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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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<!-- 2021-01-30 Sat 08:25 -->
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<!-- 2021-01-30 Sat 12:21 -->
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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 +329,151 @@ 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="#org49d04ba">1. Introduction</a></li>
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<li><a href="#org829df65">2. Numerical evaluation of the energy</a>
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<li><a href="#org8dcaa4e">1. Introduction</a></li>
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<li><a href="#orgf2d601e">2. Numerical evaluation of the energy</a>
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<ul>
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<li><a href="#org05487b8">2.1. Local energy</a>
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<li><a href="#org0b86853">2.1. Local energy</a>
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<ul>
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<li><a href="#org0a1896f">2.1.1. Exercise 1</a>
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||||
<li><a href="#orgb4f1dcb">2.1.1. Exercise 1</a>
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||||
<ul>
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||||
<li><a href="#orgeac8364">2.1.1.1. Solution</a></li>
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<li><a href="#orge1171f3">2.1.1.1. Solution</a></li>
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</ul>
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||||
</li>
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||||
<li><a href="#org54f52bf">2.1.2. Exercise 2</a>
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||||
<li><a href="#org805f102">2.1.2. Exercise 2</a>
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||||
<ul>
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||||
<li><a href="#orgec4496f">2.1.2.1. Solution</a></li>
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||||
<li><a href="#orge839f82">2.1.2.1. Solution</a></li>
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</ul>
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||||
</li>
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||||
<li><a href="#orgb2b2470">2.1.3. Exercise 3</a>
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||||
<li><a href="#org30a89b7">2.1.3. Exercise 3</a>
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||||
<ul>
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<li><a href="#orgc9384b2">2.1.3.1. Solution</a></li>
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<li><a href="#org37f3a90">2.1.3.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org27c82a8">2.1.4. Exercise 4</a>
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<li><a href="#org8a71cb9">2.1.4. Exercise 4</a>
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||||
<ul>
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<li><a href="#org3488e51">2.1.4.1. Solution</a></li>
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<li><a href="#org3127c46">2.1.4.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#orgb470382">2.1.5. Exercise 5</a>
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<li><a href="#org5ae05ca">2.1.5. Exercise 5</a>
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||||
<ul>
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<li><a href="#org95fe0ba">2.1.5.1. Solution</a></li>
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<li><a href="#orgde68619">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="#org3c5c95d">2.2. Plot of the local energy along the \(x\) axis</a>
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<li><a href="#org6b6a61c">2.2. Plot of the local energy along the \(x\) axis</a>
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<ul>
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<li><a href="#orga2b9bcf">2.2.1. Exercise</a>
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<li><a href="#orga530380">2.2.1. Exercise</a>
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<ul>
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<li><a href="#org93945d1">2.2.1.1. Solution</a></li>
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<li><a href="#orgcd9f1bf">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="#org2555573">2.3. Numerical estimation of the energy</a>
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<li><a href="#org2387d29">2.3. Numerical estimation of the energy</a>
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<ul>
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<li><a href="#org102c578">2.3.1. Exercise</a>
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<li><a href="#orgde278da">2.3.1. Exercise</a>
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<ul>
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<li><a href="#org4da0f5c">2.3.1.1. Solution</a></li>
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<li><a href="#org73954f7">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="#org74ce38e">2.4. Variance of the local energy</a>
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<li><a href="#orgd0c5f25">2.4. Variance of the local energy</a>
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<ul>
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<li><a href="#org0510da2">2.4.1. Exercise (optional)</a>
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<li><a href="#orgdd8878b">2.4.1. Exercise (optional)</a>
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<ul>
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<li><a href="#org1c5cb68">2.4.1.1. Solution</a></li>
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<li><a href="#org874aece">2.4.1.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org499a5a7">2.4.2. Exercise</a>
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<li><a href="#org55c03ac">2.4.2. Exercise</a>
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<ul>
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<li><a href="#org93fe370">2.4.2.1. Solution</a></li>
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<li><a href="#orgff8da50">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="#org6a0028a">3. Variational Monte Carlo</a>
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<li><a href="#org500fc9e">3. Variational Monte Carlo</a>
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<ul>
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<li><a href="#org8b03841">3.1. Computation of the statistical error</a>
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<li><a href="#org1395f72">3.1. Computation of the statistical error</a>
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<ul>
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<li><a href="#org5bcb81b">3.1.1. Exercise</a>
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<li><a href="#orgbf3b7c1">3.1.1. Exercise</a>
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<ul>
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<li><a href="#orgc97d0a8">3.1.1.1. Solution</a></li>
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<li><a href="#org96a3dcd">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="#org1d4a893">3.2. Uniform sampling in the box</a>
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<li><a href="#org6f903d5">3.2. Uniform sampling in the box</a>
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<ul>
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||||
<li><a href="#orgc78cc2c">3.2.1. Exercise</a>
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||||
<li><a href="#orgd5d2e4f">3.2.1. Exercise</a>
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<ul>
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<li><a href="#orgf1caad9">3.2.1.1. Solution</a></li>
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<li><a href="#orge0a3bca">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="#org9ac37db">3.3. Metropolis sampling with \(\Psi^2\)</a>
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<li><a href="#org31acf84">3.3. Metropolis sampling with \(\Psi^2\)</a>
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<ul>
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||||
<li><a href="#org5ca4e2c">3.3.1. Exercise</a>
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||||
<li><a href="#org6d54039">3.3.1. Exercise</a>
|
||||
<ul>
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||||
<li><a href="#orgde750dd">3.3.1.1. Solution</a></li>
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||||
<li><a href="#org8084209">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="#org9cb319e">3.4. Gaussian random number generator</a></li>
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||||
<li><a href="#orgc52c7d5">3.5. Generalized Metropolis algorithm</a>
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<li><a href="#org92d024f">3.4. Gaussian random number generator</a></li>
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||||
<li><a href="#org1a64e1d">3.5. Generalized Metropolis algorithm</a>
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<ul>
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||||
<li><a href="#org4689b7c">3.5.1. Exercise 1</a>
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||||
<li><a href="#org97c1b29">3.5.1. Exercise 1</a>
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||||
<ul>
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||||
<li><a href="#org26c9d33">3.5.1.1. Solution</a></li>
|
||||
<li><a href="#org19e85c9">3.5.1.1. Solution</a></li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org3e39526">3.5.2. Exercise 2</a>
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||||
<li><a href="#orgaef93b5">3.5.2. Exercise 2</a>
|
||||
<ul>
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||||
<li><a href="#orga3cf826">3.5.2.1. Solution</a></li>
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||||
<li><a href="#org17fe7c6">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="#orgebb7fc1">4. Diffusion Monte Carlo</a>
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||||
<li><a href="#org9f51d1e">4. Diffusion Monte Carlo</a>
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||||
<ul>
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||||
<li><a href="#org8cf0ab0">4.1. Schrödinger equation in imaginary time</a></li>
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||||
<li><a href="#org78a65a5">4.2. Diffusion and branching</a></li>
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||||
<li><a href="#org00d8981">4.3. Importance sampling</a>
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||||
<li><a href="#org344d9de">4.1. Schrödinger equation in imaginary time</a></li>
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<li><a href="#org06862e7">4.2. Diffusion and branching</a></li>
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||||
<li><a href="#orgdf62087">4.3. Importance sampling</a>
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||||
<ul>
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||||
<li><a href="#org024b07c">4.3.1. Appendix : Details of the Derivation</a></li>
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||||
<li><a href="#org855e049">4.3.1. Appendix : Details of the Derivation</a></li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#orgce840bc">4.4. Fixed-node DMC energy</a></li>
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||||
<li><a href="#orga97437f">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li>
|
||||
<li><a href="#org54d9e47">4.6. Hydrogen atom</a>
|
||||
<li><a href="#org6fbf6b7">4.4. Fixed-node DMC energy</a></li>
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||||
<li><a href="#org4379250">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li>
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||||
<li><a href="#org3a30f06">4.6. Hydrogen atom</a>
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||||
<ul>
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||||
<li><a href="#org77c2d07">4.6.1. Exercise</a>
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||||
<li><a href="#orgba6aa2f">4.6.1. Exercise</a>
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||||
<ul>
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||||
<li><a href="#orgc3fb998">4.6.1.1. Solution</a></li>
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||||
<li><a href="#org2311033">4.6.1.1. 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="#orgb88b04d">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
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||||
<li><a href="#org81121c2">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org622ff55">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
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||||
<li><a href="#orgf5ad8c9">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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</div>
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</div>
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<div id="outline-container-org49d04ba" class="outline-2">
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<h2 id="org49d04ba"><span class="section-number-2">1</span> Introduction</h2>
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||||
<div id="outline-container-org8dcaa4e" class="outline-2">
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||||
<h2 id="org8dcaa4e"><span class="section-number-2">1</span> Introduction</h2>
|
||||
<div class="outline-text-2" id="text-1">
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||||
<p>
|
||||
This web site contains the QMC tutorial of the 2021 LTTC winter school
|
||||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
||||
<a href="https://www.irsamc.ups-tlse.fr/lttc/Luchon">Tutorials in Theoretical Chemistry</a>.
|
||||
</p>
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||||
|
||||
@ -487,7 +487,8 @@ computes a statistical estimate of the expectation value of the energy
|
||||
associated with a given wave function, and apply this approach to the
|
||||
hydrogen atom.
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||||
Finally, we present the diffusion Monte Carlo (DMC) method which
|
||||
we use here to estimate the exact energy of the hydrogen atom and of the H<sub>2</sub> molecule.
|
||||
we use here to estimate the exact energy of the hydrogen atom and of the H<sub>2</sub> molecule,
|
||||
starting from an approximate wave function.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -510,11 +511,11 @@ coordinates, etc).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org829df65" class="outline-2">
|
||||
<h2 id="org829df65"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2>
|
||||
<div id="outline-container-orgf2d601e" class="outline-2">
|
||||
<h2 id="orgf2d601e"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
In this section we consider the Hydrogen atom with the following
|
||||
In this section, we consider the hydrogen atom with the following
|
||||
wave function:
|
||||
</p>
|
||||
|
||||
@ -525,7 +526,7 @@ wave function:
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We will first verify that, for a given value of \(a\), \(\Psi\) is an
|
||||
We will first verify that, for a particular value of \(a\), \(\Psi\) is an
|
||||
eigenfunction of the Hamiltonian
|
||||
</p>
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||||
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||||
@ -536,7 +537,7 @@ eigenfunction of the Hamiltonian
|
||||
</p>
|
||||
|
||||
<p>
|
||||
To do that, we will check if the local energy, defined as
|
||||
To do that, we will compute the local energy, defined as
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -546,31 +547,11 @@ To do that, we will check if the local energy, defined as
|
||||
</p>
|
||||
|
||||
<p>
|
||||
is constant.
|
||||
</p>
|
||||
|
||||
|
||||
<p>
|
||||
The probabilistic <i>expected value</i> of an arbitrary function \(f(x)\)
|
||||
with respect to a probability density function \(p(x)\) is given by
|
||||
and check whether it is constant.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
\[ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx. \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Recall that a probability density function \(p(x)\) is non-negative
|
||||
and integrates to one:
|
||||
</p>
|
||||
|
||||
<p>
|
||||
\[ \int_{-\infty}^\infty p(x)\,dx = 1. \]
|
||||
</p>
|
||||
|
||||
|
||||
<p>
|
||||
The electronic energy of a system is the expectation value of the
|
||||
In general, the electronic energy of a system, \(E\), can be rewritten as the expectation value of the
|
||||
local energy \(E(\mathbf{r})\) with respect to the 3N-dimensional
|
||||
electron density given by the square of the wave function:
|
||||
</p>
|
||||
@ -580,12 +561,40 @@ E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
|
||||
= \frac{\int \Psi(\mathbf{r})\, \hat{H} \Psi(\mathbf{r})\, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \\
|
||||
& = & \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
|
||||
= \frac{\int \left[\Psi(\mathbf{r})\right]^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
|
||||
= \langle E_L \rangle_{\Psi^2}
|
||||
= \langle E_L \rangle_{\Psi^2}\,,
|
||||
\end{eqnarray*}
|
||||
<p>
|
||||
where \(\mathbf{r}\) is the vector of the 3N-dimensional electronic coordinates (\(N=1\) for the hydrogen atom).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
For a small number of dimensions, one can compute \(E\) by evaluating the integrals on a grid. However,
|
||||
</p>
|
||||
|
||||
<p>
|
||||
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>
|
||||
\[ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx, \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
where probability density function \(p(x)\) is non-negative
|
||||
and integrates to one:
|
||||
</p>
|
||||
|
||||
<p>
|
||||
\[ \int_{-\infty}^\infty p(x)\,dx = 1. \]
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org05487b8" class="outline-3">
|
||||
<h3 id="org05487b8"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||
|
||||
|
||||
|
||||
<div id="outline-container-org0b86853" class="outline-3">
|
||||
<h3 id="org0b86853"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
Write all the functions of this section in a single file :
|
||||
@ -608,8 +617,8 @@ to catch the error.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0a1896f" class="outline-4">
|
||||
<h4 id="org0a1896f"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||||
<div id="outline-container-orgb4f1dcb" class="outline-4">
|
||||
<h4 id="orgb4f1dcb"><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>
|
||||
@ -653,8 +662,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgeac8364" class="outline-5">
|
||||
<h5 id="orgeac8364"><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-orge1171f3" class="outline-5">
|
||||
<h5 id="orge1171f3"><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>
|
||||
@ -694,8 +703,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org54f52bf" class="outline-4">
|
||||
<h4 id="org54f52bf"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-org805f102" class="outline-4">
|
||||
<h4 id="org805f102"><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>
|
||||
@ -730,8 +739,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgec4496f" class="outline-5">
|
||||
<h5 id="orgec4496f"><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-orge839f82" class="outline-5">
|
||||
<h5 id="orge839f82"><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>
|
||||
@ -758,8 +767,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb2b2470" class="outline-4">
|
||||
<h4 id="orgb2b2470"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||||
<div id="outline-container-org30a89b7" class="outline-4">
|
||||
<h4 id="org30a89b7"><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>
|
||||
@ -840,8 +849,8 @@ So the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc9384b2" class="outline-5">
|
||||
<h5 id="orgc9384b2"><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-org37f3a90" class="outline-5">
|
||||
<h5 id="org37f3a90"><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>
|
||||
@ -882,8 +891,8 @@ So the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org27c82a8" class="outline-4">
|
||||
<h4 id="org27c82a8"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||||
<div id="outline-container-org8a71cb9" class="outline-4">
|
||||
<h4 id="org8a71cb9"><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>
|
||||
@ -926,8 +935,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3488e51" class="outline-5">
|
||||
<h5 id="org3488e51"><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-org3127c46" class="outline-5">
|
||||
<h5 id="org3127c46"><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>
|
||||
@ -957,8 +966,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb470382" class="outline-4">
|
||||
<h4 id="orgb470382"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||||
<div id="outline-container-org5ae05ca" class="outline-4">
|
||||
<h4 id="org5ae05ca"><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>
|
||||
@ -968,8 +977,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org95fe0ba" class="outline-5">
|
||||
<h5 id="org95fe0ba"><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-orgde68619" class="outline-5">
|
||||
<h5 id="orgde68619"><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} -
|
||||
@ -989,8 +998,8 @@ equal to -0.5 atomic units.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3c5c95d" class="outline-3">
|
||||
<h3 id="org3c5c95d"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
||||
<div id="outline-container-org6b6a61c" class="outline-3">
|
||||
<h3 id="org6b6a61c"><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>
|
||||
@ -1001,8 +1010,8 @@ choose a grid which does not contain the origin.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga2b9bcf" class="outline-4">
|
||||
<h4 id="orga2b9bcf"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-orga530380" class="outline-4">
|
||||
<h4 id="orga530380"><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>
|
||||
@ -1085,8 +1094,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org93945d1" class="outline-5">
|
||||
<h5 id="org93945d1"><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-orgcd9f1bf" class="outline-5">
|
||||
<h5 id="orgcd9f1bf"><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>
|
||||
@ -1161,8 +1170,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2555573" class="outline-3">
|
||||
<h3 id="org2555573"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||||
<div id="outline-container-org2387d29" class="outline-3">
|
||||
<h3 id="org2387d29"><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 \(\mathbf{r}_i\)
|
||||
@ -1192,8 +1201,8 @@ The energy is biased because:
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org102c578" class="outline-4">
|
||||
<h4 id="org102c578"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgde278da" class="outline-4">
|
||||
<h4 id="orgde278da"><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>
|
||||
@ -1262,8 +1271,8 @@ To compile the Fortran and run it:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4da0f5c" class="outline-5">
|
||||
<h5 id="org4da0f5c"><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-org73954f7" class="outline-5">
|
||||
<h5 id="org73954f7"><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>
|
||||
@ -1378,8 +1387,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org74ce38e" class="outline-3">
|
||||
<h3 id="org74ce38e"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||||
<div id="outline-container-orgd0c5f25" class="outline-3">
|
||||
<h3 id="orgd0c5f25"><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 \(\Psi\)
|
||||
@ -1406,8 +1415,8 @@ energy can be used as a measure of the quality of a wave function.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0510da2" class="outline-4">
|
||||
<h4 id="org0510da2"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||||
<div id="outline-container-orgdd8878b" class="outline-4">
|
||||
<h4 id="orgdd8878b"><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>
|
||||
@ -1418,8 +1427,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1c5cb68" class="outline-5">
|
||||
<h5 id="org1c5cb68"><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-org874aece" class="outline-5">
|
||||
<h5 id="org874aece"><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}
|
||||
@ -1438,8 +1447,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org499a5a7" class="outline-4">
|
||||
<h4 id="org499a5a7"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||||
<div id="outline-container-org55c03ac" class="outline-4">
|
||||
<h4 id="org55c03ac"><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>
|
||||
@ -1513,8 +1522,8 @@ To compile and run:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org93fe370" class="outline-5">
|
||||
<h5 id="org93fe370"><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-orgff8da50" class="outline-5">
|
||||
<h5 id="orgff8da50"><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>
|
||||
@ -1651,8 +1660,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6a0028a" class="outline-2">
|
||||
<h2 id="org6a0028a"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||||
<div id="outline-container-org500fc9e" class="outline-2">
|
||||
<h2 id="org500fc9e"><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
|
||||
@ -1668,8 +1677,8 @@ interval.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8b03841" class="outline-3">
|
||||
<h3 id="org8b03841"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||||
<div id="outline-container-org1395f72" class="outline-3">
|
||||
<h3 id="org1395f72"><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 \(M\)
|
||||
@ -1709,8 +1718,8 @@ And the confidence interval is given by
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5bcb81b" class="outline-4">
|
||||
<h4 id="org5bcb81b"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgbf3b7c1" class="outline-4">
|
||||
<h4 id="orgbf3b7c1"><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>
|
||||
@ -1748,8 +1757,8 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc97d0a8" class="outline-5">
|
||||
<h5 id="orgc97d0a8"><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-org96a3dcd" class="outline-5">
|
||||
<h5 id="org96a3dcd"><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>
|
||||
@ -1808,8 +1817,8 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1d4a893" class="outline-3">
|
||||
<h3 id="org1d4a893"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||||
<div id="outline-container-org6f903d5" class="outline-3">
|
||||
<h3 id="org6f903d5"><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 do our first Monte Carlo calculation to compute the
|
||||
@ -1843,8 +1852,8 @@ compute the statistical error.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc78cc2c" class="outline-4">
|
||||
<h4 id="orgc78cc2c"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgd5d2e4f" class="outline-4">
|
||||
<h4 id="orgd5d2e4f"><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>
|
||||
@ -1944,8 +1953,8 @@ well as the index of the current step.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf1caad9" class="outline-5">
|
||||
<h5 id="orgf1caad9"><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-orge0a3bca" class="outline-5">
|
||||
<h5 id="orge0a3bca"><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>
|
||||
@ -2059,8 +2068,8 @@ E = -0.49518773675598715 +/- 5.2391494923686175E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9ac37db" class="outline-3">
|
||||
<h3 id="org9ac37db"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||||
<div id="outline-container-org31acf84" class="outline-3">
|
||||
<h3 id="org31acf84"><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
|
||||
@ -2148,8 +2157,8 @@ step such that the acceptance rate is close to 0.5 is a good compromise.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org5ca4e2c" class="outline-4">
|
||||
<h4 id="org5ca4e2c"><span class="section-number-4">3.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-org6d54039" class="outline-4">
|
||||
<h4 id="org6d54039"><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>
|
||||
@ -2256,8 +2265,8 @@ Can you observe a reduction in the statistical error?
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgde750dd" class="outline-5">
|
||||
<h5 id="orgde750dd"><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-org8084209" class="outline-5">
|
||||
<h5 id="org8084209"><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>
|
||||
@ -2402,8 +2411,8 @@ A = 0.51695266666666673 +/- 4.0445505648997396E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9cb319e" class="outline-3">
|
||||
<h3 id="org9cb319e"><span class="section-number-3">3.4</span> Gaussian random number generator</h3>
|
||||
<div id="outline-container-org92d024f" class="outline-3">
|
||||
<h3 id="org92d024f"><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
|
||||
@ -2465,8 +2474,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgc52c7d5" class="outline-3">
|
||||
<h3 id="orgc52c7d5"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3>
|
||||
<div id="outline-container-org1a64e1d" class="outline-3">
|
||||
<h3 id="org1a64e1d"><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,
|
||||
@ -2565,8 +2574,8 @@ The transition probability becomes:
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org4689b7c" class="outline-4">
|
||||
<h4 id="org4689b7c"><span class="section-number-4">3.5.1</span> Exercise 1</h4>
|
||||
<div id="outline-container-org97c1b29" class="outline-4">
|
||||
<h4 id="org97c1b29"><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>
|
||||
@ -2600,8 +2609,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org26c9d33" class="outline-5">
|
||||
<h5 id="org26c9d33"><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-org19e85c9" class="outline-5">
|
||||
<h5 id="org19e85c9"><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>
|
||||
@ -2634,8 +2643,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3e39526" class="outline-4">
|
||||
<h4 id="org3e39526"><span class="section-number-4">3.5.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-orgaef93b5" class="outline-4">
|
||||
<h4 id="orgaef93b5"><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>
|
||||
@ -2729,8 +2738,8 @@ Modify the previous program to introduce the drifted diffusion scheme.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga3cf826" class="outline-5">
|
||||
<h5 id="orga3cf826"><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-org17fe7c6" class="outline-5">
|
||||
<h5 id="org17fe7c6"><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>
|
||||
@ -2916,12 +2925,12 @@ A = 0.78839866666666658 +/- 3.2503783452043152E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgebb7fc1" class="outline-2">
|
||||
<h2 id="orgebb7fc1"><span class="section-number-2">4</span> Diffusion Monte Carlo   <span class="tag"><span class="solution">solution</span></span></h2>
|
||||
<div id="outline-container-org9f51d1e" class="outline-2">
|
||||
<h2 id="org9f51d1e"><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-org8cf0ab0" class="outline-3">
|
||||
<h3 id="org8cf0ab0"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
||||
<div id="outline-container-org344d9de" class="outline-3">
|
||||
<h3 id="org344d9de"><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:
|
||||
@ -2980,8 +2989,8 @@ system.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org78a65a5" class="outline-3">
|
||||
<h3 id="org78a65a5"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
||||
<div id="outline-container-org06862e7" class="outline-3">
|
||||
<h3 id="org06862e7"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
The <a href="https://en.wikipedia.org/wiki/Diffusion_equation">diffusion equation</a> of particles is given by
|
||||
@ -3035,8 +3044,8 @@ the combination of a diffusion process and a branching process.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org00d8981" class="outline-3">
|
||||
<h3 id="org00d8981"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
||||
<div id="outline-container-orgdf62087" class="outline-3">
|
||||
<h3 id="orgdf62087"><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,
|
||||
@ -3093,8 +3102,8 @@ error known as the <i>fixed node error</i>.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org024b07c" class="outline-4">
|
||||
<h4 id="org024b07c"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4>
|
||||
<div id="outline-container-org855e049" class="outline-4">
|
||||
<h4 id="org855e049"><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>
|
||||
\[
|
||||
@ -3156,8 +3165,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgce840bc" class="outline-3">
|
||||
<h3 id="orgce840bc"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3>
|
||||
<div id="outline-container-org6fbf6b7" class="outline-3">
|
||||
<h3 id="org6fbf6b7"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<p>
|
||||
Now that we have a process to sample \(\Pi(\mathbf{r},\tau) =
|
||||
@ -3209,8 +3218,8 @@ energies computed with the trial wave function.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga97437f" class="outline-3">
|
||||
<h3 id="orga97437f"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
||||
<div id="outline-container-org4379250" class="outline-3">
|
||||
<h3 id="org4379250"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
||||
<div class="outline-text-3" id="text-4-5">
|
||||
<p>
|
||||
Instead of having a variable number of particles to simulate the
|
||||
@ -3262,13 +3271,13 @@ code, so this is what we will do in the next section.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org54d9e47" class="outline-3">
|
||||
<h3 id="org54d9e47"><span class="section-number-3">4.6</span> Hydrogen atom</h3>
|
||||
<div id="outline-container-org3a30f06" class="outline-3">
|
||||
<h3 id="org3a30f06"><span class="section-number-3">4.6</span> Hydrogen atom</h3>
|
||||
<div class="outline-text-3" id="text-4-6">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org77c2d07" class="outline-4">
|
||||
<h4 id="org77c2d07"><span class="section-number-4">4.6.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgba6aa2f" class="outline-4">
|
||||
<h4 id="orgba6aa2f"><span class="section-number-4">4.6.1</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-4-6-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -3367,8 +3376,8 @@ energy of H for any value of \(a\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc3fb998" class="outline-5">
|
||||
<h5 id="orgc3fb998"><span class="section-number-5">4.6.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org2311033" class="outline-5">
|
||||
<h5 id="org2311033"><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">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -3584,8 +3593,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgb88b04d" class="outline-3">
|
||||
<h3 id="orgb88b04d"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
|
||||
<div id="outline-container-org81121c2" class="outline-3">
|
||||
<h3 id="org81121c2"><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">
|
||||
<p>
|
||||
We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the
|
||||
@ -3606,8 +3615,8 @@ the nuclei.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org622ff55" class="outline-2">
|
||||
<h2 id="org622ff55"><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-orgf5ad8c9" class="outline-2">
|
||||
<h2 id="orgf5ad8c9"><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>
|
||||
@ -3623,7 +3632,7 @@ the H\(_2\) molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.</li>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
||||
<p class="date">Created: 2021-01-30 Sat 08:25</p>
|
||||
<p class="date">Created: 2021-01-30 Sat 12:21</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
Loading…
Reference in New Issue
Block a user