy</a>
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<ul>
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<li><a href="#org1c16c21">2.1.1. Exercise 1</a>
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<li><a href="#org35672c8">2.1.1. Exercise 1</a>
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<ul>
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<li><a href="#orgf0ec491">2.1.1.1. Solution</a></li>
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||||
<li><a href="#org225c317">2.1.1.1. Solution</a></li>
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</ul>
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</li>
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||||
<li><a href="#org177e97d">2.1.2. Exercise 2</a>
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<li><a href="#orgd36eef4">2.1.2. Exercise 2</a>
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<ul>
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||||
<li><a href="#org591d468">2.1.2.1. Solution</a></li>
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<li><a href="#orgffb5984">2.1.2.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org3938b54">2.1.3. Exercise 3</a>
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<li><a href="#orgabefeb9">2.1.3. Exercise 3</a>
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<ul>
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<li><a href="#orgc0d86e9">2.1.3.1. Solution</a></li>
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<li><a href="#org3119096">2.1.3.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org6f36ec9">2.1.4. Exercise 4</a>
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<li><a href="#orge6a9288">2.1.4. Exercise 4</a>
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<ul>
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<li><a href="#org7be9be2">2.1.4.1. Solution</a></li>
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<li><a href="#org89a050a">2.1.4.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org71a1a30">2.1.5. Exercise 5</a>
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<li><a href="#org90a766f">2.1.5. Exercise 5</a>
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<ul>
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<li><a href="#orga629319">2.1.5.1. Solution</a></li>
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<li><a href="#org1669e04">2.1.5.1. Solution</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org58f173a">2.2. Plot of the local energy along the \(x\) axis</a>
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<li><a href="#orgc481fb4">2.2. Plot of the local energy along the \(x\) axis</a>
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<ul>
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<li><a href="#org74602a0">2.2.1. Exercise</a>
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<li><a href="#org8a4f299">2.2.1. Exercise</a>
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<ul>
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<li><a href="#org3998c09">2.2.1.1. Solution</a></li>
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<li><a href="#org5f96d93">2.2.1.1. Solution</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orga732d98">2.3. Numerical estimation of the energy</a>
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<li><a href="#orgbcc283c">2.3. Numerical estimation of the energy</a>
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<ul>
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<li><a href="#org66a804a">2.3.1. Exercise</a>
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<li><a href="#org5506022">2.3.1. Exercise</a>
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<ul>
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<li><a href="#orgcb5c324">2.3.1.1. Solution</a></li>
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<li><a href="#org74db5d8">2.3.1.1. Solution</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org9d5500d">2.4. Variance of the local energy</a>
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<li><a href="#org1ceafe1">2.4. Variance of the local energy</a>
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<ul>
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<li><a href="#org7ef7c25">2.4.1. Exercise (optional)</a>
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<li><a href="#orge85aa1c">2.4.1. Exercise (optional)</a>
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<ul>
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<li><a href="#org8f05b60">2.4.1.1. Solution</a></li>
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<li><a href="#org0940ecc">2.4.1.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org8d23fe3">2.4.2. Exercise</a>
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<li><a href="#orgb488446">2.4.2. Exercise</a>
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<ul>
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<li><a href="#orgf8c3f41">2.4.2.1. Solution</a></li>
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<li><a href="#org5d375ab">2.4.2.1. Solution</a></li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#org1de9633">3. Variational Monte Carlo</a>
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<li><a href="#org85e47f4">3. Variational Monte Carlo</a>
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<ul>
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<li><a href="#org9d504cf">3.1. Computation of the statistical error</a>
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<li><a href="#org17350f1">3.1. Computation of the statistical error</a>
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<ul>
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<li><a href="#org7db3239">3.1.1. Exercise</a>
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<li><a href="#org97a29b6">3.1.1. Exercise</a>
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<ul>
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||||
<li><a href="#org2355686">3.1.1.1. Solution</a></li>
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<li><a href="#orga749707">3.1.1.1. Solution</a></li>
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</ul>
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||||
</li>
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</ul>
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</li>
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<li><a href="#org064ac12">3.2. Uniform sampling in the box</a>
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<li><a href="#orgd721f86">3.2. Uniform sampling in the box</a>
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<ul>
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<li><a href="#orgb0f2034">3.2.1. Exercise</a>
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<li><a href="#org215979f">3.2.1. Exercise</a>
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<ul>
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||||
<li><a href="#org9b6a09d">3.2.1.1. Solution</a></li>
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||||
<li><a href="#orgbd3c736">3.2.1.1. Solution</a></li>
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||||
</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orgbc7bac6">3.3. Metropolis sampling with \(\Psi^2\)</a>
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<li><a href="#orgf676c01">3.3. Metropolis sampling with \(\Psi^2\)</a>
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||||
<ul>
|
||||
<li><a href="#org231d1f1">3.3.1. Exercise</a>
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||||
<li><a href="#orgc511dca">3.3.1. Exercise</a>
|
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<ul>
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||||
<li><a href="#orgdb46596">3.3.1.1. Solution</a></li>
|
||||
<li><a href="#org8b5ee4b">3.3.1.1. Solution</a></li>
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</ul>
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||||
</li>
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||||
</ul>
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</li>
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||||
<li><a href="#org3a0c9e1">3.4. Gaussian random number generator</a></li>
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||||
<li><a href="#orgd4d4844">3.5. Generalized Metropolis algorithm</a>
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<li><a href="#org542b847">3.4. Gaussian random number generator</a></li>
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||||
<li><a href="#orgce06b27">3.5. Generalized Metropolis algorithm</a>
|
||||
<ul>
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||||
<li><a href="#org88e2361">3.5.1. Exercise 1</a>
|
||||
<li><a href="#org6877da4">3.5.1. Exercise 1</a>
|
||||
<ul>
|
||||
<li><a href="#org346520f">3.5.1.1. Solution</a></li>
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||||
<li><a href="#org9f0138a">3.5.1.1. Solution</a></li>
|
||||
</ul>
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||||
</li>
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||||
<li><a href="#orgfd9abc1">3.5.2. Exercise 2</a>
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||||
<li><a href="#org36805af">3.5.2. Exercise 2</a>
|
||||
<ul>
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||||
<li><a href="#org076151d">3.5.2.1. Solution</a></li>
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||||
<li><a href="#orgd6805a2">3.5.2.1. Solution</a></li>
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||||
</ul>
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||||
</li>
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||||
</ul>
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||||
</li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org97a5ed5">4. Diffusion Monte Carlo</a>
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||||
<li><a href="#org2d2753c">4. Diffusion Monte Carlo</a>
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||||
<ul>
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<li><a href="#org1359234">4.1. Schrödinger equation in imaginary time</a></li>
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<li><a href="#org5684930">4.2. Diffusion and branching</a></li>
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||||
<li><a href="#org11122df">4.3. Importance sampling</a>
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||||
<li><a href="#org3f5bc99">4.1. Schrödinger equation in imaginary time</a></li>
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||||
<li><a href="#orgbd4d3c8">4.2. Diffusion and branching</a></li>
|
||||
<li><a href="#org11ded29">4.3. Importance sampling</a>
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||||
<ul>
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||||
<li><a href="#org3d60069">4.3.1. Appendix : Details of the Derivation</a></li>
|
||||
<li><a href="#org5669f14">4.3.1. Appendix : Details of the Derivation</a></li>
|
||||
</ul>
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||||
</li>
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||||
<li><a href="#org1171aaa">4.4. Pure Diffusion Monte Carlo (PDMC)</a></li>
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||||
<li><a href="#orga6bc5fd">4.5. Hydrogen atom</a>
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||||
<li><a href="#orged7a00f">4.4. Pure Diffusion Monte Carlo (PDMC)</a></li>
|
||||
<li><a href="#org0bcdcd6">4.5. Hydrogen atom</a>
|
||||
<ul>
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||||
<li><a href="#orgfc4c90b">4.5.1. Exercise</a>
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||||
<li><a href="#org4145b62">4.5.1. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#org156cd8e">4.5.1.1. Solution</a></li>
|
||||
<li><a href="#orgfd5af9d">4.5.1.1. Solution</a></li>
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||||
</ul>
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||||
</li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org267032c">4.6. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
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||||
<li><a href="#org74f6b7c">4.6. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
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||||
</ul>
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||||
</li>
|
||||
<li><a href="#org9cd3174">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
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||||
<li><a href="#org1b7a228">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
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||||
<li><a href="#orge8fb145">6. Schedule</a></li>
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||||
</ul>
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||||
</div>
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</div>
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<div id="outline-container-org0752ab6" class="outline-2">
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||||
<h2 id="org0752ab6"><span class="section-number-2">1</span> Introduction</h2>
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||||
<div id="outline-container-org6cc7181" class="outline-2">
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||||
<h2 id="org6cc7181"><span class="section-number-2">1</span> Introduction</h2>
|
||||
<div class="outline-text-2" id="text-1">
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||||
<p>
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||||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
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@ -496,7 +497,7 @@ starting from an approximate wave function.
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||||
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||||
<p>
|
||||
Code examples will be given in Python and Fortran. You can use
|
||||
whatever language you prefer to write the program.
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||||
whatever language you prefer to write the programs.
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||||
</p>
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||||
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||||
<p>
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||||
@ -513,8 +514,8 @@ coordinates, etc).
|
||||
</p>
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||||
</div>
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||||
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||||
<div id="outline-container-orga8b4aa7" class="outline-3">
|
||||
<h3 id="orga8b4aa7"><span class="section-number-3">1.1</span> Energy and local energy</h3>
|
||||
<div id="outline-container-org5afd847" class="outline-3">
|
||||
<h3 id="org5afd847"><span class="section-number-3">1.1</span> Energy and local energy</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<p>
|
||||
For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
|
||||
@ -543,20 +544,23 @@ E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
|
||||
\end{eqnarray*}
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||||
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||||
<p>
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||||
For few dimensions, one can easily compute \(E\) by evaluating the integrals on a grid but, for a high number of dimensions, one can resort to Monte Carlo techniques to compute \(E\).
|
||||
For few dimensions, one can easily compute \(E\) by evaluating the
|
||||
integrals on a grid but, for a high number of dimensions, one can
|
||||
resort to Monte Carlo techniques to compute \(E\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
To this aim, recall that the probabilistic <i>expected value</i> of an arbitrary function \(f(x)\)
|
||||
with respect to a probability density function \(P(x)\) is given by
|
||||
To this aim, recall that the probabilistic <i>expected value</i> of an
|
||||
arbitrary function \(f(x)\) with respect to a probability density
|
||||
function \(P(x)\) is given by
|
||||
</p>
|
||||
|
||||
<p>
|
||||
\[ \langle f \rangle_p = \int_{-\infty}^\infty P(x)\, f(x)\,dx, \]
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||||
\[ \langle f \rangle_P = \int_{-\infty}^\infty P(x)\, f(x)\,dx, \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
where a probability density function \(p(x)\) is non-negative
|
||||
where a probability density function \(P(x)\) is non-negative
|
||||
and integrates to one:
|
||||
</p>
|
||||
|
||||
@ -570,7 +574,7 @@ a probability density \(P(\mathbf{r})\) defined in 3\(N\) dimensions:
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||||
</p>
|
||||
|
||||
<p>
|
||||
\[ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r} \equiv \langle E_L \rangle_{\Psi^2}\,, \]
|
||||
\[ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r} \equiv \langle E_L \rangle_{P}\,, \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -582,7 +586,9 @@ where the probability density is given by the square of the wave function:
|
||||
</p>
|
||||
|
||||
<p>
|
||||
If we can sample \(N_{\rm MC}\) configurations \(\{\mathbf{r}\}\) distributed as \(p\), we can estimate \(E\) as the average of the local energy computed over these configurations:
|
||||
If we can sample \(N_{\rm MC}\) configurations \(\{\mathbf{r}\}\)
|
||||
distributed as \(P\), we can estimate \(E\) as the average of the local
|
||||
energy computed over these configurations:
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -592,8 +598,8 @@ If we can sample \(N_{\rm MC}\) configurations \(\{\mathbf{r}\}\) distributed as
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb06cb17" class="outline-2">
|
||||
<h2 id="orgb06cb17"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
|
||||
<div id="outline-container-org2ec7e1b" class="outline-2">
|
||||
<h2 id="org2ec7e1b"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
In this section, we consider the hydrogen atom with the following
|
||||
@ -622,8 +628,8 @@ To do that, we will compute the local energy and check whether it is constant.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9feb61d" class="outline-3">
|
||||
<h3 id="org9feb61d"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||
<div id="outline-container-orgfcc8aee" class="outline-3">
|
||||
<h3 id="orgfcc8aee"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
|
||||
@ -650,13 +656,13 @@ to catch the error.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1c16c21" class="outline-4">
|
||||
<h4 id="org1c16c21"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||||
<div id="outline-container-org35672c8" class="outline-4">
|
||||
<h4 id="org35672c8"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||||
<div class="outline-text-4" id="text-2-1-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
Write a function which computes the potential at \(\mathbf{r}\).
|
||||
The function accepts a 3-dimensional vector <code>r</code> as input arguments
|
||||
The function accepts a 3-dimensional vector <code>r</code> as input argument
|
||||
and returns the potential.
|
||||
</p>
|
||||
|
||||
@ -695,8 +701,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf0ec491" class="outline-5">
|
||||
<h5 id="orgf0ec491"><span class="section-number-5">2.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org225c317" class="outline-5">
|
||||
<h5 id="org225c317"><span class="section-number-5">2.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-1-1">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -736,8 +742,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org177e97d" class="outline-4">
|
||||
<h4 id="org177e97d"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-orgd36eef4" class="outline-4">
|
||||
<h4 id="orgd36eef4"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||||
<div class="outline-text-4" id="text-2-1-2">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -772,8 +778,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org591d468" class="outline-5">
|
||||
<h5 id="org591d468"><span class="section-number-5">2.1.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgffb5984" class="outline-5">
|
||||
<h5 id="orgffb5984"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -800,8 +806,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3938b54" class="outline-4">
|
||||
<h4 id="org3938b54"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||||
<div id="outline-container-orgabefeb9" class="outline-4">
|
||||
<h4 id="orgabefeb9"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||||
<div class="outline-text-4" id="text-2-1-3">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -813,7 +819,7 @@ local kinetic energy.
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The local kinetic energy is defined as \[-\frac{1}{2}\frac{\Delta \Psi}{\Psi}.\]
|
||||
The local kinetic energy is defined as \[T_L(\mathbf{r}) = -\frac{1}{2}\frac{\Delta \Psi(\mathbf{r})}{\Psi(\mathbf{r})}.\]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -854,7 +860,7 @@ applied to the wave function gives:
|
||||
<p>
|
||||
Therefore, the local kinetic energy is
|
||||
\[
|
||||
-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
|
||||
T_L (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
|
||||
\]
|
||||
</p>
|
||||
|
||||
@ -882,8 +888,8 @@ Therefore, the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc0d86e9" class="outline-5">
|
||||
<h5 id="orgc0d86e9"><span class="section-number-5">2.1.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org3119096" class="outline-5">
|
||||
<h5 id="org3119096"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -924,8 +930,8 @@ Therefore, the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6f36ec9" class="outline-4">
|
||||
<h4 id="org6f36ec9"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||||
<div id="outline-container-orge6a9288" class="outline-4">
|
||||
<h4 id="orge6a9288"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||||
<div class="outline-text-4" id="text-2-1-4">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -956,11 +962,27 @@ local kinetic energy.
|
||||
<p>
|
||||
<b>Fortran</b>
|
||||
</p>
|
||||
|
||||
<div class="note">
|
||||
<p>
|
||||
When you call a function in Fortran, you need to declare its
|
||||
return type.
|
||||
You might by accident choose a function name which is the
|
||||
same as an internal function of Fortran. So it is recommended to
|
||||
<b>always</b> use the keyword <code>external</code> to make sure the function you
|
||||
are calling is yours.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">e_loc</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||||
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> kinetic</span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> potential</span>
|
||||
|
||||
! <span style="color: #b22222;">TODO</span>
|
||||
|
||||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">e_loc</span>
|
||||
@ -968,8 +990,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7be9be2" class="outline-5">
|
||||
<h5 id="org7be9be2"><span class="section-number-5">2.1.4.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org89a050a" class="outline-5">
|
||||
<h5 id="org89a050a"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -988,7 +1010,8 @@ local kinetic energy.
|
||||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||||
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> kinetic, potential</span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> kinetic</span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> potential</span>
|
||||
|
||||
e_loc = kinetic(a,r) + potential(r)
|
||||
|
||||
@ -999,8 +1022,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org71a1a30" class="outline-4">
|
||||
<h4 id="org71a1a30"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||||
<div id="outline-container-org90a766f" class="outline-4">
|
||||
<h4 id="org90a766f"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||||
<div class="outline-text-4" id="text-2-1-5">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -1010,8 +1033,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga629319" class="outline-5">
|
||||
<h5 id="orga629319"><span class="section-number-5">2.1.5.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org1669e04" class="outline-5">
|
||||
<h5 id="org1669e04"><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">
|
||||
\begin{eqnarray*}
|
||||
E &=& \frac{\hat{H} \Psi}{\Psi} = - \frac{1}{2} \frac{\Delta \Psi}{\Psi} -
|
||||
@ -1031,20 +1054,38 @@ equal to -0.5 atomic units.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org58f173a" class="outline-3">
|
||||
<h3 id="org58f173a"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
||||
<div id="outline-container-orgc481fb4" class="outline-3">
|
||||
<h3 id="orgc481fb4"><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="note">
|
||||
<p>
|
||||
The potential and the kinetic energy both diverge at \(r=0\), so we
|
||||
choose a grid which does not contain the origin.
|
||||
The program you will write in this section will be written in
|
||||
another file (<code>plot_hydrogen.py</code> or <code>plot_hydrogen.f90</code> for
|
||||
example).
|
||||
It will use the functions previously defined.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In Python, you should put at the beginning of the file
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc
|
||||
</pre>
|
||||
</div>
|
||||
<p>
|
||||
to be able to use the <code>e_loc</code> function of the <code>hydrogen.py</code> file.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In Fortran, you will need to compile all the source files together:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-sh">gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org74602a0" class="outline-4">
|
||||
<h4 id="org74602a0"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-org8a4f299" class="outline-4">
|
||||
<h4 id="org8a4f299"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-2-2-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -1052,7 +1093,16 @@ For multiple values of \(a\) (0.1, 0.2, 0.5, 1., 1.5, 2.), plot the
|
||||
local energy along the \(x\) axis. In Python, you can use matplotlib
|
||||
for example. In Fortran, it is convenient to write in a text file
|
||||
the values of \(x\) and \(E_L(\mathbf{r})\) for each point, and use
|
||||
Gnuplot to plot the files.
|
||||
Gnuplot to plot the files. With Gnuplot, you will need 2 blank
|
||||
lines to separate the data corresponding to different values of \(a\).
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<div class="note">
|
||||
<p>
|
||||
The potential and the kinetic energy both diverge at \(r=0\), so we
|
||||
choose a grid which does not contain the origin to avoid numerical issues.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
@ -1127,8 +1177,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3998c09" class="outline-5">
|
||||
<h5 id="org3998c09"><span class="section-number-5">2.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org5f96d93" class="outline-5">
|
||||
<h5 id="org5f96d93"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -1203,8 +1253,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga732d98" class="outline-3">
|
||||
<h3 id="orga732d98"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||||
<div id="outline-container-orgbcc283c" class="outline-3">
|
||||
<h3 id="orgbcc283c"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
If the space is discretized in small volume elements \(\mathbf{r}_i\)
|
||||
@ -1234,8 +1284,8 @@ The energy is biased because:
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org66a804a" class="outline-4">
|
||||
<h4 id="org66a804a"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-org5506022" class="outline-4">
|
||||
<h4 id="org5506022"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-2-3-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -1304,8 +1354,8 @@ To compile the Fortran and run it:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcb5c324" class="outline-5">
|
||||
<h5 id="orgcb5c324"><span class="section-number-5">2.3.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org74db5d8" class="outline-5">
|
||||
<h5 id="org74db5d8"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -1420,8 +1470,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9d5500d" class="outline-3">
|
||||
<h3 id="org9d5500d"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||||
<div id="outline-container-org1ceafe1" class="outline-3">
|
||||
<h3 id="org1ceafe1"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
The variance of the local energy is a functional of \(\Psi\)
|
||||
@ -1448,8 +1498,8 @@ energy can be used as a measure of the quality of a wave function.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7ef7c25" class="outline-4">
|
||||
<h4 id="org7ef7c25"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||||
<div id="outline-container-orge85aa1c" class="outline-4">
|
||||
<h4 id="orge85aa1c"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||||
<div class="outline-text-4" id="text-2-4-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -1460,8 +1510,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8f05b60" class="outline-5">
|
||||
<h5 id="org8f05b60"><span class="section-number-5">2.4.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org0940ecc" class="outline-5">
|
||||
<h5 id="org0940ecc"><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">
|
||||
<p>
|
||||
\(\bar{E} = \langle E \rangle\) is a constant, so \(\langle \bar{E}
|
||||
@ -1480,8 +1530,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org8d23fe3" class="outline-4">
|
||||
<h4 id="org8d23fe3"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||||
<div id="outline-container-orgb488446" class="outline-4">
|
||||
<h4 id="orgb488446"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-2-4-2">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -1555,8 +1605,8 @@ To compile and run:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf8c3f41" class="outline-5">
|
||||
<h5 id="orgf8c3f41"><span class="section-number-5">2.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org5d375ab" class="outline-5">
|
||||
<h5 id="org5d375ab"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -1693,8 +1743,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1de9633" class="outline-2">
|
||||
<h2 id="org1de9633"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||||
<div id="outline-container-org85e47f4" class="outline-2">
|
||||
<h2 id="org85e47f4"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
Numerical integration with deterministic methods is very efficient
|
||||
@ -1710,8 +1760,8 @@ interval.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9d504cf" class="outline-3">
|
||||
<h3 id="org9d504cf"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||||
<div id="outline-container-org17350f1" class="outline-3">
|
||||
<h3 id="org17350f1"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
To compute the statistical error, you need to perform \(M\)
|
||||
@ -1751,8 +1801,8 @@ And the confidence interval is given by
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7db3239" class="outline-4">
|
||||
<h4 id="org7db3239"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||||
<div id="outline-container-org97a29b6" class="outline-4">
|
||||
<h4 id="org97a29b6"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-3-1-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -1790,8 +1840,8 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2355686" class="outline-5">
|
||||
<h5 id="org2355686"><span class="section-number-5">3.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orga749707" class="outline-5">
|
||||
<h5 id="orga749707"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -1850,8 +1900,8 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org064ac12" class="outline-3">
|
||||
<h3 id="org064ac12"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||||
<div id="outline-container-orgd721f86" class="outline-3">
|
||||
<h3 id="orgd721f86"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
We will now perform our first Monte Carlo calculation to compute the
|
||||
@ -1912,8 +1962,8 @@ compute the statistical error.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb0f2034" class="outline-4">
|
||||
<h4 id="orgb0f2034"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-org215979f" class="outline-4">
|
||||
<h4 id="org215979f"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-3-2-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -2013,8 +2063,8 @@ well as the index of the current step.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9b6a09d" class="outline-5">
|
||||
<h5 id="org9b6a09d"><span class="section-number-5">3.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgbd3c736" class="outline-5">
|
||||
<h5 id="orgbd3c736"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -2128,8 +2178,8 @@ E = -0.49518773675598715 +/- 5.2391494923686175E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbc7bac6" class="outline-3">
|
||||
<h3 id="orgbc7bac6"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||||
<div id="outline-container-orgf676c01" class="outline-3">
|
||||
<h3 id="orgf676c01"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
We will now use the square of the wave function to sample random
|
||||
@ -2268,8 +2318,8 @@ the same variable later on to store a time step.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org231d1f1" class="outline-4">
|
||||
<h4 id="org231d1f1"><span class="section-number-4">3.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgc511dca" class="outline-4">
|
||||
<h4 id="orgc511dca"><span class="section-number-4">3.3.1</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-3-3-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -2376,8 +2426,8 @@ Can you observe a reduction in the statistical error?
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdb46596" class="outline-5">
|
||||
<h5 id="orgdb46596"><span class="section-number-5">3.3.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org8b5ee4b" class="outline-5">
|
||||
<h5 id="org8b5ee4b"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -2522,8 +2572,8 @@ A = 0.51695266666666673 +/- 4.0445505648997396E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3a0c9e1" class="outline-3">
|
||||
<h3 id="org3a0c9e1"><span class="section-number-3">3.4</span> Gaussian random number generator</h3>
|
||||
<div id="outline-container-org542b847" class="outline-3">
|
||||
<h3 id="org542b847"><span class="section-number-3">3.4</span> Gaussian random number generator</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
To obtain Gaussian-distributed random numbers, you can apply the
|
||||
@ -2586,8 +2636,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd4d4844" class="outline-3">
|
||||
<h3 id="orgd4d4844"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3>
|
||||
<div id="outline-container-orgce06b27" class="outline-3">
|
||||
<h3 id="orgce06b27"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3>
|
||||
<div class="outline-text-3" id="text-3-5">
|
||||
<p>
|
||||
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
|
||||
@ -2714,8 +2764,8 @@ Evaluate \(\Psi\) and \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\) at th
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org88e2361" class="outline-4">
|
||||
<h4 id="org88e2361"><span class="section-number-4">3.5.1</span> Exercise 1</h4>
|
||||
<div id="outline-container-org6877da4" class="outline-4">
|
||||
<h4 id="org6877da4"><span class="section-number-4">3.5.1</span> Exercise 1</h4>
|
||||
<div class="outline-text-4" id="text-3-5-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -2749,8 +2799,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org346520f" class="outline-5">
|
||||
<h5 id="org346520f"><span class="section-number-5">3.5.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org9f0138a" class="outline-5">
|
||||
<h5 id="org9f0138a"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -2783,8 +2833,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfd9abc1" class="outline-4">
|
||||
<h4 id="orgfd9abc1"><span class="section-number-4">3.5.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-org36805af" class="outline-4">
|
||||
<h4 id="org36805af"><span class="section-number-4">3.5.2</span> Exercise 2</h4>
|
||||
<div class="outline-text-4" id="text-3-5-2">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -2878,8 +2928,8 @@ Modify the previous program to introduce the drift-diffusion scheme.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org076151d" class="outline-5">
|
||||
<h5 id="org076151d"><span class="section-number-5">3.5.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgd6805a2" class="outline-5">
|
||||
<h5 id="orgd6805a2"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -3065,12 +3115,12 @@ A = 0.78839866666666658 +/- 3.2503783452043152E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org97a5ed5" class="outline-2">
|
||||
<h2 id="org97a5ed5"><span class="section-number-2">4</span> Diffusion Monte Carlo   <span class="tag"><span class="solution">solution</span></span></h2>
|
||||
<div id="outline-container-org2d2753c" class="outline-2">
|
||||
<h2 id="org2d2753c"><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>
|
||||
<div id="outline-container-org1359234" class="outline-3">
|
||||
<h3 id="org1359234"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
||||
<div id="outline-container-org3f5bc99" class="outline-3">
|
||||
<h3 id="org3f5bc99"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
Consider the time-dependent Schrödinger equation:
|
||||
@ -3138,8 +3188,8 @@ system.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5684930" class="outline-3">
|
||||
<h3 id="org5684930"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
||||
<div id="outline-container-orgbd4d3c8" class="outline-3">
|
||||
<h3 id="orgbd4d3c8"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
The imaginary-time Schrödinger equation can be explicitly written in terms of the kinetic and
|
||||
@ -3236,8 +3286,8 @@ Therefore, in both cases, you are dealing with a "Bosonic" ground state.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org11122df" class="outline-3">
|
||||
<h3 id="org11122df"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
||||
<div id="outline-container-org11ded29" class="outline-3">
|
||||
<h3 id="org11ded29"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<p>
|
||||
In a molecular system, the potential is far from being constant
|
||||
@ -3333,8 +3383,8 @@ energies computed with the trial wave function.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3d60069" class="outline-4">
|
||||
<h4 id="org3d60069"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4>
|
||||
<div id="outline-container-org5669f14" class="outline-4">
|
||||
<h4 id="org5669f14"><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">
|
||||
<p>
|
||||
\[
|
||||
@ -3395,8 +3445,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1171aaa" class="outline-3">
|
||||
<h3 id="org1171aaa"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
||||
<div id="outline-container-orged7a00f" class="outline-3">
|
||||
<h3 id="orged7a00f"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<p>
|
||||
Instead of having a variable number of particles to simulate the
|
||||
@ -3477,13 +3527,13 @@ code, so this is what we will do in the next section.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga6bc5fd" class="outline-3">
|
||||
<h3 id="orga6bc5fd"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
|
||||
<div id="outline-container-org0bcdcd6" class="outline-3">
|
||||
<h3 id="org0bcdcd6"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
|
||||
<div class="outline-text-3" id="text-4-5">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfc4c90b" class="outline-4">
|
||||
<h4 id="orgfc4c90b"><span class="section-number-4">4.5.1</span> Exercise</h4>
|
||||
<div id="outline-container-org4145b62" class="outline-4">
|
||||
<h4 id="org4145b62"><span class="section-number-4">4.5.1</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-4-5-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -3582,8 +3632,8 @@ energy of H for any value of \(a\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org156cd8e" class="outline-5">
|
||||
<h5 id="org156cd8e"><span class="section-number-5">4.5.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgfd5af9d" class="outline-5">
|
||||
<h5 id="orgfd5af9d"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -3799,8 +3849,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org267032c" class="outline-3">
|
||||
<h3 id="org267032c"><span class="section-number-3">4.6</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
|
||||
<div id="outline-container-org74f6b7c" class="outline-3">
|
||||
<h3 id="org74f6b7c"><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">
|
||||
<p>
|
||||
We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the
|
||||
@ -3821,8 +3871,8 @@ the nuclei.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org9cd3174" class="outline-2">
|
||||
<h2 id="org9cd3174"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2>
|
||||
<div id="outline-container-org1b7a228" class="outline-2">
|
||||
<h2 id="org1b7a228"><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">
|
||||
<ul class="org-ul">
|
||||
<li class="off"><code>[ ]</code> Give some hints of how much time is required for each section</li>
|
||||
@ -3835,10 +3885,58 @@ the H\(_2\) molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge8fb145" class="outline-2">
|
||||
<h2 id="orge8fb145"><span class="section-number-2">6</span> Schedule</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
|
||||
|
||||
<colgroup>
|
||||
<col class="org-left" />
|
||||
|
||||
<col class="org-right" />
|
||||
</colgroup>
|
||||
<thead>
|
||||
<tr>
|
||||
<th scope="col" class="org-left"><span class="timestamp-wrapper"><span class="timestamp"><2021-02-04 Thu 09:00>–<2021-02-04 Thu 10:30></span></span></th>
|
||||
<th scope="col" class="org-right">Lecture</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp"><2021-02-04 Thu 10:45>–<2021-02-04 Thu 11:10></span></span></td>
|
||||
<td class="org-right">2.1</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp"><2021-02-04 Thu 11:10>–<2021-02-04 Thu 11:30></span></span></td>
|
||||
<td class="org-right">2.2</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp"><2021-02-04 Thu 11:30>–<2021-02-04 Thu 12:15></span></span></td>
|
||||
<td class="org-right">2.3</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp"><2021-02-04 Thu 12:15>–<2021-02-04 Thu 12:30></span></span></td>
|
||||
<td class="org-right">2.4</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-left"> </td>
|
||||
<td class="org-right"> </td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
||||
<p class="date">Created: 2021-02-01 Mon 21:10</p>
|
||||
<p class="date">Created: 2021-02-02 Tue 10:29</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
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Reference in New Issue
Block a user