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<title>Quantum Monte Carlo</title> <title>Quantum Monte Carlo</title>
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<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org954307c">1. Introduction</a> <li><a href="#org484a53e">1. Introduction</a>
<ul> <ul>
<li><a href="#orge2a394b">1.1. Energy and local energy</a></li> <li><a href="#org4798a98">1.1. Energy and local energy</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgecbb272">2. Numerical evaluation of the energy of the hydrogen atom</a> <li><a href="#orgfc53e9e">2. Numerical evaluation of the energy of the hydrogen atom</a>
<ul> <ul>
<li><a href="#org723882f">2.1. Local energy</a> <li><a href="#org57b7ef5">2.1. Local energy</a>
<ul> <ul>
<li><a href="#org6d5bae4">2.1.1. Exercise 1</a> <li><a href="#org668920d">2.1.1. Exercise 1</a>
<ul> <ul>
<li><a href="#org1e37adc">2.1.1.1. Solution</a></li> <li><a href="#org46559e7">2.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org4fe6705">2.1.2. Exercise 2</a> <li><a href="#org08f0363">2.1.2. Exercise 2</a>
<ul> <ul>
<li><a href="#org43d842e">2.1.2.1. Solution</a></li> <li><a href="#org385509d">2.1.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd8cdcb4">2.1.3. Exercise 3</a> <li><a href="#orgdbd4a06">2.1.3. Exercise 3</a>
<ul> <ul>
<li><a href="#orgfebd511">2.1.3.1. Solution</a></li> <li><a href="#orgdd8f833">2.1.3.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd51ab3d">2.1.4. Exercise 4</a> <li><a href="#orgb04b7f7">2.1.4. Exercise 4</a>
<ul> <ul>
<li><a href="#orgfd6f375">2.1.4.1. Solution</a></li> <li><a href="#orgc1c0619">2.1.4.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org6adf531">2.1.5. Exercise 5</a> <li><a href="#org3b484c9">2.1.5. Exercise 5</a>
<ul> <ul>
<li><a href="#orgfd370ec">2.1.5.1. Solution</a></li> <li><a href="#org4d9621e">2.1.5.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org5554adc">2.2. Plot of the local energy along the \(x\) axis</a> <li><a href="#org4ea3772">2.2. Plot of the local energy along the \(x\) axis</a>
<ul> <ul>
<li><a href="#org02acd53">2.2.1. Exercise</a> <li><a href="#orgaa030b2">2.2.1. Exercise</a>
<ul> <ul>
<li><a href="#org870fb66">2.2.1.1. Solution</a></li> <li><a href="#org724a0e8">2.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org6bf2cd9">2.3. Numerical estimation of the energy</a> <li><a href="#org72726f5">2.3. Numerical estimation of the energy</a>
<ul> <ul>
<li><a href="#org0ef4619">2.3.1. Exercise</a> <li><a href="#org6eb7d45">2.3.1. Exercise</a>
<ul> <ul>
<li><a href="#org06843de">2.3.1.1. Solution</a></li> <li><a href="#org59929b3">2.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org80b7c4f">2.4. Variance of the local energy</a> <li><a href="#org5a9cc67">2.4. Variance of the local energy</a>
<ul> <ul>
<li><a href="#orgae5f8a6">2.4.1. Exercise (optional)</a> <li><a href="#orgdca84d5">2.4.1. Exercise (optional)</a>
<ul> <ul>
<li><a href="#org3e32dde">2.4.1.1. <span class="done DONE">DONE</span> Solution</a></li> <li><a href="#org1c8885c">2.4.1.1. <span class="done DONE">DONE</span> Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org3f19a15">2.4.2. Exercise</a> <li><a href="#org6240c00">2.4.2. Exercise</a>
<ul> <ul>
<li><a href="#org818e157">2.4.2.1. Solution</a></li> <li><a href="#org7a641d8">2.4.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org835e03c">3. Variational Monte Carlo</a> <li><a href="#orgaf4dafc">3. Variational Monte Carlo</a>
<ul> <ul>
<li><a href="#orge3503a1">3.1. Computation of the statistical error</a> <li><a href="#orgc635579">3.1. Computation of the statistical error</a>
<ul> <ul>
<li><a href="#org3b42653">3.1.1. Exercise</a> <li><a href="#org0dbd817">3.1.1. Exercise</a>
<ul> <ul>
<li><a href="#org13e71c2">3.1.1.1. Solution</a></li> <li><a href="#org9bf5d86">3.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org7432e13">3.2. Uniform sampling in the box</a> <li><a href="#orgc075f26">3.2. Uniform sampling in the box</a>
<ul> <ul>
<li><a href="#org62499d7">3.2.1. Exercise</a> <li><a href="#org2193660">3.2.1. Exercise</a>
<ul> <ul>
<li><a href="#orgdf3af99">3.2.1.1. Solution</a></li> <li><a href="#orge564bf3">3.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgfe37817">3.3. Metropolis sampling with \(\Psi^2\)</a> <li><a href="#org49db507">3.3. Metropolis sampling with \(\Psi^2\)</a>
<ul> <ul>
<li><a href="#orgb2fcbc1">3.3.1. Optimal step size</a></li> <li><a href="#orgb9a128c">3.3.1. Optimal step size</a></li>
<li><a href="#org0b6d246">3.3.2. Exercise</a> <li><a href="#org6d05d27">3.3.2. Exercise</a>
<ul> <ul>
<li><a href="#org5b96dd9">3.3.2.1. Solution</a></li> <li><a href="#orgae06be2">3.3.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org304f80f">3.4. Generalized Metropolis algorithm</a> <li><a href="#orgce90675">3.4. Generalized Metropolis algorithm</a>
<ul> <ul>
<li><a href="#org7835ab4">3.4.1. Gaussian random number generator</a></li> <li><a href="#org6a93c6c">3.4.1. Gaussian random number generator</a></li>
<li><a href="#org5c3aeb2">3.4.2. Exercise 1</a> <li><a href="#org661046d">3.4.2. Exercise 1</a>
<ul> <ul>
<li><a href="#org27ffe6d">3.4.2.1. Solution</a></li> <li><a href="#orgfb6225c">3.4.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org3bb39ee">3.4.3. Exercise 2</a> <li><a href="#orgfe231df">3.4.3. Exercise 2</a>
<ul> <ul>
<li><a href="#org071fcac">3.4.3.1. Solution</a></li> <li><a href="#orga73cfde">3.4.3.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org8bdb78e">4. Diffusion Monte Carlo</a> <li><a href="#org0d4554c">4. Diffusion Monte Carlo</a>
<ul> <ul>
<li><a href="#org9089bcf">4.1. Schrödinger equation in imaginary time</a></li> <li><a href="#orgc93ceb9">4.1. Schrödinger equation in imaginary time</a></li>
<li><a href="#orgbce4d86">4.2. Relation to diffusion</a></li> <li><a href="#org1224530">4.2. Relation to diffusion</a></li>
<li><a href="#org2d10e2f">4.3. Importance sampling</a> <li><a href="#org393a5ec">4.3. Importance sampling</a>
<ul> <ul>
<li><a href="#orgc059c13">4.3.1. Appendix : Details of the Derivation</a></li> <li><a href="#orgecc90ea">4.3.1. Appendix : Details of the Derivation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgaee3736">4.4. Pure Diffusion Monte Carlo</a></li> <li><a href="#org292fbee">4.4. Pure Diffusion Monte Carlo</a></li>
<li><a href="#orgb75c0f2">4.5. Hydrogen atom</a> <li><a href="#orgce09db6">4.5. Hydrogen atom</a>
<ul> <ul>
<li><a href="#org9b4694b">4.5.1. Exercise</a></li> <li><a href="#orgeda527c">4.5.1. Exercise</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgadbf000">5. Project</a></li> <li><a href="#orgf4f054d">5. Project</a></li>
<li><a href="#org8c43878">6. Acknowledgments</a></li> <li><a href="#org8424122">6. Acknowledgments</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org954307c" class="outline-2"> <div id="outline-container-org484a53e" class="outline-2">
<h2 id="org954307c"><span class="section-number-2">1</span> Introduction</h2> <h2 id="org484a53e"><span class="section-number-2">1</span> Introduction</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
This website contains the QMC tutorial of the 2021 LTTC winter school This website contains the QMC tutorial of the 2021 LTTC winter school
@ -510,8 +510,8 @@ coordinates, etc).
</p> </p>
</div> </div>
<div id="outline-container-orge2a394b" class="outline-3"> <div id="outline-container-org4798a98" class="outline-3">
<h3 id="orge2a394b"><span class="section-number-3">1.1</span> Energy and local energy</h3> <h3 id="org4798a98"><span class="section-number-3">1.1</span> Energy and local energy</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
@ -594,8 +594,8 @@ energy computed over these configurations:
</div> </div>
</div> </div>
<div id="outline-container-orgecbb272" class="outline-2"> <div id="outline-container-orgfc53e9e" class="outline-2">
<h2 id="orgecbb272"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2> <h2 id="orgfc53e9e"><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"> <div class="outline-text-2" id="text-2">
<p> <p>
In this section, we consider the hydrogen atom with the following In this section, we consider the hydrogen atom with the following
@ -624,8 +624,8 @@ To do that, we will compute the local energy and check whether it is constant.
</p> </p>
</div> </div>
<div id="outline-container-org723882f" class="outline-3"> <div id="outline-container-org57b7ef5" class="outline-3">
<h3 id="org723882f"><span class="section-number-3">2.1</span> Local energy</h3> <h3 id="org57b7ef5"><span class="section-number-3">2.1</span> Local energy</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
You will now program all quantities needed to compute the local energy of the H atom for the given wave function. You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
@ -652,8 +652,8 @@ to catch the error.
</div> </div>
</div> </div>
<div id="outline-container-org6d5bae4" class="outline-4"> <div id="outline-container-org668920d" class="outline-4">
<h4 id="org6d5bae4"><span class="section-number-4">2.1.1</span> Exercise 1</h4> <h4 id="org668920d"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
<div class="outline-text-4" id="text-2-1-1"> <div class="outline-text-4" id="text-2-1-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -698,8 +698,8 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-org1e37adc" class="outline-5"> <div id="outline-container-org46559e7" class="outline-5">
<h5 id="org1e37adc"><span class="section-number-5">2.1.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="org46559e7"><span class="section-number-5">2.1.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-1-1-1"> <div class="outline-text-5" id="text-2-1-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -740,8 +740,8 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-org4fe6705" class="outline-4"> <div id="outline-container-org08f0363" class="outline-4">
<h4 id="org4fe6705"><span class="section-number-4">2.1.2</span> Exercise 2</h4> <h4 id="org08f0363"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
<div class="outline-text-4" id="text-2-1-2"> <div class="outline-text-4" id="text-2-1-2">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -776,8 +776,8 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-org43d842e" class="outline-5"> <div id="outline-container-org385509d" class="outline-5">
<h5 id="org43d842e"><span class="section-number-5">2.1.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="org385509d"><span class="section-number-5">2.1.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-1-2-1"> <div class="outline-text-5" id="text-2-1-2-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -804,8 +804,8 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-orgd8cdcb4" class="outline-4"> <div id="outline-container-orgdbd4a06" class="outline-4">
<h4 id="orgd8cdcb4"><span class="section-number-4">2.1.3</span> Exercise 3</h4> <h4 id="orgdbd4a06"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
<div class="outline-text-4" id="text-2-1-3"> <div class="outline-text-4" id="text-2-1-3">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -886,8 +886,8 @@ Therefore, the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-orgfebd511" class="outline-5"> <div id="outline-container-orgdd8f833" class="outline-5">
<h5 id="orgfebd511"><span class="section-number-5">2.1.3.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="orgdd8f833"><span class="section-number-5">2.1.3.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-1-3-1"> <div class="outline-text-5" id="text-2-1-3-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -928,8 +928,8 @@ Therefore, the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-orgd51ab3d" class="outline-4"> <div id="outline-container-orgb04b7f7" class="outline-4">
<h4 id="orgd51ab3d"><span class="section-number-4">2.1.4</span> Exercise 4</h4> <h4 id="orgb04b7f7"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
<div class="outline-text-4" id="text-2-1-4"> <div class="outline-text-4" id="text-2-1-4">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -988,8 +988,8 @@ are calling is yours.
</div> </div>
</div> </div>
<div id="outline-container-orgfd6f375" class="outline-5"> <div id="outline-container-orgc1c0619" class="outline-5">
<h5 id="orgfd6f375"><span class="section-number-5">2.1.4.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="orgc1c0619"><span class="section-number-5">2.1.4.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-1-4-1"> <div class="outline-text-5" id="text-2-1-4-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -1020,8 +1020,8 @@ are calling is yours.
</div> </div>
</div> </div>
<div id="outline-container-org6adf531" class="outline-4"> <div id="outline-container-org3b484c9" class="outline-4">
<h4 id="org6adf531"><span class="section-number-4">2.1.5</span> Exercise 5</h4> <h4 id="org3b484c9"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
<div class="outline-text-4" id="text-2-1-5"> <div class="outline-text-4" id="text-2-1-5">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -1031,8 +1031,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
</div> </div>
</div> </div>
<div id="outline-container-orgfd370ec" class="outline-5"> <div id="outline-container-org4d9621e" class="outline-5">
<h5 id="orgfd370ec"><span class="section-number-5">2.1.5.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="org4d9621e"><span class="section-number-5">2.1.5.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-1-5-1"> <div class="outline-text-5" id="text-2-1-5-1">
\begin{eqnarray*} \begin{eqnarray*}
E &=& \frac{\hat{H} \Psi}{\Psi} = - \frac{1}{2} \frac{\Delta \Psi}{\Psi} - E &=& \frac{\hat{H} \Psi}{\Psi} = - \frac{1}{2} \frac{\Delta \Psi}{\Psi} -
@ -1052,8 +1052,8 @@ equal to -0.5 atomic units.
</div> </div>
</div> </div>
<div id="outline-container-org5554adc" class="outline-3"> <div id="outline-container-org4ea3772" class="outline-3">
<h3 id="org5554adc"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3> <h3 id="org4ea3772"><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="outline-text-3" id="text-2-2">
<p> <p>
The program you will write in this section will be written in The program you will write in this section will be written in
@ -1084,8 +1084,8 @@ In Fortran, you will need to compile all the source files together:
</div> </div>
</div> </div>
<div id="outline-container-org02acd53" class="outline-4"> <div id="outline-container-orgaa030b2" class="outline-4">
<h4 id="org02acd53"><span class="section-number-4">2.2.1</span> Exercise</h4> <h4 id="orgaa030b2"><span class="section-number-4">2.2.1</span> Exercise</h4>
<div class="outline-text-4" id="text-2-2-1"> <div class="outline-text-4" id="text-2-2-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -1179,8 +1179,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
</div> </div>
</div> </div>
<div id="outline-container-org870fb66" class="outline-5"> <div id="outline-container-org724a0e8" class="outline-5">
<h5 id="org870fb66"><span class="section-number-5">2.2.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="org724a0e8"><span class="section-number-5">2.2.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-2-1-1"> <div class="outline-text-5" id="text-2-2-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -1257,8 +1257,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
</div> </div>
</div> </div>
<div id="outline-container-org6bf2cd9" class="outline-3"> <div id="outline-container-org72726f5" class="outline-3">
<h3 id="org6bf2cd9"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3> <h3 id="org72726f5"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
If the space is discretized in small volume elements \(\mathbf{r}_i\) If the space is discretized in small volume elements \(\mathbf{r}_i\)
@ -1288,8 +1288,8 @@ The energy is biased because:
</div> </div>
<div id="outline-container-org0ef4619" class="outline-4"> <div id="outline-container-org6eb7d45" class="outline-4">
<h4 id="org0ef4619"><span class="section-number-4">2.3.1</span> Exercise</h4> <h4 id="org6eb7d45"><span class="section-number-4">2.3.1</span> Exercise</h4>
<div class="outline-text-4" id="text-2-3-1"> <div class="outline-text-4" id="text-2-3-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -1360,8 +1360,8 @@ To compile the Fortran and run it:
</div> </div>
</div> </div>
<div id="outline-container-org06843de" class="outline-5"> <div id="outline-container-org59929b3" class="outline-5">
<h5 id="org06843de"><span class="section-number-5">2.3.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="org59929b3"><span class="section-number-5">2.3.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-3-1-1"> <div class="outline-text-5" id="text-2-3-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -1478,8 +1478,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
</div> </div>
</div> </div>
<div id="outline-container-org80b7c4f" class="outline-3"> <div id="outline-container-org5a9cc67" class="outline-3">
<h3 id="org80b7c4f"><span class="section-number-3">2.4</span> Variance of the local energy</h3> <h3 id="org5a9cc67"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
The variance of the local energy is a functional of \(\Psi\) The variance of the local energy is a functional of \(\Psi\)
@ -1506,8 +1506,8 @@ energy can be used as a measure of the quality of a wave function.
</p> </p>
</div> </div>
<div id="outline-container-orgae5f8a6" class="outline-4"> <div id="outline-container-orgdca84d5" class="outline-4">
<h4 id="orgae5f8a6"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4> <h4 id="orgdca84d5"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
<div class="outline-text-4" id="text-2-4-1"> <div class="outline-text-4" id="text-2-4-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -1518,8 +1518,8 @@ Prove that :
</div> </div>
</div> </div>
<div id="outline-container-org3e32dde" class="outline-5"> <div id="outline-container-org1c8885c" class="outline-5">
<h5 id="org3e32dde"><span class="section-number-5">2.4.1.1</span> <span class="done DONE">DONE</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="org1c8885c"><span class="section-number-5">2.4.1.1</span> <span class="done DONE">DONE</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-4-1-1"> <div class="outline-text-5" id="text-2-4-1-1">
<p> <p>
\(\bar{E} = \langle E \rangle\) is a constant, so \(\langle \bar{E} \(\bar{E} = \langle E \rangle\) is a constant, so \(\langle \bar{E}
@ -1538,8 +1538,8 @@ Prove that :
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org3f19a15" class="outline-4"> <div id="outline-container-org6240c00" class="outline-4">
<h4 id="org3f19a15"><span class="section-number-4">2.4.2</span> Exercise</h4> <h4 id="org6240c00"><span class="section-number-4">2.4.2</span> Exercise</h4>
<div class="outline-text-4" id="text-2-4-2"> <div class="outline-text-4" id="text-2-4-2">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -1615,8 +1615,8 @@ To compile and run:
</div> </div>
</div> </div>
<div id="outline-container-org818e157" class="outline-5"> <div id="outline-container-org7a641d8" class="outline-5">
<h5 id="org818e157"><span class="section-number-5">2.4.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="org7a641d8"><span class="section-number-5">2.4.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-2-4-2-1"> <div class="outline-text-5" id="text-2-4-2-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -1755,8 +1755,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
</div> </div>
</div> </div>
<div id="outline-container-org835e03c" class="outline-2"> <div id="outline-container-orgaf4dafc" class="outline-2">
<h2 id="org835e03c"><span class="section-number-2">3</span> Variational Monte Carlo</h2> <h2 id="orgaf4dafc"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
Numerical integration with deterministic methods is very efficient Numerical integration with deterministic methods is very efficient
@ -1772,8 +1772,8 @@ interval.
</p> </p>
</div> </div>
<div id="outline-container-orge3503a1" class="outline-3"> <div id="outline-container-orgc635579" class="outline-3">
<h3 id="orge3503a1"><span class="section-number-3">3.1</span> Computation of the statistical error</h3> <h3 id="orgc635579"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
To compute the statistical error, you need to perform \(M\) To compute the statistical error, you need to perform \(M\)
@ -1813,8 +1813,8 @@ And the confidence interval is given by
</p> </p>
</div> </div>
<div id="outline-container-org3b42653" class="outline-4"> <div id="outline-container-org0dbd817" class="outline-4">
<h4 id="org3b42653"><span class="section-number-4">3.1.1</span> Exercise</h4> <h4 id="org0dbd817"><span class="section-number-4">3.1.1</span> Exercise</h4>
<div class="outline-text-4" id="text-3-1-1"> <div class="outline-text-4" id="text-3-1-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -1854,8 +1854,8 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-org13e71c2" class="outline-5"> <div id="outline-container-org9bf5d86" class="outline-5">
<h5 id="org13e71c2"><span class="section-number-5">3.1.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="org9bf5d86"><span class="section-number-5">3.1.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-3-1-1-1"> <div class="outline-text-5" id="text-3-1-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -1916,8 +1916,8 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-org7432e13" class="outline-3"> <div id="outline-container-orgc075f26" class="outline-3">
<h3 id="org7432e13"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3> <h3 id="orgc075f26"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
We will now perform our first Monte Carlo calculation to compute the We will now perform our first Monte Carlo calculation to compute the
@ -1978,8 +1978,8 @@ compute the statistical error.
</p> </p>
</div> </div>
<div id="outline-container-org62499d7" class="outline-4"> <div id="outline-container-org2193660" class="outline-4">
<h4 id="org62499d7"><span class="section-number-4">3.2.1</span> Exercise</h4> <h4 id="org2193660"><span class="section-number-4">3.2.1</span> Exercise</h4>
<div class="outline-text-4" id="text-3-2-1"> <div class="outline-text-4" id="text-3-2-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2081,8 +2081,8 @@ well as the index of the current step.
</div> </div>
</div> </div>
<div id="outline-container-orgdf3af99" class="outline-5"> <div id="outline-container-orge564bf3" class="outline-5">
<h5 id="orgdf3af99"><span class="section-number-5">3.2.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="orge564bf3"><span class="section-number-5">3.2.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-3-2-1-1"> <div class="outline-text-5" id="text-3-2-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -2188,8 +2188,8 @@ E = -0.48084122147238995 +/- 2.4983775878329355E-003
</div> </div>
</div> </div>
<div id="outline-container-orgfe37817" class="outline-3"> <div id="outline-container-org49db507" class="outline-3">
<h3 id="orgfe37817"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3> <h3 id="org49db507"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
We will now use the square of the wave function to sample random We will now use the square of the wave function to sample random
@ -2308,8 +2308,8 @@ All samples should be kept, from both accepted <i>and</i> rejected moves.
</div> </div>
<div id="outline-container-orgb2fcbc1" class="outline-4"> <div id="outline-container-orgb9a128c" class="outline-4">
<h4 id="orgb2fcbc1"><span class="section-number-4">3.3.1</span> Optimal step size</h4> <h4 id="orgb9a128c"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
<div class="outline-text-4" id="text-3-3-1"> <div class="outline-text-4" id="text-3-3-1">
<p> <p>
If the box is infinitely small, the ratio will be very close If the box is infinitely small, the ratio will be very close
@ -2344,8 +2344,8 @@ the same variable later on to store a time step.
</div> </div>
<div id="outline-container-org0b6d246" class="outline-4"> <div id="outline-container-org6d05d27" class="outline-4">
<h4 id="org0b6d246"><span class="section-number-4">3.3.2</span> Exercise</h4> <h4 id="org6d05d27"><span class="section-number-4">3.3.2</span> Exercise</h4>
<div class="outline-text-4" id="text-3-3-2"> <div class="outline-text-4" id="text-3-3-2">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2454,8 +2454,8 @@ Can you observe a reduction in the statistical error?
</div> </div>
</div> </div>
<div id="outline-container-org5b96dd9" class="outline-5"> <div id="outline-container-orgae06be2" class="outline-5">
<h5 id="org5b96dd9"><span class="section-number-5">3.3.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="orgae06be2"><span class="section-number-5">3.3.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-3-3-2-1"> <div class="outline-text-5" id="text-3-3-2-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -2602,8 +2602,8 @@ A = 0.50762633333333318 +/- 3.4601756760043725E-004
</div> </div>
</div> </div>
<div id="outline-container-org304f80f" class="outline-3"> <div id="outline-container-orgce90675" class="outline-3">
<h3 id="org304f80f"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3> <h3 id="orgce90675"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability. One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
@ -2724,8 +2724,8 @@ The algorithm of the previous exercise is only slighlty modified as:
</ol> </ol>
</div> </div>
<div id="outline-container-org7835ab4" class="outline-4"> <div id="outline-container-org6a93c6c" class="outline-4">
<h4 id="org7835ab4"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4> <h4 id="org6a93c6c"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
<div class="outline-text-4" id="text-3-4-1"> <div class="outline-text-4" id="text-3-4-1">
<p> <p>
To obtain Gaussian-distributed random numbers, you can apply the To obtain Gaussian-distributed random numbers, you can apply the
@ -2789,8 +2789,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
</div> </div>
<div id="outline-container-org5c3aeb2" class="outline-4"> <div id="outline-container-org661046d" class="outline-4">
<h4 id="org5c3aeb2"><span class="section-number-4">3.4.2</span> Exercise 1</h4> <h4 id="org661046d"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
<div class="outline-text-4" id="text-3-4-2"> <div class="outline-text-4" id="text-3-4-2">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2832,8 +2832,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-org27ffe6d" class="outline-5"> <div id="outline-container-orgfb6225c" class="outline-5">
<h5 id="org27ffe6d"><span class="section-number-5">3.4.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="orgfb6225c"><span class="section-number-5">3.4.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-3-4-2-1"> <div class="outline-text-5" id="text-3-4-2-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -2866,8 +2866,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-org3bb39ee" class="outline-4"> <div id="outline-container-orgfe231df" class="outline-4">
<h4 id="org3bb39ee"><span class="section-number-4">3.4.3</span> Exercise 2</h4> <h4 id="orgfe231df"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
<div class="outline-text-4" id="text-3-4-3"> <div class="outline-text-4" id="text-3-4-3">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2963,8 +2963,8 @@ Modify the previous program to introduce the drift-diffusion scheme.
</div> </div>
</div> </div>
<div id="outline-container-org071fcac" class="outline-5"> <div id="outline-container-orga73cfde" class="outline-5">
<h5 id="org071fcac"><span class="section-number-5">3.4.3.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5> <h5 id="orga73cfde"><span class="section-number-5">3.4.3.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution2">solution2</span></span></h5>
<div class="outline-text-5" id="text-3-4-3-1"> <div class="outline-text-5" id="text-3-4-3-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -3152,8 +3152,8 @@ A = 0.62037333333333333 +/- 4.8970160591451110E-004
</div> </div>
</div> </div>
<div id="outline-container-org8bdb78e" class="outline-2"> <div id="outline-container-org0d4554c" class="outline-2">
<h2 id="org8bdb78e"><span class="section-number-2">4</span> Diffusion Monte Carlo</h2> <h2 id="org0d4554c"><span class="section-number-2">4</span> Diffusion Monte Carlo</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
As we have seen, Variational Monte Carlo is a powerful method to As we have seen, Variational Monte Carlo is a powerful method to
@ -3170,8 +3170,8 @@ finding a near-exact numerical solution to the Schrödinger equation.
</p> </p>
</div> </div>
<div id="outline-container-org9089bcf" class="outline-3"> <div id="outline-container-orgc93ceb9" class="outline-3">
<h3 id="org9089bcf"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3> <h3 id="orgc93ceb9"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
Consider the time-dependent Schrödinger equation: Consider the time-dependent Schrödinger equation:
@ -3239,8 +3239,8 @@ system.
</div> </div>
</div> </div>
<div id="outline-container-orgbce4d86" class="outline-3"> <div id="outline-container-org1224530" class="outline-3">
<h3 id="orgbce4d86"><span class="section-number-3">4.2</span> Relation to diffusion</h3> <h3 id="org1224530"><span class="section-number-3">4.2</span> Relation to diffusion</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
The <a href="https://en.wikipedia.org/wiki/Diffusion_equation">diffusion equation</a> of particles is given by The <a href="https://en.wikipedia.org/wiki/Diffusion_equation">diffusion equation</a> of particles is given by
@ -3320,8 +3320,8 @@ Therefore, in both cases, you are dealing with a "Bosonic" ground state.
</div> </div>
</div> </div>
<div id="outline-container-org2d10e2f" class="outline-3"> <div id="outline-container-org393a5ec" class="outline-3">
<h3 id="org2d10e2f"><span class="section-number-3">4.3</span> Importance sampling</h3> <h3 id="org393a5ec"><span class="section-number-3">4.3</span> Importance sampling</h3>
<div class="outline-text-3" id="text-4-3"> <div class="outline-text-3" id="text-4-3">
<p> <p>
In a molecular system, the potential is far from being constant In a molecular system, the potential is far from being constant
@ -3419,8 +3419,8 @@ energies computed with the trial wave function.
</p> </p>
</div> </div>
<div id="outline-container-orgc059c13" class="outline-4"> <div id="outline-container-orgecc90ea" class="outline-4">
<h4 id="orgc059c13"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4> <h4 id="orgecc90ea"><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"> <div class="outline-text-4" id="text-4-3-1">
<p> <p>
\[ \[
@ -3481,8 +3481,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
</div> </div>
</div> </div>
<div id="outline-container-orgaee3736" class="outline-3"> <div id="outline-container-org292fbee" class="outline-3">
<h3 id="orgaee3736"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo</h3> <h3 id="org292fbee"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-4-4">
<p> <p>
Instead of having a variable number of particles to simulate the Instead of having a variable number of particles to simulate the
@ -3524,7 +3524,7 @@ starting from a VMC code:
E_L(\mathbf{r}_n)\), E_L(\mathbf{r}_n)\),
and the weight \(W(\mathbf{r}_n)\) for the normalization</li> and the weight \(W(\mathbf{r}_n)\) for the normalization</li>
<li>Update \(\tau_n = \tau_{n-1} + \delta t\)</li> <li>Update \(\tau_n = \tau_{n-1} + \delta t\)</li>
<li>If \(\tau_{n} > \tau_\text{max}\), the long projection time has <li>If \(\tau_{n} > \tau_\text{max}\) (\(\tau_\text{max}\) is an input parameter), the long projection time has
been reached and we can start an new trajectory from the current been reached and we can start an new trajectory from the current
position: reset \(W(r_n) = 1\) and \(\tau_n position: reset \(W(r_n) = 1\) and \(\tau_n
= 0\)</li> = 0\)</li>
@ -3571,13 +3571,13 @@ the DMC algorithm. However, its use reduces significantly the time-step error.</
</div> </div>
<div id="outline-container-orgb75c0f2" class="outline-3"> <div id="outline-container-orgce09db6" class="outline-3">
<h3 id="orgb75c0f2"><span class="section-number-3">4.5</span> Hydrogen atom</h3> <h3 id="orgce09db6"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
<div class="outline-text-3" id="text-4-5"> <div class="outline-text-3" id="text-4-5">
</div> </div>
<div id="outline-container-org9b4694b" class="outline-4"> <div id="outline-container-orgeda527c" class="outline-4">
<h4 id="org9b4694b"><span class="section-number-4">4.5.1</span> Exercise</h4> <h4 id="orgeda527c"><span class="section-number-4">4.5.1</span> Exercise</h4>
<div class="outline-text-4" id="text-4-5-1"> <div class="outline-text-4" id="text-4-5-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -3585,7 +3585,7 @@ Modify the Metropolis VMC program into a PDMC program.
In the limit \(\delta t \rightarrow 0\), you should recover the exact In the limit \(\delta t \rightarrow 0\), you should recover the exact
energy of H for any value of \(a\), as long as the simulation is stable. energy of H for any value of \(a\), as long as the simulation is stable.
We choose here a time step of 0.05 a.u. and a fixed projection We choose here a time step of 0.05 a.u. and a fixed projection
time \(\tau\) =100 a.u. time \(\tau_{\text{max}}\) =100 a.u.
</p> </p>
</div> </div>
@ -3684,8 +3684,8 @@ time \(\tau\) =100 a.u.
<div id="outline-container-orgadbf000" class="outline-2"> <div id="outline-container-orgf4f054d" class="outline-2">
<h2 id="orgadbf000"><span class="section-number-2">5</span> Project</h2> <h2 id="orgf4f054d"><span class="section-number-2">5</span> Project</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p> <p>
Change your PDMC code for one of the following: Change your PDMC code for one of the following:
@ -3703,8 +3703,8 @@ And compute the ground state energy.
<div id="outline-container-org8c43878" class="outline-2"> <div id="outline-container-org8424122" class="outline-2">
<h2 id="org8c43878"><span class="section-number-2">6</span> Acknowledgments</h2> <h2 id="org8424122"><span class="section-number-2">6</span> Acknowledgments</h2>
<div class="outline-text-2" id="text-6"> <div class="outline-text-2" id="text-6">
<div class="figure"> <div class="figure">
@ -3724,7 +3724,7 @@ Union is not responsible for any use that might be made of such content.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Anthony Scemama, Claudia Filippi</p> <p class="author">Author: Anthony Scemama, Claudia Filippi</p>
<p class="date">Created: 2021-02-04 Thu 16:02</p> <p class="date">Created: 2021-02-04 Thu 16:27</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>