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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2021-01-13 Wed 17:26 --> <!-- 2021-01-19 Tue 09:46 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Quantum Monte Carlo</title> <title>Quantum Monte Carlo</title>
@ -257,63 +257,64 @@ 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="#org42ed47a">1. Introduction</a></li> <li><a href="#org1781353">1. Introduction</a></li>
<li><a href="#org53e00dc">2. Numerical evaluation of the energy</a> <li><a href="#org6fb8eba">2. Numerical evaluation of the energy</a>
<ul> <ul>
<li><a href="#orgcb06e2a">2.1. Local energy</a> <li><a href="#org8a01cb3">2.1. Local energy</a>
<ul> <ul>
<li><a href="#org0e5d0fc">2.1.1. Exercise 1</a></li> <li><a href="#org42e7b45">2.1.1. Exercise 1</a></li>
<li><a href="#org9b94717">2.1.2. Exercise 2</a></li> <li><a href="#org86d3136">2.1.2. Exercise 2</a></li>
<li><a href="#org2bcd29b">2.1.3. Exercise 3</a></li> <li><a href="#org4802db2">2.1.3. Exercise 3</a></li>
<li><a href="#org1203f81">2.1.4. Exercise 4</a></li> <li><a href="#org30be6bd">2.1.4. Exercise 4</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org5d34261">2.2. Plot of the local energy along the \(x\) axis</a> <li><a href="#org66f58ad">2.2. Plot of the local energy along the \(x\) axis</a>
<ul> <ul>
<li><a href="#org3cb394d">2.2.1. Exercise</a></li> <li><a href="#orgc800493">2.2.1. Exercise</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org35b332e">2.3. Numerical estimation of the energy</a> <li><a href="#org7d33595">2.3. Numerical estimation of the energy</a>
<ul> <ul>
<li><a href="#org7927f35">2.3.1. Exercise</a></li> <li><a href="#org49eff63">2.3.1. Exercise</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org565a2b8">2.4. Variance of the local energy</a> <li><a href="#org2377ff3">2.4. Variance of the local energy</a>
<ul> <ul>
<li><a href="#org773f3b3">2.4.1. Exercise</a></li> <li><a href="#orga585c67">2.4.1. Exercise (optional)</a></li>
<li><a href="#orgc835875">2.4.2. Exercise</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org55d88f8">3. Variational Monte Carlo</a> <li><a href="#org83c6a25">3. Variational Monte Carlo</a>
<ul> <ul>
<li><a href="#orgd8d3851">3.1. Computation of the statistical error</a> <li><a href="#org85df268">3.1. Computation of the statistical error</a>
<ul> <ul>
<li><a href="#org96e37d8">3.1.1. Exercise</a></li> <li><a href="#org57f2588">3.1.1. Exercise</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org406e2b4">3.2. Uniform sampling in the box</a> <li><a href="#org9818f33">3.2. Uniform sampling in the box</a>
<ul> <ul>
<li><a href="#org15e2594">3.2.1. Exercise</a></li> <li><a href="#org91d5f7e">3.2.1. Exercise</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org9158965">3.3. Gaussian sampling</a> <li><a href="#orgdc3ee3e">3.3. Gaussian sampling</a>
<ul> <ul>
<li><a href="#org4488162">3.3.1. Exercise</a></li> <li><a href="#org6fe98d4">3.3.1. Exercise</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org451cd23">3.4. Sampling with \(\Psi^2\)</a> <li><a href="#orge0e488e">3.4. Sampling with \(\Psi^2\)</a>
<ul> <ul>
<li><a href="#orgb72825c">3.4.1. Importance sampling</a></li> <li><a href="#orgcfcdbfd">3.4.1. Importance sampling</a></li>
<li><a href="#orgff63624">3.4.2. Metropolis algorithm</a></li> <li><a href="#orgc52903b">3.4.2. Metropolis algorithm</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org075fffc">4. <span class="todo TODO">TODO</span> Diffusion Monte Carlo</a> <li><a href="#orgca1fad0">4. <span class="todo TODO">TODO</span> Diffusion Monte Carlo</a>
<ul> <ul>
<li><a href="#orgd2b43b9">4.1. Hydrogen atom</a></li> <li><a href="#org8275af4">4.1. Hydrogen atom</a></li>
<li><a href="#orgc22643f">4.2. Dihydrogen</a></li> <li><a href="#org745b49b">4.2. Dihydrogen</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -321,8 +322,8 @@ for the JavaScript code in this tag.
</div> </div>
<div id="outline-container-org42ed47a" class="outline-2"> <div id="outline-container-org1781353" class="outline-2">
<h2 id="org42ed47a"><span class="section-number-2">1</span> Introduction</h2> <h2 id="org1781353"><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>
We propose different exercises to understand quantum Monte Carlo (QMC) We propose different exercises to understand quantum Monte Carlo (QMC)
@ -364,8 +365,8 @@ interpreted as a single precision value
</div> </div>
<div id="outline-container-org53e00dc" class="outline-2"> <div id="outline-container-org6fb8eba" class="outline-2">
<h2 id="org53e00dc"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2> <h2 id="org6fb8eba"><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
@ -439,13 +440,13 @@ E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
\end{eqnarray*} \end{eqnarray*}
</div> </div>
<div id="outline-container-orgcb06e2a" class="outline-3"> <div id="outline-container-org8a01cb3" class="outline-3">
<h3 id="orgcb06e2a"><span class="section-number-3">2.1</span> Local energy</h3> <h3 id="org8a01cb3"><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">
</div> </div>
<div id="outline-container-org0e5d0fc" class="outline-4"> <div id="outline-container-org42e7b45" class="outline-4">
<h4 id="org0e5d0fc"><span class="section-number-4">2.1.1</span> Exercise 1</h4> <h4 id="org42e7b45"><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>
@ -489,8 +490,8 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-org9b94717" class="outline-4"> <div id="outline-container-org86d3136" class="outline-4">
<h4 id="org9b94717"><span class="section-number-4">2.1.2</span> Exercise 2</h4> <h4 id="org86d3136"><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>
@ -525,8 +526,8 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-org2bcd29b" class="outline-4"> <div id="outline-container-org4802db2" class="outline-4">
<h4 id="org2bcd29b"><span class="section-number-4">2.1.3</span> Exercise 3</h4> <h4 id="org4802db2"><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>
@ -607,8 +608,8 @@ So the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-org1203f81" class="outline-4"> <div id="outline-container-org30be6bd" class="outline-4">
<h4 id="org1203f81"><span class="section-number-4">2.1.4</span> Exercise 4</h4> <h4 id="org30be6bd"><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>
@ -651,14 +652,14 @@ local energy.
</div> </div>
</div> </div>
<div id="outline-container-org5d34261" class="outline-3"> <div id="outline-container-org66f58ad" class="outline-3">
<h3 id="org5d34261"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3> <h3 id="org66f58ad"><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> </div>
<div id="outline-container-org3cb394d" class="outline-4"> <div id="outline-container-orgc800493" class="outline-4">
<h4 id="org3cb394d"><span class="section-number-4">2.2.1</span> Exercise</h4> <h4 id="orgc800493"><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>
@ -775,8 +776,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
</div> </div>
</div> </div>
<div id="outline-container-org35b332e" class="outline-3"> <div id="outline-container-org7d33595" class="outline-3">
<h3 id="org35b332e"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3> <h3 id="org7d33595"><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 \(\mathbf{r}_i\)
@ -806,8 +807,8 @@ The energy is biased because:
</div> </div>
<div id="outline-container-org7927f35" class="outline-4"> <div id="outline-container-org49eff63" class="outline-4">
<h4 id="org7927f35"><span class="section-number-4">2.3.1</span> Exercise</h4> <h4 id="org49eff63"><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>
@ -917,8 +918,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
</div> </div>
</div> </div>
<div id="outline-container-org565a2b8" class="outline-3"> <div id="outline-container-org2377ff3" class="outline-3">
<h3 id="org565a2b8"><span class="section-number-3">2.4</span> Variance of the local energy</h3> <h3 id="org2377ff3"><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 \(\Psi\)
@ -931,6 +932,11 @@ energy associated with \(\Psi\) around the average:
\sigma^2(E_L) = \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \left[ \sigma^2(E_L) = \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \left[
E_L(\mathbf{r}) - E \right]^2 \, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} E_L(\mathbf{r}) - E \right]^2 \, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
\] \]
which can be simplified as
</p>
<p>
\[ \sigma^2(E_L) = \langle E^2 \rangle - \langle E \rangle^2 \]
</p> </p>
<p> <p>
@ -940,12 +946,26 @@ energy can be used as a measure of the quality of a wave function.
</p> </p>
</div> </div>
<div id="outline-container-org773f3b3" class="outline-4"> <div id="outline-container-orga585c67" class="outline-4">
<h4 id="org773f3b3"><span class="section-number-4">2.4.1</span> Exercise</h4> <h4 id="orga585c67"><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>
Compute a numerical estimate of the variance of the local energy Prove that :
\[ \sigma^2(E_L) = \langle E^2 \rangle - \langle E \rangle^2 \]
</p>
</div>
</div>
</div>
<div id="outline-container-orgc835875" class="outline-4">
<h4 id="orgc835875"><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>
Add the calculation of the variance to the previous code, and
compute a numerical estimate of the variance of the local energy
in a grid of \(50\times50\times50\) points in the range in a grid of \(50\times50\times50\) points in the range
\((-5,-5,-5) \((-5,-5,-5)
\le \mathbf{r} \le (5,5,5)\) for different values of \(a\). \le \mathbf{r} \le (5,5,5)\) for different values of \(a\).
@ -967,6 +987,7 @@ in a grid of \(50\times50\times50\) points in the range
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]: <span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
<span style="color: #a0522d;">E</span> = 0. <span style="color: #a0522d;">E</span> = 0.
<span style="color: #a0522d;">E2</span> = 0.
<span style="color: #a0522d;">norm</span> = 0. <span style="color: #a0522d;">norm</span> = 0.
<span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> interval: <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[0] = x <span style="color: #a0522d;">r</span>[0] = x
@ -978,20 +999,11 @@ in a grid of \(50\times50\times50\) points in the range
<span style="color: #a0522d;">w</span> = w * w * delta <span style="color: #a0522d;">w</span> = w * w * delta
<span style="color: #a0522d;">El</span> = e_loc(a, r) <span style="color: #a0522d;">El</span> = e_loc(a, r)
<span style="color: #a0522d;">E</span> += w * El <span style="color: #a0522d;">E</span> += w * El
<span style="color: #a0522d;">E2</span> += w * El*El
<span style="color: #a0522d;">norm</span> += w <span style="color: #a0522d;">norm</span> += w
<span style="color: #a0522d;">E</span> = E / norm <span style="color: #a0522d;">E</span> = E / norm
<span style="color: #a0522d;">s2</span> = 0. <span style="color: #a0522d;">E2</span> = E2 / norm
<span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> interval: <span style="color: #a0522d;">s2</span> = E2 - E*E
<span style="color: #a0522d;">r</span>[0] = x
<span style="color: #a020f0;">for</span> y <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[1] = y
<span style="color: #a020f0;">for</span> z <span style="color: #a020f0;">in</span> interval:
<span style="color: #a0522d;">r</span>[2] = z
<span style="color: #a0522d;">w</span> = psi(a, r)
<span style="color: #a0522d;">w</span> = w * w * delta
<span style="color: #a0522d;">El</span> = e_loc(a, r)
<span style="color: #a0522d;">s2</span> += w * (El - E)**2
<span style="color: #a0522d;">s2</span> = s2 / norm
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E:10.8f} \t \sigma^2 = {s2:10.8f}"</span>) <span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E:10.8f} \t \sigma^2 = {s2:10.8f}"</span>)
</pre> </pre>
</div> </div>
@ -1083,8 +1095,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.806881
</div> </div>
<div id="outline-container-org55d88f8" class="outline-2"> <div id="outline-container-org83c6a25" class="outline-2">
<h2 id="org55d88f8"><span class="section-number-2">3</span> Variational Monte Carlo</h2> <h2 id="org83c6a25"><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
@ -1100,8 +1112,8 @@ interval.
</p> </p>
</div> </div>
<div id="outline-container-orgd8d3851" class="outline-3"> <div id="outline-container-org85df268" class="outline-3">
<h3 id="orgd8d3851"><span class="section-number-3">3.1</span> Computation of the statistical error</h3> <h3 id="org85df268"><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 \(M\)
@ -1141,8 +1153,8 @@ And the confidence interval is given by
</p> </p>
</div> </div>
<div id="outline-container-org96e37d8" class="outline-4"> <div id="outline-container-org57f2588" class="outline-4">
<h4 id="org96e37d8"><span class="section-number-4">3.1.1</span> Exercise</h4> <h4 id="org57f2588"><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>
@ -1191,8 +1203,8 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-org406e2b4" class="outline-3"> <div id="outline-container-org9818f33" class="outline-3">
<h3 id="org406e2b4"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3> <h3 id="org9818f33"><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
@ -1226,8 +1238,8 @@ statistical error.
</p> </p>
</div> </div>
<div id="outline-container-org15e2594" class="outline-4"> <div id="outline-container-org91d5f7e" class="outline-4">
<h4 id="org15e2594"><span class="section-number-4">3.2.1</span> Exercise</h4> <h4 id="org91d5f7e"><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>
@ -1337,8 +1349,8 @@ E = -0.49588321986667677 +/- 7.1758863546737969E-004
</div> </div>
</div> </div>
<div id="outline-container-org9158965" class="outline-3"> <div id="outline-container-orgdc3ee3e" class="outline-3">
<h3 id="org9158965"><span class="section-number-3">3.3</span> Gaussian sampling</h3> <h3 id="orgdc3ee3e"><span class="section-number-3">3.3</span> Gaussian sampling</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
We will now improve the sampling and allow to sample in the whole We will now improve the sampling and allow to sample in the whole
@ -1434,8 +1446,8 @@ average energy can be computed as
</div> </div>
<div id="outline-container-org4488162" class="outline-4"> <div id="outline-container-org6fe98d4" class="outline-4">
<h4 id="org4488162"><span class="section-number-4">3.3.1</span> Exercise</h4> <h4 id="org6fe98d4"><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>
@ -1546,8 +1558,8 @@ E = -0.49517104619091717 +/- 1.0685523607878961E-004
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<div id="outline-container-org451cd23" class="outline-3"> <div id="outline-container-orge0e488e" class="outline-3">
<h3 id="org451cd23"><span class="section-number-3">3.4</span> Sampling with \(\Psi^2\)</h3> <h3 id="orge0e488e"><span class="section-number-3">3.4</span> Sampling with \(\Psi^2\)</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
We will now use the square of the wave function to make the sampling: We will now use the square of the wave function to make the sampling:
@ -1572,8 +1584,8 @@ the local energies, each with a weight of 1.
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<div id="outline-container-orgb72825c" class="outline-4"> <div id="outline-container-orgcfcdbfd" class="outline-4">
<h4 id="orgb72825c"><span class="section-number-4">3.4.1</span> Importance sampling</h4> <h4 id="orgcfcdbfd"><span class="section-number-4">3.4.1</span> Importance sampling</h4>
<div class="outline-text-4" id="text-3-4-1"> <div class="outline-text-4" id="text-3-4-1">
<p> <p>
To generate the probability density \(\Psi^2\), we consider a To generate the probability density \(\Psi^2\), we consider a
@ -1686,7 +1698,7 @@ variance \(\tau\,2D\).
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<ol class="org-ol"> <ol class="org-ol">
<li><a id="org8061073"></a>Exercise 1<br /> <li><a id="org6f8bd81"></a>Exercise 1<br />
<div class="outline-text-5" id="text-3-4-1-1"> <div class="outline-text-5" id="text-3-4-1-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -1722,7 +1734,7 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
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</li> </li>
<li><a id="org22449f0"></a><span class="todo TODO">TODO</span> Exercise 2<br /> <li><a id="org4976dcc"></a><span class="todo TODO">TODO</span> Exercise 2<br />
<div class="outline-text-5" id="text-3-4-1-2"> <div class="outline-text-5" id="text-3-4-1-2">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -1834,8 +1846,8 @@ E = -0.48584030499187431 +/- 1.0411743995438257E-004
</ol> </ol>
</div> </div>
<div id="outline-container-orgff63624" class="outline-4"> <div id="outline-container-orgc52903b" class="outline-4">
<h4 id="orgff63624"><span class="section-number-4">3.4.2</span> Metropolis algorithm</h4> <h4 id="orgc52903b"><span class="section-number-4">3.4.2</span> Metropolis algorithm</h4>
<div class="outline-text-4" id="text-3-4-2"> <div class="outline-text-4" id="text-3-4-2">
<p> <p>
Discretizing the differential equation to generate the desired Discretizing the differential equation to generate the desired
@ -1896,7 +1908,7 @@ the simulation.
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<ol class="org-ol"> <ol class="org-ol">
<li><a id="orgca456a4"></a>Exercise<br /> <li><a id="orgefc66ea"></a>Exercise<br />
<div class="outline-text-5" id="text-3-4-2-1"> <div class="outline-text-5" id="text-3-4-2-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2052,17 +2064,17 @@ A = 0.78861366666666655 +/- 3.5096729498002445E-004
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<div id="outline-container-org075fffc" class="outline-2"> <div id="outline-container-orgca1fad0" class="outline-2">
<h2 id="org075fffc"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> Diffusion Monte Carlo</h2> <h2 id="orgca1fad0"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> Diffusion Monte Carlo</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
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<div id="outline-container-orgd2b43b9" class="outline-3"> <div id="outline-container-org8275af4" class="outline-3">
<h3 id="orgd2b43b9"><span class="section-number-3">4.1</span> Hydrogen atom</h3> <h3 id="org8275af4"><span class="section-number-3">4.1</span> Hydrogen atom</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
</div> </div>
<ol class="org-ol"> <ol class="org-ol">
<li><a id="orgf0b78c7"></a>Exercise<br /> <li><a id="org64b6416"></a>Exercise<br />
<div class="outline-text-5" id="text-4-1-0-1"> <div class="outline-text-5" id="text-4-1-0-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -2221,8 +2233,8 @@ A = 0.78861366666666655 +/- 3.5096729498002445E-004
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<div id="outline-container-orgc22643f" class="outline-3"> <div id="outline-container-org745b49b" class="outline-3">
<h3 id="orgc22643f"><span class="section-number-3">4.2</span> Dihydrogen</h3> <h3 id="org745b49b"><span class="section-number-3">4.2</span> Dihydrogen</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<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
@ -2244,7 +2256,7 @@ the nuclei.
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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: 2021-01-13 Wed 17:26</p> <p class="date">Created: 2021-01-19 Tue 09:46</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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</body> </body>