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<title>Quantum Monte Carlo</title> <title>Quantum Monte Carlo</title>
@ -329,164 +329,165 @@ for the JavaScript code in this tag.
<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="#orgbb97b6f">1. Introduction</a></li> <li><a href="#org49d04ba">1. Introduction</a></li>
<li><a href="#org623e4cb">2. Numerical evaluation of the energy</a> <li><a href="#org829df65">2. Numerical evaluation of the energy</a>
<ul> <ul>
<li><a href="#orgcf2eac0">2.1. Local energy</a> <li><a href="#org05487b8">2.1. Local energy</a>
<ul> <ul>
<li><a href="#org6e4be76">2.1.1. Exercise 1</a> <li><a href="#org0a1896f">2.1.1. Exercise 1</a>
<ul> <ul>
<li><a href="#org1c29fdf">2.1.1.1. Solution</a></li> <li><a href="#orgeac8364">2.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org45f3e47">2.1.2. Exercise 2</a> <li><a href="#org54f52bf">2.1.2. Exercise 2</a>
<ul> <ul>
<li><a href="#org17b1476">2.1.2.1. Solution</a></li> <li><a href="#orgec4496f">2.1.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org41fbf18">2.1.3. Exercise 3</a> <li><a href="#orgb2b2470">2.1.3. Exercise 3</a>
<ul> <ul>
<li><a href="#org58a5a88">2.1.3.1. Solution</a></li> <li><a href="#orgc9384b2">2.1.3.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org19beacc">2.1.4. Exercise 4</a> <li><a href="#org27c82a8">2.1.4. Exercise 4</a>
<ul> <ul>
<li><a href="#orgd059715">2.1.4.1. Solution</a></li> <li><a href="#org3488e51">2.1.4.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgfb98bee">2.1.5. Exercise 5</a> <li><a href="#orgb470382">2.1.5. Exercise 5</a>
<ul> <ul>
<li><a href="#orgd271963">2.1.5.1. Solution</a></li> <li><a href="#org95fe0ba">2.1.5.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org6cc9505">2.2. Plot of the local energy along the \(x\) axis</a> <li><a href="#org3c5c95d">2.2. Plot of the local energy along the \(x\) axis</a>
<ul> <ul>
<li><a href="#org969c80e">2.2.1. Exercise</a> <li><a href="#orga2b9bcf">2.2.1. Exercise</a>
<ul> <ul>
<li><a href="#org1d42b6b">2.2.1.1. Solution</a></li> <li><a href="#org93945d1">2.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org4643440">2.3. Numerical estimation of the energy</a> <li><a href="#org2555573">2.3. Numerical estimation of the energy</a>
<ul> <ul>
<li><a href="#org7934633">2.3.1. Exercise</a> <li><a href="#org102c578">2.3.1. Exercise</a>
<ul> <ul>
<li><a href="#orgce31e40">2.3.1.1. Solution</a></li> <li><a href="#org4da0f5c">2.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org6c8b12d">2.4. Variance of the local energy</a> <li><a href="#org74ce38e">2.4. Variance of the local energy</a>
<ul> <ul>
<li><a href="#orgbb0a7be">2.4.1. Exercise (optional)</a> <li><a href="#org0510da2">2.4.1. Exercise (optional)</a>
<ul> <ul>
<li><a href="#orgca7c679">2.4.1.1. Solution</a></li> <li><a href="#org1c5cb68">2.4.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org696e18d">2.4.2. Exercise</a> <li><a href="#org499a5a7">2.4.2. Exercise</a>
<ul> <ul>
<li><a href="#orgee7fe6c">2.4.2.1. Solution</a></li> <li><a href="#org93fe370">2.4.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org6d5323b">3. Variational Monte Carlo</a> <li><a href="#org6a0028a">3. Variational Monte Carlo</a>
<ul> <ul>
<li><a href="#org945932a">3.1. Computation of the statistical error</a> <li><a href="#org8b03841">3.1. Computation of the statistical error</a>
<ul> <ul>
<li><a href="#org7e965d1">3.1.1. Exercise</a> <li><a href="#org5bcb81b">3.1.1. Exercise</a>
<ul> <ul>
<li><a href="#org73d2a6e">3.1.1.1. Solution</a></li> <li><a href="#orgc97d0a8">3.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org70d91fe">3.2. Uniform sampling in the box</a> <li><a href="#org1d4a893">3.2. Uniform sampling in the box</a>
<ul> <ul>
<li><a href="#org05579fd">3.2.1. Exercise</a> <li><a href="#orgc78cc2c">3.2.1. Exercise</a>
<ul> <ul>
<li><a href="#orgc64c471">3.2.1.1. Solution</a></li> <li><a href="#orgf1caad9">3.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orga17395f">3.3. Metropolis sampling with \(\Psi^2\)</a> <li><a href="#org9ac37db">3.3. Metropolis sampling with \(\Psi^2\)</a>
<ul> <ul>
<li><a href="#org03d2480">3.3.1. Exercise</a> <li><a href="#org5ca4e2c">3.3.1. Exercise</a>
<ul> <ul>
<li><a href="#org277a0d2">3.3.1.1. Solution</a></li> <li><a href="#orgde750dd">3.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org4b2e6bb">3.4. Gaussian random number generator</a></li> <li><a href="#org9cb319e">3.4. Gaussian random number generator</a></li>
<li><a href="#orgd704542">3.5. Generalized Metropolis algorithm</a> <li><a href="#orgc52c7d5">3.5. Generalized Metropolis algorithm</a>
<ul> <ul>
<li><a href="#org107b8c2">3.5.1. Exercise 1</a> <li><a href="#org4689b7c">3.5.1. Exercise 1</a>
<ul> <ul>
<li><a href="#org8b8a874">3.5.1.1. Solution</a></li> <li><a href="#org26c9d33">3.5.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org02c9606">3.5.2. Exercise 2</a> <li><a href="#org3e39526">3.5.2. Exercise 2</a>
<ul> <ul>
<li><a href="#org4580e2f">3.5.2.1. Solution</a></li> <li><a href="#orga3cf826">3.5.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgdceacd5">4. Diffusion Monte Carlo</a> <li><a href="#orgebb7fc1">4. Diffusion Monte Carlo</a>
<ul> <ul>
<li><a href="#org3941aac">4.1. Schrödinger equation in imaginary time</a></li> <li><a href="#org8cf0ab0">4.1. Schrödinger equation in imaginary time</a></li>
<li><a href="#orge021b97">4.2. Diffusion and branching</a></li> <li><a href="#org78a65a5">4.2. Diffusion and branching</a></li>
<li><a href="#org0d37b59">4.3. Importance sampling</a> <li><a href="#org00d8981">4.3. Importance sampling</a>
<ul> <ul>
<li><a href="#orgc0fb19f">4.3.1. Appendix : Details of the Derivation</a></li> <li><a href="#org024b07c">4.3.1. Appendix : Details of the Derivation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgb90c671">4.4. Fixed-node DMC energy</a></li> <li><a href="#orgce840bc">4.4. Fixed-node DMC energy</a></li>
<li><a href="#org971d2c0">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li> <li><a href="#orga97437f">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li>
<li><a href="#orgf89d780">4.6. Hydrogen atom</a> <li><a href="#org54d9e47">4.6. Hydrogen atom</a>
<ul> <ul>
<li><a href="#orga1bc199">4.6.1. Exercise</a> <li><a href="#org77c2d07">4.6.1. Exercise</a>
<ul> <ul>
<li><a href="#orga0bccd7">4.6.1.1. Solution</a></li> <li><a href="#orgc3fb998">4.6.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org182141b">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li> <li><a href="#orgb88b04d">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
</ul> </ul>
</li> </li>
<li><a href="#org51ed78c">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li> <li><a href="#org622ff55">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-orgbb97b6f" class="outline-2"> <div id="outline-container-org49d04ba" class="outline-2">
<h2 id="orgbb97b6f"><span class="section-number-2">1</span> Introduction</h2> <h2 id="org49d04ba"><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 web site is the QMC tutorial of the LTTC winter school This web site contains the QMC tutorial of the 2021 LTTC winter school
<a href="https://www.irsamc.ups-tlse.fr/lttc/Luchon">Tutorials in Theoretical Chemistry</a>. <a href="https://www.irsamc.ups-tlse.fr/lttc/Luchon">Tutorials in Theoretical Chemistry</a>.
</p> </p>
<p> <p>
We propose different exercises to understand quantum Monte Carlo (QMC) We propose different exercises to understand quantum Monte Carlo (QMC)
methods. In the first section, we propose to compute the energy of a methods. In the first section, we start with the computation of the energy of a
hydrogen atom using numerical integration. The goal of this section is hydrogen atom using numerical integration. The goal of this section is
to introduce the <i>local energy</i>. to familarize yourself with the concept of <i>local energy</i>.
Then we introduce the variational Monte Carlo (VMC) method which Then, we introduce the variational Monte Carlo (VMC) method which
computes a statistical estimate of the expectation value of the energy computes a statistical estimate of the expectation value of the energy
associated with a given wave function. associated with a given wave function, and apply this approach to the
Finally, we introduce the diffusion Monte Carlo (DMC) method which hydrogen atom.
gives the exact energy of the hydrogen atom and of the H<sub>2</sub> molecule. Finally, we present the diffusion Monte Carlo (DMC) method which
we use here to estimate the exact energy of the hydrogen atom and of the H<sub>2</sub> molecule.
</p> </p>
<p> <p>
@ -499,7 +500,7 @@ We consider the stationary solution of the Schrödinger equation, so
the wave functions considered here are real: for an \(N\) electron the wave functions considered here are real: for an \(N\) electron
system where the electrons move in the 3-dimensional space, system where the electrons move in the 3-dimensional space,
\(\Psi : \mathbb{R}^{3N} \rightarrow \mathbb{R}\). In addition, \(\Psi\) \(\Psi : \mathbb{R}^{3N} \rightarrow \mathbb{R}\). In addition, \(\Psi\)
is defined everywhere, continuous and infinitely differentiable. is defined everywhere, continuous, and infinitely differentiable.
</p> </p>
<p> <p>
@ -509,8 +510,8 @@ coordinates, etc).
</div> </div>
</div> </div>
<div id="outline-container-org623e4cb" class="outline-2"> <div id="outline-container-org829df65" class="outline-2">
<h2 id="org623e4cb"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2> <h2 id="org829df65"><span class="section-number-2">2</span> Numerical evaluation of the energy</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
@ -583,8 +584,8 @@ E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
\end{eqnarray*} \end{eqnarray*}
</div> </div>
<div id="outline-container-orgcf2eac0" class="outline-3"> <div id="outline-container-org05487b8" class="outline-3">
<h3 id="orgcf2eac0"><span class="section-number-3">2.1</span> Local energy</h3> <h3 id="org05487b8"><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>
Write all the functions of this section in a single file : Write all the functions of this section in a single file :
@ -607,8 +608,8 @@ to catch the error.
</div> </div>
</div> </div>
<div id="outline-container-org6e4be76" class="outline-4"> <div id="outline-container-org0a1896f" class="outline-4">
<h4 id="org6e4be76"><span class="section-number-4">2.1.1</span> Exercise 1</h4> <h4 id="org0a1896f"><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>
@ -652,8 +653,8 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-org1c29fdf" class="outline-5"> <div id="outline-container-orgeac8364" class="outline-5">
<h5 id="org1c29fdf"><span class="section-number-5">2.1.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="orgeac8364"><span class="section-number-5">2.1.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -693,8 +694,8 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-org45f3e47" class="outline-4"> <div id="outline-container-org54f52bf" class="outline-4">
<h4 id="org45f3e47"><span class="section-number-4">2.1.2</span> Exercise 2</h4> <h4 id="org54f52bf"><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>
@ -729,8 +730,8 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-org17b1476" class="outline-5"> <div id="outline-container-orgec4496f" class="outline-5">
<h5 id="org17b1476"><span class="section-number-5">2.1.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="orgec4496f"><span class="section-number-5">2.1.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -757,8 +758,8 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-org41fbf18" class="outline-4"> <div id="outline-container-orgb2b2470" class="outline-4">
<h4 id="org41fbf18"><span class="section-number-4">2.1.3</span> Exercise 3</h4> <h4 id="orgb2b2470"><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>
@ -839,8 +840,8 @@ So the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-org58a5a88" class="outline-5"> <div id="outline-container-orgc9384b2" class="outline-5">
<h5 id="org58a5a88"><span class="section-number-5">2.1.3.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="orgc9384b2"><span class="section-number-5">2.1.3.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -881,8 +882,8 @@ So the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-org19beacc" class="outline-4"> <div id="outline-container-org27c82a8" class="outline-4">
<h4 id="org19beacc"><span class="section-number-4">2.1.4</span> Exercise 4</h4> <h4 id="org27c82a8"><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>
@ -925,8 +926,8 @@ local kinetic energy.
</div> </div>
</div> </div>
<div id="outline-container-orgd059715" class="outline-5"> <div id="outline-container-org3488e51" class="outline-5">
<h5 id="orgd059715"><span class="section-number-5">2.1.4.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="org3488e51"><span class="section-number-5">2.1.4.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -956,8 +957,8 @@ local kinetic energy.
</div> </div>
</div> </div>
<div id="outline-container-orgfb98bee" class="outline-4"> <div id="outline-container-orgb470382" class="outline-4">
<h4 id="orgfb98bee"><span class="section-number-4">2.1.5</span> Exercise 5</h4> <h4 id="orgb470382"><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>
@ -967,8 +968,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
</div> </div>
</div> </div>
<div id="outline-container-orgd271963" class="outline-5"> <div id="outline-container-org95fe0ba" class="outline-5">
<h5 id="orgd271963"><span class="section-number-5">2.1.5.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="org95fe0ba"><span class="section-number-5">2.1.5.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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} -
@ -988,8 +989,8 @@ equal to -0.5 atomic units.
</div> </div>
</div> </div>
<div id="outline-container-org6cc9505" class="outline-3"> <div id="outline-container-org3c5c95d" class="outline-3">
<h3 id="org6cc9505"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3> <h3 id="org3c5c95d"><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">
<div class="note"> <div class="note">
<p> <p>
@ -1000,8 +1001,8 @@ choose a grid which does not contain the origin.
</div> </div>
</div> </div>
<div id="outline-container-org969c80e" class="outline-4"> <div id="outline-container-orga2b9bcf" class="outline-4">
<h4 id="org969c80e"><span class="section-number-4">2.2.1</span> Exercise</h4> <h4 id="orga2b9bcf"><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>
@ -1084,8 +1085,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
</div> </div>
</div> </div>
<div id="outline-container-org1d42b6b" class="outline-5"> <div id="outline-container-org93945d1" class="outline-5">
<h5 id="org1d42b6b"><span class="section-number-5">2.2.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="org93945d1"><span class="section-number-5">2.2.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -1160,8 +1161,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
</div> </div>
</div> </div>
<div id="outline-container-org4643440" class="outline-3"> <div id="outline-container-org2555573" class="outline-3">
<h3 id="org4643440"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3> <h3 id="org2555573"><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 \(\mathbf{r}_i\)
@ -1191,8 +1192,8 @@ The energy is biased because:
</div> </div>
<div id="outline-container-org7934633" class="outline-4"> <div id="outline-container-org102c578" class="outline-4">
<h4 id="org7934633"><span class="section-number-4">2.3.1</span> Exercise</h4> <h4 id="org102c578"><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>
@ -1261,8 +1262,8 @@ To compile the Fortran and run it:
</div> </div>
</div> </div>
<div id="outline-container-orgce31e40" class="outline-5"> <div id="outline-container-org4da0f5c" class="outline-5">
<h5 id="orgce31e40"><span class="section-number-5">2.3.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="org4da0f5c"><span class="section-number-5">2.3.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -1377,8 +1378,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
</div> </div>
</div> </div>
<div id="outline-container-org6c8b12d" class="outline-3"> <div id="outline-container-org74ce38e" class="outline-3">
<h3 id="org6c8b12d"><span class="section-number-3">2.4</span> Variance of the local energy</h3> <h3 id="org74ce38e"><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 \(\Psi\)
@ -1405,8 +1406,8 @@ energy can be used as a measure of the quality of a wave function.
</p> </p>
</div> </div>
<div id="outline-container-orgbb0a7be" class="outline-4"> <div id="outline-container-org0510da2" class="outline-4">
<h4 id="orgbb0a7be"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4> <h4 id="org0510da2"><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>
@ -1417,8 +1418,8 @@ Prove that :
</div> </div>
</div> </div>
<div id="outline-container-orgca7c679" class="outline-5"> <div id="outline-container-org1c5cb68" class="outline-5">
<h5 id="orgca7c679"><span class="section-number-5">2.4.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="org1c5cb68"><span class="section-number-5">2.4.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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}
@ -1437,8 +1438,8 @@ Prove that :
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org696e18d" class="outline-4"> <div id="outline-container-org499a5a7" class="outline-4">
<h4 id="org696e18d"><span class="section-number-4">2.4.2</span> Exercise</h4> <h4 id="org499a5a7"><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>
@ -1512,8 +1513,8 @@ To compile and run:
</div> </div>
</div> </div>
<div id="outline-container-orgee7fe6c" class="outline-5"> <div id="outline-container-org93fe370" class="outline-5">
<h5 id="orgee7fe6c"><span class="section-number-5">2.4.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="org93fe370"><span class="section-number-5">2.4.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -1650,8 +1651,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
</div> </div>
</div> </div>
<div id="outline-container-org6d5323b" class="outline-2"> <div id="outline-container-org6a0028a" class="outline-2">
<h2 id="org6d5323b"><span class="section-number-2">3</span> Variational Monte Carlo</h2> <h2 id="org6a0028a"><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
@ -1667,8 +1668,8 @@ interval.
</p> </p>
</div> </div>
<div id="outline-container-org945932a" class="outline-3"> <div id="outline-container-org8b03841" class="outline-3">
<h3 id="org945932a"><span class="section-number-3">3.1</span> Computation of the statistical error</h3> <h3 id="org8b03841"><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 \(M\)
@ -1708,8 +1709,8 @@ And the confidence interval is given by
</p> </p>
</div> </div>
<div id="outline-container-org7e965d1" class="outline-4"> <div id="outline-container-org5bcb81b" class="outline-4">
<h4 id="org7e965d1"><span class="section-number-4">3.1.1</span> Exercise</h4> <h4 id="org5bcb81b"><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>
@ -1747,8 +1748,8 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-org73d2a6e" class="outline-5"> <div id="outline-container-orgc97d0a8" class="outline-5">
<h5 id="org73d2a6e"><span class="section-number-5">3.1.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="orgc97d0a8"><span class="section-number-5">3.1.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -1807,8 +1808,8 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-org70d91fe" class="outline-3"> <div id="outline-container-org1d4a893" class="outline-3">
<h3 id="org70d91fe"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3> <h3 id="org1d4a893"><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 do our first Monte Carlo calculation to compute the We will now do our first Monte Carlo calculation to compute the
@ -1842,8 +1843,8 @@ compute the statistical error.
</p> </p>
</div> </div>
<div id="outline-container-org05579fd" class="outline-4"> <div id="outline-container-orgc78cc2c" class="outline-4">
<h4 id="org05579fd"><span class="section-number-4">3.2.1</span> Exercise</h4> <h4 id="orgc78cc2c"><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>
@ -1943,8 +1944,8 @@ well as the index of the current step.
</div> </div>
</div> </div>
<div id="outline-container-orgc64c471" class="outline-5"> <div id="outline-container-orgf1caad9" class="outline-5">
<h5 id="orgc64c471"><span class="section-number-5">3.2.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="orgf1caad9"><span class="section-number-5">3.2.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</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>
@ -2058,8 +2059,8 @@ E = -0.49518773675598715 +/- 5.2391494923686175E-004
</div> </div>
</div> </div>
<div id="outline-container-orga17395f" class="outline-3"> <div id="outline-container-org9ac37db" class="outline-3">
<h3 id="orga17395f"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3> <h3 id="org9ac37db"><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
@ -2147,8 +2148,8 @@ step such that the acceptance rate is close to 0.5 is a good compromise.
</div> </div>
<div id="outline-container-org03d2480" class="outline-4"> <div id="outline-container-org5ca4e2c" class="outline-4">
<h4 id="org03d2480"><span class="section-number-4">3.3.1</span> Exercise</h4> <h4 id="org5ca4e2c"><span class="section-number-4">3.3.1</span> Exercise</h4>
<div class="outline-text-4" id="text-3-3-1"> <div class="outline-text-4" id="text-3-3-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2255,8 +2256,8 @@ Can you observe a reduction in the statistical error?
</div> </div>
</div> </div>
<div id="outline-container-org277a0d2" class="outline-5"> <div id="outline-container-orgde750dd" class="outline-5">
<h5 id="org277a0d2"><span class="section-number-5">3.3.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="orgde750dd"><span class="section-number-5">3.3.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5>
<div class="outline-text-5" id="text-3-3-1-1"> <div class="outline-text-5" id="text-3-3-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -2401,8 +2402,8 @@ A = 0.51695266666666673 +/- 4.0445505648997396E-004
</div> </div>
</div> </div>
<div id="outline-container-org4b2e6bb" class="outline-3"> <div id="outline-container-org9cb319e" class="outline-3">
<h3 id="org4b2e6bb"><span class="section-number-3">3.4</span> Gaussian random number generator</h3> <h3 id="org9cb319e"><span class="section-number-3">3.4</span> Gaussian random number generator</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
To obtain Gaussian-distributed random numbers, you can apply the To obtain Gaussian-distributed random numbers, you can apply the
@ -2464,8 +2465,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-orgd704542" class="outline-3"> <div id="outline-container-orgc52c7d5" class="outline-3">
<h3 id="orgd704542"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3> <h3 id="orgc52c7d5"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3>
<div class="outline-text-3" id="text-3-5"> <div class="outline-text-3" id="text-3-5">
<p> <p>
One can use more efficient numerical schemes to move the electrons, One can use more efficient numerical schemes to move the electrons,
@ -2564,8 +2565,8 @@ The transition probability becomes:
</div> </div>
<div id="outline-container-org107b8c2" class="outline-4"> <div id="outline-container-org4689b7c" class="outline-4">
<h4 id="org107b8c2"><span class="section-number-4">3.5.1</span> Exercise 1</h4> <h4 id="org4689b7c"><span class="section-number-4">3.5.1</span> Exercise 1</h4>
<div class="outline-text-4" id="text-3-5-1"> <div class="outline-text-4" id="text-3-5-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2599,8 +2600,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-org8b8a874" class="outline-5"> <div id="outline-container-org26c9d33" class="outline-5">
<h5 id="org8b8a874"><span class="section-number-5">3.5.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="org26c9d33"><span class="section-number-5">3.5.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5>
<div class="outline-text-5" id="text-3-5-1-1"> <div class="outline-text-5" id="text-3-5-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -2633,8 +2634,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-org02c9606" class="outline-4"> <div id="outline-container-org3e39526" class="outline-4">
<h4 id="org02c9606"><span class="section-number-4">3.5.2</span> Exercise 2</h4> <h4 id="org3e39526"><span class="section-number-4">3.5.2</span> Exercise 2</h4>
<div class="outline-text-4" id="text-3-5-2"> <div class="outline-text-4" id="text-3-5-2">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2728,8 +2729,8 @@ Modify the previous program to introduce the drifted diffusion scheme.
</div> </div>
</div> </div>
<div id="outline-container-org4580e2f" class="outline-5"> <div id="outline-container-orga3cf826" class="outline-5">
<h5 id="org4580e2f"><span class="section-number-5">3.5.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="orga3cf826"><span class="section-number-5">3.5.2.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5>
<div class="outline-text-5" id="text-3-5-2-1"> <div class="outline-text-5" id="text-3-5-2-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -2915,12 +2916,12 @@ A = 0.78839866666666658 +/- 3.2503783452043152E-004
</div> </div>
</div> </div>
<div id="outline-container-orgdceacd5" class="outline-2"> <div id="outline-container-orgebb7fc1" class="outline-2">
<h2 id="orgdceacd5"><span class="section-number-2">4</span> Diffusion Monte Carlo&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h2> <h2 id="orgebb7fc1"><span class="section-number-2">4</span> Diffusion Monte Carlo&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
</div> </div>
<div id="outline-container-org3941aac" class="outline-3"> <div id="outline-container-org8cf0ab0" class="outline-3">
<h3 id="org3941aac"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3> <h3 id="org8cf0ab0"><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:
@ -2979,8 +2980,8 @@ system.
</div> </div>
</div> </div>
<div id="outline-container-orge021b97" class="outline-3"> <div id="outline-container-org78a65a5" class="outline-3">
<h3 id="orge021b97"><span class="section-number-3">4.2</span> Diffusion and branching</h3> <h3 id="org78a65a5"><span class="section-number-3">4.2</span> Diffusion and branching</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
@ -3034,8 +3035,8 @@ the combination of a diffusion process and a branching process.
</div> </div>
</div> </div>
<div id="outline-container-org0d37b59" class="outline-3"> <div id="outline-container-org00d8981" class="outline-3">
<h3 id="org0d37b59"><span class="section-number-3">4.3</span> Importance sampling</h3> <h3 id="org00d8981"><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,
@ -3092,8 +3093,8 @@ error known as the <i>fixed node error</i>.
</p> </p>
</div> </div>
<div id="outline-container-orgc0fb19f" class="outline-4"> <div id="outline-container-org024b07c" class="outline-4">
<h4 id="orgc0fb19f"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4> <h4 id="org024b07c"><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>
\[ \[
@ -3155,8 +3156,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
</div> </div>
<div id="outline-container-orgb90c671" class="outline-3"> <div id="outline-container-orgce840bc" class="outline-3">
<h3 id="orgb90c671"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3> <h3 id="orgce840bc"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-4-4">
<p> <p>
Now that we have a process to sample \(\Pi(\mathbf{r},\tau) = Now that we have a process to sample \(\Pi(\mathbf{r},\tau) =
@ -3208,8 +3209,8 @@ energies computed with the trial wave function.
</div> </div>
</div> </div>
<div id="outline-container-org971d2c0" class="outline-3"> <div id="outline-container-orga97437f" class="outline-3">
<h3 id="org971d2c0"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3> <h3 id="orga97437f"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3>
<div class="outline-text-3" id="text-4-5"> <div class="outline-text-3" id="text-4-5">
<p> <p>
Instead of having a variable number of particles to simulate the Instead of having a variable number of particles to simulate the
@ -3261,13 +3262,13 @@ code, so this is what we will do in the next section.
</div> </div>
</div> </div>
<div id="outline-container-orgf89d780" class="outline-3"> <div id="outline-container-org54d9e47" class="outline-3">
<h3 id="orgf89d780"><span class="section-number-3">4.6</span> Hydrogen atom</h3> <h3 id="org54d9e47"><span class="section-number-3">4.6</span> Hydrogen atom</h3>
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<h4 id="orga1bc199"><span class="section-number-4">4.6.1</span> Exercise</h4> <h4 id="org77c2d07"><span class="section-number-4">4.6.1</span> Exercise</h4>
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<p> <p>
@ -3366,8 +3367,8 @@ energy of H for any value of \(a\).
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<h5 id="orga0bccd7"><span class="section-number-5">4.6.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5> <h5 id="orgc3fb998"><span class="section-number-5">4.6.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5>
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<p> <p>
<b>Python</b> <b>Python</b>
@ -3583,8 +3584,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
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<h3 id="org182141b"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3> <h3 id="orgb88b04d"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
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<p> <p>
We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the
@ -3605,8 +3606,8 @@ the nuclei.
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<h2 id="org51ed78c"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2> <h2 id="org622ff55"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2>
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<li class="off"><code>[&#xa0;]</code> Give some hints of how much time is required for each section</li> <li class="off"><code>[&#xa0;]</code> Give some hints of how much time is required for each section</li>
@ -3622,7 +3623,7 @@ the H\(_2\) molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.</li>
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<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: 2021-01-29 Fri 12:24</p> <p class="date">Created: 2021-01-30 Sat 08:25</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>
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