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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2021-01-25 Mon 22:54 -->
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<!-- 2021-01-25 Mon 23:23 -->
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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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@ -257,128 +257,135 @@ 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="#orgf7a78e4">1. Introduction</a></li>
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<li><a href="#org5696f8b">2. Numerical evaluation of the energy</a>
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<li><a href="#orga35005a">1. Introduction</a></li>
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<li><a href="#org99b226a">2. Numerical evaluation of the energy</a>
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<ul>
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<li><a href="#org05580cd">2.1. Local energy</a>
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<li><a href="#orgb54c6e8">2.1. Local energy</a>
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<ul>
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||||
<li><a href="#org23947c1">2.1.1. Exercise 1</a></li>
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<li><a href="#orgd624a8b">2.1.2. Exercise 2</a>
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<li><a href="#orgd1fb793">2.1.1. Exercise 1</a></li>
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<li><a href="#org3860cee">2.1.2. Exercise 2</a>
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<ul>
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<li><a href="#org1661138">2.1.2.1. Python</a></li>
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<li><a href="#org50c3aad">2.1.2.2. Python</a></li>
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<li><a href="#org3f11dde">2.1.2.3. Fortran</a></li>
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<li><a href="#org92d265e">2.1.2.4. Fortran</a></li>
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||||
<li><a href="#orgc1f4406">2.1.2.1. Python</a></li>
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<li><a href="#orgec28d8a">2.1.2.2. Python</a></li>
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<li><a href="#org7eaea85">2.1.2.3. Fortran</a></li>
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<li><a href="#orgc042d6f">2.1.2.4. Fortran</a></li>
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</ul>
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</li>
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<li><a href="#orgc68ea75">2.1.3. Exercise 3</a>
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<li><a href="#orga7e8e23">2.1.3. Exercise 3</a>
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<ul>
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<li><a href="#orgaa63204">2.1.3.1. Python</a></li>
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<li><a href="#orge262e44">2.1.3.2. Python</a></li>
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<li><a href="#org0ab4bfb">2.1.3.3. Fortran</a></li>
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<li><a href="#org9a6c1fb">2.1.3.4. Fortran</a></li>
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<li><a href="#orgc8636eb">2.1.3.1. Python</a></li>
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<li><a href="#org4aa281b">2.1.3.2. Python</a></li>
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<li><a href="#org35818d2">2.1.3.3. Fortran</a></li>
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<li><a href="#org6f79896">2.1.3.4. Fortran</a></li>
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</ul>
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</li>
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<li><a href="#org547c9c4">2.1.4. Exercise 4</a>
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<li><a href="#org7169dad">2.1.4. Exercise 4</a>
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<ul>
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<li><a href="#org74c97bf">2.1.4.1. Python</a></li>
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<li><a href="#orgcc44043">2.1.4.2. Python</a></li>
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<li><a href="#orgce2c2e5">2.1.4.3. Fortran</a></li>
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<li><a href="#org941820f">2.1.4.4. Fortran</a></li>
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<li><a href="#org49311f6">2.1.4.1. Python</a></li>
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<li><a href="#org64701c8">2.1.4.2. Python</a></li>
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<li><a href="#orgcf1f8bb">2.1.4.3. Fortran</a></li>
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<li><a href="#orga709c70">2.1.4.4. Fortran</a></li>
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</ul>
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</li>
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<li><a href="#orgbdaff18">2.1.5. Exercise 5</a>
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<li><a href="#orgabff35a">2.1.5. Exercise 5</a>
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<ul>
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<li><a href="#org159140e">2.1.5.1. Python</a></li>
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<li><a href="#org1e4cde2">2.1.5.2. Python</a></li>
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<li><a href="#org02d18b8">2.1.5.3. Fortran</a></li>
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<li><a href="#org63ad259">2.1.5.4. Fortran</a></li>
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||||
<li><a href="#org7d1aba4">2.1.5.1. Python</a></li>
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<li><a href="#orgb2b4bad">2.1.5.2. Python</a></li>
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<li><a href="#org6e7a546">2.1.5.3. Fortran</a></li>
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<li><a href="#org53213b5">2.1.5.4. Fortran</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="#orgdd79233">2.2. Plot of the local energy along the \(x\) axis</a>
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<li><a href="#orgbeab5ec">2.2. Plot of the local energy along the \(x\) axis</a>
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<ul>
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<li><a href="#orgd250551">2.2.1. Exercise</a>
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<li><a href="#org3510925">2.2.1. Exercise</a>
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||||
<ul>
|
||||
<li><a href="#orgb5fae5f">2.2.1.1. Python</a></li>
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||||
<li><a href="#orgb2f1970">2.2.1.2. Python</a></li>
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<li><a href="#orga11ec40">2.2.1.3. Fortran</a></li>
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||||
<li><a href="#org6eafd17">2.2.1.4. Fortran</a></li>
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||||
<li><a href="#orge574bc5">2.2.1.1. Python</a></li>
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<li><a href="#org580bac9">2.2.1.2. Python</a></li>
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<li><a href="#org03b17a4">2.2.1.3. Fortran</a></li>
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<li><a href="#org9a6dc16">2.2.1.4. Fortran</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="#org554689a">2.3. Numerical estimation of the energy</a>
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||||
<li><a href="#orgbf0209f">2.3. Numerical estimation of the energy</a>
|
||||
<ul>
|
||||
<li><a href="#org6369add">2.3.1. Exercise</a>
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||||
<li><a href="#org3958454">2.3.1. Exercise</a>
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||||
<ul>
|
||||
<li><a href="#orgf3840f2">2.3.1.1. Python</a></li>
|
||||
<li><a href="#orgf5e3ecd">2.3.1.2. Python</a></li>
|
||||
<li><a href="#org826f9f4">2.3.1.3. Fortran</a></li>
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||||
<li><a href="#org97c81ba">2.3.1.4. Fortran</a></li>
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||||
<li><a href="#org30bade1">2.3.1.1. Python</a></li>
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||||
<li><a href="#org77b7b0a">2.3.1.2. Python</a></li>
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||||
<li><a href="#orga93a5dd">2.3.1.3. Fortran</a></li>
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||||
<li><a href="#org6dd2871">2.3.1.4. Fortran</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="#org4eb04ef">2.4. Variance of the local energy</a>
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||||
<li><a href="#orgeedbee0">2.4. Variance of the local energy</a>
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||||
<ul>
|
||||
<li><a href="#org9443a40">2.4.1. Exercise (optional)</a></li>
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||||
<li><a href="#org5106671">2.4.2. Exercise</a>
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||||
<li><a href="#org4acda64">2.4.1. Exercise (optional)</a></li>
|
||||
<li><a href="#org6854abf">2.4.2. Exercise</a>
|
||||
<ul>
|
||||
<li><a href="#org5767c40">2.4.2.1. Python</a></li>
|
||||
<li><a href="#org32c72a0">2.4.2.2. Python</a></li>
|
||||
<li><a href="#org1b3f7f4">2.4.2.3. Fortran</a></li>
|
||||
<li><a href="#org7f15713">2.4.2.4. Fortran</a></li>
|
||||
<li><a href="#org0d826af">2.4.2.1. Python</a></li>
|
||||
<li><a href="#org576fa85">2.4.2.2. Python</a></li>
|
||||
<li><a href="#org307c660">2.4.2.3. Fortran</a></li>
|
||||
<li><a href="#org119dad1">2.4.2.4. Fortran</a></li>
|
||||
</ul>
|
||||
</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="#org6414cd7">3. <span class="todo TODO">TODO</span> Variational Monte Carlo</a>
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||||
<li><a href="#org0c2fa9b">3. Variational Monte Carlo</a>
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||||
<ul>
|
||||
<li><a href="#org76f4f47">3.1. <span class="todo TODO">TODO</span> Computation of the statistical error</a>
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||||
<li><a href="#org4e85021">3.1. Computation of the statistical error</a>
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||||
<ul>
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||||
<li><a href="#orgc801982">3.1.1. Exercise</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org8b118ec">3.2. <span class="todo TODO">TODO</span> Uniform sampling in the box</a>
|
||||
<li><a href="#orga3687ff">3.1.1. Exercise</a>
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||||
<ul>
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||||
<li><a href="#org925cb8e">3.2.1. Exercise</a></li>
|
||||
</ul>
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||||
</li>
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||||
<li><a href="#org8c24125">3.3. <span class="todo TODO">TODO</span> Metropolis sampling with \(\Psi^2\)</a>
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||||
<ul>
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||||
<li><a href="#org99bc8b5">3.3.1. Exercise</a></li>
|
||||
</ul>
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||||
</li>
|
||||
<li><a href="#org85cb077">3.4. <span class="todo TODO">TODO</span> Gaussian random number generator</a></li>
|
||||
<li><a href="#org3f7062f">3.5. <span class="todo TODO">TODO</span> Generalized Metropolis algorithm</a>
|
||||
<ul>
|
||||
<li><a href="#orgc5b23ac">3.5.1. Exercise 1</a></li>
|
||||
<li><a href="#org876a231">3.5.2. Exercise 2</a></li>
|
||||
<li><a href="#org568b394">3.1.1.1. Python</a></li>
|
||||
<li><a href="#org3d69b65">3.1.1.2. Python</a></li>
|
||||
<li><a href="#org314b0e7">3.1.1.3. Fortran</a></li>
|
||||
<li><a href="#org0f8bdcf">3.1.1.4. Fortran</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="#org86be009">4. <span class="todo TODO">TODO</span> Diffusion Monte Carlo</a>
|
||||
<li><a href="#orgb0930a1">3.2. <span class="todo TODO">TODO</span> Uniform sampling in the box</a>
|
||||
<ul>
|
||||
<li><a href="#orgbbfe631">4.1. <span class="todo TODO">TODO</span> Hydrogen atom</a>
|
||||
<ul>
|
||||
<li><a href="#org574137c">4.1.1. Exercise</a></li>
|
||||
<li><a href="#org1ecf130">3.2.1. Exercise</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgdb34f3d">4.2. <span class="todo TODO">TODO</span> Dihydrogen</a></li>
|
||||
<li><a href="#org5d0adc4">3.3. <span class="todo TODO">TODO</span> Metropolis sampling with \(\Psi^2\)</a>
|
||||
<ul>
|
||||
<li><a href="#orgb3a648e">3.3.1. Exercise</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgc3ba7b8">5. <span class="todo TODO">TODO</span> <code>[0/1]</code> Last things to do</a></li>
|
||||
<li><a href="#orgb5348ef">3.4. <span class="todo TODO">TODO</span> Gaussian random number generator</a></li>
|
||||
<li><a href="#org932bae5">3.5. <span class="todo TODO">TODO</span> Generalized Metropolis algorithm</a>
|
||||
<ul>
|
||||
<li><a href="#org460f3fe">3.5.1. Exercise 1</a></li>
|
||||
<li><a href="#orgd39a7d7">3.5.2. Exercise 2</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgaf5996c">4. <span class="todo TODO">TODO</span> Diffusion Monte Carlo</a>
|
||||
<ul>
|
||||
<li><a href="#org12ab3c5">4.1. <span class="todo TODO">TODO</span> Hydrogen atom</a>
|
||||
<ul>
|
||||
<li><a href="#org3b3e8ae">4.1.1. Exercise</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgf0f2402">4.2. <span class="todo TODO">TODO</span> Dihydrogen</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org1c017d7">5. <span class="todo TODO">TODO</span> <code>[0/1]</code> Last things to do</a></li>
|
||||
</ul>
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||||
</div>
|
||||
</div>
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||||
|
||||
<div id="outline-container-orgf7a78e4" class="outline-2">
|
||||
<h2 id="orgf7a78e4"><span class="section-number-2">1</span> Introduction</h2>
|
||||
<div id="outline-container-orga35005a" class="outline-2">
|
||||
<h2 id="orga35005a"><span class="section-number-2">1</span> Introduction</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
We propose different exercises to understand quantum Monte Carlo (QMC)
|
||||
@ -413,8 +420,8 @@ coordinates, etc).
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org5696f8b" class="outline-2">
|
||||
<h2 id="org5696f8b"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2>
|
||||
<div id="outline-container-org99b226a" class="outline-2">
|
||||
<h2 id="org99b226a"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
In this section we consider the Hydrogen atom with the following
|
||||
@ -487,18 +494,36 @@ E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
|
||||
\end{eqnarray*}
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org05580cd" class="outline-3">
|
||||
<h3 id="org05580cd"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||
<div id="outline-container-orgb54c6e8" class="outline-3">
|
||||
<h3 id="orgb54c6e8"><span class="section-number-3">2.1</span> Local energy</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
Write all the functions of this section in a single file :
|
||||
<code>hydrogen.py</code> if you use Python, or <code>hydrogen.f90</code> is you use
|
||||
Fortran.
|
||||
</p>
|
||||
|
||||
<div class="note">
|
||||
<ul class="org-ul">
|
||||
<li>When computing a square root in \(\mathbb{R}\), <b>always</b> make sure
|
||||
that the argument of the square root is non-negative.</li>
|
||||
<li>When you divide, <b>always</b> make sure that you will not divide by zero</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
If a <i>floating-point exception</i> can occur, you should make a test
|
||||
to catch the error.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org23947c1" class="outline-4">
|
||||
<h4 id="org23947c1"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||||
<p>
|
||||
#+end<sub>note</sub>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd1fb793" class="outline-4">
|
||||
<h4 id="orgd1fb793"><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>
|
||||
@ -509,8 +534,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd624a8b" class="outline-4">
|
||||
<h4 id="orgd624a8b"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-org3860cee" class="outline-4">
|
||||
<h4 id="org3860cee"><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>
|
||||
@ -529,8 +554,8 @@ and returns the potential.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1661138" class="outline-5">
|
||||
<h5 id="org1661138"><span class="section-number-5">2.1.2.1</span> Python</h5>
|
||||
<div id="outline-container-orgc1f4406" class="outline-5">
|
||||
<h5 id="orgc1f4406"><span class="section-number-5">2.1.2.1</span> Python</h5>
|
||||
<div class="outline-text-5" id="text-2-1-2-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||||
@ -542,21 +567,23 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org50c3aad" class="outline-5">
|
||||
<h5 id="org50c3aad"><span class="section-number-5">2.1.2.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgec28d8a" class="outline-5">
|
||||
<h5 id="orgec28d8a"><span class="section-number-5">2.1.2.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-2-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||||
|
||||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">potential</span>(r):
|
||||
<span style="color: #a020f0;">return</span> -1. / np.sqrt(np.dot(r,r))
|
||||
<span style="color: #a0522d;">distance</span> = np.sqrt(np.dot(r,r))
|
||||
<span style="color: #a020f0;">assert</span> (distance > 0)
|
||||
<span style="color: #a020f0;">return</span> -1. / distance
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3f11dde" class="outline-5">
|
||||
<h5 id="org3f11dde"><span class="section-number-5">2.1.2.3</span> Fortran</h5>
|
||||
<div id="outline-container-org7eaea85" class="outline-5">
|
||||
<h5 id="org7eaea85"><span class="section-number-5">2.1.2.3</span> Fortran</h5>
|
||||
<div class="outline-text-5" id="text-2-1-2-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">potential</span><span style="color: #000000; background-color: #ffffff;">(r)</span>
|
||||
@ -569,14 +596,20 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org92d265e" class="outline-5">
|
||||
<h5 id="org92d265e"><span class="section-number-5">2.1.2.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgc042d6f" class="outline-5">
|
||||
<h5 id="orgc042d6f"><span class="section-number-5">2.1.2.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-2-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">potential</span><span style="color: #000000; background-color: #ffffff;">(r)</span>
|
||||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> r(3)</span>
|
||||
potential = -1.d0 / dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
|
||||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> distance</span>
|
||||
distance = dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
|
||||
<span style="color: #a020f0;">if</span> (distance > 0.d0) <span style="color: #a020f0;">then</span>
|
||||
potential = -1.d0 / dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
|
||||
<span style="color: #a020f0;">else</span>
|
||||
<span style="color: #a020f0;">stop</span> <span style="color: #8b2252;">'potential at r=0.d0 diverges'</span>
|
||||
<span style="color: #a020f0;">end if</span>
|
||||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">potential</span>
|
||||
</pre>
|
||||
</div>
|
||||
@ -584,8 +617,8 @@ and returns the potential.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc68ea75" class="outline-4">
|
||||
<h4 id="orgc68ea75"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||||
<div id="outline-container-orga7e8e23" class="outline-4">
|
||||
<h4 id="orga7e8e23"><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>
|
||||
@ -597,9 +630,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgaa63204" class="outline-5">
|
||||
<h5 id="orgaa63204"><span class="section-number-5">2.1.3.1</span> Python</h5>
|
||||
<div id="outline-container-orgc8636eb" class="outline-5">
|
||||
<h5 id="orgc8636eb"><span class="section-number-5">2.1.3.1</span> Python</h5>
|
||||
<div class="outline-text-5" id="text-2-1-3-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">psi</span>(a, r):
|
||||
@ -609,8 +641,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge262e44" class="outline-5">
|
||||
<h5 id="orge262e44"><span class="section-number-5">2.1.3.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org4aa281b" class="outline-5">
|
||||
<h5 id="org4aa281b"><span class="section-number-5">2.1.3.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-3-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">psi</span>(a, r):
|
||||
@ -620,8 +652,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0ab4bfb" class="outline-5">
|
||||
<h5 id="org0ab4bfb"><span class="section-number-5">2.1.3.3</span> Fortran</h5>
|
||||
<div id="outline-container-org35818d2" class="outline-5">
|
||||
<h5 id="org35818d2"><span class="section-number-5">2.1.3.3</span> Fortran</h5>
|
||||
<div class="outline-text-5" id="text-2-1-3-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">psi</span><span style="color: #000000; background-color: #ffffff;">(a, r)</span>
|
||||
@ -634,8 +666,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9a6c1fb" class="outline-5">
|
||||
<h5 id="org9a6c1fb"><span class="section-number-5">2.1.3.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org6f79896" class="outline-5">
|
||||
<h5 id="org6f79896"><span class="section-number-5">2.1.3.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-3-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">psi</span><span style="color: #000000; background-color: #ffffff;">(a, r)</span>
|
||||
@ -649,8 +681,8 @@ input arguments, and returns a scalar.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org547c9c4" class="outline-4">
|
||||
<h4 id="org547c9c4"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||||
<div id="outline-container-org7169dad" class="outline-4">
|
||||
<h4 id="org7169dad"><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>
|
||||
@ -708,8 +740,8 @@ So the local kinetic energy is
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org74c97bf" class="outline-5">
|
||||
<h5 id="org74c97bf"><span class="section-number-5">2.1.4.1</span> Python</h5>
|
||||
<div id="outline-container-org49311f6" class="outline-5">
|
||||
<h5 id="org49311f6"><span class="section-number-5">2.1.4.1</span> Python</h5>
|
||||
<div class="outline-text-5" id="text-2-1-4-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">kinetic</span>(a,r):
|
||||
@ -719,19 +751,21 @@ So the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcc44043" class="outline-5">
|
||||
<h5 id="orgcc44043"><span class="section-number-5">2.1.4.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org64701c8" class="outline-5">
|
||||
<h5 id="org64701c8"><span class="section-number-5">2.1.4.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-4-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">kinetic</span>(a,r):
|
||||
<span style="color: #a020f0;">return</span> -0.5 * (a**2 - (2.*a)/np.sqrt(np.dot(r,r)))
|
||||
<span style="color: #a0522d;">distance</span> = np.sqrt(np.dot(r,r))
|
||||
<span style="color: #a020f0;">assert</span> (distance > 0.)
|
||||
<span style="color: #a020f0;">return</span> -0.5 * (a**2 - (2.*a)/distance)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgce2c2e5" class="outline-5">
|
||||
<h5 id="orgce2c2e5"><span class="section-number-5">2.1.4.3</span> Fortran</h5>
|
||||
<div id="outline-container-orgcf1f8bb" class="outline-5">
|
||||
<h5 id="orgcf1f8bb"><span class="section-number-5">2.1.4.3</span> Fortran</h5>
|
||||
<div class="outline-text-5" id="text-2-1-4-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">kinetic</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||||
@ -744,15 +778,20 @@ So the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org941820f" class="outline-5">
|
||||
<h5 id="org941820f"><span class="section-number-5">2.1.4.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orga709c70" class="outline-5">
|
||||
<h5 id="orga709c70"><span class="section-number-5">2.1.4.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-4-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">kinetic</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||||
kinetic = -0.5d0 * (a*a - (2.d0*a) / <span style="color: #a020f0;">&</span>
|
||||
dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ) )
|
||||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> distance</span>
|
||||
distance = dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
|
||||
<span style="color: #a020f0;">if</span> (distance > 0.d0) <span style="color: #a020f0;">then</span>
|
||||
kinetic = -0.5d0 * (a*a - (2.d0*a) / distance)
|
||||
<span style="color: #a020f0;">else</span>
|
||||
<span style="color: #a020f0;">stop</span> <span style="color: #8b2252;">'kinetic energy diverges at r=0'</span>
|
||||
<span style="color: #a020f0;">end if</span>
|
||||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">kinetic</span>
|
||||
</pre>
|
||||
</div>
|
||||
@ -760,8 +799,8 @@ So the local kinetic energy is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbdaff18" class="outline-4">
|
||||
<h4 id="orgbdaff18"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||||
<div id="outline-container-orgabff35a" class="outline-4">
|
||||
<h4 id="orgabff35a"><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>
|
||||
@ -781,8 +820,8 @@ local kinetic energy.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org159140e" class="outline-5">
|
||||
<h5 id="org159140e"><span class="section-number-5">2.1.5.1</span> Python</h5>
|
||||
<div id="outline-container-org7d1aba4" class="outline-5">
|
||||
<h5 id="org7d1aba4"><span class="section-number-5">2.1.5.1</span> Python</h5>
|
||||
<div class="outline-text-5" id="text-2-1-5-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">e_loc</span>(a,r):
|
||||
@ -792,8 +831,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1e4cde2" class="outline-5">
|
||||
<h5 id="org1e4cde2"><span class="section-number-5">2.1.5.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-orgb2b4bad" class="outline-5">
|
||||
<h5 id="orgb2b4bad"><span class="section-number-5">2.1.5.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-5-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">e_loc</span>(a,r):
|
||||
@ -803,8 +842,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org02d18b8" class="outline-5">
|
||||
<h5 id="org02d18b8"><span class="section-number-5">2.1.5.3</span> Fortran</h5>
|
||||
<div id="outline-container-org6e7a546" class="outline-5">
|
||||
<h5 id="org6e7a546"><span class="section-number-5">2.1.5.3</span> Fortran</h5>
|
||||
<div class="outline-text-5" id="text-2-1-5-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">e_loc</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||||
@ -817,8 +856,8 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org63ad259" class="outline-5">
|
||||
<h5 id="org63ad259"><span class="section-number-5">2.1.5.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org53213b5" class="outline-5">
|
||||
<h5 id="org53213b5"><span class="section-number-5">2.1.5.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-1-5-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">e_loc</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||||
@ -834,13 +873,20 @@ local kinetic energy.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdd79233" class="outline-3">
|
||||
<h3 id="orgdd79233"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
||||
<div id="outline-container-orgbeab5ec" class="outline-3">
|
||||
<h3 id="orgbeab5ec"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<div class="note">
|
||||
<p>
|
||||
The potential and the kinetic energy both diverge at \(r=0\), so we
|
||||
choose a grid which does not contain the origin.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd250551" class="outline-4">
|
||||
<h4 id="orgd250551"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-org3510925" class="outline-4">
|
||||
<h4 id="org3510925"><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>
|
||||
@ -854,8 +900,8 @@ Gnuplot to plot the files.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb5fae5f" class="outline-5">
|
||||
<h5 id="orgb5fae5f"><span class="section-number-5">2.2.1.1</span> Python</h5>
|
||||
<div id="outline-container-orge574bc5" class="outline-5">
|
||||
<h5 id="orge574bc5"><span class="section-number-5">2.2.1.1</span> Python</h5>
|
||||
<div class="outline-text-5" id="text-2-2-1-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||||
@ -876,8 +922,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb2f1970" class="outline-5">
|
||||
<h5 id="orgb2f1970"><span class="section-number-5">2.2.1.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org580bac9" class="outline-5">
|
||||
<h5 id="org580bac9"><span class="section-number-5">2.2.1.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-2-1-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||||
@ -906,8 +952,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga11ec40" class="outline-5">
|
||||
<h5 id="orga11ec40"><span class="section-number-5">2.2.1.3</span> Fortran</h5>
|
||||
<div id="outline-container-org03b17a4" class="outline-5">
|
||||
<h5 id="org03b17a4"><span class="section-number-5">2.2.1.3</span> Fortran</h5>
|
||||
<div class="outline-text-5" id="text-2-2-1-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">plot</span>
|
||||
@ -957,8 +1003,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6eafd17" class="outline-5">
|
||||
<h5 id="org6eafd17"><span class="section-number-5">2.2.1.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org9a6dc16" class="outline-5">
|
||||
<h5 id="org9a6dc16"><span class="section-number-5">2.2.1.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-2-1-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">plot</span>
|
||||
@ -1029,8 +1075,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org554689a" class="outline-3">
|
||||
<h3 id="org554689a"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||||
<div id="outline-container-orgbf0209f" class="outline-3">
|
||||
<h3 id="orgbf0209f"><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 \(\mathbf{r}_i\)
|
||||
@ -1060,8 +1106,8 @@ The energy is biased because:
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org6369add" class="outline-4">
|
||||
<h4 id="org6369add"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-org3958454" class="outline-4">
|
||||
<h4 id="org3958454"><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>
|
||||
@ -1073,8 +1119,8 @@ Compute a numerical estimate of the energy in a grid of
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf3840f2" class="outline-5">
|
||||
<h5 id="orgf3840f2"><span class="section-number-5">2.3.1.1</span> Python</h5>
|
||||
<div id="outline-container-org30bade1" class="outline-5">
|
||||
<h5 id="org30bade1"><span class="section-number-5">2.3.1.1</span> Python</h5>
|
||||
<div class="outline-text-5" id="text-2-3-1-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||||
@ -1094,8 +1140,8 @@ Compute a numerical estimate of the energy in a grid of
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf5e3ecd" class="outline-5">
|
||||
<h5 id="orgf5e3ecd"><span class="section-number-5">2.3.1.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org77b7b0a" class="outline-5">
|
||||
<h5 id="org77b7b0a"><span class="section-number-5">2.3.1.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-3-1-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||||
@ -1127,8 +1173,8 @@ Compute a numerical estimate of the energy in a grid of
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org826f9f4" class="outline-5">
|
||||
<h5 id="org826f9f4"><span class="section-number-5">2.3.1.3</span> Fortran</h5>
|
||||
<div id="outline-container-orga93a5dd" class="outline-5">
|
||||
<h5 id="orga93a5dd"><span class="section-number-5">2.3.1.3</span> Fortran</h5>
|
||||
<div class="outline-text-5" id="text-2-3-1-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">energy_hydrogen</span>
|
||||
@ -1165,8 +1211,8 @@ To compile the Fortran and run it:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org97c81ba" class="outline-5">
|
||||
<h5 id="org97c81ba"><span class="section-number-5">2.3.1.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org6dd2871" class="outline-5">
|
||||
<h5 id="org6dd2871"><span class="section-number-5">2.3.1.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-3-1-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">energy_hydrogen</span>
|
||||
@ -1234,8 +1280,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4eb04ef" class="outline-3">
|
||||
<h3 id="org4eb04ef"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||||
<div id="outline-container-orgeedbee0" class="outline-3">
|
||||
<h3 id="orgeedbee0"><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 \(\Psi\)
|
||||
@ -1262,8 +1308,8 @@ energy can be used as a measure of the quality of a wave function.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9443a40" class="outline-4">
|
||||
<h4 id="org9443a40"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||||
<div id="outline-container-org4acda64" class="outline-4">
|
||||
<h4 id="org4acda64"><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>
|
||||
@ -1275,8 +1321,8 @@ Prove that :
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5106671" class="outline-4">
|
||||
<h4 id="org5106671"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||||
<div id="outline-container-org6854abf" class="outline-4">
|
||||
<h4 id="org6854abf"><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>
|
||||
@ -1290,8 +1336,8 @@ in a grid of \(50\times50\times50\) points in the range
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5767c40" class="outline-5">
|
||||
<h5 id="org5767c40"><span class="section-number-5">2.4.2.1</span> Python</h5>
|
||||
<div id="outline-container-org0d826af" class="outline-5">
|
||||
<h5 id="org0d826af"><span class="section-number-5">2.4.2.1</span> Python</h5>
|
||||
<div class="outline-text-5" id="text-2-4-2-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||||
@ -1310,8 +1356,8 @@ in a grid of \(50\times50\times50\) points in the range
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org32c72a0" class="outline-5">
|
||||
<h5 id="org32c72a0"><span class="section-number-5">2.4.2.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org576fa85" class="outline-5">
|
||||
<h5 id="org576fa85"><span class="section-number-5">2.4.2.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-4-2-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||||
@ -1347,8 +1393,8 @@ in a grid of \(50\times50\times50\) points in the range
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1b3f7f4" class="outline-5">
|
||||
<h5 id="org1b3f7f4"><span class="section-number-5">2.4.2.3</span> Fortran</h5>
|
||||
<div id="outline-container-org307c660" class="outline-5">
|
||||
<h5 id="org307c660"><span class="section-number-5">2.4.2.3</span> Fortran</h5>
|
||||
<div class="outline-text-5" id="text-2-4-2-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">variance_hydrogen</span>
|
||||
@ -1390,8 +1436,8 @@ To compile and run:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7f15713" class="outline-5">
|
||||
<h5 id="org7f15713"><span class="section-number-5">2.4.2.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div id="outline-container-org119dad1" class="outline-5">
|
||||
<h5 id="org119dad1"><span class="section-number-5">2.4.2.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-2-4-2-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">variance_hydrogen</span>
|
||||
@ -1467,8 +1513,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.806881
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org6414cd7" class="outline-2">
|
||||
<h2 id="org6414cd7"><span class="section-number-2">3</span> <span class="todo TODO">TODO</span> Variational Monte Carlo</h2>
|
||||
<div id="outline-container-org0c2fa9b" class="outline-2">
|
||||
<h2 id="org0c2fa9b"><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
|
||||
@ -1484,14 +1530,14 @@ interval.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org76f4f47" class="outline-3">
|
||||
<h3 id="org76f4f47"><span class="section-number-3">3.1</span> <span class="todo TODO">TODO</span> Computation of the statistical error</h3>
|
||||
<div id="outline-container-org4e85021" class="outline-3">
|
||||
<h3 id="org4e85021"><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 \(M\)
|
||||
independent Monte Carlo calculations. You will obtain \(M\) different
|
||||
estimates of the energy, which are expected to have a Gaussian
|
||||
distribution by the central limit theorem.
|
||||
distribution according to the <a href="https://en.wikipedia.org/wiki/Central_limit_theorem">Central Limit Theorem</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1525,8 +1571,8 @@ And the confidence interval is given by
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc801982" class="outline-4">
|
||||
<h4 id="orgc801982"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||||
<div id="outline-container-orga3687ff" class="outline-4">
|
||||
<h4 id="orga3687ff"><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>
|
||||
@ -1534,25 +1580,61 @@ Write a function returning the average and statistical error of an
|
||||
input array.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
<b>Python</b>
|
||||
</p>
|
||||
<div id="outline-container-org568b394" class="outline-5">
|
||||
<h5 id="org568b394"><span class="section-number-5">3.1.1.1</span> Python</h5>
|
||||
<div class="outline-text-5" id="text-3-1-1-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">from</span> math <span style="color: #a020f0;">import</span> sqrt
|
||||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">ave_error</span>(arr):
|
||||
#<span style="color: #b22222;">TODO</span>
|
||||
<span style="color: #a020f0;">return</span> (average, error)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3d69b65" class="outline-5">
|
||||
<h5 id="org3d69b65"><span class="section-number-5">3.1.1.2</span> Python   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-3-1-1-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-python"><span style="color: #a020f0;">from</span> math <span style="color: #a020f0;">import</span> sqrt
|
||||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">ave_error</span>(arr):
|
||||
<span style="color: #a0522d;">M</span> = <span style="color: #483d8b;">len</span>(arr)
|
||||
<span style="color: #a020f0;">assert</span> (M>1)
|
||||
<span style="color: #a0522d;">average</span> = <span style="color: #483d8b;">sum</span>(arr)/M
|
||||
<span style="color: #a0522d;">variance</span> = 1./(M-1) * <span style="color: #483d8b;">sum</span>( [ (x - average)**2 <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> arr ] )
|
||||
<span style="color: #a020f0;">return</span> (average, sqrt(variance/M))
|
||||
<span style="color: #a020f0;">assert</span>(M>0)
|
||||
<span style="color: #a020f0;">if</span> M == 1:
|
||||
<span style="color: #a020f0;">return</span> (arr[0], 0.)
|
||||
<span style="color: #a020f0;">else</span>:
|
||||
<span style="color: #a0522d;">average</span> = <span style="color: #483d8b;">sum</span>(arr)/M
|
||||
<span style="color: #a0522d;">variance</span> = 1./(M-1) * <span style="color: #483d8b;">sum</span>( [ (x - average)**2 <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> arr ] )
|
||||
<span style="color: #a0522d;">error</span> = sqrt(variance/M)
|
||||
<span style="color: #a020f0;">return</span> (average, error)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
<b>Fortran</b>
|
||||
</p>
|
||||
<div id="outline-container-org314b0e7" class="outline-5">
|
||||
<h5 id="org314b0e7"><span class="section-number-5">3.1.1.3</span> Fortran</h5>
|
||||
<div class="outline-text-5" id="text-3-1-1-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">ave_error</span>(x,n,ave,err)
|
||||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||||
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> n </span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> x(n) </span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> ave, err</span>
|
||||
! <span style="color: #b22222;">TODO</span>
|
||||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">ave_error</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0f8bdcf" class="outline-5">
|
||||
<h5 id="org0f8bdcf"><span class="section-number-5">3.1.1.4</span> Fortran   <span class="tag"><span class="solution">solution</span></span></h5>
|
||||
<div class="outline-text-5" id="text-3-1-1-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">ave_error</span>(x,n,ave,err)
|
||||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||||
@ -1560,7 +1642,9 @@ input array.
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> x(n) </span>
|
||||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> ave, err</span>
|
||||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> variance</span>
|
||||
<span style="color: #a020f0;">if</span> (n == 1) <span style="color: #a020f0;">then</span>
|
||||
<span style="color: #a020f0;">if</span> (n < 1) <span style="color: #a020f0;">then</span>
|
||||
<span style="color: #a020f0;">stop</span> <span style="color: #8b2252;">'n<1 in ave_error'</span>
|
||||
<span style="color: #a020f0;">else if</span> (n == 1) <span style="color: #a020f0;">then</span>
|
||||
ave = x(1)
|
||||
err = 0.d0
|
||||
<span style="color: #a020f0;">else</span>
|
||||
@ -1574,9 +1658,10 @@ input array.
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8b118ec" class="outline-3">
|
||||
<h3 id="org8b118ec"><span class="section-number-3">3.2</span> <span class="todo TODO">TODO</span> Uniform sampling in the box</h3>
|
||||
<div id="outline-container-orgb0930a1" class="outline-3">
|
||||
<h3 id="orgb0930a1"><span class="section-number-3">3.2</span> <span class="todo TODO">TODO</span> Uniform sampling in the box</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
We will now do our first Monte Carlo calculation to compute the
|
||||
@ -1610,8 +1695,8 @@ statistical error.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org925cb8e" class="outline-4">
|
||||
<h4 id="org925cb8e"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||||
<div id="outline-container-org1ecf130" class="outline-4">
|
||||
<h4 id="org1ecf130"><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>
|
||||
@ -1721,8 +1806,8 @@ E = -0.49588321986667677 +/- 7.1758863546737969E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8c24125" class="outline-3">
|
||||
<h3 id="org8c24125"><span class="section-number-3">3.3</span> <span class="todo TODO">TODO</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||||
<div id="outline-container-org5d0adc4" class="outline-3">
|
||||
<h3 id="org5d0adc4"><span class="section-number-3">3.3</span> <span class="todo TODO">TODO</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
|
||||
@ -1809,8 +1894,8 @@ step such that the acceptance rate is close to 0.5 is a good compromise.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org99bc8b5" class="outline-4">
|
||||
<h4 id="org99bc8b5"><span class="section-number-4">3.3.1</span> Exercise</h4>
|
||||
<div id="outline-container-orgb3a648e" class="outline-4">
|
||||
<h4 id="orgb3a648e"><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>
|
||||
@ -1940,8 +2025,8 @@ A = 0.51737800000000000 +/- 4.1827406733181444E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org85cb077" class="outline-3">
|
||||
<h3 id="org85cb077"><span class="section-number-3">3.4</span> <span class="todo TODO">TODO</span> Gaussian random number generator</h3>
|
||||
<div id="outline-container-orgb5348ef" class="outline-3">
|
||||
<h3 id="orgb5348ef"><span class="section-number-3">3.4</span> <span class="todo TODO">TODO</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
|
||||
@ -1995,8 +2080,8 @@ following sections.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3f7062f" class="outline-3">
|
||||
<h3 id="org3f7062f"><span class="section-number-3">3.5</span> <span class="todo TODO">TODO</span> Generalized Metropolis algorithm</h3>
|
||||
<div id="outline-container-org932bae5" class="outline-3">
|
||||
<h3 id="org932bae5"><span class="section-number-3">3.5</span> <span class="todo TODO">TODO</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.
|
||||
@ -2095,8 +2180,8 @@ The transition probability becomes:
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgc5b23ac" class="outline-4">
|
||||
<h4 id="orgc5b23ac"><span class="section-number-4">3.5.1</span> Exercise 1</h4>
|
||||
<div id="outline-container-org460f3fe" class="outline-4">
|
||||
<h4 id="org460f3fe"><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>
|
||||
@ -2132,8 +2217,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org876a231" class="outline-4">
|
||||
<h4 id="org876a231"><span class="section-number-4">3.5.2</span> Exercise 2</h4>
|
||||
<div id="outline-container-orgd39a7d7" class="outline-4">
|
||||
<h4 id="orgd39a7d7"><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>
|
||||
@ -2285,17 +2370,17 @@ A = 0.78861366666666655 +/- 3.5096729498002445E-004
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org86be009" class="outline-2">
|
||||
<h2 id="org86be009"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> Diffusion Monte Carlo</h2>
|
||||
<div id="outline-container-orgaf5996c" class="outline-2">
|
||||
<h2 id="orgaf5996c"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> Diffusion Monte Carlo</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbbfe631" class="outline-3">
|
||||
<h3 id="orgbbfe631"><span class="section-number-3">4.1</span> <span class="todo TODO">TODO</span> Hydrogen atom</h3>
|
||||
<div id="outline-container-org12ab3c5" class="outline-3">
|
||||
<h3 id="org12ab3c5"><span class="section-number-3">4.1</span> <span class="todo TODO">TODO</span> Hydrogen atom</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
</div>
|
||||
<div id="outline-container-org574137c" class="outline-4">
|
||||
<h4 id="org574137c"><span class="section-number-4">4.1.1</span> Exercise</h4>
|
||||
<div id="outline-container-org3b3e8ae" class="outline-4">
|
||||
<h4 id="org3b3e8ae"><span class="section-number-4">4.1.1</span> Exercise</h4>
|
||||
<div class="outline-text-4" id="text-4-1-1">
|
||||
<div class="exercise">
|
||||
<p>
|
||||
@ -2453,8 +2538,8 @@ A = 0.78861366666666655 +/- 3.5096729498002445E-004
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgdb34f3d" class="outline-3">
|
||||
<h3 id="orgdb34f3d"><span class="section-number-3">4.2</span> <span class="todo TODO">TODO</span> Dihydrogen</h3>
|
||||
<div id="outline-container-orgf0f2402" class="outline-3">
|
||||
<h3 id="orgf0f2402"><span class="section-number-3">4.2</span> <span class="todo TODO">TODO</span> Dihydrogen</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the
|
||||
@ -2475,8 +2560,8 @@ the nuclei.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgc3ba7b8" class="outline-2">
|
||||
<h2 id="orgc3ba7b8"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/1]</code> Last things to do</h2>
|
||||
<div id="outline-container-org1c017d7" class="outline-2">
|
||||
<h2 id="org1c017d7"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/1]</code> Last things to do</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<ul class="org-ul">
|
||||
<li class="off"><code>[ ]</code> Prepare 4 questions for the exam: multiple-choice questions
|
||||
@ -2491,7 +2576,7 @@ the H\(_2\) molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.</li>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
||||
<p class="date">Created: 2021-01-25 Mon 22:54</p>
|
||||
<p class="date">Created: 2021-01-25 Mon 23:23</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