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<head> <head>
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<title>Quantum Monte Carlo</title> <title>Quantum Monte Carlo</title>
@ -329,148 +329,147 @@ 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="#org8dcaa4e">1. Introduction</a></li> <li><a href="#org0ddeb79">1. Introduction</a>
<li><a href="#orgf2d601e">2. Numerical evaluation of the energy</a>
<ul> <ul>
<li><a href="#org0b86853">2.1. Local energy</a> <li><a href="#org6f0b456">1.1. Local energy</a>
<ul> <ul>
<li><a href="#orgb4f1dcb">2.1.1. Exercise 1</a> <li><a href="#org78005c6">1.1.1. Exercise 1</a>
<ul> <ul>
<li><a href="#orge1171f3">2.1.1.1. Solution</a></li> <li><a href="#org69e8d62">1.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org805f102">2.1.2. Exercise 2</a> <li><a href="#orgeb36ece">1.1.2. Exercise 2</a>
<ul> <ul>
<li><a href="#orge839f82">2.1.2.1. Solution</a></li> <li><a href="#orgad4cf40">1.1.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org30a89b7">2.1.3. Exercise 3</a> <li><a href="#orgde512b1">1.1.3. Exercise 3</a>
<ul> <ul>
<li><a href="#org37f3a90">2.1.3.1. Solution</a></li> <li><a href="#org3c2172c">1.1.3.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8a71cb9">2.1.4. Exercise 4</a> <li><a href="#org81e2a70">1.1.4. Exercise 4</a>
<ul> <ul>
<li><a href="#org3127c46">2.1.4.1. Solution</a></li> <li><a href="#orgfc45098">1.1.4.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org5ae05ca">2.1.5. Exercise 5</a> <li><a href="#org6d8d5bf">1.1.5. Exercise 5</a>
<ul> <ul>
<li><a href="#orgde68619">2.1.5.1. Solution</a></li> <li><a href="#org29c7bdc">1.1.5.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org6b6a61c">2.2. Plot of the local energy along the \(x\) axis</a> <li><a href="#org8a94425">1.2. Plot of the local energy along the \(x\) axis</a>
<ul> <ul>
<li><a href="#orga530380">2.2.1. Exercise</a> <li><a href="#org6f8bb22">1.2.1. Exercise</a>
<ul> <ul>
<li><a href="#orgcd9f1bf">2.2.1.1. Solution</a></li> <li><a href="#orgde055e6">1.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org2387d29">2.3. Numerical estimation of the energy</a> <li><a href="#org1a0dc00">1.3. Numerical estimation of the energy</a>
<ul> <ul>
<li><a href="#orgde278da">2.3.1. Exercise</a> <li><a href="#org8e504ef">1.3.1. Exercise</a>
<ul> <ul>
<li><a href="#org73954f7">2.3.1.1. Solution</a></li> <li><a href="#orgae9b7e6">1.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgd0c5f25">2.4. Variance of the local energy</a> <li><a href="#org18a28d8">1.4. Variance of the local energy</a>
<ul> <ul>
<li><a href="#orgdd8878b">2.4.1. Exercise (optional)</a> <li><a href="#org1e571a6">1.4.1. Exercise (optional)</a>
<ul> <ul>
<li><a href="#org874aece">2.4.1.1. Solution</a></li> <li><a href="#orga8d0d0c">1.4.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org55c03ac">2.4.2. Exercise</a> <li><a href="#org8e32fec">1.4.2. Exercise</a>
<ul> <ul>
<li><a href="#orgff8da50">2.4.2.1. Solution</a></li> <li><a href="#org8c29b0e">1.4.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org500fc9e">3. Variational Monte Carlo</a> <li><a href="#orgad22bd9">2. Variational Monte Carlo</a>
<ul> <ul>
<li><a href="#org1395f72">3.1. Computation of the statistical error</a> <li><a href="#orgf4230ee">2.1. Computation of the statistical error</a>
<ul> <ul>
<li><a href="#orgbf3b7c1">3.1.1. Exercise</a> <li><a href="#org0a476b8">2.1.1. Exercise</a>
<ul> <ul>
<li><a href="#org96a3dcd">3.1.1.1. Solution</a></li> <li><a href="#org22fa572">2.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org6f903d5">3.2. Uniform sampling in the box</a> <li><a href="#org51ecf49">2.2. Uniform sampling in the box</a>
<ul> <ul>
<li><a href="#orgd5d2e4f">3.2.1. Exercise</a> <li><a href="#org8683a22">2.2.1. Exercise</a>
<ul> <ul>
<li><a href="#orge0a3bca">3.2.1.1. Solution</a></li> <li><a href="#org580c059">2.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org31acf84">3.3. Metropolis sampling with \(\Psi^2\)</a> <li><a href="#org0ce09e0">2.3. Metropolis sampling with \(\Psi^2\)</a>
<ul> <ul>
<li><a href="#org6d54039">3.3.1. Exercise</a> <li><a href="#org5cdc0da">2.3.1. Exercise</a>
<ul> <ul>
<li><a href="#org8084209">3.3.1.1. Solution</a></li> <li><a href="#orgb3affe7">2.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org92d024f">3.4. Gaussian random number generator</a></li> <li><a href="#org1d0503e">2.4. Gaussian random number generator</a></li>
<li><a href="#org1a64e1d">3.5. Generalized Metropolis algorithm</a> <li><a href="#org3c33d0c">2.5. Generalized Metropolis algorithm</a>
<ul> <ul>
<li><a href="#org97c1b29">3.5.1. Exercise 1</a> <li><a href="#orgdcf7766">2.5.1. Exercise 1</a>
<ul> <ul>
<li><a href="#org19e85c9">3.5.1.1. Solution</a></li> <li><a href="#org0468289">2.5.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgaef93b5">3.5.2. Exercise 2</a> <li><a href="#orgf25990d">2.5.2. Exercise 2</a>
<ul> <ul>
<li><a href="#org17fe7c6">3.5.2.1. Solution</a></li> <li><a href="#org1b40299">2.5.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org9f51d1e">4. Diffusion Monte Carlo</a> <li><a href="#org7159b49">3. Diffusion Monte Carlo</a>
<ul> <ul>
<li><a href="#org344d9de">4.1. Schrödinger equation in imaginary time</a></li> <li><a href="#org851f123">3.1. Schrödinger equation in imaginary time</a></li>
<li><a href="#org06862e7">4.2. Diffusion and branching</a></li> <li><a href="#orgb662256">3.2. Diffusion and branching</a></li>
<li><a href="#orgdf62087">4.3. Importance sampling</a> <li><a href="#org0ed0c07">3.3. Importance sampling</a>
<ul> <ul>
<li><a href="#org855e049">4.3.1. Appendix : Details of the Derivation</a></li> <li><a href="#org5a760ca">3.3.1. Appendix : Details of the Derivation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org6fbf6b7">4.4. Fixed-node DMC energy</a></li> <li><a href="#orgf9fe967">3.4. Fixed-node DMC energy</a></li>
<li><a href="#org4379250">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li> <li><a href="#orgbd478f8">3.5. Pure Diffusion Monte Carlo (PDMC)</a></li>
<li><a href="#org3a30f06">4.6. Hydrogen atom</a> <li><a href="#org2828b37">3.6. Hydrogen atom</a>
<ul> <ul>
<li><a href="#orgba6aa2f">4.6.1. Exercise</a> <li><a href="#orge0e4678">3.6.1. Exercise</a>
<ul> <ul>
<li><a href="#org2311033">4.6.1.1. Solution</a></li> <li><a href="#org14a27f3">3.6.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org81121c2">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li> <li><a href="#orge51396e">3.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgf5ad8c9">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li> <li><a href="#org6c71552">4. <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-org8dcaa4e" class="outline-2"> <div id="outline-container-org0ddeb79" class="outline-2">
<h2 id="org8dcaa4e"><span class="section-number-2">1</span> Introduction</h2> <h2 id="org0ddeb79"><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
@ -508,12 +507,88 @@ is defined everywhere, continuous, and infinitely differentiable.
All the quantities are expressed in <i>atomic units</i> (energies, All the quantities are expressed in <i>atomic units</i> (energies,
coordinates, etc). coordinates, etc).
</p> </p>
</div>
</div>
<div id="outline-container-orgf2d601e" class="outline-2"> <p>
<h2 id="orgf2d601e"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2> ** Energy and local energy
<div class="outline-text-2" id="text-2"> </p>
<p>
For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
</p>
<p>
\[
E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
\]
</p>
<p>
where \(\mathbf{r}\) denotes the 3N-dimensional electronic coordinates.
</p>
<p>
The electronic energy of a system, \(E\), can be rewritten in terms of the
local energy \(E_L(\mathbf{r})\) as
</p>
\begin{eqnarray*}
E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
= \frac{\int \Psi(\mathbf{r})\, \hat{H} \Psi(\mathbf{r})\, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \\
& = & \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
= \frac{\int \left[\Psi(\mathbf{r})\right]^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
\end{eqnarray*}
<p>
For few dimensions, one can easily compute \(E\) by evaluating the integrals on a grid but, for a high number of dimensions, one can resort to Monte Carlo techniques to compute \(E\).
</p>
<p>
To this aim, recall that the probabilistic <i>expected value</i> of an arbitrary function \(f(x)\)
with respect to a probability density function \(p(x)\) is given by
</p>
<p>
\[ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx, \]
</p>
<p>
where a probability density function \(p(x)\) is non-negative
and integrates to one:
</p>
<p>
\[ \int_{-\infty}^\infty p(x)\,dx = 1. \]
</p>
<p>
Similarly, we can view the the energy of a system, \(E\), as the expected value of the local energy with respect to
a probability density \(p(\mathbf{r}}\) defined in 3\(N\) dimensions:
</p>
<p>
\[ E = \int E_L(\mathbf{r}) p(\mathbf{r})\,d\mathbf{r}} \equiv \langle E_L \rangle_{\Psi^2}\,, \]
</p>
<p>
where the probability density is given by the square of the wave function:
</p>
<p>
\[ p(\mathbf{r}) = \frac{|Psi(\mathbf{r}|^2){\int \left |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,. \]
</p>
<p>
If we can sample configurations \(\{\mathbf{r}\}\) distributed as \(p\), we can estimate \(E\) as the average of the local energy computed over these configurations:
</p>
<p>
$$ E &asymp; \frac{1}{M} &sum;<sub>i=1</sub><sup>M</sup> E<sub>L</sub>(\mathbf{r}<sub>i</sub>} \,.
</p>
<ul class="org-ul">
<li>Numerical evaluation of the energy of the hydrogen atoms</li>
</ul>
<p> <p>
In this section, we consider the hydrogen atom with the following In this section, we consider the hydrogen atom with the following
wave function: wave function:
@ -540,62 +615,15 @@ eigenfunction of the Hamiltonian
To do that, we will compute the local energy, defined as To do that, we will compute the local energy, defined as
</p> </p>
<p>
\[
E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
\]
</p>
<p> <p>
and check whether it is constant. and check whether it is constant.
</p> </p>
<p>
In general, the electronic energy of a system, \(E\), can be rewritten as the expectation value of the
local energy \(E(\mathbf{r})\) with respect to the 3N-dimensional
electron density given by the square of the wave function:
</p>
\begin{eqnarray*}
E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
= \frac{\int \Psi(\mathbf{r})\, \hat{H} \Psi(\mathbf{r})\, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \\
& = & \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
= \frac{\int \left[\Psi(\mathbf{r})\right]^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
= \langle E_L \rangle_{\Psi^2}\,,
\end{eqnarray*}
<p>
where \(\mathbf{r}\) is the vector of the 3N-dimensional electronic coordinates (\(N=1\) for the hydrogen atom).
</p>
<p>
For a small number of dimensions, one can compute \(E\) by evaluating the integrals on a grid. However,
</p>
<p>
The probabilistic <i>expected value</i> of an arbitrary function \(f(x)\)
with respect to a probability density function \(p(x)\) is given by
</p>
<p>
\[ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx, \]
</p>
<p>
where probability density function \(p(x)\) is non-negative
and integrates to one:
</p>
<p>
\[ \int_{-\infty}^\infty p(x)\,dx = 1. \]
</p>
</div> </div>
<div id="outline-container-org6f0b456" class="outline-3">
<h3 id="org6f0b456"><span class="section-number-3">1.1</span> Local energy</h3>
<div class="outline-text-3" id="text-1-1">
<div id="outline-container-org0b86853" class="outline-3">
<h3 id="org0b86853"><span class="section-number-3">2.1</span> Local energy</h3>
<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 :
<code>hydrogen.py</code> if you use Python, or <code>hydrogen.f90</code> is you use <code>hydrogen.py</code> if you use Python, or <code>hydrogen.f90</code> is you use
@ -617,9 +645,9 @@ to catch the error.
</div> </div>
</div> </div>
<div id="outline-container-orgb4f1dcb" class="outline-4"> <div id="outline-container-org78005c6" class="outline-4">
<h4 id="orgb4f1dcb"><span class="section-number-4">2.1.1</span> Exercise 1</h4> <h4 id="org78005c6"><span class="section-number-4">1.1.1</span> Exercise 1</h4>
<div class="outline-text-4" id="text-2-1-1"> <div class="outline-text-4" id="text-1-1-1">
<div class="exercise"> <div class="exercise">
<p> <p>
Write a function which computes the potential at \(\mathbf{r}\). Write a function which computes the potential at \(\mathbf{r}\).
@ -662,9 +690,9 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-orge1171f3" class="outline-5"> <div id="outline-container-org69e8d62" class="outline-5">
<h5 id="orge1171f3"><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="org69e8d62"><span class="section-number-5">1.1.1.1</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-1-1-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -703,9 +731,9 @@ and returns the potential.
</div> </div>
</div> </div>
<div id="outline-container-org805f102" class="outline-4"> <div id="outline-container-orgeb36ece" class="outline-4">
<h4 id="org805f102"><span class="section-number-4">2.1.2</span> Exercise 2</h4> <h4 id="orgeb36ece"><span class="section-number-4">1.1.2</span> Exercise 2</h4>
<div class="outline-text-4" id="text-2-1-2"> <div class="outline-text-4" id="text-1-1-2">
<div class="exercise"> <div class="exercise">
<p> <p>
Write a function which computes the wave function at \(\mathbf{r}\). Write a function which computes the wave function at \(\mathbf{r}\).
@ -739,9 +767,9 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-orge839f82" class="outline-5"> <div id="outline-container-orgad4cf40" class="outline-5">
<h5 id="orge839f82"><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="orgad4cf40"><span class="section-number-5">1.1.2.1</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-1-1-2-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -767,9 +795,9 @@ input arguments, and returns a scalar.
</div> </div>
</div> </div>
<div id="outline-container-org30a89b7" class="outline-4"> <div id="outline-container-orgde512b1" class="outline-4">
<h4 id="org30a89b7"><span class="section-number-4">2.1.3</span> Exercise 3</h4> <h4 id="orgde512b1"><span class="section-number-4">1.1.3</span> Exercise 3</h4>
<div class="outline-text-4" id="text-2-1-3"> <div class="outline-text-4" id="text-1-1-3">
<div class="exercise"> <div class="exercise">
<p> <p>
Write a function which computes the local kinetic energy at \(\mathbf{r}\). Write a function which computes the local kinetic energy at \(\mathbf{r}\).
@ -849,9 +877,9 @@ So the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-org37f3a90" class="outline-5"> <div id="outline-container-org3c2172c" class="outline-5">
<h5 id="org37f3a90"><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="org3c2172c"><span class="section-number-5">1.1.3.1</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-1-1-3-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -891,9 +919,9 @@ So the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-org8a71cb9" class="outline-4"> <div id="outline-container-org81e2a70" class="outline-4">
<h4 id="org8a71cb9"><span class="section-number-4">2.1.4</span> Exercise 4</h4> <h4 id="org81e2a70"><span class="section-number-4">1.1.4</span> Exercise 4</h4>
<div class="outline-text-4" id="text-2-1-4"> <div class="outline-text-4" id="text-1-1-4">
<div class="exercise"> <div class="exercise">
<p> <p>
Write a function which computes the local energy at \(\mathbf{r}\), Write a function which computes the local energy at \(\mathbf{r}\),
@ -935,9 +963,9 @@ local kinetic energy.
</div> </div>
</div> </div>
<div id="outline-container-org3127c46" class="outline-5"> <div id="outline-container-orgfc45098" class="outline-5">
<h5 id="org3127c46"><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="orgfc45098"><span class="section-number-5">1.1.4.1</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-1-1-4-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -966,9 +994,9 @@ local kinetic energy.
</div> </div>
</div> </div>
<div id="outline-container-org5ae05ca" class="outline-4"> <div id="outline-container-org6d8d5bf" class="outline-4">
<h4 id="org5ae05ca"><span class="section-number-4">2.1.5</span> Exercise 5</h4> <h4 id="org6d8d5bf"><span class="section-number-4">1.1.5</span> Exercise 5</h4>
<div class="outline-text-4" id="text-2-1-5"> <div class="outline-text-4" id="text-1-1-5">
<div class="exercise"> <div class="exercise">
<p> <p>
Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(\hat{H}\). Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(\hat{H}\).
@ -977,9 +1005,9 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
</div> </div>
</div> </div>
<div id="outline-container-orgde68619" class="outline-5"> <div id="outline-container-org29c7bdc" class="outline-5">
<h5 id="orgde68619"><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="org29c7bdc"><span class="section-number-5">1.1.5.1</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-1-1-5-1">
\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} -
\frac{1}{|\mathbf{r}|} \\ \frac{1}{|\mathbf{r}|} \\
@ -998,9 +1026,9 @@ equal to -0.5 atomic units.
</div> </div>
</div> </div>
<div id="outline-container-org6b6a61c" class="outline-3"> <div id="outline-container-org8a94425" class="outline-3">
<h3 id="org6b6a61c"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3> <h3 id="org8a94425"><span class="section-number-3">1.2</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-1-2">
<div class="note"> <div class="note">
<p> <p>
The potential and the kinetic energy both diverge at \(r=0\), so we The potential and the kinetic energy both diverge at \(r=0\), so we
@ -1010,9 +1038,9 @@ choose a grid which does not contain the origin.
</div> </div>
</div> </div>
<div id="outline-container-orga530380" class="outline-4"> <div id="outline-container-org6f8bb22" class="outline-4">
<h4 id="orga530380"><span class="section-number-4">2.2.1</span> Exercise</h4> <h4 id="org6f8bb22"><span class="section-number-4">1.2.1</span> Exercise</h4>
<div class="outline-text-4" id="text-2-2-1"> <div class="outline-text-4" id="text-1-2-1">
<div class="exercise"> <div class="exercise">
<p> <p>
For multiple values of \(a\) (0.1, 0.2, 0.5, 1., 1.5, 2.), plot the For multiple values of \(a\) (0.1, 0.2, 0.5, 1., 1.5, 2.), plot the
@ -1094,9 +1122,9 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
</div> </div>
</div> </div>
<div id="outline-container-orgcd9f1bf" class="outline-5"> <div id="outline-container-orgde055e6" class="outline-5">
<h5 id="orgcd9f1bf"><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="orgde055e6"><span class="section-number-5">1.2.1.1</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-1-2-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -1170,9 +1198,9 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
</div> </div>
</div> </div>
<div id="outline-container-org2387d29" class="outline-3"> <div id="outline-container-org1a0dc00" class="outline-3">
<h3 id="org2387d29"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3> <h3 id="org1a0dc00"><span class="section-number-3">1.3</span> Numerical estimation of the energy</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-1-3">
<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 \(\langle E_L \rangle_{\Psi^2}\)
@ -1201,9 +1229,9 @@ The energy is biased because:
</div> </div>
<div id="outline-container-orgde278da" class="outline-4"> <div id="outline-container-org8e504ef" class="outline-4">
<h4 id="orgde278da"><span class="section-number-4">2.3.1</span> Exercise</h4> <h4 id="org8e504ef"><span class="section-number-4">1.3.1</span> Exercise</h4>
<div class="outline-text-4" id="text-2-3-1"> <div class="outline-text-4" id="text-1-3-1">
<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 in a grid of
@ -1271,9 +1299,9 @@ To compile the Fortran and run it:
</div> </div>
</div> </div>
<div id="outline-container-org73954f7" class="outline-5"> <div id="outline-container-orgae9b7e6" class="outline-5">
<h5 id="org73954f7"><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="orgae9b7e6"><span class="section-number-5">1.3.1.1</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-1-3-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -1387,9 +1415,9 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002
</div> </div>
</div> </div>
<div id="outline-container-orgd0c5f25" class="outline-3"> <div id="outline-container-org18a28d8" class="outline-3">
<h3 id="orgd0c5f25"><span class="section-number-3">2.4</span> Variance of the local energy</h3> <h3 id="org18a28d8"><span class="section-number-3">1.4</span> Variance of the local energy</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-1-4">
<p> <p>
The variance of the local energy is a functional of \(\Psi\) The variance of the local energy is a functional of \(\Psi\)
which measures the magnitude of the fluctuations of the local which measures the magnitude of the fluctuations of the local
@ -1415,9 +1443,9 @@ energy can be used as a measure of the quality of a wave function.
</p> </p>
</div> </div>
<div id="outline-container-orgdd8878b" class="outline-4"> <div id="outline-container-org1e571a6" class="outline-4">
<h4 id="orgdd8878b"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4> <h4 id="org1e571a6"><span class="section-number-4">1.4.1</span> Exercise (optional)</h4>
<div class="outline-text-4" id="text-2-4-1"> <div class="outline-text-4" id="text-1-4-1">
<div class="exercise"> <div class="exercise">
<p> <p>
Prove that : Prove that :
@ -1427,9 +1455,9 @@ Prove that :
</div> </div>
</div> </div>
<div id="outline-container-org874aece" class="outline-5"> <div id="outline-container-orga8d0d0c" class="outline-5">
<h5 id="org874aece"><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="orga8d0d0c"><span class="section-number-5">1.4.1.1</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-1-4-1-1">
<p> <p>
\(\bar{E} = \langle E \rangle\) is a constant, so \(\langle \bar{E} \(\bar{E} = \langle E \rangle\) is a constant, so \(\langle \bar{E}
\rangle = \bar{E}\) . \rangle = \bar{E}\) .
@ -1447,9 +1475,9 @@ Prove that :
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org55c03ac" class="outline-4"> <div id="outline-container-org8e32fec" class="outline-4">
<h4 id="org55c03ac"><span class="section-number-4">2.4.2</span> Exercise</h4> <h4 id="org8e32fec"><span class="section-number-4">1.4.2</span> Exercise</h4>
<div class="outline-text-4" id="text-2-4-2"> <div class="outline-text-4" id="text-1-4-2">
<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
@ -1522,9 +1550,9 @@ To compile and run:
</div> </div>
</div> </div>
<div id="outline-container-orgff8da50" class="outline-5"> <div id="outline-container-org8c29b0e" class="outline-5">
<h5 id="orgff8da50"><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="org8c29b0e"><span class="section-number-5">1.4.2.1</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-1-4-2-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -1660,9 +1688,9 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
</div> </div>
</div> </div>
<div id="outline-container-org500fc9e" class="outline-2"> <div id="outline-container-orgad22bd9" class="outline-2">
<h2 id="org500fc9e"><span class="section-number-2">3</span> Variational Monte Carlo</h2> <h2 id="orgad22bd9"><span class="section-number-2">2</span> Variational Monte Carlo</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-2">
<p> <p>
Numerical integration with deterministic methods is very efficient Numerical integration with deterministic methods is very efficient
in low dimensions. When the number of dimensions becomes large, in low dimensions. When the number of dimensions becomes large,
@ -1677,9 +1705,9 @@ interval.
</p> </p>
</div> </div>
<div id="outline-container-org1395f72" class="outline-3"> <div id="outline-container-orgf4230ee" class="outline-3">
<h3 id="org1395f72"><span class="section-number-3">3.1</span> Computation of the statistical error</h3> <h3 id="orgf4230ee"><span class="section-number-3">2.1</span> Computation of the statistical error</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
To compute the statistical error, you need to perform \(M\) To compute the statistical error, you need to perform \(M\)
independent Monte Carlo calculations. You will obtain \(M\) different independent Monte Carlo calculations. You will obtain \(M\) different
@ -1718,9 +1746,9 @@ And the confidence interval is given by
</p> </p>
</div> </div>
<div id="outline-container-orgbf3b7c1" class="outline-4"> <div id="outline-container-org0a476b8" class="outline-4">
<h4 id="orgbf3b7c1"><span class="section-number-4">3.1.1</span> Exercise</h4> <h4 id="org0a476b8"><span class="section-number-4">2.1.1</span> Exercise</h4>
<div class="outline-text-4" id="text-3-1-1"> <div class="outline-text-4" id="text-2-1-1">
<div class="exercise"> <div class="exercise">
<p> <p>
Write a function returning the average and statistical error of an Write a function returning the average and statistical error of an
@ -1757,9 +1785,9 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-org96a3dcd" class="outline-5"> <div id="outline-container-org22fa572" class="outline-5">
<h5 id="org96a3dcd"><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="org22fa572"><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-3-1-1-1"> <div class="outline-text-5" id="text-2-1-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -1817,9 +1845,9 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-org6f903d5" class="outline-3"> <div id="outline-container-org51ecf49" class="outline-3">
<h3 id="org6f903d5"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3> <h3 id="org51ecf49"><span class="section-number-3">2.2</span> Uniform sampling in the box</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-2-2">
<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
energy of the hydrogen atom. energy of the hydrogen atom.
@ -1852,9 +1880,9 @@ compute the statistical error.
</p> </p>
</div> </div>
<div id="outline-container-orgd5d2e4f" class="outline-4"> <div id="outline-container-org8683a22" class="outline-4">
<h4 id="orgd5d2e4f"><span class="section-number-4">3.2.1</span> Exercise</h4> <h4 id="org8683a22"><span class="section-number-4">2.2.1</span> Exercise</h4>
<div class="outline-text-4" id="text-3-2-1"> <div class="outline-text-4" id="text-2-2-1">
<div class="exercise"> <div class="exercise">
<p> <p>
Parameterize the wave function with \(a=0.9\). Perform 30 Parameterize the wave function with \(a=0.9\). Perform 30
@ -1953,9 +1981,9 @@ well as the index of the current step.
</div> </div>
</div> </div>
<div id="outline-container-orge0a3bca" class="outline-5"> <div id="outline-container-org580c059" class="outline-5">
<h5 id="orge0a3bca"><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="org580c059"><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-3-2-1-1"> <div class="outline-text-5" id="text-2-2-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -2068,9 +2096,9 @@ E = -0.49518773675598715 +/- 5.2391494923686175E-004
</div> </div>
</div> </div>
<div id="outline-container-org31acf84" class="outline-3"> <div id="outline-container-org0ce09e0" class="outline-3">
<h3 id="org31acf84"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3> <h3 id="org0ce09e0"><span class="section-number-3">2.3</span> Metropolis sampling with \(\Psi^2\)</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-2-3">
<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
points distributed with the probability density points distributed with the probability density
@ -2157,9 +2185,9 @@ step such that the acceptance rate is close to 0.5 is a good compromise.
</div> </div>
<div id="outline-container-org6d54039" class="outline-4"> <div id="outline-container-org5cdc0da" class="outline-4">
<h4 id="org6d54039"><span class="section-number-4">3.3.1</span> Exercise</h4> <h4 id="org5cdc0da"><span class="section-number-4">2.3.1</span> Exercise</h4>
<div class="outline-text-4" id="text-3-3-1"> <div class="outline-text-4" id="text-2-3-1">
<div class="exercise"> <div class="exercise">
<p> <p>
Modify the program of the previous section to compute the energy, Modify the program of the previous section to compute the energy,
@ -2265,9 +2293,9 @@ Can you observe a reduction in the statistical error?
</div> </div>
</div> </div>
<div id="outline-container-org8084209" class="outline-5"> <div id="outline-container-orgb3affe7" class="outline-5">
<h5 id="org8084209"><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="orgb3affe7"><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-3-3-1-1"> <div class="outline-text-5" id="text-2-3-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -2411,9 +2439,9 @@ A = 0.51695266666666673 +/- 4.0445505648997396E-004
</div> </div>
</div> </div>
<div id="outline-container-org92d024f" class="outline-3"> <div id="outline-container-org1d0503e" class="outline-3">
<h3 id="org92d024f"><span class="section-number-3">3.4</span> Gaussian random number generator</h3> <h3 id="org1d0503e"><span class="section-number-3">2.4</span> Gaussian random number generator</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
To obtain Gaussian-distributed random numbers, you can apply the To obtain Gaussian-distributed random numbers, you can apply the
<a href="https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform">Box Muller transform</a> to uniform random numbers: <a href="https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform">Box Muller transform</a> to uniform random numbers:
@ -2474,9 +2502,9 @@ 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-org1a64e1d" class="outline-3"> <div id="outline-container-org3c33d0c" class="outline-3">
<h3 id="org1a64e1d"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3> <h3 id="org3c33d0c"><span class="section-number-3">2.5</span> Generalized Metropolis algorithm</h3>
<div class="outline-text-3" id="text-3-5"> <div class="outline-text-3" id="text-2-5">
<p> <p>
One can use more efficient numerical schemes to move the electrons, One can use more efficient numerical schemes to move the electrons,
but the Metropolis accepation step has to be adapted accordingly: but the Metropolis accepation step has to be adapted accordingly:
@ -2574,9 +2602,9 @@ The transition probability becomes:
</div> </div>
<div id="outline-container-org97c1b29" class="outline-4"> <div id="outline-container-orgdcf7766" class="outline-4">
<h4 id="org97c1b29"><span class="section-number-4">3.5.1</span> Exercise 1</h4> <h4 id="orgdcf7766"><span class="section-number-4">2.5.1</span> Exercise 1</h4>
<div class="outline-text-4" id="text-3-5-1"> <div class="outline-text-4" id="text-2-5-1">
<div class="exercise"> <div class="exercise">
<p> <p>
Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\). Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\).
@ -2609,9 +2637,9 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-org19e85c9" class="outline-5"> <div id="outline-container-org0468289" class="outline-5">
<h5 id="org19e85c9"><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="org0468289"><span class="section-number-5">2.5.1.1</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-2-5-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -2643,9 +2671,9 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-orgaef93b5" class="outline-4"> <div id="outline-container-orgf25990d" class="outline-4">
<h4 id="orgaef93b5"><span class="section-number-4">3.5.2</span> Exercise 2</h4> <h4 id="orgf25990d"><span class="section-number-4">2.5.2</span> Exercise 2</h4>
<div class="outline-text-4" id="text-3-5-2"> <div class="outline-text-4" id="text-2-5-2">
<div class="exercise"> <div class="exercise">
<p> <p>
Modify the previous program to introduce the drifted diffusion scheme. Modify the previous program to introduce the drifted diffusion scheme.
@ -2738,9 +2766,9 @@ Modify the previous program to introduce the drifted diffusion scheme.
</div> </div>
</div> </div>
<div id="outline-container-org17fe7c6" class="outline-5"> <div id="outline-container-org1b40299" class="outline-5">
<h5 id="org17fe7c6"><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="org1b40299"><span class="section-number-5">2.5.2.1</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-2-5-2-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -2925,13 +2953,13 @@ A = 0.78839866666666658 +/- 3.2503783452043152E-004
</div> </div>
</div> </div>
<div id="outline-container-org9f51d1e" class="outline-2"> <div id="outline-container-org7159b49" class="outline-2">
<h2 id="org9f51d1e"><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="org7159b49"><span class="section-number-2">3</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-3">
</div> </div>
<div id="outline-container-org344d9de" class="outline-3"> <div id="outline-container-org851f123" class="outline-3">
<h3 id="org344d9de"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3> <h3 id="org851f123"><span class="section-number-3">3.1</span> Schrödinger equation in imaginary time</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
Consider the time-dependent Schrödinger equation: Consider the time-dependent Schrödinger equation:
</p> </p>
@ -2989,9 +3017,9 @@ system.
</div> </div>
</div> </div>
<div id="outline-container-org06862e7" class="outline-3"> <div id="outline-container-orgb662256" class="outline-3">
<h3 id="org06862e7"><span class="section-number-3">4.2</span> Diffusion and branching</h3> <h3 id="orgb662256"><span class="section-number-3">3.2</span> Diffusion and branching</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-3-2">
<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
</p> </p>
@ -3044,9 +3072,9 @@ the combination of a diffusion process and a branching process.
</div> </div>
</div> </div>
<div id="outline-container-orgdf62087" class="outline-3"> <div id="outline-container-org0ed0c07" class="outline-3">
<h3 id="orgdf62087"><span class="section-number-3">4.3</span> Importance sampling</h3> <h3 id="org0ed0c07"><span class="section-number-3">3.3</span> Importance sampling</h3>
<div class="outline-text-3" id="text-4-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
In a molecular system, the potential is far from being constant, In a molecular system, the potential is far from being constant,
and diverges at inter-particle coalescence points. Hence, when the and diverges at inter-particle coalescence points. Hence, when the
@ -3102,9 +3130,9 @@ error known as the <i>fixed node error</i>.
</p> </p>
</div> </div>
<div id="outline-container-org855e049" class="outline-4"> <div id="outline-container-org5a760ca" class="outline-4">
<h4 id="org855e049"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4> <h4 id="org5a760ca"><span class="section-number-4">3.3.1</span> Appendix : Details of the Derivation</h4>
<div class="outline-text-4" id="text-4-3-1"> <div class="outline-text-4" id="text-3-3-1">
<p> <p>
\[ \[
-\frac{\partial \psi(\mathbf{r},\tau)}{\partial \tau} \Psi_T(\mathbf{r}) = -\frac{\partial \psi(\mathbf{r},\tau)}{\partial \tau} \Psi_T(\mathbf{r}) =
@ -3165,9 +3193,9 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
</div> </div>
<div id="outline-container-org6fbf6b7" class="outline-3"> <div id="outline-container-orgf9fe967" class="outline-3">
<h3 id="org6fbf6b7"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3> <h3 id="orgf9fe967"><span class="section-number-3">3.4</span> Fixed-node DMC energy</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
Now that we have a process to sample \(\Pi(\mathbf{r},\tau) = Now that we have a process to sample \(\Pi(\mathbf{r},\tau) =
\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})\), we can compute the exact \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})\), we can compute the exact
@ -3218,9 +3246,9 @@ energies computed with the trial wave function.
</div> </div>
</div> </div>
<div id="outline-container-org4379250" class="outline-3"> <div id="outline-container-orgbd478f8" class="outline-3">
<h3 id="org4379250"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3> <h3 id="orgbd478f8"><span class="section-number-3">3.5</span> Pure Diffusion Monte Carlo (PDMC)</h3>
<div class="outline-text-3" id="text-4-5"> <div class="outline-text-3" id="text-3-5">
<p> <p>
Instead of having a variable number of particles to simulate the Instead of having a variable number of particles to simulate the
branching process, one can choose to sample \([\Psi_T(\mathbf{r})]^2\) instead of branching process, one can choose to sample \([\Psi_T(\mathbf{r})]^2\) instead of
@ -3271,14 +3299,14 @@ code, so this is what we will do in the next section.
</div> </div>
</div> </div>
<div id="outline-container-org3a30f06" class="outline-3"> <div id="outline-container-org2828b37" class="outline-3">
<h3 id="org3a30f06"><span class="section-number-3">4.6</span> Hydrogen atom</h3> <h3 id="org2828b37"><span class="section-number-3">3.6</span> Hydrogen atom</h3>
<div class="outline-text-3" id="text-4-6"> <div class="outline-text-3" id="text-3-6">
</div> </div>
<div id="outline-container-orgba6aa2f" class="outline-4"> <div id="outline-container-orge0e4678" class="outline-4">
<h4 id="orgba6aa2f"><span class="section-number-4">4.6.1</span> Exercise</h4> <h4 id="orge0e4678"><span class="section-number-4">3.6.1</span> Exercise</h4>
<div class="outline-text-4" id="text-4-6-1"> <div class="outline-text-4" id="text-3-6-1">
<div class="exercise"> <div class="exercise">
<p> <p>
Modify the Metropolis VMC program to introduce the PDMC weight. Modify the Metropolis VMC program to introduce the PDMC weight.
@ -3376,9 +3404,9 @@ energy of H for any value of \(a\).
</div> </div>
</div> </div>
<div id="outline-container-org2311033" class="outline-5"> <div id="outline-container-org14a27f3" class="outline-5">
<h5 id="org2311033"><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="org14a27f3"><span class="section-number-5">3.6.1.1</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-3-6-1-1">
<p> <p>
<b>Python</b> <b>Python</b>
</p> </p>
@ -3593,9 +3621,9 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
</div> </div>
<div id="outline-container-org81121c2" class="outline-3"> <div id="outline-container-orge51396e" class="outline-3">
<h3 id="org81121c2"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3> <h3 id="orge51396e"><span class="section-number-3">3.7</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-3-7">
<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
\(1s\) orbitals of the hydrogen atoms: \(1s\) orbitals of the hydrogen atoms:
@ -3615,9 +3643,9 @@ the nuclei.
</div> </div>
<div id="outline-container-orgf5ad8c9" class="outline-2"> <div id="outline-container-org6c71552" class="outline-2">
<h2 id="orgf5ad8c9"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</h2> <h2 id="org6c71552"><span class="section-number-2">4</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-4">
<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>
<li class="off"><code>[&#xa0;]</code> Prepare 4 questions for the exam: multiple-choice questions <li class="off"><code>[&#xa0;]</code> Prepare 4 questions for the exam: multiple-choice questions
@ -3632,7 +3660,7 @@ the H\(_2\) molecule at $R$=1.4010 bohr. Answer: 0.17406 a.u.</li>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Anthony Scemama, Claudia Filippi</p> <p class="author">Author: Anthony Scemama, Claudia Filippi</p>
<p class="date">Created: 2021-01-30 Sat 12:21</p> <p class="date">Created: 2021-01-30 Sat 21:41</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>