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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2021-02-02 Tue 16:01 -->
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<!-- 2021-02-02 Tue 16:05 -->
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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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@ -329,153 +329,153 @@ 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="#orgc4e758b">1. Introduction</a>
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<li><a href="#orgf35fcfa">1. Introduction</a>
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<ul>
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<li><a href="#orgdba9547">1.1. Energy and local energy</a></li>
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<li><a href="#orgff4a500">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="#orge64c748">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<li><a href="#orge5391c1">2. Numerical evaluation of the energy of the hydrogen atom</a>
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<ul>
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<li><a href="#org683b2fd">2.1. Local energy</a>
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<li><a href="#orgc557da6">2.1. Local energy</a>
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<ul>
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<li><a href="#org1f3996c">2.1.1. Exercise 1</a>
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<li><a href="#orga4e6b54">2.1.1. Exercise 1</a>
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||||
<ul>
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||||
<li><a href="#org01b5503">2.1.1.1. Solution</a></li>
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||||
<li><a href="#org801cff3">2.1.1.1. Solution</a></li>
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</ul>
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||||
</li>
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||||
<li><a href="#org9fd3548">2.1.2. Exercise 2</a>
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<li><a href="#org4677e85">2.1.2. Exercise 2</a>
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||||
<ul>
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||||
<li><a href="#orgafc07f4">2.1.2.1. Solution</a></li>
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<li><a href="#orgbc257d9">2.1.2.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org4e056ec">2.1.3. Exercise 3</a>
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<li><a href="#orgc8a1a2b">2.1.3. Exercise 3</a>
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<ul>
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||||
<li><a href="#org41a05f4">2.1.3.1. Solution</a></li>
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<li><a href="#org46a48e1">2.1.3.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org3447ad1">2.1.4. Exercise 4</a>
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<li><a href="#org49b162e">2.1.4. Exercise 4</a>
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||||
<ul>
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||||
<li><a href="#org9062ca6">2.1.4.1. Solution</a></li>
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<li><a href="#org11b7ba8">2.1.4.1. Solution</a></li>
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</ul>
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</li>
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<li><a href="#org46514dd">2.1.5. Exercise 5</a>
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<li><a href="#orgea20a4d">2.1.5. Exercise 5</a>
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<ul>
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<li><a href="#org644dc09">2.1.5.1. Solution</a></li>
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<li><a href="#orga8ca1f7">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="#org8c99b58">2.2. Plot of the local energy along the \(x\) axis</a>
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<li><a href="#org1aceedb">2.2. Plot of the local energy along the \(x\) axis</a>
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<ul>
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<li><a href="#orge8967d0">2.2.1. Exercise</a>
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<li><a href="#orgfdfeaeb">2.2.1. Exercise</a>
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<ul>
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<li><a href="#org93bb898">2.2.1.1. Solution</a></li>
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<li><a href="#org7b8e758">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="#orgd2d5d22">2.3. Numerical estimation of the energy</a>
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<li><a href="#orgb878d1c">2.3. Numerical estimation of the energy</a>
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<ul>
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<li><a href="#org9b6bf44">2.3.1. Exercise</a>
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<li><a href="#orga3c04f6">2.3.1. Exercise</a>
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<ul>
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<li><a href="#org0346611">2.3.1.1. Solution</a></li>
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<li><a href="#org3ff01c5">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="#org82d4afa">2.4. Variance of the local energy</a>
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<li><a href="#org61a9a5a">2.4. Variance of the local energy</a>
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<ul>
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<li><a href="#org475ea22">2.4.1. Exercise (optional)</a>
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<li><a href="#org3a9cd19">2.4.1. Exercise (optional)</a>
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||||
<ul>
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<li><a href="#org9072218">2.4.1.1. Solution</a></li>
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<li><a href="#orge49d691">2.4.1.1. Solution</a></li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org350c442">2.4.2. Exercise</a>
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||||
<li><a href="#orgc95b8fc">2.4.2. Exercise</a>
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||||
<ul>
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||||
<li><a href="#orgea0025a">2.4.2.1. Solution</a></li>
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||||
<li><a href="#org0cf5abe">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="#org95128a7">3. Variational Monte Carlo</a>
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<li><a href="#org03c7717">3. Variational Monte Carlo</a>
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<ul>
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<li><a href="#orgc3d4a1f">3.1. Computation of the statistical error</a>
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<li><a href="#org7092837">3.1. Computation of the statistical error</a>
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<ul>
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||||
<li><a href="#orgf14af1b">3.1.1. Exercise</a>
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||||
<li><a href="#orgd7c44e9">3.1.1. Exercise</a>
|
||||
<ul>
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||||
<li><a href="#orgd2188aa">3.1.1.1. Solution</a></li>
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||||
<li><a href="#org7408e00">3.1.1.1. Solution</a></li>
|
||||
</ul>
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||||
</li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org3da75b5">3.2. Uniform sampling in the box</a>
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<li><a href="#org05b64e8">3.2. Uniform sampling in the box</a>
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||||
<ul>
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||||
<li><a href="#org0fd4d67">3.2.1. Exercise</a>
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||||
<li><a href="#org5169b03">3.2.1. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#org086f36b">3.2.1.1. Solution</a></li>
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||||
<li><a href="#org57eabbc">3.2.1.1. Solution</a></li>
|
||||
</ul>
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||||
</li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#org1a74b53">3.3. Metropolis sampling with \(\Psi^2\)</a>
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||||
<li><a href="#org4ee2da0">3.3. Metropolis sampling with \(\Psi^2\)</a>
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||||
<ul>
|
||||
<li><a href="#org3e04f82">3.3.1. Optimal step size</a></li>
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||||
<li><a href="#org149f494">3.3.2. Exercise</a>
|
||||
<li><a href="#org521c913">3.3.1. Optimal step size</a></li>
|
||||
<li><a href="#org50b36aa">3.3.2. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#orgfae5a0d">3.3.2.1. Solution</a></li>
|
||||
<li><a href="#org9783fbe">3.3.2.1. Solution</a></li>
|
||||
</ul>
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||||
</li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#orgdff9465">3.4. Generalized Metropolis algorithm</a>
|
||||
<li><a href="#orgf7e192c">3.4. Generalized Metropolis algorithm</a>
|
||||
<ul>
|
||||
<li><a href="#orgf0dbb7c">3.4.1. Gaussian random number generator</a></li>
|
||||
<li><a href="#org4e17623">3.4.2. Exercise 1</a>
|
||||
<li><a href="#org868c86a">3.4.1. Gaussian random number generator</a></li>
|
||||
<li><a href="#org3da25b4">3.4.2. Exercise 1</a>
|
||||
<ul>
|
||||
<li><a href="#org43f8745">3.4.2.1. Solution</a></li>
|
||||
<li><a href="#orgdaf9219">3.4.2.1. Solution</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgb6bef1a">3.4.3. Exercise 2</a>
|
||||
<li><a href="#org5a4e90b">3.4.3. Exercise 2</a>
|
||||
<ul>
|
||||
<li><a href="#org63c7646">3.4.3.1. Solution</a></li>
|
||||
<li><a href="#orgb2b31bc">3.4.3.1. 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>
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||||
<li><a href="#orgbf8947f">4. Diffusion Monte Carlo</a>
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||||
<li><a href="#orgefd778a">4. Diffusion Monte Carlo</a>
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||||
<ul>
|
||||
<li><a href="#orgdbd4d14">4.1. Schrödinger equation in imaginary time</a></li>
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||||
<li><a href="#org2fdf8aa">4.2. Diffusion and branching</a></li>
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||||
<li><a href="#org921cddb">4.3. Importance sampling</a>
|
||||
<li><a href="#org481ff44">4.1. Schrödinger equation in imaginary time</a></li>
|
||||
<li><a href="#orgc28941f">4.2. Diffusion and branching</a></li>
|
||||
<li><a href="#org19f6b36">4.3. Importance sampling</a>
|
||||
<ul>
|
||||
<li><a href="#orgf907fde">4.3.1. Appendix : Details of the Derivation</a></li>
|
||||
<li><a href="#orgd63eaab">4.3.1. Appendix : Details of the Derivation</a></li>
|
||||
</ul>
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||||
</li>
|
||||
<li><a href="#orgbce88fc">4.4. Pure Diffusion Monte Carlo (PDMC)</a></li>
|
||||
<li><a href="#orgc1e3399">4.5. Hydrogen atom</a>
|
||||
<li><a href="#org498bc42">4.4. Pure Diffusion Monte Carlo (PDMC)</a></li>
|
||||
<li><a href="#org82aee19">4.5. Hydrogen atom</a>
|
||||
<ul>
|
||||
<li><a href="#org3d7eb95">4.5.1. Exercise</a>
|
||||
<li><a href="#org28fd5e4">4.5.1. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#org810f81c">4.5.1.1. Solution</a></li>
|
||||
<li><a href="#org89b211a">4.5.1.1. Solution</a></li>
|
||||
</ul>
|
||||
</li>
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||||
</ul>
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||||
</li>
|
||||
<li><a href="#org384931d">4.6. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
|
||||
<li><a href="#org3cb42a6">4.6. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org8a56f38">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
|
||||
<li><a href="#orgcb541e2">6. Schedule</a></li>
|
||||
<li><a href="#orgb2c8cf6">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
|
||||
<li><a href="#org43d0643">6. Schedule</a></li>
|
||||
</ul>
|
||||
</div>
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||||
</div>
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||||
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||||
<div id="outline-container-orgc4e758b" class="outline-2">
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||||
<h2 id="orgc4e758b"><span class="section-number-2">1</span> Introduction</h2>
|
||||
<div id="outline-container-orgf35fcfa" class="outline-2">
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||||
<h2 id="orgf35fcfa"><span class="section-number-2">1</span> Introduction</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
||||
@ -515,8 +515,8 @@ coordinates, etc).
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdba9547" class="outline-3">
|
||||
<h3 id="orgdba9547"><span class="section-number-3">1.1</span> Energy and local energy</h3>
|
||||
<div id="outline-container-orgff4a500" class="outline-3">
|
||||
<h3 id="orgff4a500"><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
|
||||
@ -599,8 +599,8 @@ energy computed over these configurations:
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||||
</div>
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||||
</div>
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||||
|
||||
<div id="outline-container-orge64c748" class="outline-2">
|
||||
<h2 id="orge64c748"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
|
||||
<div id="outline-container-orge5391c1" class="outline-2">
|
||||
<h2 id="orge5391c1"><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
|
||||
@ -629,8 +629,8 @@ To do that, we will compute the local energy and check whether it is constant.
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||||
</p>
|
||||
</div>
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||||
|
||||
<div id="outline-container-org683b2fd" class="outline-3">
|
||||
<h3 id="org683b2fd"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||
<div id="outline-container-orgc557da6" class="outline-3">
|
||||
<h3 id="orgc557da6"><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.
|
||||
@ -657,8 +657,8 @@ to catch the error.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1f3996c" class="outline-4">
|
||||
<h4 id="org1f3996c"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||||
<div id="outline-container-orga4e6b54" class="outline-4">
|
||||
<h4 id="orga4e6b54"><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>
|
||||
@ -703,8 +703,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org01b5503" class="outline-5">
|
||||
<h5 id="org01b5503"><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-org801cff3" class="outline-5">
|
||||
<h5 id="org801cff3"><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>
|
||||
@ -745,8 +745,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9fd3548" class="outline-4">
|
||||
<h4 id="org9fd3548"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-org4677e85" class="outline-4">
|
||||
<h4 id="org4677e85"><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>
|
||||
@ -781,8 +781,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgafc07f4" class="outline-5">
|
||||
<h5 id="orgafc07f4"><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-orgbc257d9" class="outline-5">
|
||||
<h5 id="orgbc257d9"><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>
|
||||
@ -809,8 +809,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4e056ec" class="outline-4">
|
||||
<h4 id="org4e056ec"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||||
<div id="outline-container-orgc8a1a2b" class="outline-4">
|
||||
<h4 id="orgc8a1a2b"><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>
|
||||
@ -891,8 +891,8 @@ Therefore, the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org41a05f4" class="outline-5">
|
||||
<h5 id="org41a05f4"><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-org46a48e1" class="outline-5">
|
||||
<h5 id="org46a48e1"><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>
|
||||
@ -933,8 +933,8 @@ Therefore, the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3447ad1" class="outline-4">
|
||||
<h4 id="org3447ad1"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||||
<div id="outline-container-org49b162e" class="outline-4">
|
||||
<h4 id="org49b162e"><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>
|
||||
@ -993,8 +993,8 @@ are calling is yours.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9062ca6" class="outline-5">
|
||||
<h5 id="org9062ca6"><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-org11b7ba8" class="outline-5">
|
||||
<h5 id="org11b7ba8"><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>
|
||||
@ -1025,8 +1025,8 @@ are calling is yours.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org46514dd" class="outline-4">
|
||||
<h4 id="org46514dd"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||||
<div id="outline-container-orgea20a4d" class="outline-4">
|
||||
<h4 id="orgea20a4d"><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>
|
||||
@ -1036,8 +1036,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org644dc09" class="outline-5">
|
||||
<h5 id="org644dc09"><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-orga8ca1f7" class="outline-5">
|
||||
<h5 id="orga8ca1f7"><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} -
|
||||
@ -1057,8 +1057,8 @@ equal to -0.5 atomic units.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8c99b58" class="outline-3">
|
||||
<h3 id="org8c99b58"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
||||
<div id="outline-container-org1aceedb" class="outline-3">
|
||||
<h3 id="org1aceedb"><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">
|
||||
<p>
|
||||
The program you will write in this section will be written in
|
||||
@ -1089,8 +1089,8 @@ In Fortran, you will need to compile all the source files together:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge8967d0" class="outline-4">
|
||||
<h4 id="orge8967d0"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgfdfeaeb" class="outline-4">
|
||||
<h4 id="orgfdfeaeb"><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>
|
||||
@ -1184,8 +1184,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org93bb898" class="outline-5">
|
||||
<h5 id="org93bb898"><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-org7b8e758" class="outline-5">
|
||||
<h5 id="org7b8e758"><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>
|
||||
@ -1262,8 +1262,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd2d5d22" class="outline-3">
|
||||
<h3 id="orgd2d5d22"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||||
<div id="outline-container-orgb878d1c" class="outline-3">
|
||||
<h3 id="orgb878d1c"><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 \(\mathbf{r}_i\)
|
||||
@ -1293,8 +1293,8 @@ The energy is biased because:
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org9b6bf44" class="outline-4">
|
||||
<h4 id="org9b6bf44"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-orga3c04f6" class="outline-4">
|
||||
<h4 id="orga3c04f6"><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>
|
||||
@ -1365,8 +1365,8 @@ To compile the Fortran and run it:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0346611" class="outline-5">
|
||||
<h5 id="org0346611"><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-org3ff01c5" class="outline-5">
|
||||
<h5 id="org3ff01c5"><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>
|
||||
@ -1483,8 +1483,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org82d4afa" class="outline-3">
|
||||
<h3 id="org82d4afa"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||||
<div id="outline-container-org61a9a5a" class="outline-3">
|
||||
<h3 id="org61a9a5a"><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 \(\Psi\)
|
||||
@ -1511,8 +1511,8 @@ energy can be used as a measure of the quality of a wave function.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org475ea22" class="outline-4">
|
||||
<h4 id="org475ea22"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||||
<div id="outline-container-org3a9cd19" class="outline-4">
|
||||
<h4 id="org3a9cd19"><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>
|
||||
@ -1523,8 +1523,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9072218" class="outline-5">
|
||||
<h5 id="org9072218"><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-orge49d691" class="outline-5">
|
||||
<h5 id="orge49d691"><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}
|
||||
@ -1543,8 +1543,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org350c442" class="outline-4">
|
||||
<h4 id="org350c442"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||||
<div id="outline-container-orgc95b8fc" class="outline-4">
|
||||
<h4 id="orgc95b8fc"><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>
|
||||
@ -1620,8 +1620,8 @@ To compile and run:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgea0025a" class="outline-5">
|
||||
<h5 id="orgea0025a"><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-org0cf5abe" class="outline-5">
|
||||
<h5 id="org0cf5abe"><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>
|
||||
@ -1760,8 +1760,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org95128a7" class="outline-2">
|
||||
<h2 id="org95128a7"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||||
<div id="outline-container-org03c7717" class="outline-2">
|
||||
<h2 id="org03c7717"><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
|
||||
@ -1777,8 +1777,8 @@ interval.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc3d4a1f" class="outline-3">
|
||||
<h3 id="orgc3d4a1f"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||||
<div id="outline-container-org7092837" class="outline-3">
|
||||
<h3 id="org7092837"><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 \(M\)
|
||||
@ -1818,8 +1818,8 @@ And the confidence interval is given by
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf14af1b" class="outline-4">
|
||||
<h4 id="orgf14af1b"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgd7c44e9" class="outline-4">
|
||||
<h4 id="orgd7c44e9"><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>
|
||||
@ -1859,8 +1859,8 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd2188aa" class="outline-5">
|
||||
<h5 id="orgd2188aa"><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-org7408e00" class="outline-5">
|
||||
<h5 id="org7408e00"><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>
|
||||
@ -1921,8 +1921,8 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3da75b5" class="outline-3">
|
||||
<h3 id="org3da75b5"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||||
<div id="outline-container-org05b64e8" class="outline-3">
|
||||
<h3 id="org05b64e8"><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
|
||||
@ -1983,8 +1983,8 @@ compute the statistical error.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0fd4d67" class="outline-4">
|
||||
<h4 id="org0fd4d67"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-org5169b03" class="outline-4">
|
||||
<h4 id="org5169b03"><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>
|
||||
@ -2086,8 +2086,8 @@ well as the index of the current step.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org086f36b" class="outline-5">
|
||||
<h5 id="org086f36b"><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-org57eabbc" class="outline-5">
|
||||
<h5 id="org57eabbc"><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>
|
||||
@ -2193,8 +2193,8 @@ E = -0.48084122147238995 +/- 2.4983775878329355E-003
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1a74b53" class="outline-3">
|
||||
<h3 id="org1a74b53"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||||
<div id="outline-container-org4ee2da0" class="outline-3">
|
||||
<h3 id="org4ee2da0"><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
|
||||
@ -2313,8 +2313,8 @@ All samples should be kept, from both accepted <i>and</i> rejected moves.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org3e04f82" class="outline-4">
|
||||
<h4 id="org3e04f82"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
|
||||
<div id="outline-container-org521c913" class="outline-4">
|
||||
<h4 id="org521c913"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
|
||||
<div class="outline-text-4" id="text-3-3-1">
|
||||
<p>
|
||||
If the box is infinitely small, the ratio will be very close
|
||||
@ -2349,8 +2349,8 @@ the same variable later on to store a time step.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org149f494" class="outline-4">
|
||||
<h4 id="org149f494"><span class="section-number-4">3.3.2</span> Exercise</h4>
|
||||
<div id="outline-container-org50b36aa" class="outline-4">
|
||||
<h4 id="org50b36aa"><span class="section-number-4">3.3.2</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-3-3-2">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -2459,8 +2459,8 @@ Can you observe a reduction in the statistical error?
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfae5a0d" class="outline-5">
|
||||
<h5 id="orgfae5a0d"><span class="section-number-5">3.3.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org9783fbe" class="outline-5">
|
||||
<h5 id="org9783fbe"><span class="section-number-5">3.3.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-3-3-2-1">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -2607,8 +2607,8 @@ A = 0.50762633333333318 +/- 3.4601756760043725E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdff9465" class="outline-3">
|
||||
<h3 id="orgdff9465"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
|
||||
<div id="outline-container-orgf7e192c" class="outline-3">
|
||||
<h3 id="orgf7e192c"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
|
||||
@ -2729,8 +2729,8 @@ The algorithm of the previous exercise is only slighlty modified as:
|
||||
</ol>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf0dbb7c" class="outline-4">
|
||||
<h4 id="orgf0dbb7c"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
|
||||
<div id="outline-container-org868c86a" class="outline-4">
|
||||
<h4 id="org868c86a"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
|
||||
<div class="outline-text-4" id="text-3-4-1">
|
||||
<p>
|
||||
To obtain Gaussian-distributed random numbers, you can apply the
|
||||
@ -2794,8 +2794,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org4e17623" class="outline-4">
|
||||
<h4 id="org4e17623"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
|
||||
<div id="outline-container-org3da25b4" class="outline-4">
|
||||
<h4 id="org3da25b4"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
|
||||
<div class="outline-text-4" id="text-3-4-2">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -2837,8 +2837,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org43f8745" class="outline-5">
|
||||
<h5 id="org43f8745"><span class="section-number-5">3.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgdaf9219" class="outline-5">
|
||||
<h5 id="orgdaf9219"><span class="section-number-5">3.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-3-4-2-1">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -2871,8 +2871,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb6bef1a" class="outline-4">
|
||||
<h4 id="orgb6bef1a"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
|
||||
<div id="outline-container-org5a4e90b" class="outline-4">
|
||||
<h4 id="org5a4e90b"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
|
||||
<div class="outline-text-4" id="text-3-4-3">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -2968,8 +2968,8 @@ Modify the previous program to introduce the drift-diffusion scheme.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org63c7646" class="outline-5">
|
||||
<h5 id="org63c7646"><span class="section-number-5">3.4.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgb2b31bc" class="outline-5">
|
||||
<h5 id="orgb2b31bc"><span class="section-number-5">3.4.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-3-4-3-1">
|
||||
<p>
|
||||
<b>Python</b>
|
||||
@ -3157,12 +3157,12 @@ A = 0.62037333333333333 +/- 4.8970160591451110E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbf8947f" class="outline-2">
|
||||
<h2 id="orgbf8947f"><span class="section-number-2">4</span> Diffusion Monte Carlo   <span class="tag"><span class="solution">solution</span></span></h2>
|
||||
<div id="outline-container-orgefd778a" class="outline-2">
|
||||
<h2 id="orgefd778a"><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-orgdbd4d14" class="outline-3">
|
||||
<h3 id="orgdbd4d14"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
||||
<div id="outline-container-org481ff44" class="outline-3">
|
||||
<h3 id="org481ff44"><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:
|
||||
@ -3230,8 +3230,8 @@ system.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2fdf8aa" class="outline-3">
|
||||
<h3 id="org2fdf8aa"><span class="section-number-3">4.2</span> Diffusion and branching</h3>
|
||||
<div id="outline-container-orgc28941f" class="outline-3">
|
||||
<h3 id="orgc28941f"><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
|
||||
@ -3328,8 +3328,8 @@ Therefore, in both cases, you are dealing with a "Bosonic" ground state.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org921cddb" class="outline-3">
|
||||
<h3 id="org921cddb"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
||||
<div id="outline-container-org19f6b36" class="outline-3">
|
||||
<h3 id="org19f6b36"><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
|
||||
@ -3425,8 +3425,8 @@ energies computed with the trial wave function.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf907fde" class="outline-4">
|
||||
<h4 id="orgf907fde"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4>
|
||||
<div id="outline-container-orgd63eaab" class="outline-4">
|
||||
<h4 id="orgd63eaab"><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>
|
||||
\[
|
||||
@ -3487,8 +3487,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbce88fc" class="outline-3">
|
||||
<h3 id="orgbce88fc"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo (PDMC)</h3>
|
||||
<div id="outline-container-org498bc42" class="outline-3">
|
||||
<h3 id="org498bc42"><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
|
||||
@ -3514,23 +3514,20 @@ The algorithm can be rather easily built on top of your VMC code:
|
||||
</p>
|
||||
|
||||
<ol class="org-ol">
|
||||
<li><p>
|
||||
Compute a new position \(\mathbf{r'} = \mathbf{r}_n +
|
||||
\delta t\, \frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi\)
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Evaluate \(\Psi\) and \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\) at the new position
|
||||
</p></li>
|
||||
<li>Compute the ratio \(A = \frac{T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) P(\mathbf{r}_{n+1})}{T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) P(\mathbf{r}_{n})}\)</li>
|
||||
<li>Start with \(W=1\)</li>
|
||||
<li>Evaluate the local energy at \(\mathbf{r}_{n}\) and accumulate it</li>
|
||||
<li>Compute the weight \(w(\mathbf{r}_n)\)</li>
|
||||
<li>Update \(W\)</li>
|
||||
<li>Compute a new position \(\mathbf{r'} = \mathbf{r}_n +
|
||||
\delta t\, \frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi\)</li>
|
||||
<li>Evaluate \(\Psi(\mathbf{r}')\) and \(\frac{\nabla \Psi(\mathbf{r'})}{\Psi(\mathbf{r'})}\) at the new position</li>
|
||||
<li>Compute the ratio \(A = \frac{T(\mathbf{r}' \rightarrow \mathbf{r}_{n}) P(\mathbf{r}')}{T(\mathbf{r}_{n} \rightarrow \mathbf{r}') P(\mathbf{r}_{n})}\)</li>
|
||||
<li>Draw a uniform random number \(v \in [0,1]\)</li>
|
||||
<li>if \(v \le A\), accept the move : set \(\mathbf{r}_{n+1} = \mathbf{r'}\)</li>
|
||||
<li>else, reject the move : set \(\mathbf{r}_{n+1} = \mathbf{r}_n\)</li>
|
||||
<li>evaluate the local energy at \(\mathbf{r}_{n+1}\)</li>
|
||||
<li>compute the weight \(w(\mathbf{r}_i)\)</li>
|
||||
<li>update \(W\)</li>
|
||||
</ol>
|
||||
|
||||
|
||||
<p>
|
||||
Some comments are needed:
|
||||
</p>
|
||||
@ -3567,13 +3564,13 @@ code, so this is what we will do in the next section.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc1e3399" class="outline-3">
|
||||
<h3 id="orgc1e3399"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
|
||||
<div id="outline-container-org82aee19" class="outline-3">
|
||||
<h3 id="org82aee19"><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-org3d7eb95" class="outline-4">
|
||||
<h4 id="org3d7eb95"><span class="section-number-4">4.5.1</span> Exercise</h4>
|
||||
<div id="outline-container-org28fd5e4" class="outline-4">
|
||||
<h4 id="org28fd5e4"><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>
|
||||
@ -3672,8 +3669,8 @@ energy of H for any value of \(a\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org810f81c" class="outline-5">
|
||||
<h5 id="org810f81c"><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-org89b211a" class="outline-5">
|
||||
<h5 id="org89b211a"><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>
|
||||
@ -3891,8 +3888,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org384931d" class="outline-3">
|
||||
<h3 id="org384931d"><span class="section-number-3">4.6</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
|
||||
<div id="outline-container-org3cb42a6" class="outline-3">
|
||||
<h3 id="org3cb42a6"><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
|
||||
@ -3913,8 +3910,8 @@ the nuclei.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org8a56f38" class="outline-2">
|
||||
<h2 id="org8a56f38"><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-orgb2c8cf6" class="outline-2">
|
||||
<h2 id="orgb2c8cf6"><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>
|
||||
@ -3928,8 +3925,8 @@ the H\(_2\) molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.</li>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcb541e2" class="outline-2">
|
||||
<h2 id="orgcb541e2"><span class="section-number-2">6</span> Schedule</h2>
|
||||
<div id="outline-container-org43d0643" class="outline-2">
|
||||
<h2 id="org43d0643"><span class="section-number-2">6</span> Schedule</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
|
||||
@ -3993,7 +3990,7 @@ 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: 2021-02-02 Tue 16:01</p>
|
||||
<p class="date">Created: 2021-02-02 Tue 16:05</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