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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="#org6777e9e">1. Introduction</a> <li><a href="#orge2a6b44">1. Introduction</a>
<ul> <ul>
<li><a href="#org4f28a82">1.1. Energy and local energy</a></li> <li><a href="#org0f16b33">1.1. Energy and local energy</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org911b777">2. Numerical evaluation of the energy of the hydrogen atom</a> <li><a href="#orgf8de501">2. Numerical evaluation of the energy of the hydrogen atom</a>
<ul> <ul>
<li><a href="#org924f091">2.1. Local energy</a> <li><a href="#org13e906f">2.1. Local energy</a>
<ul> <ul>
<li><a href="#org79cde61">2.1.1. Exercise 1</a> <li><a href="#orgd072c5b">2.1.1. Exercise 1</a>
<ul> <ul>
<li><a href="#orge822000">2.1.1.1. Solution</a></li> <li><a href="#org959a351">2.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org58189bc">2.1.2. Exercise 2</a> <li><a href="#org08e784d">2.1.2. Exercise 2</a>
<ul> <ul>
<li><a href="#org184891c">2.1.2.1. Solution</a></li> <li><a href="#orgb8d093c">2.1.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org798c2a6">2.1.3. Exercise 3</a> <li><a href="#orgc2112e2">2.1.3. Exercise 3</a>
<ul> <ul>
<li><a href="#orgf70cacc">2.1.3.1. Solution</a></li> <li><a href="#orgcfbe3ee">2.1.3.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd4cc122">2.1.4. Exercise 4</a> <li><a href="#org19dde43">2.1.4. Exercise 4</a>
<ul> <ul>
<li><a href="#org2ded1e4">2.1.4.1. Solution</a></li> <li><a href="#orgde229be">2.1.4.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgca45d7a">2.1.5. Exercise 5</a> <li><a href="#orgecbce75">2.1.5. Exercise 5</a>
<ul> <ul>
<li><a href="#org74ebe9b">2.1.5.1. Solution</a></li> <li><a href="#org541ad24">2.1.5.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgaa52522">2.2. Plot of the local energy along the \(x\) axis</a> <li><a href="#orgce16153">2.2. Plot of the local energy along the \(x\) axis</a>
<ul> <ul>
<li><a href="#org7dd8ed4">2.2.1. Exercise</a> <li><a href="#orgee4e8d1">2.2.1. Exercise</a>
<ul> <ul>
<li><a href="#org526da2a">2.2.1.1. Solution</a></li> <li><a href="#org7652a96">2.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org604128a">2.3. Numerical estimation of the energy</a> <li><a href="#org8523d1c">2.3. Numerical estimation of the energy</a>
<ul> <ul>
<li><a href="#orgd398af3">2.3.1. Exercise</a> <li><a href="#org35bb3fb">2.3.1. Exercise</a>
<ul> <ul>
<li><a href="#orga1c4b7e">2.3.1.1. Solution</a></li> <li><a href="#org1bfc07a">2.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org62fcf61">2.4. Variance of the local energy</a> <li><a href="#orgf845070">2.4. Variance of the local energy</a>
<ul> <ul>
<li><a href="#org1ebf660">2.4.1. Exercise (optional)</a> <li><a href="#orgf56984c">2.4.1. Exercise (optional)</a>
<ul> <ul>
<li><a href="#orgb0a8731">2.4.1.1. Solution</a></li> <li><a href="#org67a0d97">2.4.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8197a34">2.4.2. Exercise</a> <li><a href="#orgd31720a">2.4.2. Exercise</a>
<ul> <ul>
<li><a href="#org62f416d">2.4.2.1. Solution</a></li> <li><a href="#org5b88542">2.4.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgd879d7c">3. Variational Monte Carlo</a> <li><a href="#orgee08883">3. Variational Monte Carlo</a>
<ul> <ul>
<li><a href="#org62254aa">3.1. Computation of the statistical error</a> <li><a href="#org61b40b7">3.1. Computation of the statistical error</a>
<ul> <ul>
<li><a href="#org35368a2">3.1.1. Exercise</a> <li><a href="#orgab3cffd">3.1.1. Exercise</a>
<ul> <ul>
<li><a href="#orgf1b0dd9">3.1.1.1. Solution</a></li> <li><a href="#orgdc05008">3.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgc3cc175">3.2. Uniform sampling in the box</a> <li><a href="#org31e93ba">3.2. Uniform sampling in the box</a>
<ul> <ul>
<li><a href="#org605c779">3.2.1. Exercise</a> <li><a href="#org355c3a1">3.2.1. Exercise</a>
<ul> <ul>
<li><a href="#org1d0eb6e">3.2.1.1. Solution</a></li> <li><a href="#org64e4ca3">3.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org6feb28a">3.3. Metropolis sampling with \(\Psi^2\)</a> <li><a href="#org2b8b15f">3.3. Metropolis sampling with \(\Psi^2\)</a>
<ul> <ul>
<li><a href="#orgd212d29">3.3.1. Exercise</a> <li><a href="#org4b23b03">3.3.1. Exercise</a>
<ul> <ul>
<li><a href="#org959ea46">3.3.1.1. Solution</a></li> <li><a href="#orgca36c12">3.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org4036058">3.4. Gaussian random number generator</a></li> <li><a href="#org2ecccd9">3.4. Gaussian random number generator</a></li>
<li><a href="#orgb0dd016">3.5. Generalized Metropolis algorithm</a> <li><a href="#org911b3c9">3.5. Generalized Metropolis algorithm</a>
<ul> <ul>
<li><a href="#org93fecc0">3.5.1. Exercise 1</a> <li><a href="#org63393d2">3.5.1. Exercise 1</a>
<ul> <ul>
<li><a href="#orgae49be1">3.5.1.1. Solution</a></li> <li><a href="#org2b1ae2a">3.5.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org3a2cc38">3.5.2. Exercise 2</a> <li><a href="#orgd03861d">3.5.2. Exercise 2</a>
<ul> <ul>
<li><a href="#org7075a5c">3.5.2.1. Solution</a></li> <li><a href="#org75ee22c">3.5.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgbfcd1f7">4. Diffusion Monte Carlo</a> <li><a href="#org623ac9b">4. Diffusion Monte Carlo</a>
<ul> <ul>
<li><a href="#org1cefc43">4.1. Schrödinger equation in imaginary time</a></li> <li><a href="#org75d615c">4.1. Schrödinger equation in imaginary time</a></li>
<li><a href="#org99d2c1e">4.2. Diffusion and branching</a></li> <li><a href="#org1497029">4.2. Diffusion and branching</a></li>
<li><a href="#org06c76e9">4.3. Importance sampling</a> <li><a href="#orgefdcd47">4.3. Importance sampling</a>
<ul> <ul>
<li><a href="#orgf070b5b">4.3.1. Appendix : Details of the Derivation</a></li> <li><a href="#org0b3840e">4.3.1. Appendix : Details of the Derivation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org9e034a9">4.4. Fixed-node DMC energy</a></li> <li><a href="#org7d0e033">4.4. Fixed-node DMC energy</a></li>
<li><a href="#org7a6a278">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li> <li><a href="#org3696228">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li>
<li><a href="#orgb3846a0">4.6. Hydrogen atom</a> <li><a href="#org66fb772">4.6. Hydrogen atom</a>
<ul> <ul>
<li><a href="#org1c8423d">4.6.1. Exercise</a> <li><a href="#org9d98668">4.6.1. Exercise</a>
<ul> <ul>
<li><a href="#org220ff7f">4.6.1.1. Solution</a></li> <li><a href="#orgaf39a8c">4.6.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org47e0d2d">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li> <li><a href="#org7ecf66b">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
</ul> </ul>
</li> </li>
<li><a href="#orga58154e">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li> <li><a href="#org8bfdf8b">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-org6777e9e" class="outline-2"> <div id="outline-container-orge2a6b44" class="outline-2">
<h2 id="org6777e9e"><span class="section-number-2">1</span> Introduction</h2> <h2 id="orge2a6b44"><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
@ -514,8 +514,8 @@ coordinates, etc).
</p> </p>
</div> </div>
<div id="outline-container-org4f28a82" class="outline-3"> <div id="outline-container-org0f16b33" class="outline-3">
<h3 id="org4f28a82"><span class="section-number-3">1.1</span> Energy and local energy</h3> <h3 id="org0f16b33"><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
@ -593,8 +593,8 @@ $$ E &asymp; \frac{1}{M} &sum;<sub>i=1</sub><sup>M</sup> E<sub>L</sub>(\mathbf{r
</div> </div>
</div> </div>
<div id="outline-container-org911b777" class="outline-2"> <div id="outline-container-orgf8de501" class="outline-2">
<h2 id="org911b777"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2> <h2 id="orgf8de501"><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
@ -623,8 +623,8 @@ To do that, we will compute the local energy and check whether it is constant.
</p> </p>
</div> </div>
<div id="outline-container-org924f091" class="outline-3"> <div id="outline-container-org13e906f" class="outline-3">
<h3 id="org924f091"><span class="section-number-3">2.1</span> Local energy</h3> <h3 id="org13e906f"><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.
@ -651,8 +651,8 @@ to catch the error.
</div> </div>
</div> </div>
<div id="outline-container-org79cde61" class="outline-4"> <div id="outline-container-orgd072c5b" class="outline-4">
<h4 id="org79cde61"><span class="section-number-4">2.1.1</span> Exercise 1</h4> <h4 id="orgd072c5b"><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>
@ -696,8 +696,8 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-orge822000" class="outline-5"> <div id="outline-container-org959a351" class="outline-5">
<h5 id="orge822000"><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="org959a351"><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>
@ -737,8 +737,8 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-org58189bc" class="outline-4"> <div id="outline-container-org08e784d" class="outline-4">
<h4 id="org58189bc"><span class="section-number-4">2.1.2</span> Exercise 2</h4> <h4 id="org08e784d"><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>
@ -773,8 +773,8 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-org184891c" class="outline-5"> <div id="outline-container-orgb8d093c" class="outline-5">
<h5 id="org184891c"><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="orgb8d093c"><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>
@ -801,8 +801,8 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-org798c2a6" class="outline-4"> <div id="outline-container-orgc2112e2" class="outline-4">
<h4 id="org798c2a6"><span class="section-number-4">2.1.3</span> Exercise 3</h4> <h4 id="orgc2112e2"><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>
@ -848,12 +848,12 @@ applied to the wave function gives:
<p> <p>
\[ \[
\Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r}) \Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r})\,.
\] \]
</p> </p>
<p> <p>
So the local kinetic energy is Therefore, the local kinetic energy is
\[ \[
-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right) -\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
\] \]
@ -883,8 +883,8 @@ So the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-orgf70cacc" class="outline-5"> <div id="outline-container-orgcfbe3ee" class="outline-5">
<h5 id="orgf70cacc"><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="orgcfbe3ee"><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>
@ -925,8 +925,8 @@ So the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-orgd4cc122" class="outline-4"> <div id="outline-container-org19dde43" class="outline-4">
<h4 id="orgd4cc122"><span class="section-number-4">2.1.4</span> Exercise 4</h4> <h4 id="org19dde43"><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>
@ -969,8 +969,8 @@ local kinetic energy.
</div> </div>
</div> </div>
<div id="outline-container-org2ded1e4" class="outline-5"> <div id="outline-container-orgde229be" class="outline-5">
<h5 id="org2ded1e4"><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="orgde229be"><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>
@ -1000,8 +1000,8 @@ local kinetic energy.
</div> </div>
</div> </div>
<div id="outline-container-orgca45d7a" class="outline-4"> <div id="outline-container-orgecbce75" class="outline-4">
<h4 id="orgca45d7a"><span class="section-number-4">2.1.5</span> Exercise 5</h4> <h4 id="orgecbce75"><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>
@ -1011,8 +1011,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
</div> </div>
</div> </div>
<div id="outline-container-org74ebe9b" class="outline-5"> <div id="outline-container-org541ad24" class="outline-5">
<h5 id="org74ebe9b"><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="org541ad24"><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} -
@ -1032,8 +1032,8 @@ equal to -0.5 atomic units.
</div> </div>
</div> </div>
<div id="outline-container-orgaa52522" class="outline-3"> <div id="outline-container-orgce16153" class="outline-3">
<h3 id="orgaa52522"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3> <h3 id="orgce16153"><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>
@ -1044,8 +1044,8 @@ choose a grid which does not contain the origin.
</div> </div>
</div> </div>
<div id="outline-container-org7dd8ed4" class="outline-4"> <div id="outline-container-orgee4e8d1" class="outline-4">
<h4 id="org7dd8ed4"><span class="section-number-4">2.2.1</span> Exercise</h4> <h4 id="orgee4e8d1"><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>
@ -1128,8 +1128,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
</div> </div>
</div> </div>
<div id="outline-container-org526da2a" class="outline-5"> <div id="outline-container-org7652a96" class="outline-5">
<h5 id="org526da2a"><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="org7652a96"><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>
@ -1204,14 +1204,14 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
</div> </div>
</div> </div>
<div id="outline-container-org604128a" class="outline-3"> <div id="outline-container-org8523d1c" class="outline-3">
<h3 id="org604128a"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3> <h3 id="org8523d1c"><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 \(\mathbf{r}_i\)
of size \(\delta \mathbf{r}\), the expression of \(\langle E_L \rangle_{\Psi^2}\) of size \(\delta \mathbf{r}\), the expression of \(\langle E_L \rangle_{\Psi^2}\)
becomes a weighted average of the local energy, where the weights becomes a weighted average of the local energy, where the weights
are the values of the probability density at \(\mathbf{r}_i\) are the values of the wave function square at \(\mathbf{r}_i\)
multiplied by the volume element: multiplied by the volume element:
</p> </p>
@ -1235,12 +1235,12 @@ The energy is biased because:
</div> </div>
<div id="outline-container-orgd398af3" class="outline-4"> <div id="outline-container-org35bb3fb" class="outline-4">
<h4 id="orgd398af3"><span class="section-number-4">2.3.1</span> Exercise</h4> <h4 id="org35bb3fb"><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>
Compute a numerical estimate of the energy in a grid of Compute a numerical estimate of the energy using a grid of
\(50\times50\times50\) points in the range \((-5,-5,-5) \le \(50\times50\times50\) points in the range \((-5,-5,-5) \le
\mathbf{r} \le (5,5,5)\). \mathbf{r} \le (5,5,5)\).
</p> </p>
@ -1305,8 +1305,8 @@ To compile the Fortran and run it:
</div> </div>
</div> </div>
<div id="outline-container-orga1c4b7e" class="outline-5"> <div id="outline-container-org1bfc07a" class="outline-5">
<h5 id="orga1c4b7e"><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="org1bfc07a"><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>
@ -1421,8 +1421,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
</div> </div>
</div> </div>
<div id="outline-container-org62fcf61" class="outline-3"> <div id="outline-container-orgf845070" class="outline-3">
<h3 id="org62fcf61"><span class="section-number-3">2.4</span> Variance of the local energy</h3> <h3 id="orgf845070"><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 \(\Psi\)
@ -1449,8 +1449,8 @@ energy can be used as a measure of the quality of a wave function.
</p> </p>
</div> </div>
<div id="outline-container-org1ebf660" class="outline-4"> <div id="outline-container-orgf56984c" class="outline-4">
<h4 id="org1ebf660"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4> <h4 id="orgf56984c"><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>
@ -1461,8 +1461,8 @@ Prove that :
</div> </div>
</div> </div>
<div id="outline-container-orgb0a8731" class="outline-5"> <div id="outline-container-org67a0d97" class="outline-5">
<h5 id="orgb0a8731"><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="org67a0d97"><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}
@ -1481,13 +1481,13 @@ Prove that :
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org8197a34" class="outline-4"> <div id="outline-container-orgd31720a" class="outline-4">
<h4 id="org8197a34"><span class="section-number-4">2.4.2</span> Exercise</h4> <h4 id="orgd31720a"><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>
Add the calculation of the variance to the previous code, and Add the calculation of the variance to the previous code, and
compute a numerical estimate of the variance of the local energy in compute a numerical estimate of the variance of the local energy using
a grid of \(50\times50\times50\) points in the range \((-5,-5,-5) \le a grid of \(50\times50\times50\) points in the range \((-5,-5,-5) \le
\mathbf{r} \le (5,5,5)\) for different values of \(a\). \mathbf{r} \le (5,5,5)\) for different values of \(a\).
</p> </p>
@ -1556,8 +1556,8 @@ To compile and run:
</div> </div>
</div> </div>
<div id="outline-container-org62f416d" class="outline-5"> <div id="outline-container-org5b88542" class="outline-5">
<h5 id="org62f416d"><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="org5b88542"><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>
@ -1694,8 +1694,8 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
</div> </div>
</div> </div>
<div id="outline-container-orgd879d7c" class="outline-2"> <div id="outline-container-orgee08883" class="outline-2">
<h2 id="orgd879d7c"><span class="section-number-2">3</span> Variational Monte Carlo</h2> <h2 id="orgee08883"><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
@ -1711,8 +1711,8 @@ interval.
</p> </p>
</div> </div>
<div id="outline-container-org62254aa" class="outline-3"> <div id="outline-container-org61b40b7" class="outline-3">
<h3 id="org62254aa"><span class="section-number-3">3.1</span> Computation of the statistical error</h3> <h3 id="org61b40b7"><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 \(M\)
@ -1752,8 +1752,8 @@ And the confidence interval is given by
</p> </p>
</div> </div>
<div id="outline-container-org35368a2" class="outline-4"> <div id="outline-container-orgab3cffd" class="outline-4">
<h4 id="org35368a2"><span class="section-number-4">3.1.1</span> Exercise</h4> <h4 id="orgab3cffd"><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>
@ -1791,8 +1791,8 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-orgf1b0dd9" class="outline-5"> <div id="outline-container-orgdc05008" class="outline-5">
<h5 id="orgf1b0dd9"><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="orgdc05008"><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>
@ -1851,8 +1851,8 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-orgc3cc175" class="outline-3"> <div id="outline-container-org31e93ba" class="outline-3">
<h3 id="orgc3cc175"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3> <h3 id="org31e93ba"><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
@ -1886,8 +1886,8 @@ compute the statistical error.
</p> </p>
</div> </div>
<div id="outline-container-org605c779" class="outline-4"> <div id="outline-container-org355c3a1" class="outline-4">
<h4 id="org605c779"><span class="section-number-4">3.2.1</span> Exercise</h4> <h4 id="org355c3a1"><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>
@ -1987,8 +1987,8 @@ well as the index of the current step.
</div> </div>
</div> </div>
<div id="outline-container-org1d0eb6e" class="outline-5"> <div id="outline-container-org64e4ca3" class="outline-5">
<h5 id="org1d0eb6e"><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="org64e4ca3"><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>
@ -2102,8 +2102,8 @@ E = -0.49518773675598715 +/- 5.2391494923686175E-004
</div> </div>
</div> </div>
<div id="outline-container-org6feb28a" class="outline-3"> <div id="outline-container-org2b8b15f" class="outline-3">
<h3 id="org6feb28a"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3> <h3 id="org2b8b15f"><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
@ -2191,8 +2191,8 @@ step such that the acceptance rate is close to 0.5 is a good compromise.
</div> </div>
<div id="outline-container-orgd212d29" class="outline-4"> <div id="outline-container-org4b23b03" class="outline-4">
<h4 id="orgd212d29"><span class="section-number-4">3.3.1</span> Exercise</h4> <h4 id="org4b23b03"><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>
@ -2299,8 +2299,8 @@ Can you observe a reduction in the statistical error?
</div> </div>
</div> </div>
<div id="outline-container-org959ea46" class="outline-5"> <div id="outline-container-orgca36c12" class="outline-5">
<h5 id="org959ea46"><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="orgca36c12"><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>
@ -2445,8 +2445,8 @@ A = 0.51695266666666673 +/- 4.0445505648997396E-004
</div> </div>
</div> </div>
<div id="outline-container-org4036058" class="outline-3"> <div id="outline-container-org2ecccd9" class="outline-3">
<h3 id="org4036058"><span class="section-number-3">3.4</span> Gaussian random number generator</h3> <h3 id="org2ecccd9"><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
@ -2508,8 +2508,8 @@ 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-orgb0dd016" class="outline-3"> <div id="outline-container-org911b3c9" class="outline-3">
<h3 id="orgb0dd016"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3> <h3 id="org911b3c9"><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,
@ -2608,8 +2608,8 @@ The transition probability becomes:
</div> </div>
<div id="outline-container-org93fecc0" class="outline-4"> <div id="outline-container-org63393d2" class="outline-4">
<h4 id="org93fecc0"><span class="section-number-4">3.5.1</span> Exercise 1</h4> <h4 id="org63393d2"><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>
@ -2643,8 +2643,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-orgae49be1" class="outline-5"> <div id="outline-container-org2b1ae2a" class="outline-5">
<h5 id="orgae49be1"><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="org2b1ae2a"><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>
@ -2677,8 +2677,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-org3a2cc38" class="outline-4"> <div id="outline-container-orgd03861d" class="outline-4">
<h4 id="org3a2cc38"><span class="section-number-4">3.5.2</span> Exercise 2</h4> <h4 id="orgd03861d"><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>
@ -2772,8 +2772,8 @@ Modify the previous program to introduce the drifted diffusion scheme.
</div> </div>
</div> </div>
<div id="outline-container-org7075a5c" class="outline-5"> <div id="outline-container-org75ee22c" class="outline-5">
<h5 id="org7075a5c"><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="org75ee22c"><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>
@ -2959,12 +2959,12 @@ A = 0.78839866666666658 +/- 3.2503783452043152E-004
</div> </div>
</div> </div>
<div id="outline-container-orgbfcd1f7" class="outline-2"> <div id="outline-container-org623ac9b" class="outline-2">
<h2 id="orgbfcd1f7"><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="org623ac9b"><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-org1cefc43" class="outline-3"> <div id="outline-container-org75d615c" class="outline-3">
<h3 id="org1cefc43"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3> <h3 id="org75d615c"><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:
@ -3023,8 +3023,8 @@ system.
</div> </div>
</div> </div>
<div id="outline-container-org99d2c1e" class="outline-3"> <div id="outline-container-org1497029" class="outline-3">
<h3 id="org99d2c1e"><span class="section-number-3">4.2</span> Diffusion and branching</h3> <h3 id="org1497029"><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
@ -3078,8 +3078,8 @@ the combination of a diffusion process and a branching process.
</div> </div>
</div> </div>
<div id="outline-container-org06c76e9" class="outline-3"> <div id="outline-container-orgefdcd47" class="outline-3">
<h3 id="org06c76e9"><span class="section-number-3">4.3</span> Importance sampling</h3> <h3 id="orgefdcd47"><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,
@ -3136,8 +3136,8 @@ error known as the <i>fixed node error</i>.
</p> </p>
</div> </div>
<div id="outline-container-orgf070b5b" class="outline-4"> <div id="outline-container-org0b3840e" class="outline-4">
<h4 id="orgf070b5b"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4> <h4 id="org0b3840e"><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>
\[ \[
@ -3199,8 +3199,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
</div> </div>
<div id="outline-container-org9e034a9" class="outline-3"> <div id="outline-container-org7d0e033" class="outline-3">
<h3 id="org9e034a9"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3> <h3 id="org7d0e033"><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) =
@ -3252,8 +3252,8 @@ energies computed with the trial wave function.
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<div id="outline-container-org7a6a278" class="outline-3"> <div id="outline-container-org3696228" class="outline-3">
<h3 id="org7a6a278"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3> <h3 id="org3696228"><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
@ -3305,13 +3305,13 @@ code, so this is what we will do in the next section.
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<div id="outline-container-orgb3846a0" class="outline-3"> <div id="outline-container-org66fb772" class="outline-3">
<h3 id="orgb3846a0"><span class="section-number-3">4.6</span> Hydrogen atom</h3> <h3 id="org66fb772"><span class="section-number-3">4.6</span> Hydrogen atom</h3>
<div class="outline-text-3" id="text-4-6"> <div class="outline-text-3" id="text-4-6">
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<div id="outline-container-org1c8423d" class="outline-4"> <div id="outline-container-org9d98668" class="outline-4">
<h4 id="org1c8423d"><span class="section-number-4">4.6.1</span> Exercise</h4> <h4 id="org9d98668"><span class="section-number-4">4.6.1</span> Exercise</h4>
<div class="outline-text-4" id="text-4-6-1"> <div class="outline-text-4" id="text-4-6-1">
<div class="exercise"> <div class="exercise">
<p> <p>
@ -3410,8 +3410,8 @@ energy of H for any value of \(a\).
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<div id="outline-container-org220ff7f" class="outline-5"> <div id="outline-container-orgaf39a8c" class="outline-5">
<h5 id="org220ff7f"><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="orgaf39a8c"><span class="section-number-5">4.6.1.1</span> Solution&#xa0;&#xa0;&#xa0;<span class="tag"><span class="solution">solution</span></span></h5>
<div class="outline-text-5" id="text-4-6-1-1"> <div class="outline-text-5" id="text-4-6-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
@ -3627,8 +3627,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
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<div id="outline-container-org47e0d2d" class="outline-3"> <div id="outline-container-org7ecf66b" class="outline-3">
<h3 id="org47e0d2d"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3> <h3 id="org7ecf66b"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3>
<div class="outline-text-3" id="text-4-7"> <div class="outline-text-3" id="text-4-7">
<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
@ -3649,8 +3649,8 @@ the nuclei.
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<div id="outline-container-orga58154e" class="outline-2"> <div id="outline-container-org8bfdf8b" class="outline-2">
<h2 id="orga58154e"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2> <h2 id="org8bfdf8b"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<ul class="org-ul"> <ul class="org-ul">
<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>
@ -3666,7 +3666,7 @@ 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: 2021-01-30 Sat 22:00</p> <p class="date">Created: 2021-01-30 Sat 22:30</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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