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<head>
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Quantum Monte Carlo</title>
@ -329,75 +329,75 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org33fbe61">1. Introduction</a>
<li><a href="#org5ef0488">1. Introduction</a>
<ul>
<li><a href="#org38ead28">1.1. Energy and local energy</a></li>
<li><a href="#org900f1d4">1.1. Energy and local energy</a></li>
</ul>
</li>
<li><a href="#org0472c34">2. Numerical evaluation of the energy of the hydrogen atom</a>
<li><a href="#orgeb42fdf">2. Numerical evaluation of the energy of the hydrogen atom</a>
<ul>
<li><a href="#org16d7c63">2.1. Local energy</a>
<li><a href="#org4a6f916">2.1. Local energy</a>
<ul>
<li><a href="#orgf4fd067">2.1.1. Exercise 1</a></li>
<li><a href="#org513eaf4">2.1.2. Exercise 2</a></li>
<li><a href="#org6a7eab4">2.1.3. Exercise 3</a></li>
<li><a href="#orga7d4d97">2.1.4. Exercise 4</a></li>
<li><a href="#org9c9442f">2.1.5. Exercise 5</a></li>
<li><a href="#orgd3ba702">2.1.1. Exercise 1</a></li>
<li><a href="#org0939e15">2.1.2. Exercise 2</a></li>
<li><a href="#org8603375">2.1.3. Exercise 3</a></li>
<li><a href="#org4d5cd01">2.1.4. Exercise 4</a></li>
<li><a href="#org3af2b6e">2.1.5. Exercise 5</a></li>
</ul>
</li>
<li><a href="#orgec27611">2.2. Plot of the local energy along the \(x\) axis</a>
<li><a href="#org5ab9959">2.2. Plot of the local energy along the \(x\) axis</a>
<ul>
<li><a href="#orgd135288">2.2.1. Exercise</a></li>
<li><a href="#orga7ce1d2">2.2.1. Exercise</a></li>
</ul>
</li>
<li><a href="#orgba4342b">2.3. Numerical estimation of the energy</a>
<li><a href="#orgd0d619f">2.3. Numerical estimation of the energy</a>
<ul>
<li><a href="#org9550bb5">2.3.1. Exercise</a></li>
<li><a href="#org1229a69">2.3.1. Exercise</a></li>
</ul>
</li>
<li><a href="#orgf5b4c7e">2.4. Variance of the local energy</a>
<li><a href="#orgad1ee8c">2.4. Variance of the local energy</a>
<ul>
<li><a href="#org8f630a1">2.4.1. Exercise (optional)</a></li>
<li><a href="#org262ec70">2.4.2. Exercise</a></li>
<li><a href="#org0afc4b1">2.4.1. Exercise (optional)</a></li>
<li><a href="#org10f6a19">2.4.2. Exercise</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org5f1ee99">3. Variational Monte Carlo</a>
<li><a href="#org0de2b52">3. Variational Monte Carlo</a>
<ul>
<li><a href="#org352609f">3.1. Computation of the statistical error</a>
<li><a href="#org4abd3b8">3.1. Computation of the statistical error</a>
<ul>
<li><a href="#orge016510">3.1.1. Exercise</a></li>
<li><a href="#org5a1c95b">3.1.1. Exercise</a></li>
</ul>
</li>
<li><a href="#org0db1d68">3.2. Uniform sampling in the box</a>
<li><a href="#org8a01ad5">3.2. Uniform sampling in the box</a>
<ul>
<li><a href="#org22b0b1c">3.2.1. Exercise</a></li>
<li><a href="#orgae0d75a">3.2.1. Exercise</a></li>
</ul>
</li>
<li><a href="#orgbbd4395">3.3. Metropolis sampling with \(\Psi^2\)</a>
<li><a href="#org23b5713">3.3. Metropolis sampling with \(\Psi^2\)</a>
<ul>
<li><a href="#org1ce147d">3.3.1. Optimal step size</a></li>
<li><a href="#orgba66664">3.3.2. Exercise</a></li>
<li><a href="#org06c5905">3.3.1. Optimal step size</a></li>
<li><a href="#org654010b">3.3.2. Exercise</a></li>
</ul>
</li>
<li><a href="#org9850980">3.4. Generalized Metropolis algorithm</a>
<li><a href="#org215f308">3.4. Generalized Metropolis algorithm</a>
<ul>
<li><a href="#org0bd3830">3.4.1. Gaussian random number generator</a></li>
<li><a href="#org0d8efaf">3.4.2. Exercise 1</a></li>
<li><a href="#orgce8adcc">3.4.3. Exercise 2</a></li>
<li><a href="#org31cb975">3.4.1. Gaussian random number generator</a></li>
<li><a href="#org48820db">3.4.2. Exercise 1</a></li>
<li><a href="#org1ce793c">3.4.3. Exercise 2</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org3818d44">4. Project</a></li>
<li><a href="#org03cad4e">5. Acknowledgments</a></li>
<li><a href="#orgc4c9346">4. Project</a></li>
<li><a href="#org79362c8">5. Acknowledgments</a></li>
</ul>
</div>
</div>
<div id="outline-container-org33fbe61" class="outline-2">
<h2 id="org33fbe61"><span class="section-number-2">1</span> Introduction</h2>
<div id="outline-container-org5ef0488" class="outline-2">
<h2 id="org5ef0488"><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
@ -437,8 +437,8 @@ coordinates, etc).
</p>
</div>
<div id="outline-container-org38ead28" class="outline-3">
<h3 id="org38ead28"><span class="section-number-3">1.1</span> Energy and local energy</h3>
<div id="outline-container-org900f1d4" class="outline-3">
<h3 id="org900f1d4"><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
@ -521,8 +521,8 @@ energy computed over these configurations:
</div>
</div>
<div id="outline-container-org0472c34" class="outline-2">
<h2 id="org0472c34"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
<div id="outline-container-orgeb42fdf" class="outline-2">
<h2 id="orgeb42fdf"><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
@ -551,8 +551,8 @@ To do that, we will compute the local energy and check whether it is constant.
</p>
</div>
<div id="outline-container-org16d7c63" class="outline-3">
<h3 id="org16d7c63"><span class="section-number-3">2.1</span> Local energy</h3>
<div id="outline-container-org4a6f916" class="outline-3">
<h3 id="org4a6f916"><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.
@ -579,8 +579,8 @@ to catch the error.
</div>
</div>
<div id="outline-container-orgf4fd067" class="outline-4">
<h4 id="orgf4fd067"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
<div id="outline-container-orgd3ba702" class="outline-4">
<h4 id="orgd3ba702"><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>
@ -626,8 +626,8 @@ and returns the potential.
</div>
</div>
<div id="outline-container-org513eaf4" class="outline-4">
<h4 id="org513eaf4"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
<div id="outline-container-org0939e15" class="outline-4">
<h4 id="org0939e15"><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>
@ -663,8 +663,8 @@ input arguments, and returns a scalar.
</div>
</div>
<div id="outline-container-org6a7eab4" class="outline-4">
<h4 id="org6a7eab4"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
<div id="outline-container-org8603375" class="outline-4">
<h4 id="org8603375"><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>
@ -746,8 +746,8 @@ Therefore, the local kinetic energy is
</div>
</div>
<div id="outline-container-orga7d4d97" class="outline-4">
<h4 id="orga7d4d97"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
<div id="outline-container-org4d5cd01" class="outline-4">
<h4 id="org4d5cd01"><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>
@ -807,8 +807,8 @@ are calling is yours.
</div>
</div>
<div id="outline-container-org9c9442f" class="outline-4">
<h4 id="org9c9442f"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
<div id="outline-container-org3af2b6e" class="outline-4">
<h4 id="org3af2b6e"><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>
@ -820,8 +820,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
</div>
</div>
<div id="outline-container-orgec27611" class="outline-3">
<h3 id="orgec27611"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
<div id="outline-container-org5ab9959" class="outline-3">
<h3 id="org5ab9959"><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
@ -852,8 +852,8 @@ In Fortran, you will need to compile all the source files together:
</div>
</div>
<div id="outline-container-orgd135288" class="outline-4">
<h4 id="orgd135288"><span class="section-number-4">2.2.1</span> Exercise</h4>
<div id="outline-container-orga7ce1d2" class="outline-4">
<h4 id="orga7ce1d2"><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>
@ -949,8 +949,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
</div>
</div>
<div id="outline-container-orgba4342b" class="outline-3">
<h3 id="orgba4342b"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
<div id="outline-container-orgd0d619f" class="outline-3">
<h3 id="orgd0d619f"><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 \(\mathbf{r}_i\)
@ -980,8 +980,8 @@ The energy is biased because:
</div>
<div id="outline-container-org9550bb5" class="outline-4">
<h4 id="org9550bb5"><span class="section-number-4">2.3.1</span> Exercise</h4>
<div id="outline-container-org1229a69" class="outline-4">
<h4 id="org1229a69"><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>
@ -1054,8 +1054,8 @@ To compile the Fortran and run it:
</div>
</div>
<div id="outline-container-orgf5b4c7e" class="outline-3">
<h3 id="orgf5b4c7e"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
<div id="outline-container-orgad1ee8c" class="outline-3">
<h3 id="orgad1ee8c"><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 \(\Psi\)
@ -1082,8 +1082,8 @@ energy can be used as a measure of the quality of a wave function.
</p>
</div>
<div id="outline-container-org8f630a1" class="outline-4">
<h4 id="org8f630a1"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
<div id="outline-container-org0afc4b1" class="outline-4">
<h4 id="org0afc4b1"><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>
@ -1094,8 +1094,8 @@ Prove that :
</div>
</div>
</div>
<div id="outline-container-org262ec70" class="outline-4">
<h4 id="org262ec70"><span class="section-number-4">2.4.2</span> Exercise</h4>
<div id="outline-container-org10f6a19" class="outline-4">
<h4 id="org10f6a19"><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>
@ -1174,8 +1174,8 @@ To compile and run:
</div>
</div>
<div id="outline-container-org5f1ee99" class="outline-2">
<h2 id="org5f1ee99"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
<div id="outline-container-org0de2b52" class="outline-2">
<h2 id="org0de2b52"><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
@ -1191,8 +1191,8 @@ interval.
</p>
</div>
<div id="outline-container-org352609f" class="outline-3">
<h3 id="org352609f"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
<div id="outline-container-org4abd3b8" class="outline-3">
<h3 id="org4abd3b8"><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 \(M\)
@ -1232,8 +1232,8 @@ And the confidence interval is given by
</p>
</div>
<div id="outline-container-orge016510" class="outline-4">
<h4 id="orge016510"><span class="section-number-4">3.1.1</span> Exercise</h4>
<div id="outline-container-org5a1c95b" class="outline-4">
<h4 id="org5a1c95b"><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>
@ -1275,8 +1275,8 @@ input array.
</div>
</div>
<div id="outline-container-org0db1d68" class="outline-3">
<h3 id="org0db1d68"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
<div id="outline-container-org8a01ad5" class="outline-3">
<h3 id="org8a01ad5"><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
@ -1337,8 +1337,8 @@ compute the statistical error.
</p>
</div>
<div id="outline-container-org22b0b1c" class="outline-4">
<h4 id="org22b0b1c"><span class="section-number-4">3.2.1</span> Exercise</h4>
<div id="outline-container-orgae0d75a" class="outline-4">
<h4 id="orgae0d75a"><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>
@ -1442,8 +1442,8 @@ well as the index of the current step.
</div>
</div>
<div id="outline-container-orgbbd4395" class="outline-3">
<h3 id="orgbbd4395"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
<div id="outline-container-org23b5713" class="outline-3">
<h3 id="org23b5713"><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
@ -1562,8 +1562,8 @@ All samples should be kept, from both accepted <i>and</i> rejected moves.
</div>
<div id="outline-container-org1ce147d" class="outline-4">
<h4 id="org1ce147d"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
<div id="outline-container-org06c5905" class="outline-4">
<h4 id="org06c5905"><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
@ -1598,8 +1598,8 @@ the same variable later on to store a time step.
</div>
<div id="outline-container-orgba66664" class="outline-4">
<h4 id="orgba66664"><span class="section-number-4">3.3.2</span> Exercise</h4>
<div id="outline-container-org654010b" class="outline-4">
<h4 id="org654010b"><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>
@ -1710,8 +1710,8 @@ Can you observe a reduction in the statistical error?
</div>
</div>
<div id="outline-container-org9850980" class="outline-3">
<h3 id="org9850980"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
<div id="outline-container-org215f308" class="outline-3">
<h3 id="org215f308"><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.
@ -1832,8 +1832,8 @@ The algorithm of the previous exercise is only slighlty modified as:
</ol>
</div>
<div id="outline-container-org0bd3830" class="outline-4">
<h4 id="org0bd3830"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
<div id="outline-container-org31cb975" class="outline-4">
<h4 id="org31cb975"><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
@ -1897,8 +1897,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
</div>
<div id="outline-container-org0d8efaf" class="outline-4">
<h4 id="org0d8efaf"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
<div id="outline-container-org48820db" class="outline-4">
<h4 id="org48820db"><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>
@ -1941,8 +1941,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div>
</div>
<div id="outline-container-orgce8adcc" class="outline-4">
<h4 id="orgce8adcc"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
<div id="outline-container-org1ce793c" class="outline-4">
<h4 id="org1ce793c"><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>
@ -2041,8 +2041,8 @@ Modify the previous program to introduce the drift-diffusion scheme.
</div>
</div>
<div id="outline-container-org3818d44" class="outline-2">
<h2 id="org3818d44"><span class="section-number-2">4</span> Project</h2>
<div id="outline-container-orgc4c9346" class="outline-2">
<h2 id="orgc4c9346"><span class="section-number-2">4</span> Project</h2>
<div class="outline-text-2" id="text-4">
<p>
Change your PDMC code for one of the following:
@ -2059,8 +2059,8 @@ And compute the ground state energy.
</div>
<div id="outline-container-org03cad4e" class="outline-2">
<h2 id="org03cad4e"><span class="section-number-2">5</span> Acknowledgments</h2>
<div id="outline-container-org79362c8" class="outline-2">
<h2 id="org79362c8"><span class="section-number-2">5</span> Acknowledgments</h2>
<div class="outline-text-2" id="text-5">
<div class="figure">
@ -2080,7 +2080,7 @@ Union is not responsible for any use that might be made of such content.
</div>
<div id="postamble" class="status">
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
<p class="date">Created: 2021-02-03 Wed 21:14</p>
<p class="date">Created: 2021-02-04 Thu 07:47</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>