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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 08:08 -->
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<!-- 2021-02-01 Mon 08:59 -->
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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="#orge8bc398">1. Introduction</a>
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<li><a href="#org9f02c0e">1. Introduction</a>
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<ul>
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<li><a href="#org706146e">1.1. Energy and local energy</a></li>
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<li><a href="#org0992d05">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="#org8eba34e">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<li><a href="#org4b43e4d">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<ul>
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<li><a href="#orgb6798fe">2.1. Local energy</a>
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<li><a href="#orge3325c8">2.1. Local energy</a>
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<ul>
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<li><a href="#org309f3a3">2.1.1. Exercise 1</a>
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<li><a href="#orgc1ad1db">2.1.1. Exercise 1</a>
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||||
<ul>
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||||
<li><a href="#orged35f0c">2.1.1.1. Solution</a></li>
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||||
<li><a href="#orgb248cd2">2.1.1.1. Solution</a></li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org9cb5b69">2.1.2. Exercise 2</a>
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<li><a href="#orgab084dd">2.1.2. Exercise 2</a>
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||||
<ul>
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||||
<li><a href="#org9650566">2.1.2.1. Solution</a></li>
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<li><a href="#org614b3ba">2.1.2.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org3c48519">2.1.3. Exercise 3</a>
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<li><a href="#org4675fef">2.1.3. Exercise 3</a>
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<ul>
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<li><a href="#org705631c">2.1.3.1. Solution</a></li>
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<li><a href="#org36b6bcb">2.1.3.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#orgd94ed87">2.1.4. Exercise 4</a>
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<li><a href="#orgf3e0fa2">2.1.4. Exercise 4</a>
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<ul>
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||||
<li><a href="#orgd9baa77">2.1.4.1. Solution</a></li>
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<li><a href="#orgd4a5933">2.1.4.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#orgd1d6cba">2.1.5. Exercise 5</a>
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<li><a href="#org4007ce9">2.1.5. Exercise 5</a>
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<ul>
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<li><a href="#org6c2caf1">2.1.5.1. Solution</a></li>
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<li><a href="#orgfdf081d">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="#orgf3480bd">2.2. Plot of the local energy along the \(x\) axis</a>
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<li><a href="#org4fac075">2.2. Plot of the local energy along the \(x\) axis</a>
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<ul>
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<li><a href="#org75d9a33">2.2.1. Exercise</a>
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<li><a href="#org44aab77">2.2.1. Exercise</a>
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<ul>
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<li><a href="#orgb512433">2.2.1.1. Solution</a></li>
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<li><a href="#org0e6f4ed">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="#orgd8457b5">2.3. Numerical estimation of the energy</a>
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<li><a href="#org591bfd4">2.3. Numerical estimation of the energy</a>
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<ul>
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<li><a href="#orgd00b45f">2.3.1. Exercise</a>
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<li><a href="#org2183ef6">2.3.1. Exercise</a>
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<ul>
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<li><a href="#org8ef4dfd">2.3.1.1. Solution</a></li>
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<li><a href="#org0fad15d">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="#org54576bb">2.4. Variance of the local energy</a>
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<li><a href="#org9947984">2.4. Variance of the local energy</a>
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<ul>
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<li><a href="#org77249ac">2.4.1. Exercise (optional)</a>
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<li><a href="#org548400e">2.4.1. Exercise (optional)</a>
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||||
<ul>
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<li><a href="#org229d0cf">2.4.1.1. Solution</a></li>
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<li><a href="#orge43c657">2.4.1.1. Solution</a></li>
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</ul>
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||||
</li>
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||||
<li><a href="#orga7192e3">2.4.2. Exercise</a>
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||||
<li><a href="#orga2f5f44">2.4.2. Exercise</a>
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<ul>
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<li><a href="#org6ffe540">2.4.2.1. Solution</a></li>
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<li><a href="#org23ddbdd">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="#org7c5ed18">3. Variational Monte Carlo</a>
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<li><a href="#org436d73e">3. Variational Monte Carlo</a>
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<ul>
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<li><a href="#orgbd75310">3.1. Computation of the statistical error</a>
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<li><a href="#orgea24b87">3.1. Computation of the statistical error</a>
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<ul>
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<li><a href="#org776529e">3.1.1. Exercise</a>
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||||
<li><a href="#orgfe75700">3.1.1. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#orgd623403">3.1.1.1. Solution</a></li>
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||||
<li><a href="#orgcb05f71">3.1.1.1. Solution</a></li>
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||||
</ul>
|
||||
</li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#orged02a3d">3.2. Uniform sampling in the box</a>
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<li><a href="#org27459de">3.2. Uniform sampling in the box</a>
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||||
<ul>
|
||||
<li><a href="#orgafe912c">3.2.1. Exercise</a>
|
||||
<li><a href="#org06e6e66">3.2.1. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#org6d2da6a">3.2.1.1. Solution</a></li>
|
||||
<li><a href="#orgb739fbc">3.2.1.1. Solution</a></li>
|
||||
</ul>
|
||||
</li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org24dd766">3.3. Metropolis sampling with \(\Psi^2\)</a>
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||||
<li><a href="#org42c2fcd">3.3. Metropolis sampling with \(\Psi^2\)</a>
|
||||
<ul>
|
||||
<li><a href="#org38a53b5">3.3.1. Exercise</a>
|
||||
<li><a href="#orge5d434a">3.3.1. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#org0478678">3.3.1.1. Solution</a></li>
|
||||
<li><a href="#org7c6d44c">3.3.1.1. Solution</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
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||||
</li>
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||||
<li><a href="#org9fbaf17">3.4. Gaussian random number generator</a></li>
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||||
<li><a href="#org157d686">3.5. Generalized Metropolis algorithm</a>
|
||||
<li><a href="#orga212348">3.4. Gaussian random number generator</a></li>
|
||||
<li><a href="#orga845819">3.5. Generalized Metropolis algorithm</a>
|
||||
<ul>
|
||||
<li><a href="#org664cd9b">3.5.1. Exercise 1</a>
|
||||
<li><a href="#org1180a91">3.5.1. Exercise 1</a>
|
||||
<ul>
|
||||
<li><a href="#org5e7f9b6">3.5.1.1. Solution</a></li>
|
||||
<li><a href="#orgd5804e5">3.5.1.1. Solution</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org7fc8051">3.5.2. Exercise 2</a>
|
||||
<li><a href="#orgd78ceb7">3.5.2. Exercise 2</a>
|
||||
<ul>
|
||||
<li><a href="#orgf51edc7">3.5.2.1. Solution</a></li>
|
||||
<li><a href="#org25cdc99">3.5.2.1. Solution</a></li>
|
||||
</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>
|
||||
<li><a href="#org05f3f4f">4. Diffusion Monte Carlo</a>
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||||
<li><a href="#orge79baf6">4. Diffusion Monte Carlo</a>
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||||
<ul>
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||||
<li><a href="#org2360326">4.1. Schrödinger equation in imaginary time</a></li>
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||||
<li><a href="#org269a713">4.2. Diffusion and branching</a></li>
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||||
<li><a href="#org420b954">4.3. Importance sampling</a>
|
||||
<li><a href="#org7088239">4.1. Schrödinger equation in imaginary time</a></li>
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||||
<li><a href="#orgae407e7">4.2. Diffusion and branching</a></li>
|
||||
<li><a href="#org46761e4">4.3. Importance sampling</a>
|
||||
<ul>
|
||||
<li><a href="#org24db661">4.3.1. Appendix : Details of the Derivation</a></li>
|
||||
<li><a href="#org502e85d">4.3.1. Appendix : Details of the Derivation</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org5d1b87c">4.4. Pure Diffusion Monte Carlo (PDMC)</a></li>
|
||||
<li><a href="#org8fce010">4.5. Hydrogen atom</a>
|
||||
<li><a href="#orgf1b4507">4.4. Pure Diffusion Monte Carlo (PDMC)</a></li>
|
||||
<li><a href="#org88c5828">4.5. Hydrogen atom</a>
|
||||
<ul>
|
||||
<li><a href="#org1ae20b2">4.5.1. Exercise</a>
|
||||
<li><a href="#org143cee7">4.5.1. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#orgd9f1564">4.5.1.1. Solution</a></li>
|
||||
<li><a href="#org0dc2cc6">4.5.1.1. Solution</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org711f9e1">4.6. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
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||||
<li><a href="#org1cb1587">4.6. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
|
||||
</ul>
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||||
</li>
|
||||
<li><a href="#org4874bf9">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
|
||||
<li><a href="#orgb377e8a">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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||||
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||||
<div id="outline-container-orge8bc398" class="outline-2">
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||||
<h2 id="orge8bc398"><span class="section-number-2">1</span> Introduction</h2>
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||||
<div id="outline-container-org9f02c0e" class="outline-2">
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||||
<h2 id="org9f02c0e"><span class="section-number-2">1</span> Introduction</h2>
|
||||
<div class="outline-text-2" id="text-1">
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||||
<p>
|
||||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
||||
@ -513,8 +513,8 @@ coordinates, etc).
|
||||
</p>
|
||||
</div>
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||||
|
||||
<div id="outline-container-org706146e" class="outline-3">
|
||||
<h3 id="org706146e"><span class="section-number-3">1.1</span> Energy and local energy</h3>
|
||||
<div id="outline-container-org0992d05" class="outline-3">
|
||||
<h3 id="org0992d05"><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
|
||||
@ -592,8 +592,8 @@ $$ E ≈ \frac{1}{N<sub>\rm MC</sub>} ∑<sub>i=1</sub><sup>N<sub>\rm MC</
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||||
</div>
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||||
</div>
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||||
|
||||
<div id="outline-container-org8eba34e" class="outline-2">
|
||||
<h2 id="org8eba34e"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
|
||||
<div id="outline-container-org4b43e4d" class="outline-2">
|
||||
<h2 id="org4b43e4d"><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 +622,8 @@ To do that, we will compute the local energy and check whether it is constant.
|
||||
</p>
|
||||
</div>
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||||
|
||||
<div id="outline-container-orgb6798fe" class="outline-3">
|
||||
<h3 id="orgb6798fe"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||
<div id="outline-container-orge3325c8" class="outline-3">
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||||
<h3 id="orge3325c8"><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.
|
||||
@ -650,8 +650,8 @@ to catch the error.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org309f3a3" class="outline-4">
|
||||
<h4 id="org309f3a3"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||||
<div id="outline-container-orgc1ad1db" class="outline-4">
|
||||
<h4 id="orgc1ad1db"><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>
|
||||
@ -695,8 +695,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orged35f0c" class="outline-5">
|
||||
<h5 id="orged35f0c"><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-orgb248cd2" class="outline-5">
|
||||
<h5 id="orgb248cd2"><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 +736,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9cb5b69" class="outline-4">
|
||||
<h4 id="org9cb5b69"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-orgab084dd" class="outline-4">
|
||||
<h4 id="orgab084dd"><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 +772,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9650566" class="outline-5">
|
||||
<h5 id="org9650566"><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-org614b3ba" class="outline-5">
|
||||
<h5 id="org614b3ba"><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 +800,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3c48519" class="outline-4">
|
||||
<h4 id="org3c48519"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||||
<div id="outline-container-org4675fef" class="outline-4">
|
||||
<h4 id="org4675fef"><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>
|
||||
@ -882,8 +882,8 @@ Therefore, the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org705631c" class="outline-5">
|
||||
<h5 id="org705631c"><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-org36b6bcb" class="outline-5">
|
||||
<h5 id="org36b6bcb"><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 +924,8 @@ Therefore, the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd94ed87" class="outline-4">
|
||||
<h4 id="orgd94ed87"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||||
<div id="outline-container-orgf3e0fa2" class="outline-4">
|
||||
<h4 id="orgf3e0fa2"><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>
|
||||
@ -968,8 +968,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd9baa77" class="outline-5">
|
||||
<h5 id="orgd9baa77"><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-orgd4a5933" class="outline-5">
|
||||
<h5 id="orgd4a5933"><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>
|
||||
@ -999,8 +999,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd1d6cba" class="outline-4">
|
||||
<h4 id="orgd1d6cba"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||||
<div id="outline-container-org4007ce9" class="outline-4">
|
||||
<h4 id="org4007ce9"><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 +1010,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6c2caf1" class="outline-5">
|
||||
<h5 id="org6c2caf1"><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-orgfdf081d" class="outline-5">
|
||||
<h5 id="orgfdf081d"><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} -
|
||||
@ -1031,8 +1031,8 @@ equal to -0.5 atomic units.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf3480bd" class="outline-3">
|
||||
<h3 id="orgf3480bd"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
||||
<div id="outline-container-org4fac075" class="outline-3">
|
||||
<h3 id="org4fac075"><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>
|
||||
@ -1043,8 +1043,8 @@ choose a grid which does not contain the origin.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org75d9a33" class="outline-4">
|
||||
<h4 id="org75d9a33"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-org44aab77" class="outline-4">
|
||||
<h4 id="org44aab77"><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>
|
||||
@ -1127,8 +1127,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb512433" class="outline-5">
|
||||
<h5 id="orgb512433"><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-org0e6f4ed" class="outline-5">
|
||||
<h5 id="org0e6f4ed"><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 +1203,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd8457b5" class="outline-3">
|
||||
<h3 id="orgd8457b5"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||||
<div id="outline-container-org591bfd4" class="outline-3">
|
||||
<h3 id="org591bfd4"><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 +1234,8 @@ The energy is biased because:
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgd00b45f" class="outline-4">
|
||||
<h4 id="orgd00b45f"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-org2183ef6" class="outline-4">
|
||||
<h4 id="org2183ef6"><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 +1304,8 @@ To compile the Fortran and run it:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8ef4dfd" class="outline-5">
|
||||
<h5 id="org8ef4dfd"><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-org0fad15d" class="outline-5">
|
||||
<h5 id="org0fad15d"><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 +1420,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org54576bb" class="outline-3">
|
||||
<h3 id="org54576bb"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||||
<div id="outline-container-org9947984" class="outline-3">
|
||||
<h3 id="org9947984"><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 +1448,8 @@ energy can be used as a measure of the quality of a wave function.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org77249ac" class="outline-4">
|
||||
<h4 id="org77249ac"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||||
<div id="outline-container-org548400e" class="outline-4">
|
||||
<h4 id="org548400e"><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 +1460,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org229d0cf" class="outline-5">
|
||||
<h5 id="org229d0cf"><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-orge43c657" class="outline-5">
|
||||
<h5 id="orge43c657"><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 +1480,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orga7192e3" class="outline-4">
|
||||
<h4 id="orga7192e3"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||||
<div id="outline-container-orga2f5f44" class="outline-4">
|
||||
<h4 id="orga2f5f44"><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 +1555,8 @@ To compile and run:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6ffe540" class="outline-5">
|
||||
<h5 id="org6ffe540"><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-org23ddbdd" class="outline-5">
|
||||
<h5 id="org23ddbdd"><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 +1693,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7c5ed18" class="outline-2">
|
||||
<h2 id="org7c5ed18"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||||
<div id="outline-container-org436d73e" class="outline-2">
|
||||
<h2 id="org436d73e"><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 +1710,8 @@ interval.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbd75310" class="outline-3">
|
||||
<h3 id="orgbd75310"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||||
<div id="outline-container-orgea24b87" class="outline-3">
|
||||
<h3 id="orgea24b87"><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 +1751,8 @@ And the confidence interval is given by
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org776529e" class="outline-4">
|
||||
<h4 id="org776529e"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgfe75700" class="outline-4">
|
||||
<h4 id="orgfe75700"><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 +1790,8 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd623403" class="outline-5">
|
||||
<h5 id="orgd623403"><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-orgcb05f71" class="outline-5">
|
||||
<h5 id="orgcb05f71"><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 +1850,8 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orged02a3d" class="outline-3">
|
||||
<h3 id="orged02a3d"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||||
<div id="outline-container-org27459de" class="outline-3">
|
||||
<h3 id="org27459de"><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 +1912,8 @@ compute the statistical error.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgafe912c" class="outline-4">
|
||||
<h4 id="orgafe912c"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-org06e6e66" class="outline-4">
|
||||
<h4 id="org06e6e66"><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 +2013,8 @@ well as the index of the current step.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6d2da6a" class="outline-5">
|
||||
<h5 id="org6d2da6a"><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-orgb739fbc" class="outline-5">
|
||||
<h5 id="orgb739fbc"><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 +2128,8 @@ E = -0.49518773675598715 +/- 5.2391494923686175E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org24dd766" class="outline-3">
|
||||
<h3 id="org24dd766"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||||
<div id="outline-container-org42c2fcd" class="outline-3">
|
||||
<h3 id="org42c2fcd"><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 +2268,8 @@ the same variable later on to store a time step.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org38a53b5" class="outline-4">
|
||||
<h4 id="org38a53b5"><span class="section-number-4">3.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-orge5d434a" class="outline-4">
|
||||
<h4 id="orge5d434a"><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 +2376,8 @@ Can you observe a reduction in the statistical error?
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0478678" class="outline-5">
|
||||
<h5 id="org0478678"><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-org7c6d44c" class="outline-5">
|
||||
<h5 id="org7c6d44c"><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 +2522,8 @@ A = 0.51695266666666673 +/- 4.0445505648997396E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9fbaf17" class="outline-3">
|
||||
<h3 id="org9fbaf17"><span class="section-number-3">3.4</span> Gaussian random number generator</h3>
|
||||
<div id="outline-container-orga212348" class="outline-3">
|
||||
<h3 id="orga212348"><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 +2586,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org157d686" class="outline-3">
|
||||
<h3 id="org157d686"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3>
|
||||
<div id="outline-container-orga845819" class="outline-3">
|
||||
<h3 id="orga845819"><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.
|
||||
@ -2719,8 +2719,8 @@ Evaluate \(\Psi\) and \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\) at th
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org664cd9b" class="outline-4">
|
||||
<h4 id="org664cd9b"><span class="section-number-4">3.5.1</span> Exercise 1</h4>
|
||||
<div id="outline-container-org1180a91" class="outline-4">
|
||||
<h4 id="org1180a91"><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>
|
||||
@ -2754,8 +2754,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5e7f9b6" class="outline-5">
|
||||
<h5 id="org5e7f9b6"><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-orgd5804e5" class="outline-5">
|
||||
<h5 id="orgd5804e5"><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>
|
||||
@ -2788,8 +2788,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7fc8051" class="outline-4">
|
||||
<h4 id="org7fc8051"><span class="section-number-4">3.5.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-orgd78ceb7" class="outline-4">
|
||||
<h4 id="orgd78ceb7"><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>
|
||||
@ -2883,8 +2883,8 @@ Modify the previous program to introduce the drift-diffusion scheme.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf51edc7" class="outline-5">
|
||||
<h5 id="orgf51edc7"><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-org25cdc99" class="outline-5">
|
||||
<h5 id="org25cdc99"><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>
|
||||
@ -3070,12 +3070,12 @@ A = 0.78839866666666658 +/- 3.2503783452043152E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org05f3f4f" class="outline-2">
|
||||
<h2 id="org05f3f4f"><span class="section-number-2">4</span> Diffusion Monte Carlo   <span class="tag"><span class="solution">solution</span></span></h2>
|
||||
<div id="outline-container-orge79baf6" class="outline-2">
|
||||
<h2 id="orge79baf6"><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-org2360326" class="outline-3">
|
||||
<h3 id="org2360326"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
||||
<div id="outline-container-org7088239" class="outline-3">
|
||||
<h3 id="org7088239"><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:
|
||||
@ -3143,8 +3143,8 @@ system.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org269a713" class="outline-3">
|
||||
<h3 id="org269a713"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
||||
<div id="outline-container-orgae407e7" class="outline-3">
|
||||
<h3 id="orgae407e7"><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
|
||||
@ -3244,8 +3244,8 @@ Therefore, in both cases, you are dealing with a "Bosonic" ground state.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org420b954" class="outline-3">
|
||||
<h3 id="org420b954"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
||||
<div id="outline-container-org46761e4" class="outline-3">
|
||||
<h3 id="org46761e4"><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
|
||||
@ -3341,8 +3341,8 @@ energies computed with the trial wave function.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org24db661" class="outline-4">
|
||||
<h4 id="org24db661"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4>
|
||||
<div id="outline-container-org502e85d" class="outline-4">
|
||||
<h4 id="org502e85d"><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>
|
||||
\[
|
||||
@ -3403,8 +3403,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5d1b87c" class="outline-3">
|
||||
<h3 id="org5d1b87c"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
||||
<div id="outline-container-orgf1b4507" class="outline-3">
|
||||
<h3 id="orgf1b4507"><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
|
||||
@ -3455,13 +3455,13 @@ code, so this is what we will do in the next section.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8fce010" class="outline-3">
|
||||
<h3 id="org8fce010"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
|
||||
<div id="outline-container-org88c5828" class="outline-3">
|
||||
<h3 id="org88c5828"><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-org1ae20b2" class="outline-4">
|
||||
<h4 id="org1ae20b2"><span class="section-number-4">4.5.1</span> Exercise</h4>
|
||||
<div id="outline-container-org143cee7" class="outline-4">
|
||||
<h4 id="org143cee7"><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>
|
||||
@ -3479,7 +3479,7 @@ energy of H for any value of \(a\).
|
||||
<pre class="src src-python"><span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||||
|
||||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a, nmax, dt, Eref):
|
||||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a, nmax, dt, tau, Eref):
|
||||
# <span style="color: #b22222;">TODO</span>
|
||||
|
||||
# <span style="color: #b22222;">Run simulation</span>
|
||||
@ -3488,7 +3488,7 @@ energy of H for any value of \(a\).
|
||||
<span style="color: #a0522d;">dt</span> = 0.01
|
||||
<span style="color: #a0522d;">E_ref</span> = -0.5
|
||||
|
||||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a, nmax, dt, E_ref) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a, nmax, dt, tau, E_ref) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||||
|
||||
# <span style="color: #b22222;">Energy</span>
|
||||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||||
@ -3560,8 +3560,8 @@ energy of H for any value of \(a\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd9f1564" class="outline-5">
|
||||
<h5 id="orgd9f1564"><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-org0dc2cc6" class="outline-5">
|
||||
<h5 id="org0dc2cc6"><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>
|
||||
@ -3777,8 +3777,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org711f9e1" class="outline-3">
|
||||
<h3 id="org711f9e1"><span class="section-number-3">4.6</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
|
||||
<div id="outline-container-org1cb1587" class="outline-3">
|
||||
<h3 id="org1cb1587"><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
|
||||
@ -3799,8 +3799,8 @@ the nuclei.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org4874bf9" class="outline-2">
|
||||
<h2 id="org4874bf9"><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-orgb377e8a" class="outline-2">
|
||||
<h2 id="orgb377e8a"><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>
|
||||
@ -3816,7 +3816,7 @@ 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: 2021-02-01 Mon 08:08</p>
|
||||
<p class="date">Created: 2021-02-01 Mon 08:59</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