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<title>Quantum Monte Carlo</title> <title>Quantum Monte Carlo</title>
@ -329,152 +329,152 @@ 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="#orge2a6b44">1. Introduction</a> <li><a href="#org0a42dc6">1. Introduction</a>
<ul> <ul>
<li><a href="#org0f16b33">1.1. Energy and local energy</a></li> <li><a href="#org6174a0a">1.1. Energy and local energy</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgf8de501">2. Numerical evaluation of the energy of the hydrogen atom</a> <li><a href="#orge35e3b0">2. Numerical evaluation of the energy of the hydrogen atom</a>
<ul> <ul>
<li><a href="#org13e906f">2.1. Local energy</a> <li><a href="#org536ab35">2.1. Local energy</a>
<ul> <ul>
<li><a href="#orgd072c5b">2.1.1. Exercise 1</a> <li><a href="#orge1d3531">2.1.1. Exercise 1</a>
<ul> <ul>
<li><a href="#org959a351">2.1.1.1. Solution</a></li> <li><a href="#orge29b2d5">2.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org08e784d">2.1.2. Exercise 2</a> <li><a href="#org0082181">2.1.2. Exercise 2</a>
<ul> <ul>
<li><a href="#orgb8d093c">2.1.2.1. Solution</a></li> <li><a href="#orgcb5ffb6">2.1.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgc2112e2">2.1.3. Exercise 3</a> <li><a href="#org950428b">2.1.3. Exercise 3</a>
<ul> <ul>
<li><a href="#orgcfbe3ee">2.1.3.1. Solution</a></li> <li><a href="#orgb3b7bff">2.1.3.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org19dde43">2.1.4. Exercise 4</a> <li><a href="#orgd1768b7">2.1.4. Exercise 4</a>
<ul> <ul>
<li><a href="#orgde229be">2.1.4.1. Solution</a></li> <li><a href="#orgca62f3b">2.1.4.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgecbce75">2.1.5. Exercise 5</a> <li><a href="#orgd9f5f66">2.1.5. Exercise 5</a>
<ul> <ul>
<li><a href="#org541ad24">2.1.5.1. Solution</a></li> <li><a href="#org1e5b04a">2.1.5.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgce16153">2.2. Plot of the local energy along the \(x\) axis</a> <li><a href="#org8d93b3a">2.2. Plot of the local energy along the \(x\) axis</a>
<ul> <ul>
<li><a href="#orgee4e8d1">2.2.1. Exercise</a> <li><a href="#org73c7953">2.2.1. Exercise</a>
<ul> <ul>
<li><a href="#org7652a96">2.2.1.1. Solution</a></li> <li><a href="#org0fcc683">2.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org8523d1c">2.3. Numerical estimation of the energy</a> <li><a href="#org997c814">2.3. Numerical estimation of the energy</a>
<ul> <ul>
<li><a href="#org35bb3fb">2.3.1. Exercise</a> <li><a href="#org7a24d0a">2.3.1. Exercise</a>
<ul> <ul>
<li><a href="#org1bfc07a">2.3.1.1. Solution</a></li> <li><a href="#org44d22a3">2.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgf845070">2.4. Variance of the local energy</a> <li><a href="#org361114f">2.4. Variance of the local energy</a>
<ul> <ul>
<li><a href="#orgf56984c">2.4.1. Exercise (optional)</a> <li><a href="#org2c40196">2.4.1. Exercise (optional)</a>
<ul> <ul>
<li><a href="#org67a0d97">2.4.1.1. Solution</a></li> <li><a href="#orgbecaeef">2.4.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd31720a">2.4.2. Exercise</a> <li><a href="#orgf1e8e3d">2.4.2. Exercise</a>
<ul> <ul>
<li><a href="#org5b88542">2.4.2.1. Solution</a></li> <li><a href="#org8fc9a13">2.4.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#orgee08883">3. Variational Monte Carlo</a> <li><a href="#org7aa6a0a">3. Variational Monte Carlo</a>
<ul> <ul>
<li><a href="#org61b40b7">3.1. Computation of the statistical error</a> <li><a href="#org2087616">3.1. Computation of the statistical error</a>
<ul> <ul>
<li><a href="#orgab3cffd">3.1.1. Exercise</a> <li><a href="#orgaeae4c2">3.1.1. Exercise</a>
<ul> <ul>
<li><a href="#orgdc05008">3.1.1.1. Solution</a></li> <li><a href="#orgdfc7b53">3.1.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org31e93ba">3.2. Uniform sampling in the box</a> <li><a href="#orge7d91c3">3.2. Uniform sampling in the box</a>
<ul> <ul>
<li><a href="#org355c3a1">3.2.1. Exercise</a> <li><a href="#org96d1bb2">3.2.1. Exercise</a>
<ul> <ul>
<li><a href="#org64e4ca3">3.2.1.1. Solution</a></li> <li><a href="#orgc8c3acf">3.2.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org2b8b15f">3.3. Metropolis sampling with \(\Psi^2\)</a> <li><a href="#org0e66655">3.3. Metropolis sampling with \(\Psi^2\)</a>
<ul> <ul>
<li><a href="#org4b23b03">3.3.1. Exercise</a> <li><a href="#org9692143">3.3.1. Exercise</a>
<ul> <ul>
<li><a href="#orgca36c12">3.3.1.1. Solution</a></li> <li><a href="#org88f43cd">3.3.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org2ecccd9">3.4. Gaussian random number generator</a></li> <li><a href="#org6c65f73">3.4. Gaussian random number generator</a></li>
<li><a href="#org911b3c9">3.5. Generalized Metropolis algorithm</a> <li><a href="#org70ec84a">3.5. Generalized Metropolis algorithm</a>
<ul> <ul>
<li><a href="#org63393d2">3.5.1. Exercise 1</a> <li><a href="#org6e34709">3.5.1. Exercise 1</a>
<ul> <ul>
<li><a href="#org2b1ae2a">3.5.1.1. Solution</a></li> <li><a href="#orgcf485b0">3.5.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd03861d">3.5.2. Exercise 2</a> <li><a href="#orge3ee834">3.5.2. Exercise 2</a>
<ul> <ul>
<li><a href="#org75ee22c">3.5.2.1. Solution</a></li> <li><a href="#org2d51c0d">3.5.2.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org623ac9b">4. Diffusion Monte Carlo</a> <li><a href="#org58ca27b">4. Diffusion Monte Carlo</a>
<ul> <ul>
<li><a href="#org75d615c">4.1. Schrödinger equation in imaginary time</a></li> <li><a href="#orgde86ff6">4.1. Schrödinger equation in imaginary time</a></li>
<li><a href="#org1497029">4.2. Diffusion and branching</a></li> <li><a href="#org02ca0c2">4.2. Diffusion and branching</a></li>
<li><a href="#orgefdcd47">4.3. Importance sampling</a> <li><a href="#org1a44a46">4.3. Importance sampling</a>
<ul> <ul>
<li><a href="#org0b3840e">4.3.1. Appendix : Details of the Derivation</a></li> <li><a href="#org25d9eba">4.3.1. Appendix : Details of the Derivation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org7d0e033">4.4. Fixed-node DMC energy</a></li> <li><a href="#org4206909">4.4. Fixed-node DMC energy</a></li>
<li><a href="#org3696228">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li> <li><a href="#orgbc4844d">4.5. Pure Diffusion Monte Carlo (PDMC)</a></li>
<li><a href="#org66fb772">4.6. Hydrogen atom</a> <li><a href="#org0e0bf5c">4.6. Hydrogen atom</a>
<ul> <ul>
<li><a href="#org9d98668">4.6.1. Exercise</a> <li><a href="#org044109b">4.6.1. Exercise</a>
<ul> <ul>
<li><a href="#orgaf39a8c">4.6.1.1. Solution</a></li> <li><a href="#orgf0ab8a1">4.6.1.1. Solution</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org7ecf66b">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li> <li><a href="#orgce119bb">4.7. <span class="todo TODO">TODO</span> H<sub>2</sub></a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8bfdf8b">5. <span class="todo TODO">TODO</span> <code>[0/3]</code> Last things to do</a></li> <li><a href="#org02606e3">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-orge2a6b44" class="outline-2"> <div id="outline-container-org0a42dc6" class="outline-2">
<h2 id="orge2a6b44"><span class="section-number-2">1</span> Introduction</h2> <h2 id="org0a42dc6"><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-org0f16b33" class="outline-3"> <div id="outline-container-org6174a0a" class="outline-3">
<h3 id="org0f16b33"><span class="section-number-3">1.1</span> Energy and local energy</h3> <h3 id="org6174a0a"><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
@ -549,11 +549,11 @@ For few dimensions, one can easily compute \(E\) by evaluating the integrals on
<p> <p>
To this aim, recall that the probabilistic <i>expected value</i> of an arbitrary function \(f(x)\) 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 with respect to a probability density function \(P(x)\) is given by
</p> </p>
<p> <p>
\[ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx, \] \[ \langle f \rangle_p = \int_{-\infty}^\infty P(x)\, f(x)\,dx, \]
</p> </p>
<p> <p>
@ -562,16 +562,16 @@ and integrates to one:
</p> </p>
<p> <p>
\[ \int_{-\infty}^\infty p(x)\,dx = 1. \] \[ \int_{-\infty}^\infty P(x)\,dx = 1. \]
</p> </p>
<p> <p>
Similarly, we can view the the energy of a system, \(E\), as the expected value of the local energy with respect to 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: a probability density \(P(\mathbf{r}}\) defined in 3\(N\) dimensions:
</p> </p>
<p> <p>
\[ E = \int E_L(\mathbf{r}) p(\mathbf{r})\,d\mathbf{r}} \equiv \langle E_L \rangle_{\Psi^2}\,, \] \[ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r}} \equiv \langle E_L \rangle_{\Psi^2}\,, \]
</p> </p>
<p> <p>
@ -579,22 +579,22 @@ where the probability density is given by the square of the wave function:
</p> </p>
<p> <p>
\[ p(\mathbf{r}) = \frac{|Psi(\mathbf{r}|^2){\int \left |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,. \] \[ P(\mathbf{r}) = \frac{|Psi(\mathbf{r}|^2){\int \left |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,. \]
</p> </p>
<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: If we can sample \(N_{\rm MC}\) configurations \(\{\mathbf{r}\}\) distributed as \(p\), we can estimate \(E\) as the average of the local energy computed over these configurations:
</p> </p>
<p> <p>
$$ E &asymp; \frac{1}{M} &sum;<sub>i=1</sub><sup>M</sup> E<sub>L</sub>(\mathbf{r}<sub>i</sub>} \,. $$ E &asymp; \frac{1}{N<sub>\rm MC</sub>} &sum;<sub>i=1</sub><sup>N<sub>\rm MC</sub></sup> E<sub>L</sub>(\mathbf{r}<sub>i</sub>} \,.
</p> </p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgf8de501" class="outline-2"> <div id="outline-container-orge35e3b0" class="outline-2">
<h2 id="orgf8de501"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2> <h2 id="orge35e3b0"><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-org13e906f" class="outline-3"> <div id="outline-container-org536ab35" class="outline-3">
<h3 id="org13e906f"><span class="section-number-3">2.1</span> Local energy</h3> <h3 id="org536ab35"><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-orgd072c5b" class="outline-4"> <div id="outline-container-orge1d3531" class="outline-4">
<h4 id="orgd072c5b"><span class="section-number-4">2.1.1</span> Exercise 1</h4> <h4 id="orge1d3531"><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-org959a351" class="outline-5"> <div id="outline-container-orge29b2d5" class="outline-5">
<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> <h5 id="orge29b2d5"><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-org08e784d" class="outline-4"> <div id="outline-container-org0082181" class="outline-4">
<h4 id="org08e784d"><span class="section-number-4">2.1.2</span> Exercise 2</h4> <h4 id="org0082181"><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-orgb8d093c" class="outline-5"> <div id="outline-container-orgcb5ffb6" class="outline-5">
<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> <h5 id="orgcb5ffb6"><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-orgc2112e2" class="outline-4"> <div id="outline-container-org950428b" class="outline-4">
<h4 id="orgc2112e2"><span class="section-number-4">2.1.3</span> Exercise 3</h4> <h4 id="org950428b"><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>
@ -883,8 +883,8 @@ Therefore, the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-orgcfbe3ee" class="outline-5"> <div id="outline-container-orgb3b7bff" class="outline-5">
<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> <h5 id="orgb3b7bff"><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 @@ Therefore, the local kinetic energy is
</div> </div>
</div> </div>
<div id="outline-container-org19dde43" class="outline-4"> <div id="outline-container-orgd1768b7" class="outline-4">
<h4 id="org19dde43"><span class="section-number-4">2.1.4</span> Exercise 4</h4> <h4 id="orgd1768b7"><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-orgde229be" class="outline-5"> <div id="outline-container-orgca62f3b" class="outline-5">
<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> <h5 id="orgca62f3b"><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-orgecbce75" class="outline-4"> <div id="outline-container-orgd9f5f66" class="outline-4">
<h4 id="orgecbce75"><span class="section-number-4">2.1.5</span> Exercise 5</h4> <h4 id="orgd9f5f66"><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-org541ad24" class="outline-5"> <div id="outline-container-org1e5b04a" class="outline-5">
<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> <h5 id="org1e5b04a"><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-orgce16153" class="outline-3"> <div id="outline-container-org8d93b3a" class="outline-3">
<h3 id="orgce16153"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3> <h3 id="org8d93b3a"><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-orgee4e8d1" class="outline-4"> <div id="outline-container-org73c7953" class="outline-4">
<h4 id="orgee4e8d1"><span class="section-number-4">2.2.1</span> Exercise</h4> <h4 id="org73c7953"><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-org7652a96" class="outline-5"> <div id="outline-container-org0fcc683" class="outline-5">
<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> <h5 id="org0fcc683"><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>
@ -1204,8 +1204,8 @@ plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
</div> </div>
</div> </div>
<div id="outline-container-org8523d1c" class="outline-3"> <div id="outline-container-org997c814" class="outline-3">
<h3 id="org8523d1c"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3> <h3 id="org997c814"><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 \(\mathbf{r}_i\)
@ -1235,8 +1235,8 @@ The energy is biased because:
</div> </div>
<div id="outline-container-org35bb3fb" class="outline-4"> <div id="outline-container-org7a24d0a" class="outline-4">
<h4 id="org35bb3fb"><span class="section-number-4">2.3.1</span> Exercise</h4> <h4 id="org7a24d0a"><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>
@ -1305,8 +1305,8 @@ To compile the Fortran and run it:
</div> </div>
</div> </div>
<div id="outline-container-org1bfc07a" class="outline-5"> <div id="outline-container-org44d22a3" class="outline-5">
<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> <h5 id="org44d22a3"><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-orgf845070" class="outline-3"> <div id="outline-container-org361114f" class="outline-3">
<h3 id="orgf845070"><span class="section-number-3">2.4</span> Variance of the local energy</h3> <h3 id="org361114f"><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-orgf56984c" class="outline-4"> <div id="outline-container-org2c40196" class="outline-4">
<h4 id="orgf56984c"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4> <h4 id="org2c40196"><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-org67a0d97" class="outline-5"> <div id="outline-container-orgbecaeef" class="outline-5">
<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> <h5 id="orgbecaeef"><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}
@ -1481,8 +1481,8 @@ Prove that :
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgd31720a" class="outline-4"> <div id="outline-container-orgf1e8e3d" class="outline-4">
<h4 id="orgd31720a"><span class="section-number-4">2.4.2</span> Exercise</h4> <h4 id="orgf1e8e3d"><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>
@ -1556,8 +1556,8 @@ To compile and run:
</div> </div>
</div> </div>
<div id="outline-container-org5b88542" class="outline-5"> <div id="outline-container-org8fc9a13" class="outline-5">
<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> <h5 id="org8fc9a13"><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>
@ -1694,31 +1694,31 @@ a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814
</div> </div>
</div> </div>
<div id="outline-container-orgee08883" class="outline-2"> <div id="outline-container-org7aa6a0a" class="outline-2">
<h2 id="orgee08883"><span class="section-number-2">3</span> Variational Monte Carlo</h2> <h2 id="org7aa6a0a"><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
in low dimensions. When the number of dimensions becomes large, in low dimensions. When the number of dimensions becomes large,
instead of computing the average energy as a numerical integration instead of computing the average energy as a numerical integration
on a grid, it is usually more efficient to do a Monte Carlo sampling. on a grid, it is usually more efficient to use Monte Carlo sampling.
</p> </p>
<p> <p>
Moreover, a Monte Carlo sampling will alow us to remove the bias due Moreover, Monte Carlo sampling will alow us to remove the bias due
to the discretization of space, and compute a statistical confidence to the discretization of space, and compute a statistical confidence
interval. interval.
</p> </p>
</div> </div>
<div id="outline-container-org61b40b7" class="outline-3"> <div id="outline-container-org2087616" class="outline-3">
<h3 id="org61b40b7"><span class="section-number-3">3.1</span> Computation of the statistical error</h3> <h3 id="org2087616"><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 \(M\)
independent Monte Carlo calculations. You will obtain \(M\) different independent Monte Carlo calculations. You will obtain \(M\) different
estimates of the energy, which are expected to have a Gaussian estimates of the energy, which are expected to have a Gaussian
distribution according to the <a href="https://en.wikipedia.org/wiki/Central_limit_theorem">Central Limit Theorem</a>. distribution for large \(M\), according to the <a href="https://en.wikipedia.org/wiki/Central_limit_theorem">Central Limit Theorem</a>.
</p> </p>
<p> <p>
@ -1752,8 +1752,8 @@ And the confidence interval is given by
</p> </p>
</div> </div>
<div id="outline-container-orgab3cffd" class="outline-4"> <div id="outline-container-orgaeae4c2" class="outline-4">
<h4 id="orgab3cffd"><span class="section-number-4">3.1.1</span> Exercise</h4> <h4 id="orgaeae4c2"><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-orgdc05008" class="outline-5"> <div id="outline-container-orgdfc7b53" class="outline-5">
<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> <h5 id="orgdfc7b53"><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>
@ -1851,16 +1851,44 @@ input array.
</div> </div>
</div> </div>
<div id="outline-container-org31e93ba" class="outline-3"> <div id="outline-container-orge7d91c3" class="outline-3">
<h3 id="org31e93ba"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3> <h3 id="orge7d91c3"><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 perform our first Monte Carlo calculation to compute the
energy of the hydrogen atom. energy of the hydrogen atom.
</p> </p>
<p> <p>
At every Monte Carlo iteration: Consider again the expression of the energy
</p>
\begin{eqnarray*}
E & = & \frac{\int E_L(\mathbf{r})\left[\Psi(\mathbf{r})\right]^2\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}\,.
\end{eqnarray*}
<p>
Clearly, the square of the wave function is a good choice of probability density to sample but we will start with something simpler and rewrite the energy as
</p>
\begin{eqnarray*}
E & = & \frac{\int E_L(\mathbf{r})\frac{|\Psi(\mathbf{r})|^2}{p(\mathbf{r})}p(\mathbf{r})\, \,d\mathbf{r}}{\int \frac{|\Psi(\mathbf{r})|^2 }{p(\mathbf{r})}p(\mathbf{r})d\mathbf{r}}\,.
\end{eqnarray*}
<p>
Here, we will sample a uniform probability \(p(\mathbf{r})\) in a cube of volume \(L^3\) centered at the origin:
</p>
<p>
\[ p(\mathbf{r}) = \frac{1}{L^3}\,, \]
</p>
<p>
and zero outside the cube.
</p>
<p>
One Monte Carlo run will consist of \(N_{\rm MC}\) Monte Carlo iterations. At every Monte Carlo iteration:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
@ -1873,9 +1901,8 @@ result in a variable <code>energy</code></li>
</ul> </ul>
<p> <p>
One Monte Carlo run will consist of \(N\) Monte Carlo iterations. Once all the Once all the iterations have been computed, the run returns the average energy
iterations have been computed, the run returns the average energy \(\bar{E}_k\) over the \(N_{\rm MC}\) iterations of the run.
\(\bar{E}_k\) over the \(N\) iterations of the run.
</p> </p>
<p> <p>
@ -1886,8 +1913,8 @@ compute the statistical error.
</p> </p>
</div> </div>
<div id="outline-container-org355c3a1" class="outline-4"> <div id="outline-container-org96d1bb2" class="outline-4">
<h4 id="org355c3a1"><span class="section-number-4">3.2.1</span> Exercise</h4> <h4 id="org96d1bb2"><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 +2014,8 @@ well as the index of the current step.
</div> </div>
</div> </div>
<div id="outline-container-org64e4ca3" class="outline-5"> <div id="outline-container-orgc8c3acf" class="outline-5">
<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> <h5 id="orgc8c3acf"><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>
@ -2102,30 +2129,29 @@ E = -0.49518773675598715 +/- 5.2391494923686175E-004
</div> </div>
</div> </div>
<div id="outline-container-org2b8b15f" class="outline-3"> <div id="outline-container-org0e66655" class="outline-3">
<h3 id="org2b8b15f"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3> <h3 id="org0e66655"><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
points distributed with the probability density points distributed with the probability density
\[ \[
P(\mathbf{r}) = \left[\Psi(\mathbf{r})\right]^2 P(\mathbf{r}) = \frac{|Psi(\mathbf{r}|^2){\int \left |\Psi(\mathbf{r})|^2 d\mathbf{r}}
\] \]
</p> </p>
<p> <p>
The expression of the average energy is now simplified as the average of The expression of the average energy is now simplified as the average of
the local energies, since the weights are taken care of by the the local energies, since the weights are taken care of by the
sampling : sampling:
</p> </p>
<p> <p>
\[ \[
E \approx \frac{1}{M}\sum_{i=1}^M E_L(\mathbf{r}_i) E \approx \frac{1}{N_{\rm MC}}\sum_{i=1}^{N_{\rm MC} E_L(\mathbf{r}_i)
\] \]
</p> </p>
<p> <p>
To sample a chosen probability density, an efficient method is the To sample a chosen probability density, an efficient method is the
<a href="https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm">Metropolis-Hastings sampling algorithm</a>. Starting from a random <a href="https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm">Metropolis-Hastings sampling algorithm</a>. Starting from a random
@ -2191,8 +2217,8 @@ step such that the acceptance rate is close to 0.5 is a good compromise.
</div> </div>
<div id="outline-container-org4b23b03" class="outline-4"> <div id="outline-container-org9692143" class="outline-4">
<h4 id="org4b23b03"><span class="section-number-4">3.3.1</span> Exercise</h4> <h4 id="org9692143"><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 +2325,8 @@ Can you observe a reduction in the statistical error?
</div> </div>
</div> </div>
<div id="outline-container-orgca36c12" class="outline-5"> <div id="outline-container-org88f43cd" class="outline-5">
<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> <h5 id="org88f43cd"><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 +2471,8 @@ A = 0.51695266666666673 +/- 4.0445505648997396E-004
</div> </div>
</div> </div>
<div id="outline-container-org2ecccd9" class="outline-3"> <div id="outline-container-org6c65f73" class="outline-3">
<h3 id="org2ecccd9"><span class="section-number-3">3.4</span> Gaussian random number generator</h3> <h3 id="org6c65f73"><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 +2534,8 @@ 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-org911b3c9" class="outline-3"> <div id="outline-container-org70ec84a" class="outline-3">
<h3 id="org911b3c9"><span class="section-number-3">3.5</span> Generalized Metropolis algorithm</h3> <h3 id="org70ec84a"><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 +2634,8 @@ The transition probability becomes:
</div> </div>
<div id="outline-container-org63393d2" class="outline-4"> <div id="outline-container-org6e34709" class="outline-4">
<h4 id="org63393d2"><span class="section-number-4">3.5.1</span> Exercise 1</h4> <h4 id="org6e34709"><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 +2669,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-org2b1ae2a" class="outline-5"> <div id="outline-container-orgcf485b0" class="outline-5">
<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> <h5 id="orgcf485b0"><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 +2703,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div> </div>
</div> </div>
<div id="outline-container-orgd03861d" class="outline-4"> <div id="outline-container-orge3ee834" class="outline-4">
<h4 id="orgd03861d"><span class="section-number-4">3.5.2</span> Exercise 2</h4> <h4 id="orge3ee834"><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 +2798,8 @@ Modify the previous program to introduce the drifted diffusion scheme.
</div> </div>
</div> </div>
<div id="outline-container-org75ee22c" class="outline-5"> <div id="outline-container-org2d51c0d" class="outline-5">
<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> <h5 id="org2d51c0d"><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 +2985,12 @@ A = 0.78839866666666658 +/- 3.2503783452043152E-004
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</div> </div>
<div id="outline-container-org623ac9b" class="outline-2"> <div id="outline-container-org58ca27b" class="outline-2">
<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> <h2 id="org58ca27b"><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">
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<div id="outline-container-org75d615c" class="outline-3"> <div id="outline-container-orgde86ff6" class="outline-3">
<h3 id="org75d615c"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3> <h3 id="orgde86ff6"><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 +3049,8 @@ system.
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</div> </div>
<div id="outline-container-org1497029" class="outline-3"> <div id="outline-container-org02ca0c2" class="outline-3">
<h3 id="org1497029"><span class="section-number-3">4.2</span> Diffusion and branching</h3> <h3 id="org02ca0c2"><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 +3104,8 @@ the combination of a diffusion process and a branching process.
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</div> </div>
<div id="outline-container-orgefdcd47" class="outline-3"> <div id="outline-container-org1a44a46" class="outline-3">
<h3 id="orgefdcd47"><span class="section-number-3">4.3</span> Importance sampling</h3> <h3 id="org1a44a46"><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 +3162,8 @@ error known as the <i>fixed node error</i>.
</p> </p>
</div> </div>
<div id="outline-container-org0b3840e" class="outline-4"> <div id="outline-container-org25d9eba" class="outline-4">
<h4 id="org0b3840e"><span class="section-number-4">4.3.1</span> Appendix : Details of the Derivation</h4> <h4 id="org25d9eba"><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 +3225,8 @@ Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
</div> </div>
<div id="outline-container-org7d0e033" class="outline-3"> <div id="outline-container-org4206909" class="outline-3">
<h3 id="org7d0e033"><span class="section-number-3">4.4</span> Fixed-node DMC energy</h3> <h3 id="org4206909"><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 +3278,8 @@ energies computed with the trial wave function.
</div> </div>
</div> </div>
<div id="outline-container-org3696228" class="outline-3"> <div id="outline-container-orgbc4844d" class="outline-3">
<h3 id="org3696228"><span class="section-number-3">4.5</span> Pure Diffusion Monte Carlo (PDMC)</h3> <h3 id="orgbc4844d"><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 +3331,13 @@ code, so this is what we will do in the next section.
</div> </div>
</div> </div>
<div id="outline-container-org66fb772" class="outline-3"> <div id="outline-container-org0e0bf5c" class="outline-3">
<h3 id="org66fb772"><span class="section-number-3">4.6</span> Hydrogen atom</h3> <h3 id="org0e0bf5c"><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">
</div> </div>
<div id="outline-container-org9d98668" class="outline-4"> <div id="outline-container-org044109b" class="outline-4">
<h4 id="org9d98668"><span class="section-number-4">4.6.1</span> Exercise</h4> <h4 id="org044109b"><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 +3436,8 @@ energy of H for any value of \(a\).
</div> </div>
</div> </div>
<div id="outline-container-orgaf39a8c" class="outline-5"> <div id="outline-container-orgf0ab8a1" class="outline-5">
<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> <h5 id="orgf0ab8a1"><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 +3653,8 @@ A = 0.98788066666666663 +/- 7.2889356133441110E-005
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<div id="outline-container-org7ecf66b" class="outline-3"> <div id="outline-container-orgce119bb" class="outline-3">
<h3 id="org7ecf66b"><span class="section-number-3">4.7</span> <span class="todo TODO">TODO</span> H<sub>2</sub></h3> <h3 id="orgce119bb"><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 +3675,8 @@ the nuclei.
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<div id="outline-container-org8bfdf8b" class="outline-2"> <div id="outline-container-org02606e3" class="outline-2">
<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> <h2 id="org02606e3"><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 +3692,7 @@ 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 22:30</p> <p class="date">Created: 2021-01-31 Sun 08:40</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>