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<h1 class="title">Quantum Monte Carlo</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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||
<li><a href="#org8147bc4">1. Introduction</a>
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<ul>
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||
<li><a href="#org0c90626">1.1. Energy and local energy</a></li>
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||
</ul>
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||
</li>
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<li><a href="#org236e915">2. Numerical evaluation of the energy of the hydrogen atom</a>
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||
<ul>
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<li><a href="#org20fce44">2.1. Local energy</a>
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||
<ul>
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||
<li><a href="#org6aa2eea">2.1.1. Exercise 1</a>
|
||
<ul>
|
||
<li><a href="#orgea8540d">2.1.1.1. Solution</a></li>
|
||
</ul>
|
||
</li>
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||
<li><a href="#orga99da73">2.1.2. Exercise 2</a>
|
||
<ul>
|
||
<li><a href="#orgf3fa97e">2.1.2.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#orgb83dff4">2.1.3. Exercise 3</a>
|
||
<ul>
|
||
<li><a href="#orge819a18">2.1.3.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#orgd73f763">2.1.4. Exercise 4</a>
|
||
<ul>
|
||
<li><a href="#org37ddd7b">2.1.4.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org0448ef8">2.1.5. Exercise 5</a>
|
||
<ul>
|
||
<li><a href="#orgc6d6285">2.1.5.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
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||
<li><a href="#orgacb0e57">2.2. Plot of the local energy along the \(x\) axis</a>
|
||
<ul>
|
||
<li><a href="#orgb0b40fb">2.2.1. Exercise</a>
|
||
<ul>
|
||
<li><a href="#orgd3b3255">2.2.1.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
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||
<li><a href="#orgf2d1b9f">2.3. Numerical estimation of the energy</a>
|
||
<ul>
|
||
<li><a href="#org37958cd">2.3.1. Exercise</a>
|
||
<ul>
|
||
<li><a href="#org0a6757b">2.3.1.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org3a5c93e">2.4. Variance of the local energy</a>
|
||
<ul>
|
||
<li><a href="#orgf25a4df">2.4.1. Exercise (optional)</a>
|
||
<ul>
|
||
<li><a href="#org7cfa378">2.4.1.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org82674e9">2.4.2. Exercise</a>
|
||
<ul>
|
||
<li><a href="#org984e04b">2.4.2.1. Solution</a></li>
|
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</ul>
|
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</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
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||
<li><a href="#orged64ccd">3. Variational Monte Carlo</a>
|
||
<ul>
|
||
<li><a href="#org01e38cc">3.1. Computation of the statistical error</a>
|
||
<ul>
|
||
<li><a href="#org0dbdc24">3.1.1. Exercise</a>
|
||
<ul>
|
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<li><a href="#orgdfc0533">3.1.1.1. Solution</a></li>
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</ul>
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</ul>
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</li>
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<li><a href="#orgfd318d6">3.2. Uniform sampling in the box</a>
|
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<li><a href="#org74432b8">3.2.1. Exercise</a>
|
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<ul>
|
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<li><a href="#org99a7b2f">3.2.1.1. Solution</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orgcddcdcf">3.3. Metropolis sampling with \(\Psi^2\)</a>
|
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<ul>
|
||
<li><a href="#org8b3f6fe">3.3.1. Optimal step size</a></li>
|
||
<li><a href="#orgd7c523f">3.3.2. Exercise</a>
|
||
<ul>
|
||
<li><a href="#orgac64351">3.3.2.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org4f2ffad">3.4. Generalized Metropolis algorithm</a>
|
||
<ul>
|
||
<li><a href="#org6290597">3.4.1. Gaussian random number generator</a></li>
|
||
<li><a href="#orgc3a15c4">3.4.2. Exercise 1</a>
|
||
<ul>
|
||
<li><a href="#org1bb7357">3.4.2.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org3594157">3.4.3. Exercise 2</a>
|
||
<ul>
|
||
<li><a href="#orgfe0293b">3.4.3.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org3154c2b">4. Diffusion Monte Carlo</a>
|
||
<ul>
|
||
<li><a href="#org7b4dacf">4.1. Schrödinger equation in imaginary time</a></li>
|
||
<li><a href="#orgd350691">4.2. Relation to diffusion</a></li>
|
||
<li><a href="#org78b3d2b">4.3. Importance sampling</a>
|
||
<ul>
|
||
<li><a href="#org7548f1f">4.3.1. Appendix : Details of the Derivation</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org4a7db09">4.4. Pure Diffusion Monte Carlo</a></li>
|
||
<li><a href="#orgb1e1f31">4.5. Hydrogen atom</a>
|
||
<ul>
|
||
<li><a href="#org10928e0">4.5.1. Exercise</a>
|
||
<ul>
|
||
<li><a href="#org834cb8c">4.5.1.1. Solution</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#org3be55e0">5. Project</a></li>
|
||
<li><a href="#orgfa0dc42">6. Acknowledgments</a></li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org8147bc4" class="outline-2">
|
||
<h2 id="org8147bc4"><span class="section-number-2">1</span> Introduction</h2>
|
||
<div class="outline-text-2" id="text-1">
|
||
<p>
|
||
This website contains the QMC tutorial of the 2021 LTTC winter school
|
||
<a href="https://www.irsamc.ups-tlse.fr/lttc/Luchon">Tutorials in Theoretical Chemistry</a>.
|
||
</p>
|
||
|
||
<p>
|
||
We propose different exercises to understand quantum Monte Carlo (QMC)
|
||
methods. In the first section, we start with the computation of the energy of a
|
||
hydrogen atom using numerical integration. The goal of this section is
|
||
to familarize yourself with the concept of <i>local energy</i>.
|
||
Then, we introduce the variational Monte Carlo (VMC) method which
|
||
computes a statistical estimate of the expectation value of the energy
|
||
associated with a given wave function, and apply this approach to the
|
||
hydrogen atom.
|
||
Finally, we present the diffusion Monte Carlo (DMC) method which
|
||
we use here to estimate the exact energy of the hydrogen atom and of the H<sub>2</sub> molecule,
|
||
starting from an approximate wave function.
|
||
</p>
|
||
|
||
<p>
|
||
Code examples will be given in Python3 and Fortran. You can use
|
||
whatever language you prefer to write the programs.
|
||
</p>
|
||
|
||
<p>
|
||
We consider the stationary solution of the Schrödinger equation, so
|
||
the wave functions considered here are real: for an \(N\) electron
|
||
system where the electrons move in the 3-dimensional space,
|
||
\(\Psi : \mathbb{R}^{3N} \rightarrow \mathbb{R}\). In addition, \(\Psi\)
|
||
is defined everywhere, continuous, and infinitely differentiable.
|
||
</p>
|
||
|
||
<p>
|
||
All the quantities are expressed in <i>atomic units</i> (energies,
|
||
coordinates, etc).
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org0c90626" class="outline-3">
|
||
<h3 id="org0c90626"><span class="section-number-3">1.1</span> Energy and local energy</h3>
|
||
<div class="outline-text-3" id="text-1-1">
|
||
<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 |\Psi(\mathbf{r}) |^2 d\mathbf{r}} \\
|
||
& = & \frac{\int |\Psi(\mathbf{r})|^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^2 d\mathbf{r}}
|
||
= \frac{\int |\Psi(\mathbf{r})|^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^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>
|
||
\[ E = \int E_L(\mathbf{r}) P(\mathbf{r})\,d\mathbf{r} \equiv \langle E_L \rangle_{P}\,, \]
|
||
</p>
|
||
|
||
<p>
|
||
where the probability density is given by the square of the wave function:
|
||
</p>
|
||
|
||
<p>
|
||
\[ P(\mathbf{r}) = \frac{|\Psi(\mathbf{r})|^2}{\int |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,. \]
|
||
</p>
|
||
|
||
<p>
|
||
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>
|
||
\[ E \approx \frac{1}{N_{\rm MC}} \sum_{i=1}^{N_{\rm MC}} E_L(\mathbf{r}_i) \,. \]
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org236e915" class="outline-2">
|
||
<h2 id="org236e915"><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">
|
||
<p>
|
||
In this section, we consider the hydrogen atom with the following
|
||
wave function:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\Psi(\mathbf{r}) = \exp(-a |\mathbf{r}|)
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
We will first verify that, for a particular value of \(a\), \(\Psi\) is an
|
||
eigenfunction of the Hamiltonian
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\hat{H} = \hat{T} + \hat{V} = - \frac{1}{2} \Delta - \frac{1}{|\mathbf{r}|}
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
To do that, we will compute the local energy and check whether it is constant.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org20fce44" class="outline-3">
|
||
<h3 id="org20fce44"><span class="section-number-3">2.1</span> Local energy</h3>
|
||
<div class="outline-text-3" id="text-2-1">
|
||
<p>
|
||
You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
|
||
</p>
|
||
|
||
<p>
|
||
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
|
||
Fortran.
|
||
</p>
|
||
|
||
<div class="note">
|
||
<ul class="org-ul">
|
||
<li>When computing a square root in \(\mathbb{R}\), <b>always</b> make sure
|
||
that the argument of the square root is non-negative.</li>
|
||
<li>When you divide, <b>always</b> make sure that you will not divide by zero</li>
|
||
</ul>
|
||
|
||
<p>
|
||
If a <i>floating-point exception</i> can occur, you should make a test
|
||
to catch the error.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org6aa2eea" class="outline-4">
|
||
<h4 id="org6aa2eea"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
|
||
<div class="outline-text-4" id="text-2-1-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Write a function which computes the potential at \(\mathbf{r}\).
|
||
The function accepts a 3-dimensional vector <code>r</code> as input argument
|
||
and returns the potential.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
\(\mathbf{r}=\left( \begin{array}{c} x \\ y\\ z\end{array} \right)\), so
|
||
\[
|
||
V(\mathbf{r}) = -\frac{1}{\sqrt{x^2 + y^2 + z^2}}
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
<span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">potential</span>(r):
|
||
# <span style="color: #b22222;">TODO</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">potential</span><span style="color: #000000; background-color: #ffffff;">(r)</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> r(3)</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">potential</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgea8540d" class="outline-5">
|
||
<h5 id="orgea8540d"><span class="section-number-5">2.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-1-1-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
<span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">potential</span>(r):
|
||
<span style="color: #a0522d;">distance</span> = np.sqrt(np.dot(r,r))
|
||
<span style="color: #a020f0;">assert</span> (distance > 0)
|
||
<span style="color: #a020f0;">return</span> -1. / distance
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">potential</span><span style="color: #000000; background-color: #ffffff;">(r)</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> r(3)</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> distance</span>
|
||
|
||
distance = dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
|
||
|
||
<span style="color: #a020f0;">if</span> (distance > 0.d0) <span style="color: #a020f0;">then</span>
|
||
potential = -1.d0 / distance
|
||
<span style="color: #a020f0;">else</span>
|
||
<span style="color: #a020f0;">stop</span> <span style="color: #8b2252;">'potential at r=0.d0 diverges'</span>
|
||
<span style="color: #a020f0;">end if</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">potential</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orga99da73" class="outline-4">
|
||
<h4 id="orga99da73"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
|
||
<div class="outline-text-4" id="text-2-1-2">
|
||
<div class="exercise">
|
||
<p>
|
||
Write a function which computes the wave function at \(\mathbf{r}\).
|
||
The function accepts a scalar <code>a</code> and a 3-dimensional vector <code>r</code> as
|
||
input arguments, and returns a scalar.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">psi</span>(a, r):
|
||
# <span style="color: #b22222;">TODO</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">psi</span><span style="color: #000000; background-color: #ffffff;">(a, r)</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">psi</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgf3fa97e" class="outline-5">
|
||
<h5 id="orgf3fa97e"><span class="section-number-5">2.1.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-1-2-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">psi</span>(a, r):
|
||
<span style="color: #a020f0;">return</span> np.exp(-a*np.sqrt(np.dot(r,r)))
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">psi</span><span style="color: #000000; background-color: #ffffff;">(a, r)</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||
|
||
psi = dexp(-a * dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ))
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">psi</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgb83dff4" class="outline-4">
|
||
<h4 id="orgb83dff4"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
|
||
<div class="outline-text-4" id="text-2-1-3">
|
||
<div class="exercise">
|
||
<p>
|
||
Write a function which computes the local kinetic energy at \(\mathbf{r}\).
|
||
The function accepts <code>a</code> and <code>r</code> as input arguments and returns the
|
||
local kinetic energy.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
The local kinetic energy is defined as \[T_L(\mathbf{r}) = -\frac{1}{2}\frac{\Delta \Psi(\mathbf{r})}{\Psi(\mathbf{r})}.\]
|
||
</p>
|
||
|
||
<p>
|
||
We differentiate \(\Psi\) with respect to \(x\):
|
||
</p>
|
||
|
||
<p>
|
||
\[ \Psi(\mathbf{r}) = \exp(-a\,|\mathbf{r}|) \]
|
||
\[\frac{\partial \Psi}{\partial x}
|
||
= \frac{\partial \Psi}{\partial |\mathbf{r}|} \frac{\partial |\mathbf{r}|}{\partial x}
|
||
= - \frac{a\,x}{|\mathbf{r}|} \Psi(\mathbf{r}) \]
|
||
</p>
|
||
|
||
<p>
|
||
and we differentiate a second time:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\frac{\partial^2 \Psi}{\partial x^2} =
|
||
\left( \frac{a^2\,x^2}{|\mathbf{r}|^2} -
|
||
\frac{a(y^2+z^2)}{|\mathbf{r}|^{3}} \right) \Psi(\mathbf{r}).
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
The Laplacian operator \(\Delta = \frac{\partial^2}{\partial x^2} +
|
||
\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\)
|
||
applied to the wave function gives:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r})\,.
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
Therefore, the local kinetic energy is
|
||
\[
|
||
T_L (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">kinetic</span>(a,r):
|
||
# <span style="color: #b22222;">TODO</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">kinetic</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">kinetic</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orge819a18" class="outline-5">
|
||
<h5 id="orge819a18"><span class="section-number-5">2.1.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-1-3-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">kinetic</span>(a,r):
|
||
<span style="color: #a0522d;">distance</span> = np.sqrt(np.dot(r,r))
|
||
<span style="color: #a020f0;">assert</span> (distance > 0.)
|
||
|
||
<span style="color: #a020f0;">return</span> a * (1./distance - 0.5 * a)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">kinetic</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> distance</span>
|
||
|
||
distance = dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
|
||
|
||
<span style="color: #a020f0;">if</span> (distance > 0.d0) <span style="color: #a020f0;">then</span>
|
||
|
||
kinetic = a * (1.d0 / distance - 0.5d0 * a)
|
||
|
||
<span style="color: #a020f0;">else</span>
|
||
<span style="color: #a020f0;">stop</span> <span style="color: #8b2252;">'kinetic energy diverges at r=0'</span>
|
||
<span style="color: #a020f0;">end if</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">kinetic</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd73f763" class="outline-4">
|
||
<h4 id="orgd73f763"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
|
||
<div class="outline-text-4" id="text-2-1-4">
|
||
<div class="exercise">
|
||
<p>
|
||
Write a function which computes the local energy at \(\mathbf{r}\),
|
||
using the previously defined functions.
|
||
The function accepts <code>a</code> and <code>r</code> as input arguments and returns the
|
||
local kinetic energy.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
\[
|
||
E_L(\mathbf{r}) = -\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) + V(\mathbf{r})
|
||
\]
|
||
</p>
|
||
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">e_loc</span>(a,r):
|
||
#<span style="color: #b22222;">TODO</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
|
||
<div class="note">
|
||
<p>
|
||
When you call a function in Fortran, you need to declare its
|
||
return type.
|
||
You might by accident choose a function name which is the
|
||
same as an internal function of Fortran. So it is recommended to
|
||
<b>always</b> use the keyword <code>external</code> to make sure the function you
|
||
are calling is yours.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">e_loc</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> kinetic</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> potential</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">e_loc</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org37ddd7b" class="outline-5">
|
||
<h5 id="org37ddd7b"><span class="section-number-5">2.1.4.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-1-4-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">e_loc</span>(a,r):
|
||
<span style="color: #a020f0;">return</span> kinetic(a,r) + potential(r)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #228b22;">double precision </span><span style="color: #a020f0;">function</span><span style="color: #a0522d;"> </span><span style="color: #0000ff;">e_loc</span><span style="color: #000000; background-color: #ffffff;">(a,r)</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> kinetic</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> potential</span>
|
||
|
||
e_loc = kinetic(a,r) + potential(r)
|
||
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">e_loc</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org0448ef8" class="outline-4">
|
||
<h4 id="org0448ef8"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
|
||
<div class="outline-text-4" id="text-2-1-5">
|
||
<div class="exercise">
|
||
<p>
|
||
Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(\hat{H}\).
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgc6d6285" class="outline-5">
|
||
<h5 id="orgc6d6285"><span class="section-number-5">2.1.5.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-1-5-1">
|
||
\begin{eqnarray*}
|
||
E &=& \frac{\hat{H} \Psi}{\Psi} = - \frac{1}{2} \frac{\Delta \Psi}{\Psi} -
|
||
\frac{1}{|\mathbf{r}|} \\
|
||
&=& -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right) -
|
||
\frac{1}{|\mathbf{r}|} \\
|
||
&=&
|
||
-\frac{1}{2} a^2 + \frac{a-1}{\mathbf{|r|}}
|
||
\end{eqnarray*}
|
||
|
||
<p>
|
||
\(a=1\) cancels the \(1/|r|\) term, and makes the energy constant and
|
||
equal to -0.5 atomic units.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgacb0e57" class="outline-3">
|
||
<h3 id="orgacb0e57"><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">
|
||
<p>
|
||
The program you will write in this section will be written in
|
||
another file (<code>plot_hydrogen.py</code> or <code>plot_hydrogen.f90</code> for
|
||
example).
|
||
It will use the functions previously defined.
|
||
</p>
|
||
|
||
<p>
|
||
In Python, you should put at the beginning of the file
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc
|
||
</pre>
|
||
</div>
|
||
<p>
|
||
to be able to use the <code>e_loc</code> function of the <code>hydrogen.py</code> file.
|
||
</p>
|
||
|
||
<p>
|
||
In Fortran, you will need to compile all the source files together:
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgb0b40fb" class="outline-4">
|
||
<h4 id="orgb0b40fb"><span class="section-number-4">2.2.1</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-2-2-1">
|
||
<div class="exercise">
|
||
<p>
|
||
For multiple values of \(a\) (0.1, 0.2, 0.5, 1., 1.5, 2.), plot the
|
||
local energy along the \(x\) axis. In Python, you can use matplotlib
|
||
for example. In Fortran, it is convenient to write in a text file
|
||
the values of \(x\) and \(E_L(\mathbf{r})\) for each point, and use
|
||
Gnuplot to plot the files. With Gnuplot, you will need 2 blank
|
||
lines to separate the data corresponding to different values of \(a\).
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="note">
|
||
<p>
|
||
The potential and the kinetic energy both diverge at \(r=0\), so we
|
||
choose a grid which does not contain the origin to avoid numerical issues.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||
<span style="color: #a020f0;">import</span> matplotlib.pyplot <span style="color: #a020f0;">as</span> plt
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc
|
||
|
||
<span style="color: #a0522d;">x</span>=np.linspace(-5,5)
|
||
plt.figure(figsize=(10,5))
|
||
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
plt.tight_layout()
|
||
plt.legend()
|
||
plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">plot</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), dx</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, j</span>
|
||
|
||
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
x(i) = -5.d0 + (i-1)*dx
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">plot</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
To compile and run:
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
|
||
./plot_hydrogen > data
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
To plot the data using Gnuplot:
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-gnuplot">set grid
|
||
set xrange [-5:5]
|
||
set yrange [-2:1]
|
||
plot './data' index 0 using 1:2 with lines title 'a=0.1', \
|
||
'./data' index 1 using 1:2 with lines title 'a=0.2', \
|
||
'./data' index 2 using 1:2 with lines title 'a=0.5', \
|
||
'./data' index 3 using 1:2 with lines title 'a=1.0', \
|
||
'./data' index 4 using 1:2 with lines title 'a=1.5', \
|
||
'./data' index 5 using 1:2 with lines title 'a=2.0'
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd3b3255" class="outline-5">
|
||
<h5 id="orgd3b3255"><span class="section-number-5">2.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-2-1-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||
<span style="color: #a020f0;">import</span> matplotlib.pyplot <span style="color: #a020f0;">as</span> plt
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc
|
||
|
||
<span style="color: #a0522d;">x</span>=np.linspace(-5,5)
|
||
plt.figure(figsize=(10,5))
|
||
|
||
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 1., 1.5, 2.]:
|
||
<span style="color: #a0522d;">y</span>=np.array([ e_loc(a, np.array([t,0.,0.]) ) <span style="color: #a020f0;">for</span> t <span style="color: #a020f0;">in</span> x])
|
||
plt.plot(x,y,label=f<span style="color: #8b2252;">"a={a}"</span>)
|
||
|
||
plt.tight_layout()
|
||
plt.legend()
|
||
plt.savefig(<span style="color: #8b2252;">"plot_py.png"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div class="figure">
|
||
<p><img src="./plot_py.png" alt="plot_py.png" />
|
||
</p>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">plot</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), energy, dx, r(3), a(6)</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, j</span>
|
||
|
||
a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
|
||
|
||
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
x(i) = -5.d0 + (i-1)*dx
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
r(:) = 0.d0
|
||
|
||
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'# a='</span>, a(j)
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
r(1) = x(i)
|
||
energy = e_loc( a(j), r )
|
||
<span style="color: #a020f0;">print</span> *, x(i), energy
|
||
<span style="color: #a020f0;">end do</span>
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">''</span>
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">''</span>
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">plot</span>
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div class="figure">
|
||
<p><img src="plot.png" alt="plot.png" />
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgf2d1b9f" class="outline-3">
|
||
<h3 id="orgf2d1b9f"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
|
||
<div class="outline-text-3" id="text-2-3">
|
||
<p>
|
||
If the space is discretized in small volume elements \(\mathbf{r}_i\)
|
||
of size \(\delta \mathbf{r}\), the expression of \(\langle E_L \rangle_{\Psi^2}\)
|
||
becomes a weighted average of the local energy, where the weights
|
||
are the values of the wave function square at \(\mathbf{r}_i\)
|
||
multiplied by the volume element:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\langle E \rangle_{\Psi^2} \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\;
|
||
w_i = \left|\Psi(\mathbf{r}_i)\right|^2 \delta \mathbf{r}
|
||
\]
|
||
</p>
|
||
|
||
<div class="note">
|
||
<p>
|
||
The energy is biased because:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>The volume elements are not infinitely small (discretization error)</li>
|
||
<li>The energy is evaluated only inside the box (incompleteness of the space)</li>
|
||
</ul>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-org37958cd" class="outline-4">
|
||
<h4 id="org37958cd"><span class="section-number-4">2.3.1</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-2-3-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Compute a numerical estimate of the energy using a grid of
|
||
\(50\times50\times50\) points in the range \((-5,-5,-5) \le
|
||
\mathbf{r} \le (5,5,5)\).
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc, psi
|
||
|
||
<span style="color: #a0522d;">interval</span> = np.linspace(-5,5,num=50)
|
||
<span style="color: #a0522d;">delta</span> = (interval[1]-interval[0])**3
|
||
|
||
<span style="color: #a0522d;">r</span> = np.array([0.,0.,0.])
|
||
|
||
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
|
||
# <span style="color: #b22222;">TODO</span>
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E}"</span>)
|
||
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">energy_hydrogen</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), w, delta, energy, dx, r(3), a(6), norm</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, k, l, j</span>
|
||
|
||
a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
|
||
|
||
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
x(i) = -5.d0 + (i-1)*dx
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'a = '</span>, a(j), <span style="color: #8b2252;">' E = '</span>, energy
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">energy_hydrogen</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
To compile the Fortran and run it:
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen
|
||
./energy_hydrogen
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org0a6757b" class="outline-5">
|
||
<h5 id="org0a6757b"><span class="section-number-5">2.3.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-3-1-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc, psi
|
||
|
||
<span style="color: #a0522d;">interval</span> = np.linspace(-5,5,num=50)
|
||
<span style="color: #a0522d;">delta</span> = (interval[1]-interval[0])**3
|
||
|
||
<span style="color: #a0522d;">r</span> = np.array([0.,0.,0.])
|
||
|
||
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
|
||
<span style="color: #a0522d;">E</span> = 0.
|
||
<span style="color: #a0522d;">norm</span> = 0.
|
||
|
||
<span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> interval:
|
||
<span style="color: #a0522d;">r</span>[0] = x
|
||
<span style="color: #a020f0;">for</span> y <span style="color: #a020f0;">in</span> interval:
|
||
<span style="color: #a0522d;">r</span>[1] = y
|
||
<span style="color: #a020f0;">for</span> z <span style="color: #a020f0;">in</span> interval:
|
||
<span style="color: #a0522d;">r</span>[2] = z
|
||
|
||
<span style="color: #a0522d;">w</span> = psi(a,r)
|
||
<span style="color: #a0522d;">w</span> = w * w * delta
|
||
|
||
<span style="color: #a0522d;">E</span> += w * e_loc(a,r)
|
||
<span style="color: #a0522d;">norm</span> += w
|
||
|
||
<span style="color: #a0522d;">E</span> = E / norm
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E}"</span>)
|
||
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
a = 0.1 E = -0.24518438948809218
|
||
a = 0.2 E = -0.26966057967803525
|
||
a = 0.5 E = -0.3856357612517407
|
||
a = 0.9 E = -0.49435709786716214
|
||
a = 1.0 E = -0.5
|
||
a = 1.5 E = -0.39242967082602226
|
||
a = 2.0 E = -0.08086980667844901
|
||
|
||
</pre>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">energy_hydrogen</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), w, delta, energy, dx, r(3), a(6), norm</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, k, l, j</span>
|
||
|
||
a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
|
||
|
||
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
x(i) = -5.d0 + (i-1)*dx
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
delta = dx**3
|
||
|
||
r(:) = 0.d0
|
||
|
||
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
|
||
energy = 0.d0
|
||
norm = 0.d0
|
||
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
r(1) = x(i)
|
||
|
||
<span style="color: #a020f0;">do</span> k=1,<span style="color: #a020f0;">size</span>(x)
|
||
r(2) = x(k)
|
||
|
||
<span style="color: #a020f0;">do</span> l=1,<span style="color: #a020f0;">size</span>(x)
|
||
r(3) = x(l)
|
||
|
||
w = psi(a(j),r)
|
||
w = w * w * delta
|
||
|
||
energy = energy + w * e_loc(a(j), r)
|
||
norm = norm + w
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
energy = energy / norm
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'a = '</span>, a(j), <span style="color: #8b2252;">' E = '</span>, energy
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">energy_hydrogen</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
a = 0.10000000000000001 E = -0.24518438948809140
|
||
a = 0.20000000000000001 E = -0.26966057967803236
|
||
a = 0.50000000000000000 E = -0.38563576125173815
|
||
a = 1.0000000000000000 E = -0.50000000000000000
|
||
a = 1.5000000000000000 E = -0.39242967082602065
|
||
a = 2.0000000000000000 E = -8.0869806678448772E-002
|
||
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3a5c93e" class="outline-3">
|
||
<h3 id="org3a5c93e"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
|
||
<div class="outline-text-3" id="text-2-4">
|
||
<p>
|
||
The variance of the local energy is a functional of \(\Psi\)
|
||
which measures the magnitude of the fluctuations of the local
|
||
energy associated with \(\Psi\) around its average:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\sigma^2(E_L) = \frac{\int |\Psi(\mathbf{r})|^2\, \left[
|
||
E_L(\mathbf{r}) - E \right]^2 \, d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^2 d\mathbf{r}}
|
||
\]
|
||
which can be simplified as
|
||
</p>
|
||
|
||
<p>
|
||
\[ \sigma^2(E_L) = \langle E_L^2 \rangle_{\Psi^2} - \langle E_L \rangle_{\Psi^2}^2.\]
|
||
</p>
|
||
|
||
<p>
|
||
If the local energy is constant (i.e. \(\Psi\) is an eigenfunction of
|
||
\(\hat{H}\)) the variance is zero, so the variance of the local
|
||
energy can be used as a measure of the quality of a wave function.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-orgf25a4df" class="outline-4">
|
||
<h4 id="orgf25a4df"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
|
||
<div class="outline-text-4" id="text-2-4-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Prove that :
|
||
\[\langle \left( E - \langle E \rangle_{\Psi^2} \right)^2\rangle_{\Psi^2} = \langle E^2 \rangle_{\Psi^2} - \langle E \rangle_{\Psi^2}^2 \]
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org7cfa378" class="outline-5">
|
||
<h5 id="org7cfa378"><span class="section-number-5">2.4.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-4-1-1">
|
||
<p>
|
||
\(\bar{E} = \langle E \rangle\) is a constant, so \(\langle \bar{E}
|
||
\rangle = \bar{E}\) .
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
\langle (E - \bar{E})^2 \rangle & = &
|
||
\langle E^2 - 2 E \bar{E} + \bar{E}^2 \rangle \\
|
||
&=& \langle E^2 \rangle - 2 \langle E \bar{E} \rangle + \langle \bar{E}^2 \rangle \\
|
||
&=& \langle E^2 \rangle - 2 \langle E \rangle \bar{E} + \bar{E}^2 \\
|
||
&=& \langle E^2 \rangle - 2 \bar{E}^2 + \bar{E}^2 \\
|
||
&=& \langle E^2 \rangle - \bar{E}^2 \\
|
||
&=& \langle E^2 \rangle - \langle E \rangle^2 \\
|
||
\end{eqnarray*}
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-org82674e9" class="outline-4">
|
||
<h4 id="org82674e9"><span class="section-number-4">2.4.2</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-2-4-2">
|
||
<div class="exercise">
|
||
<p>
|
||
Add the calculation of the variance to the previous code, and
|
||
compute a numerical estimate of the variance of the local energy using
|
||
a grid of \(50\times50\times50\) points in the range \((-5,-5,-5) \le
|
||
\mathbf{r} \le (5,5,5)\) for different values of \(a\).
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np <span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc, psi
|
||
|
||
<span style="color: #a0522d;">interval</span> = np.linspace(-5,5,num=50)
|
||
|
||
<span style="color: #a0522d;">delta</span> = (interval[1]-interval[0])**3
|
||
|
||
<span style="color: #a0522d;">r</span> = np.array([0.,0.,0.])
|
||
|
||
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
|
||
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E:10.8f} \t \sigma^2 = {s2:10.8f}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">variance_hydrogen</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), w, delta, energy, energy2</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> dx, r(3), a(6), norm, e_tmp, s2</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, k, l, j</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
|
||
a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
|
||
|
||
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
x(i) = -5.d0 + (i-1)*dx
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'a = '</span>, a(j), <span style="color: #8b2252;">' E = '</span>, energy
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">variance_hydrogen</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
To compile and run:
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
|
||
./variance_hydrogen
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org984e04b" class="outline-5">
|
||
<h5 id="org984e04b"><span class="section-number-5">2.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-2-4-2-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">import</span> numpy <span style="color: #a020f0;">as</span> np
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> e_loc, psi
|
||
|
||
<span style="color: #a0522d;">interval</span> = np.linspace(-5,5,num=50)
|
||
|
||
<span style="color: #a0522d;">delta</span> = (interval[1]-interval[0])**3
|
||
|
||
<span style="color: #a0522d;">r</span> = np.array([0.,0.,0.])
|
||
|
||
<span style="color: #a020f0;">for</span> a <span style="color: #a020f0;">in</span> [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
|
||
<span style="color: #a0522d;">E</span> = 0.
|
||
<span style="color: #a0522d;">E2</span> = 0.
|
||
<span style="color: #a0522d;">norm</span> = 0.
|
||
|
||
<span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> interval:
|
||
<span style="color: #a0522d;">r</span>[0] = x
|
||
|
||
<span style="color: #a020f0;">for</span> y <span style="color: #a020f0;">in</span> interval:
|
||
<span style="color: #a0522d;">r</span>[1] = y
|
||
|
||
<span style="color: #a020f0;">for</span> z <span style="color: #a020f0;">in</span> interval:
|
||
<span style="color: #a0522d;">r</span>[2] = z
|
||
|
||
<span style="color: #a0522d;">w</span> = psi(a,r)
|
||
<span style="color: #a0522d;">w</span> = w * w * delta
|
||
|
||
<span style="color: #a0522d;">e_tmp</span> = e_loc(a,r)
|
||
<span style="color: #a0522d;">E</span> += w * e_tmp
|
||
<span style="color: #a0522d;">E2</span> += w * e_tmp * e_tmp
|
||
<span style="color: #a0522d;">norm</span> += w
|
||
|
||
<span style="color: #a0522d;">E</span> = E / norm
|
||
<span style="color: #a0522d;">E2</span> = E2 / norm
|
||
|
||
<span style="color: #a0522d;">s2</span> = E2 - E**2
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E:10.8f} \t \sigma^2 = {s2:10.8f}"</span>)
|
||
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
a = 0.1 E = -0.24518439 \sigma^2 = 0.02696522
|
||
a = 0.2 E = -0.26966058 \sigma^2 = 0.03719707
|
||
a = 0.5 E = -0.38563576 \sigma^2 = 0.05318597
|
||
a = 0.9 E = -0.49435710 \sigma^2 = 0.00577812
|
||
a = 1.0 E = -0.50000000 \sigma^2 = 0.00000000
|
||
a = 1.5 E = -0.39242967 \sigma^2 = 0.31449671
|
||
a = 2.0 E = -0.08086981 \sigma^2 = 1.80688143
|
||
|
||
</pre>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">program</span> <span style="color: #0000ff;">variance_hydrogen</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> x(50), w, delta, energy, energy2</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> dx, r(3), a(6), norm, e_tmp, s2</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i, k, l, j</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
|
||
a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
|
||
|
||
dx = 10.d0/(<span style="color: #a020f0;">size</span>(x)-1)
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
x(i) = -5.d0 + (i-1)*dx
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
delta = dx**3
|
||
|
||
r(:) = 0.d0
|
||
|
||
<span style="color: #a020f0;">do</span> j=1,<span style="color: #a020f0;">size</span>(a)
|
||
energy = 0.d0
|
||
energy2 = 0.d0
|
||
norm = 0.d0
|
||
|
||
<span style="color: #a020f0;">do</span> i=1,<span style="color: #a020f0;">size</span>(x)
|
||
r(1) = x(i)
|
||
|
||
<span style="color: #a020f0;">do</span> k=1,<span style="color: #a020f0;">size</span>(x)
|
||
r(2) = x(k)
|
||
|
||
<span style="color: #a020f0;">do</span> l=1,<span style="color: #a020f0;">size</span>(x)
|
||
r(3) = x(l)
|
||
|
||
w = psi(a(j),r)
|
||
w = w * w * delta
|
||
|
||
e_tmp = e_loc(a(j), r)
|
||
|
||
energy = energy + w * e_tmp
|
||
energy2 = energy2 + w * e_tmp * e_tmp
|
||
norm = norm + w
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
energy = energy / norm
|
||
energy2 = energy2 / norm
|
||
|
||
s2 = energy2 - energy*energy
|
||
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'a = '</span>, a(j), <span style="color: #8b2252;">' E = '</span>, energy, <span style="color: #8b2252;">' s2 = '</span>, s2
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">variance_hydrogen</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
a = 0.10000000000000001 E = -0.24518438948809140 s2 = 2.6965218719722767E-002
|
||
a = 0.20000000000000001 E = -0.26966057967803236 s2 = 3.7197072370201284E-002
|
||
a = 0.50000000000000000 E = -0.38563576125173815 s2 = 5.3185967578480653E-002
|
||
a = 1.0000000000000000 E = -0.50000000000000000 s2 = 0.0000000000000000
|
||
a = 1.5000000000000000 E = -0.39242967082602065 s2 = 0.31449670909172917
|
||
a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814270846534
|
||
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orged64ccd" class="outline-2">
|
||
<h2 id="orged64ccd"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
|
||
<div class="outline-text-2" id="text-3">
|
||
<p>
|
||
Numerical integration with deterministic methods is very efficient
|
||
in low dimensions. When the number of dimensions becomes large,
|
||
instead of computing the average energy as a numerical integration
|
||
on a grid, it is usually more efficient to use Monte Carlo sampling.
|
||
</p>
|
||
|
||
<p>
|
||
Moreover, Monte Carlo sampling will allow us to remove the bias due
|
||
to the discretization of space, and compute a statistical confidence
|
||
interval.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org01e38cc" class="outline-3">
|
||
<h3 id="org01e38cc"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
|
||
<div class="outline-text-3" id="text-3-1">
|
||
<p>
|
||
To compute the statistical error, you need to perform \(M\)
|
||
independent Monte Carlo calculations. You will obtain \(M\) different
|
||
estimates of the energy, which are expected to have a Gaussian
|
||
distribution for large \(M\), according to the <a href="https://en.wikipedia.org/wiki/Central_limit_theorem">Central Limit Theorem</a>.
|
||
</p>
|
||
|
||
<p>
|
||
The estimate of the energy is
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
E = \frac{1}{M} \sum_{i=1}^M E_i
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
The variance of the average energies can be computed as
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\sigma^2 = \frac{1}{M-1} \sum_{i=1}^{M} (E_i - E)^2
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
And the confidence interval is given by
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
E \pm \delta E, \text{ where } \delta E = \frac{\sigma}{\sqrt{M}}
|
||
\]
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org0dbdc24" class="outline-4">
|
||
<h4 id="org0dbdc24"><span class="section-number-4">3.1.1</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-3-1-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Write a function returning the average and statistical error of an
|
||
input array.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> math <span style="color: #a020f0;">import</span> sqrt
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">ave_error</span>(arr):
|
||
#<span style="color: #b22222;">TODO</span>
|
||
<span style="color: #a020f0;">return</span> (average, error)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">ave_error</span>(x,n,ave,err)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> n </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> x(n) </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">ave_error</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgdfc0533" class="outline-5">
|
||
<h5 id="orgdfc0533"><span class="section-number-5">3.1.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-3-1-1-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> math <span style="color: #a020f0;">import</span> sqrt
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">ave_error</span>(arr):
|
||
<span style="color: #a0522d;">M</span> = <span style="color: #483d8b;">len</span>(arr)
|
||
<span style="color: #a020f0;">assert</span>(M>0)
|
||
|
||
<span style="color: #a020f0;">if</span> M == 1:
|
||
<span style="color: #a0522d;">average</span> = arr[0]
|
||
<span style="color: #a0522d;">error</span> = 0.
|
||
|
||
<span style="color: #a020f0;">else</span>:
|
||
<span style="color: #a0522d;">average</span> = <span style="color: #483d8b;">sum</span>(arr)/M
|
||
<span style="color: #a0522d;">variance</span> = 1./(M-1) * <span style="color: #483d8b;">sum</span>( [ (x - average)**2 <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> arr ] )
|
||
<span style="color: #a0522d;">error</span> = sqrt(variance/M)
|
||
|
||
<span style="color: #a020f0;">return</span> (average, error)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">ave_error</span>(x,n,ave,err)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
|
||
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> n </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> x(n) </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> variance</span>
|
||
|
||
<span style="color: #a020f0;">if</span> (n < 1) <span style="color: #a020f0;">then</span>
|
||
<span style="color: #a020f0;">stop</span> <span style="color: #8b2252;">'n<1 in ave_error'</span>
|
||
|
||
<span style="color: #a020f0;">else if</span> (n == 1) <span style="color: #a020f0;">then</span>
|
||
ave = x(1)
|
||
err = 0.d0
|
||
|
||
<span style="color: #a020f0;">else</span>
|
||
ave = <span style="color: #a020f0;">sum</span>(x(:)) / <span style="color: #a020f0;">dble</span>(n)
|
||
|
||
variance = <span style="color: #a020f0;">sum</span>((x(:) - ave)**2) / <span style="color: #a020f0;">dble</span>(n-1)
|
||
err = dsqrt(variance/<span style="color: #a020f0;">dble</span>(n))
|
||
|
||
<span style="color: #a020f0;">endif</span>
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">ave_error</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgfd318d6" class="outline-3">
|
||
<h3 id="orgfd318d6"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
|
||
<div class="outline-text-3" id="text-3-2">
|
||
<p>
|
||
We will now perform our first Monte Carlo calculation to compute the
|
||
energy of the hydrogen atom.
|
||
</p>
|
||
|
||
<p>
|
||
Consider again the expression of the energy
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
E & = & \frac{\int E_L(\mathbf{r})|\Psi(\mathbf{r})|^2\,d\mathbf{r}}{\int |\Psi(\mathbf{r}) |^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>
|
||
|
||
<ul class="org-ul">
|
||
<li>Draw a random point \(\mathbf{r}_i\) in the box \((-5,-5,-5) \le
|
||
(x,y,z) \le (5,5,5)\)</li>
|
||
<li>Compute \(|\Psi(\mathbf{r}_i)|^2\) and accumulate the result in a
|
||
variable <code>normalization</code></li>
|
||
<li>Compute \(|\Psi(\mathbf{r}_i)|^2 \times E_L(\mathbf{r}_i)\), and accumulate the
|
||
result in a variable <code>energy</code></li>
|
||
</ul>
|
||
|
||
<p>
|
||
Once all the iterations have been computed, the run returns the average energy
|
||
\(\bar{E}_k\) over the \(N_{\rm MC}\) iterations of the run.
|
||
</p>
|
||
|
||
<p>
|
||
To compute the statistical error, perform \(M\) independent runs. The
|
||
final estimate of the energy will be the average over the
|
||
\(\bar{E}_k\), and the variance of the \(\bar{E}_k\) will be used to
|
||
compute the statistical error.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org74432b8" class="outline-4">
|
||
<h4 id="org74432b8"><span class="section-number-4">3.2.1</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-3-2-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Parameterize the wave function with \(a=1.2\). Perform 30
|
||
independent Monte Carlo runs, each with 100 000 Monte Carlo
|
||
steps. Store the final energies of each run and use this array to
|
||
compute the average energy and the associated error bar.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="note">
|
||
<p>
|
||
To draw a uniform random number in Python, you can use
|
||
the <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.uniform.html"><code>random.uniform</code></a> function of Numpy.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a, nmax):
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
|
||
#<span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="note">
|
||
<p>
|
||
To draw a uniform random number in Fortran, you can use
|
||
the <a href="https://gcc.gnu.org/onlinedocs/gfortran/RANDOM_005fNUMBER.html"><code>RANDOM_NUMBER</code></a> subroutine.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="note">
|
||
<p>
|
||
When running Monte Carlo calculations, the number of steps is
|
||
usually very large. We expect <code>nmax</code> to be possibly larger than 2
|
||
billion, so we use 8-byte integers (<code>integer*8</code>) to represent it, as
|
||
well as the index of the current step.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">uniform_montecarlo</span>(a,nmax,energy)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> norm, r(3), w</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">uniform_montecarlo</span>
|
||
|
||
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 1.2d0</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
|
||
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
|
||
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
!<span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
|
||
./qmc_uniform
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org99a7b2f" class="outline-5">
|
||
<h5 id="org99a7b2f"><span class="section-number-5">3.2.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-3-2-1-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a, nmax):
|
||
<span style="color: #a0522d;">energy</span> = 0.
|
||
<span style="color: #a0522d;">normalization</span> = 0.
|
||
|
||
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
|
||
<span style="color: #a0522d;">r</span> = np.random.uniform(-5., 5., (3))
|
||
|
||
<span style="color: #a0522d;">w</span> = psi(a,r)
|
||
<span style="color: #a0522d;">w</span> = w*w
|
||
|
||
<span style="color: #a0522d;">energy</span> += w * e_loc(a,r)
|
||
<span style="color: #a0522d;">normalization</span> += w
|
||
|
||
<span style="color: #a020f0;">return</span> energy / normalization
|
||
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
|
||
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,nmax) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
|
||
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
E = -0.48363807880008725 +/- 0.002330876047368999
|
||
|
||
</pre>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">uniform_montecarlo</span>(a,nmax,energy)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> norm, r(3), w</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
|
||
energy = 0.d0
|
||
norm = 0.d0
|
||
|
||
<span style="color: #a020f0;">do</span> istep = 1,nmax
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(r)
|
||
r(:) = -5.d0 + 10.d0*r(:)
|
||
|
||
w = psi(a,r)
|
||
w = w*w
|
||
|
||
energy = energy + w * e_loc(a,r)
|
||
norm = norm + w
|
||
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
energy = energy / norm
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">uniform_montecarlo</span>
|
||
|
||
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 1.2d0</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
|
||
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
|
||
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
<span style="color: #a020f0;">do</span> irun=1,nruns
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">uniform_montecarlo</span>(a, nmax, X(irun))
|
||
<span style="color: #a020f0;">enddo</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X, nruns, ave, err)
|
||
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
E = -0.48084122147238995 +/- 2.4983775878329355E-003
|
||
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgcddcdcf" class="outline-3">
|
||
<h3 id="orgcddcdcf"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
|
||
<div class="outline-text-3" id="text-3-3">
|
||
<p>
|
||
We will now use the square of the wave function to sample random
|
||
points distributed with the probability density
|
||
\[
|
||
P(\mathbf{r}) = \frac{|\Psi(\mathbf{r})|^2}{\int |\Psi(\mathbf{r})|^2 d\mathbf{r}}\,.
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
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
|
||
sampling:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
E \approx \frac{1}{N_{\rm MC}}\sum_{i=1}^{N_{\rm MC}} E_L(\mathbf{r}_i)\,.
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
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
|
||
initial position \(\mathbf{r}_0\), we will realize a random walk:
|
||
</p>
|
||
|
||
<p>
|
||
\[ \mathbf{r}_0 \rightarrow \mathbf{r}_1 \rightarrow \mathbf{r}_2 \ldots \rightarrow \mathbf{r}_{N_{\rm MC}}\,, \]
|
||
</p>
|
||
|
||
<p>
|
||
according to the following algorithm.
|
||
</p>
|
||
|
||
<p>
|
||
At every step, we propose a new move according to a transition probability \(T(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1})\) of our choice.
|
||
</p>
|
||
|
||
<p>
|
||
For simplicity, we will move the electron in a 3-dimensional box of side \(2\delta L\) centered at the current position
|
||
of the electron:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta L \, \mathbf{u}
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
where \(\delta L\) is a fixed constant, and
|
||
\(\mathbf{u}\) is a uniform random number in a 3-dimensional box
|
||
\((-1,-1,-1) \le \mathbf{u} \le (1,1,1)\).
|
||
</p>
|
||
|
||
<p>
|
||
After having moved the electron, we add the
|
||
accept/reject step that guarantees that the distribution of the
|
||
\(\mathbf{r}_n\) is \(\Psi^2\). This amounts to accepting the move with
|
||
probability
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
A(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1}) = \min\left(1,\frac{T(\mathbf{r}_{n+1}\rightarrow\mathbf{r}_{n}) P(\mathbf{r}_{n+1})}{T(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1})P(\mathbf{r}_{n})}\right)\,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
which, for our choice of transition probability, becomes
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
A(\mathbf{r}_{n}\rightarrow\mathbf{r}_{n+1}) = \min\left(1,\frac{P(\mathbf{r}_{n+1})}{P(\mathbf{r}_{n})}\right)= \min\left(1,\frac{|\Psi(\mathbf{r}_{n+1})|^2}{|\Psi(\mathbf{r}_{n})|^2}\right)\,.
|
||
\]
|
||
</p>
|
||
|
||
<div class="exercise">
|
||
<p>
|
||
Explain why the transition probability cancels out in the
|
||
expression of \(A\).
|
||
</p>
|
||
|
||
</div>
|
||
<p>
|
||
Also note that we do not need to compute the norm of the wave function!
|
||
</p>
|
||
|
||
<p>
|
||
The algorithm is summarized as follows:
|
||
</p>
|
||
|
||
<ol class="org-ol">
|
||
<li>Evaluate the local energy at \(\mathbf{r}_n\) and accumulate it</li>
|
||
<li>Compute a new position \(\mathbf{r'} = \mathbf{r}_n + \delta L\, \mathbf{u}\)</li>
|
||
<li>Evaluate \(\Psi(\mathbf{r}')\) at the new position</li>
|
||
<li>Compute the ratio \(A = \frac{\left|\Psi(\mathbf{r'})\right|^2}{\left|\Psi(\mathbf{r}_{n})\right|^2}\)</li>
|
||
<li>Draw a uniform random number \(v \in [0,1]\)</li>
|
||
<li>if \(v \le A\), accept the move : set \(\mathbf{r}_{n+1} = \mathbf{r'}\)</li>
|
||
<li>else, reject the move : set \(\mathbf{r}_{n+1} = \mathbf{r}_n\)</li>
|
||
</ol>
|
||
|
||
<div class="note">
|
||
<p>
|
||
A common error is to remove the rejected samples from the
|
||
calculation of the average. <b>Don't do it!</b>
|
||
</p>
|
||
|
||
<p>
|
||
All samples should be kept, from both accepted <i>and</i> rejected moves.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-org8b3f6fe" class="outline-4">
|
||
<h4 id="org8b3f6fe"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
|
||
<div class="outline-text-4" id="text-3-3-1">
|
||
<p>
|
||
If the box is infinitely small, the ratio will be very close
|
||
to one and all the steps will be accepted. However, the moves will be
|
||
very correlated and you will visit the configurational space very slowly.
|
||
</p>
|
||
|
||
<p>
|
||
On the other hand, if you propose too large moves, the number of
|
||
accepted steps will decrease because the ratios might become
|
||
small. If the number of accepted steps is close to zero, then the
|
||
space is not well sampled either.
|
||
</p>
|
||
|
||
<p>
|
||
The size of the move should be adjusted so that it is as large as
|
||
possible, keeping the number of accepted steps not too small. To
|
||
achieve that, we define the acceptance rate as the number of
|
||
accepted steps over the total number of steps. Adjusting the time
|
||
step such that the acceptance rate is close to 0.5 is a good
|
||
compromise for the current problem.
|
||
</p>
|
||
|
||
<div class="note">
|
||
<p>
|
||
Below, we use the symbol \(\delta t\) to denote \(\delta L\) since we will use
|
||
the same variable later on to store a time step.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-orgd7c523f" class="outline-4">
|
||
<h4 id="orgd7c523f"><span class="section-number-4">3.3.2</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-3-3-2">
|
||
<div class="exercise">
|
||
<p>
|
||
Modify the program of the previous section to compute the energy,
|
||
sampled with \(\Psi^2\).
|
||
</p>
|
||
|
||
<p>
|
||
Compute also the acceptance rate, so that you can adapt the time
|
||
step in order to have an acceptance rate close to 0.5.
|
||
</p>
|
||
|
||
<p>
|
||
Can you observe a reduction in the statistical error?
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,nmax,dt):
|
||
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">return</span> energy/nmax, N_accep/nmax
|
||
|
||
|
||
# <span style="color: #b22222;">Run simulation</span>
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
<span style="color: #a0522d;">dt</span> = #<span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a,nmax,dt) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
|
||
# <span style="color: #b22222;">Energy</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
|
||
|
||
# <span style="color: #b22222;">Acceptance rate</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (_, x) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"A = {A} +/- {deltaA}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">metropolis_montecarlo</span>(a,nmax,dt,energy,accep)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> dt </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> accep</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_accep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3), psi_old, psi_new</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> v, ratio</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi, gaussian</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">metropolis_montecarlo</span>
|
||
|
||
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 1.2d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> dt = </span>! <span style="color: #b22222;">TODO</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
|
||
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
|
||
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns), Y(nruns)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
<span style="color: #a020f0;">do</span> irun=1,nruns
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">metropolis_montecarlo</span>(a,nmax,dt,X(irun),Y(irun))
|
||
<span style="color: #a020f0;">enddo</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(Y,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'A = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
|
||
./qmc_metropolis
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgac64351" class="outline-5">
|
||
<h5 id="orgac64351"><span class="section-number-5">3.3.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-3-3-2-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,nmax,dt):
|
||
<span style="color: #a0522d;">energy</span> = 0.
|
||
<span style="color: #a0522d;">N_accep</span> = 0
|
||
|
||
<span style="color: #a0522d;">r_old</span> = np.random.uniform(-dt, dt, (3))
|
||
<span style="color: #a0522d;">psi_old</span> = psi(a,r_old)
|
||
|
||
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
|
||
<span style="color: #a0522d;">energy</span> += e_loc(a,r_old)
|
||
|
||
<span style="color: #a0522d;">r_new</span> = r_old + np.random.uniform(-dt,dt,(3))
|
||
<span style="color: #a0522d;">psi_new</span> = psi(a,r_new)
|
||
|
||
<span style="color: #a0522d;">ratio</span> = (psi_new / psi_old)**2
|
||
|
||
<span style="color: #a020f0;">if</span> np.random.uniform() <= ratio:
|
||
<span style="color: #a0522d;">N_accep</span> += 1
|
||
|
||
<span style="color: #a0522d;">r_old</span> = r_new
|
||
<span style="color: #a0522d;">psi_old</span> = psi_new
|
||
|
||
<span style="color: #a020f0;">return</span> energy/nmax, N_accep/nmax
|
||
|
||
# <span style="color: #b22222;">Run simulation</span>
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
<span style="color: #a0522d;">dt</span> = 1.0
|
||
|
||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a,nmax,dt) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
|
||
# <span style="color: #b22222;">Energy</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
|
||
|
||
# <span style="color: #b22222;">Acceptance rate</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (_, x) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"A = {A} +/- {deltaA}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
E = -0.4802595860693983 +/- 0.0005124420418289207
|
||
A = 0.5074913333333334 +/- 0.000350889422714878
|
||
|
||
</pre>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">metropolis_montecarlo</span>(a,nmax,dt,energy,accep)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> dt</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> accep</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3), psi_old, psi_new</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> v, ratio</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_accep</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi, gaussian</span>
|
||
|
||
energy = 0.d0
|
||
n_accep = 0_8
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(r_old)
|
||
r_old(:) = dt * (2.d0*r_old(:) - 1.d0)
|
||
psi_old = psi(a,r_old)
|
||
|
||
<span style="color: #a020f0;">do</span> istep = 1,nmax
|
||
energy = energy + e_loc(a,r_old)
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(r_new)
|
||
r_new(:) = r_old(:) + dt*(2.d0*r_new(:) - 1.d0)
|
||
|
||
psi_new = psi(a,r_new)
|
||
|
||
ratio = (psi_new / psi_old)**2
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(v)
|
||
|
||
<span style="color: #a020f0;">if</span> (v <= ratio) <span style="color: #a020f0;">then</span>
|
||
|
||
n_accep = n_accep + 1_8
|
||
|
||
r_old(:) = r_new(:)
|
||
psi_old = psi_new
|
||
|
||
<span style="color: #a020f0;">endif</span>
|
||
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
energy = energy / <span style="color: #a020f0;">dble</span>(nmax)
|
||
accep = <span style="color: #a020f0;">dble</span>(n_accep) / <span style="color: #a020f0;">dble</span>(nmax)
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">metropolis_montecarlo</span>
|
||
|
||
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 1.2d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> dt = 1.0d0</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
|
||
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
|
||
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns), Y(nruns)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
<span style="color: #a020f0;">do</span> irun=1,nruns
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">metropolis_montecarlo</span>(a,nmax,dt,X(irun),Y(irun))
|
||
<span style="color: #a020f0;">enddo</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(Y,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'A = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
E = -0.47948142754168033 +/- 4.8410177863919307E-004
|
||
A = 0.50762633333333318 +/- 3.4601756760043725E-004
|
||
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org4f2ffad" class="outline-3">
|
||
<h3 id="org4f2ffad"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
|
||
<div class="outline-text-3" id="text-3-4">
|
||
<p>
|
||
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
|
||
</p>
|
||
|
||
<p>
|
||
The Metropolis acceptance step has to be adapted accordingly to ensure that the detailed balance condition is satisfied. This means that
|
||
the acceptance probability \(A\) is chosen so that it is consistent with the
|
||
probability of leaving \(\mathbf{r}_n\) and the probability of
|
||
entering \(\mathbf{r}_{n+1}\):
|
||
</p>
|
||
|
||
<p>
|
||
\[ A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \min \left( 1,
|
||
\frac{T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) P(\mathbf{r}_{n+1})}
|
||
{T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) P(\mathbf{r}_{n})}
|
||
\right)
|
||
\]
|
||
where \(T(\mathbf{r}_n \rightarrow \mathbf{r}_{n+1})\) is the
|
||
probability of transition from \(\mathbf{r}_n\) to
|
||
\(\mathbf{r}_{n+1}\).
|
||
</p>
|
||
|
||
<p>
|
||
In the previous example, we were using uniform sampling in a box centered
|
||
at the current position. Hence, the transition probability was symmetric
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n})
|
||
= \text{constant}\,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
so the expression of \(A\) was simplified to the ratios of the squared
|
||
wave functions.
|
||
</p>
|
||
|
||
<p>
|
||
Now, if instead of drawing uniform random numbers, we
|
||
choose to draw Gaussian random numbers with zero mean and variance
|
||
\(\delta t\), the transition probability becomes:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
|
||
\frac{1}{(2\pi\,\delta t)^{3/2}} \exp \left[ - \frac{\left(
|
||
\mathbf{r}_{n+1} - \mathbf{r}_{n} \right)^2}{2\delta t} \right]\,.
|
||
\]
|
||
</p>
|
||
|
||
|
||
<p>
|
||
Furthermore, to sample the density even better, we can "push" the electrons
|
||
into in the regions of high probability, and "pull" them away from
|
||
the low-probability regions. This will increase the
|
||
acceptance ratios and improve the sampling.
|
||
</p>
|
||
|
||
<p>
|
||
To do this, we can use the gradient of the probability density
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\frac{\nabla [ \Psi^2 ]}{\Psi^2} = 2 \frac{\nabla \Psi}{\Psi}\,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
and add the so-called drift vector, \(\frac{\nabla \Psi}{\Psi}\), so that the numerical scheme becomes a
|
||
drifted diffusion with transition probability:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
|
||
\frac{1}{(2\pi\,\delta t)^{3/2}} \exp \left[ - \frac{\left(
|
||
\mathbf{r}_{n+1} - \mathbf{r}_{n} - \delta t\frac{\nabla
|
||
\Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{2\,\delta t} \right]\,.
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
The corresponding move is proposed as
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta t\, \frac{\nabla
|
||
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi \,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
where \(\chi\) is a Gaussian random variable with zero mean and
|
||
variance \(\delta t\).
|
||
</p>
|
||
|
||
|
||
|
||
<p>
|
||
The algorithm of the previous exercise is only slighlty modified as:
|
||
</p>
|
||
|
||
<ol class="org-ol">
|
||
<li>Evaluate the local energy at \(\mathbf{r}_{n}\) and accumulate it</li>
|
||
<li>Compute a new position \(\mathbf{r'} = \mathbf{r}_n +
|
||
\delta t\, \frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi\)</li>
|
||
<li>Evaluate \(\Psi(\mathbf{r}')\) and \(\frac{\nabla \Psi(\mathbf{r'})}{\Psi(\mathbf{r'})}\) at the new position</li>
|
||
<li>Compute the ratio \(A = \frac{T(\mathbf{r}' \rightarrow \mathbf{r}_{n}) P(\mathbf{r}')}{T(\mathbf{r}_{n} \rightarrow \mathbf{r}') P(\mathbf{r}_{n})}\)</li>
|
||
<li>Draw a uniform random number \(v \in [0,1]\)</li>
|
||
<li>if \(v \le A\), accept the move : set \(\mathbf{r}_{n+1} = \mathbf{r'}\)</li>
|
||
<li>else, reject the move : set \(\mathbf{r}_{n+1} = \mathbf{r}_n\)</li>
|
||
</ol>
|
||
</div>
|
||
|
||
<div id="outline-container-org6290597" class="outline-4">
|
||
<h4 id="org6290597"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
|
||
<div class="outline-text-4" id="text-3-4-1">
|
||
<p>
|
||
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:
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
z_1 &=& \sqrt{-2 \ln u_1} \cos(2 \pi u_2) \\
|
||
z_2 &=& \sqrt{-2 \ln u_1} \sin(2 \pi u_2)
|
||
\end{eqnarray*}
|
||
|
||
<p>
|
||
Below is a Fortran implementation returning a Gaussian-distributed
|
||
n-dimensional vector \(\mathbf{z}\). This will be useful for the
|
||
following sections.
|
||
</p>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">random_gauss</span>(z,n)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">integer</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> n</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> z(n)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> u(n+1)</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> two_pi = 2.d0*dacos(-1.d0)</span>
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> i</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(u)
|
||
|
||
<span style="color: #a020f0;">if</span> (<span style="color: #a020f0;">iand</span>(n,1) == 0) <span style="color: #a020f0;">then</span>
|
||
! <span style="color: #b22222;">n is even</span>
|
||
<span style="color: #a020f0;">do</span> i=1,n,2
|
||
z(i) = dsqrt(-2.d0*dlog(u(i)))
|
||
z(i+1) = z(i) * dsin( two_pi*u(i+1) )
|
||
z(i) = z(i) * dcos( two_pi*u(i+1) )
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
<span style="color: #a020f0;">else</span>
|
||
! <span style="color: #b22222;">n is odd</span>
|
||
<span style="color: #a020f0;">do</span> i=1,n-1,2
|
||
z(i) = dsqrt(-2.d0*dlog(u(i)))
|
||
z(i+1) = z(i) * dsin( two_pi*u(i+1) )
|
||
z(i) = z(i) * dcos( two_pi*u(i+1) )
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
z(n) = dsqrt(-2.d0*dlog(u(n)))
|
||
z(n) = z(n) * dcos( two_pi*u(n+1) )
|
||
|
||
<span style="color: #a020f0;">end if</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">random_gauss</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
In Python, you can use the <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html"><code>random.normal</code></a> function of Numpy.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-orgc3a15c4" class="outline-4">
|
||
<h4 id="orgc3a15c4"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
|
||
<div class="outline-text-4" id="text-3-4-2">
|
||
<div class="exercise">
|
||
<p>
|
||
If you use Fortran, copy/paste the <code>random_gauss</code> function in
|
||
a Fortran file.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="exercise">
|
||
<p>
|
||
Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\).
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">drift</span>(a,r):
|
||
# <span style="color: #b22222;">TODO</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">drift</span>(a,r,b)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> b(3)</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">drift</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org1bb7357" class="outline-5">
|
||
<h5 id="org1bb7357"><span class="section-number-5">3.4.2.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-3-4-2-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">def</span> <span style="color: #0000ff;">drift</span>(a,r):
|
||
<span style="color: #a0522d;">ar_inv</span> = -a/np.sqrt(np.dot(r,r))
|
||
<span style="color: #a020f0;">return</span> r * ar_inv
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">drift</span>(a,r,b)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, r(3)</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> b(3)</span>
|
||
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ar_inv</span>
|
||
|
||
ar_inv = -a / dsqrt(r(1)*r(1) + r(2)*r(2) + r(3)*r(3))
|
||
b(:) = r(:) * ar_inv
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">drift</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3594157" class="outline-4">
|
||
<h4 id="org3594157"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
|
||
<div class="outline-text-4" id="text-3-4-3">
|
||
<div class="exercise">
|
||
<p>
|
||
Modify the previous program to introduce the drift-diffusion scheme.
|
||
(This is a necessary step for the next section on diffusion Monte Carlo).
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,nmax,dt):
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
# <span style="color: #b22222;">Run simulation</span>
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
<span style="color: #a0522d;">dt</span> = # <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a,nmax,dt) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
|
||
# <span style="color: #b22222;">Energy</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
|
||
|
||
# <span style="color: #b22222;">Acceptance rate</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (_, x) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"A = {A} +/- {deltaA}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,dt,nmax,energy,accep)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, dt</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy, accep</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_accep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> sq_dt, chi(3)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> psi_old, psi_new</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> d_old(3), d_new(3)</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>
|
||
|
||
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 1.2d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> dt = </span>! <span style="color: #b22222;">TODO</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
|
||
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
|
||
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns), accep(nruns)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
<span style="color: #a020f0;">do</span> irun=1,nruns
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,dt,nmax,X(irun),accep(irun))
|
||
<span style="color: #a020f0;">enddo</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(accep,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'A = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
|
||
./vmc_metropolis
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgfe0293b" class="outline-5">
|
||
<h5 id="orgfe0293b"><span class="section-number-5">3.4.3.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-3-4-3-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,nmax,dt):
|
||
<span style="color: #a0522d;">sq_dt</span> = np.sqrt(dt)
|
||
|
||
<span style="color: #a0522d;">energy</span> = 0.
|
||
<span style="color: #a0522d;">N_accep</span> = 0
|
||
|
||
<span style="color: #a0522d;">r_old</span> = np.random.normal(loc=0., scale=1.0, size=(3))
|
||
<span style="color: #a0522d;">d_old</span> = drift(a,r_old)
|
||
<span style="color: #a0522d;">d2_old</span> = np.dot(d_old,d_old)
|
||
<span style="color: #a0522d;">psi_old</span> = psi(a,r_old)
|
||
|
||
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
|
||
<span style="color: #a0522d;">chi</span> = np.random.normal(loc=0., scale=1.0, size=(3))
|
||
|
||
<span style="color: #a0522d;">energy</span> += e_loc(a,r_old)
|
||
|
||
<span style="color: #a0522d;">r_new</span> = r_old + dt * d_old + sq_dt * chi
|
||
<span style="color: #a0522d;">d_new</span> = drift(a,r_new)
|
||
<span style="color: #a0522d;">d2_new</span> = np.dot(d_new,d_new)
|
||
<span style="color: #a0522d;">psi_new</span> = psi(a,r_new)
|
||
|
||
# <span style="color: #b22222;">Metropolis</span>
|
||
<span style="color: #a0522d;">prod</span> = np.dot((d_new + d_old), (r_new - r_old))
|
||
<span style="color: #a0522d;">argexpo</span> = 0.5 * (d2_new - d2_old)*dt + prod
|
||
|
||
<span style="color: #a0522d;">q</span> = psi_new / psi_old
|
||
<span style="color: #a0522d;">q</span> = np.exp(-argexpo) * q*q
|
||
|
||
<span style="color: #a020f0;">if</span> np.random.uniform() <= q:
|
||
<span style="color: #a0522d;">N_accep</span> += 1
|
||
|
||
<span style="color: #a0522d;">r_old</span> = r_new
|
||
<span style="color: #a0522d;">d_old</span> = d_new
|
||
<span style="color: #a0522d;">d2_old</span> = d2_new
|
||
<span style="color: #a0522d;">psi_old</span> = psi_new
|
||
|
||
<span style="color: #a020f0;">return</span> energy/nmax, N_accep/nmax
|
||
|
||
|
||
# <span style="color: #b22222;">Run simulation</span>
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
<span style="color: #a0522d;">dt</span> = 1.0
|
||
|
||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a,nmax,dt) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
|
||
# <span style="color: #b22222;">Energy</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
|
||
|
||
# <span style="color: #b22222;">Acceptance rate</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (_, x) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"A = {A} +/- {deltaA}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
E = -0.48034171558629885 +/- 0.0005286038561061781
|
||
A = 0.6210380000000001 +/- 0.0005457375900937905
|
||
|
||
</pre>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,dt,nmax,energy,accep)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, dt</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy, accep</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_accep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> sq_dt, chi(3), d2_old, prod, u</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> psi_old, psi_new, d2_new, argexpo, q</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> d_old(3), d_new(3)</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
|
||
sq_dt = dsqrt(dt)
|
||
|
||
! <span style="color: #b22222;">Initialization</span>
|
||
energy = 0.d0
|
||
n_accep = 0_8
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(r_old,3)
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_old,d_old)
|
||
d2_old = d_old(1)*d_old(1) + <span style="color: #a020f0;">&</span>
|
||
d_old(2)*d_old(2) + <span style="color: #a020f0;">&</span>
|
||
d_old(3)*d_old(3)
|
||
|
||
psi_old = psi(a,r_old)
|
||
|
||
<span style="color: #a020f0;">do</span> istep = 1,nmax
|
||
energy = energy + e_loc(a,r_old)
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(chi,3)
|
||
r_new(:) = r_old(:) + dt*d_old(:) + chi(:)*sq_dt
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_new,d_new)
|
||
d2_new = d_new(1)*d_new(1) + <span style="color: #a020f0;">&</span>
|
||
d_new(2)*d_new(2) + <span style="color: #a020f0;">&</span>
|
||
d_new(3)*d_new(3)
|
||
|
||
psi_new = psi(a,r_new)
|
||
|
||
! <span style="color: #b22222;">Metropolis</span>
|
||
prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + <span style="color: #a020f0;">&</span>
|
||
(d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + <span style="color: #a020f0;">&</span>
|
||
(d_new(3) + d_old(3))*(r_new(3) - r_old(3))
|
||
|
||
argexpo = 0.5d0 * (d2_new - d2_old)*dt + prod
|
||
|
||
q = psi_new / psi_old
|
||
q = dexp(-argexpo) * q*q
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(u)
|
||
|
||
<span style="color: #a020f0;">if</span> (u <= q) <span style="color: #a020f0;">then</span>
|
||
|
||
n_accep = n_accep + 1_8
|
||
|
||
r_old(:) = r_new(:)
|
||
d_old(:) = d_new(:)
|
||
d2_old = d2_new
|
||
psi_old = psi_new
|
||
|
||
<span style="color: #a020f0;">end if</span>
|
||
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
energy = energy / <span style="color: #a020f0;">dble</span>(nmax)
|
||
accep = <span style="color: #a020f0;">dble</span>(n_accep) / <span style="color: #a020f0;">dble</span>(nmax)
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">variational_montecarlo</span>
|
||
|
||
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 1.2d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> dt = 1.0d0</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
|
||
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
|
||
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns), accep(nruns)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
<span style="color: #a020f0;">do</span> irun=1,nruns
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">variational_montecarlo</span>(a,dt,nmax,X(irun),accep(irun))
|
||
<span style="color: #a020f0;">enddo</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(accep,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'A = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
E = -0.47940635575542706 +/- 5.5613594433433764E-004
|
||
A = 0.62037333333333333 +/- 4.8970160591451110E-004
|
||
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3154c2b" class="outline-2">
|
||
<h2 id="org3154c2b"><span class="section-number-2">4</span> Diffusion Monte Carlo</h2>
|
||
<div class="outline-text-2" id="text-4">
|
||
<p>
|
||
As we have seen, Variational Monte Carlo is a powerful method to
|
||
compute integrals in large dimensions. It is often used in cases
|
||
where the expression of the wave function is such that the integrals
|
||
can't be evaluated (multi-centered Slater-type orbitals, correlation
|
||
factors, etc).
|
||
</p>
|
||
|
||
<p>
|
||
Diffusion Monte Carlo is different. It goes beyond the computation
|
||
of the integrals associated with an input wave function, and aims at
|
||
finding a near-exact numerical solution to the Schrödinger equation.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org7b4dacf" class="outline-3">
|
||
<h3 id="org7b4dacf"><span class="section-number-3">4.1</span> Schrödinger equation in imaginary time</h3>
|
||
<div class="outline-text-3" id="text-4-1">
|
||
<p>
|
||
Consider the time-dependent Schrödinger equation:
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
i\frac{\partial \Psi(\mathbf{r},t)}{\partial t} = (\hat{H} -E_{\rm ref}) \Psi(\mathbf{r},t)\,.
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
where we introduced a shift in the energy, \(E_{\rm ref}\), for reasons which will become apparent below.
|
||
</p>
|
||
|
||
<p>
|
||
We can expand a given starting wave function, \(\Psi(\mathbf{r},0)\), in the basis of the eigenstates
|
||
of the time-independent Hamiltonian, \(\Phi_k\), with energies \(E_k\):
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\Psi(\mathbf{r},0) = \sum_k a_k\, \Phi_k(\mathbf{r}).
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
The solution of the Schrödinger equation at time \(t\) is
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\Psi(\mathbf{r},t) = \sum_k a_k \exp \left( -i\, (E_k-E_{\rm ref})\, t \right) \Phi_k(\mathbf{r}).
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
Now, if we replace the time variable \(t\) by an imaginary time variable
|
||
\(\tau=i\,t\), we obtain
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
-\frac{\partial \psi(\mathbf{r}, \tau)}{\partial \tau} = (\hat{H} -E_{\rm ref}) \psi(\mathbf{r}, \tau)
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
where \(\psi(\mathbf{r},\tau) = \Psi(\mathbf{r},-i\,\tau)\)
|
||
and
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
\psi(\mathbf{r},\tau) &=& \sum_k a_k \exp( -(E_k-E_{\rm ref})\, \tau) \Phi_k(\mathbf{r})\\
|
||
&=& \exp(-(E_0-E_{\rm ref})\, \tau)\sum_k a_k \exp( -(E_k-E_0)\, \tau) \Phi_k(\mathbf{r})\,.
|
||
\end{eqnarray*}
|
||
|
||
<p>
|
||
For large positive values of \(\tau\), \(\psi\) is dominated by the
|
||
\(k=0\) term, namely, the lowest eigenstate. If we adjust \(E_{\rm ref}\) to the running estimate of \(E_0\),
|
||
we can expect that simulating the differetial equation in
|
||
imaginary time will converge to the exact ground state of the
|
||
system.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd350691" class="outline-3">
|
||
<h3 id="orgd350691"><span class="section-number-3">4.2</span> Relation to diffusion</h3>
|
||
<div class="outline-text-3" id="text-4-2">
|
||
<p>
|
||
The <a href="https://en.wikipedia.org/wiki/Diffusion_equation">diffusion equation</a> of particles is given by
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\frac{\partial \psi(\mathbf{r},t)}{\partial t} = D\, \Delta \psi(\mathbf{r},t)
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
where \(D\) is the diffusion coefficient. When the imaginary-time
|
||
Schrödinger equation is written in terms of the kinetic energy and
|
||
potential,
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\frac{\partial \psi(\mathbf{r}, \tau)}{\partial \tau} =
|
||
\left(\frac{1}{2}\Delta - [V(\mathbf{r}) -E_{\rm ref}]\right) \psi(\mathbf{r}, \tau)\,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
it can be identified as the combination of:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>a diffusion equation (Laplacian)</li>
|
||
<li>an equation whose solution is an exponential (potential)</li>
|
||
</ul>
|
||
|
||
<p>
|
||
The diffusion equation can be simulated by a Brownian motion:
|
||
</p>
|
||
|
||
<p>
|
||
\[ \mathbf{r}_{n+1} = \mathbf{r}_{n} + \sqrt{\delta t}\, \chi \]
|
||
</p>
|
||
|
||
<p>
|
||
where \(\chi\) is a Gaussian random variable, and the potential term
|
||
can be simulated by creating or destroying particles over time (a
|
||
so-called branching process) or by simply considering it as a
|
||
cumulative multiplicative weight along the diffusion trajectory
|
||
(pure Diffusion Monte Carlo):
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\prod_i \exp \left( - (V(\mathbf{r}_i) - E_{\text{ref}}) \delta t \right).
|
||
\]
|
||
</p>
|
||
|
||
|
||
<p>
|
||
We note that the ground-state wave function of a Fermionic system is
|
||
antisymmetric and changes sign. Therefore, its interpretation as a probability
|
||
distribution is somewhat problematic. In fact, mathematically, since
|
||
the Bosonic ground state is lower in energy than the Fermionic one, for
|
||
large \(\tau\), the system will evolve towards the Bosonic solution.
|
||
</p>
|
||
|
||
<p>
|
||
For the systems you will study, this is not an issue:
|
||
</p>
|
||
|
||
<ul class="org-ul">
|
||
<li>Hydrogen atom: You only have one electron!</li>
|
||
<li>Two-electron system (\(H_2\) or He): The ground-wave function is
|
||
antisymmetric in the spin variables but symmetric in the space ones.</li>
|
||
</ul>
|
||
|
||
<p>
|
||
Therefore, in both cases, you are dealing with a "Bosonic" ground state.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org78b3d2b" class="outline-3">
|
||
<h3 id="org78b3d2b"><span class="section-number-3">4.3</span> Importance sampling</h3>
|
||
<div class="outline-text-3" id="text-4-3">
|
||
<p>
|
||
In a molecular system, the potential is far from being constant
|
||
and, in fact, diverges at the inter-particle coalescence points. Hence,
|
||
it results in very large fluctuations of the erm weight associated with
|
||
the potental, making the calculations impossible in practice.
|
||
Fortunately, if we multiply the Schrödinger equation by a chosen
|
||
<i>trial wave function</i> \(\Psi_T(\mathbf{r})\) (Hartree-Fock, Kohn-Sham
|
||
determinant, CI wave function, <i>etc</i>), one obtains
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
-\frac{\partial \psi(\mathbf{r},\tau)}{\partial \tau} \Psi_T(\mathbf{r}) =
|
||
\left[ -\frac{1}{2} \Delta \psi(\mathbf{r},\tau) + V(\mathbf{r}) \psi(\mathbf{r},\tau) \right] \Psi_T(\mathbf{r})
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
Defining \(\Pi(\mathbf{r},\tau) = \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})\), (see appendix for details)
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
-\frac{\partial \Pi(\mathbf{r},\tau)}{\partial \tau}
|
||
= -\frac{1}{2} \Delta \Pi(\mathbf{r},\tau) +
|
||
\nabla \left[ \Pi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}
|
||
\right] + (E_L(\mathbf{r})-E_{\rm ref})\Pi(\mathbf{r},\tau)
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
The new "kinetic energy" can be simulated by the drift-diffusion
|
||
scheme presented in the previous section (VMC).
|
||
The new "potential" is the local energy, which has smaller fluctuations
|
||
when \(\Psi_T\) gets closer to the exact wave function.
|
||
This term can be simulated by
|
||
\[
|
||
\prod_i \exp \left( - (E_L(\mathbf{r}_i) - E_{\text{ref}}) \delta t \right).
|
||
\]
|
||
where \(E_{\rm ref}\) is the constant we had introduced above, which is adjusted to
|
||
an estimate of the average energy to keep the weights close to one.
|
||
</p>
|
||
|
||
<p>
|
||
This equation generates the <i>N</i>-electron density \(\Pi\), which is the
|
||
product of the ground state solution with the trial wave
|
||
function. You may then ask: how can we compute the total energy of
|
||
the system?
|
||
</p>
|
||
|
||
<p>
|
||
To this aim, we use the <i>mixed estimator</i> of the energy:
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
E(\tau) &=& \frac{\langle \psi(\tau) | \hat{H} | \Psi_T \rangle}{\langle \psi(\tau) | \Psi_T \rangle}\\
|
||
&=& \frac{\int \psi(\mathbf{r},\tau) \hat{H} \Psi_T(\mathbf{r}) d\mathbf{r}}
|
||
{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}} \\
|
||
&=& \frac{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) E_L(\mathbf{r}) d\mathbf{r}}
|
||
{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}} \,.
|
||
\end{eqnarray*}
|
||
|
||
<p>
|
||
For large \(\tau\), we have that
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
\Pi(\mathbf{r},\tau) =\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \rightarrow \Phi_0(\mathbf{r}) \Psi_T(\mathbf{r})\,,
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
and, using that \(\hat{H}\) is Hermitian and that \(\Phi_0\) is an
|
||
eigenstate of the Hamiltonian, we obtain for large \(\tau\)
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
E(\tau) = \frac{\langle \psi_\tau | \hat{H} | \Psi_T \rangle}
|
||
{\langle \psi_\tau | \Psi_T \rangle}
|
||
= \frac{\langle \Psi_T | \hat{H} | \psi_\tau \rangle}
|
||
{\langle \Psi_T | \psi_\tau \rangle}
|
||
\rightarrow E_0 \frac{\langle \Psi_T | \Phi_0 \rangle}
|
||
{\langle \Psi_T | \Phi_0 \rangle}
|
||
= E_0
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
Therefore, we can compute the energy within DMC by generating the
|
||
density \(\Pi\) with random walks, and simply averaging the local
|
||
energies computed with the trial wave function.
|
||
</p>
|
||
</div>
|
||
|
||
<div id="outline-container-org7548f1f" class="outline-4">
|
||
<h4 id="org7548f1f"><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">
|
||
<p>
|
||
\[
|
||
-\frac{\partial \psi(\mathbf{r},\tau)}{\partial \tau} \Psi_T(\mathbf{r}) =
|
||
\left[ -\frac{1}{2} \Delta \psi(\mathbf{r},\tau) + V(\mathbf{r}) \psi(\mathbf{r},\tau) \right] \Psi_T(\mathbf{r})
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
-\frac{\partial \big[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big]}{\partial \tau}
|
||
= -\frac{1}{2} \Big( \Delta \big[
|
||
\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big] -
|
||
\psi(\mathbf{r},\tau) \Delta \Psi_T(\mathbf{r}) - 2
|
||
\nabla \psi(\mathbf{r},\tau) \nabla \Psi_T(\mathbf{r}) \Big) + V(\mathbf{r}) \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
-\frac{\partial \big[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big]}{\partial \tau}
|
||
= -\frac{1}{2} \Delta \big[\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big] +
|
||
\frac{1}{2} \psi(\mathbf{r},\tau) \Delta \Psi_T(\mathbf{r}) +
|
||
\Psi_T(\mathbf{r})\nabla \psi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})} + V(\mathbf{r}) \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
-\frac{\partial \big[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big]}{\partial \tau}
|
||
= -\frac{1}{2} \Delta \big[\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big] +
|
||
\psi(\mathbf{r},\tau) \Delta \Psi_T(\mathbf{r}) +
|
||
\Psi_T(\mathbf{r})\nabla \psi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})} + E_L(\mathbf{r}) \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||
\]
|
||
\[
|
||
-\frac{\partial \big[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big]}{\partial \tau}
|
||
= -\frac{1}{2} \Delta \big[\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) \big] +
|
||
\nabla \left[ \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||
\frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}
|
||
\right] + E_L(\mathbf{r}) \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})
|
||
\]
|
||
</p>
|
||
|
||
<p>
|
||
Defining \(\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau)
|
||
\Psi_T(\mathbf{r})\),
|
||
</p>
|
||
|
||
<p>
|
||
\[
|
||
-\frac{\partial \Pi(\mathbf{r},\tau)}{\partial \tau}
|
||
= -\frac{1}{2} \Delta \Pi(\mathbf{r},\tau) +
|
||
\nabla \left[ \Pi(\mathbf{r},\tau) \frac{\nabla \Psi_T(\mathbf{r})}{\Psi_T(\mathbf{r})}
|
||
\right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau)
|
||
\]
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org4a7db09" class="outline-3">
|
||
<h3 id="org4a7db09"><span class="section-number-3">4.4</span> Pure Diffusion Monte Carlo</h3>
|
||
<div class="outline-text-3" id="text-4-4">
|
||
<p>
|
||
Instead of having a variable number of particles to simulate the
|
||
branching process as in the <i>Diffusion Monte Carlo</i> (DMC) algorithm, we
|
||
use variant called <i>pure Diffusion Monte Carlo</i> (PDMC) where
|
||
the potential term is considered as a cumulative product of weights:
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
W(\mathbf{r}_n, \tau) = \prod_{i=1}^{n} \exp \left( -\delta t\,
|
||
(E_L(\mathbf{r}_i) - E_{\text{ref}}) \right) =
|
||
\prod_{i=1}^{n} w(\mathbf{r}_i)
|
||
\end{eqnarray*}
|
||
|
||
<p>
|
||
where \(\mathbf{r}_i\) are the coordinates along the trajectory and
|
||
we introduced a <i>time-step variable</i> \(\delta t\) to discretize the
|
||
integral.
|
||
</p>
|
||
|
||
<p>
|
||
The PDMC algorithm is less stable than the DMC algorithm: it
|
||
requires to have a value of \(E_\text{ref}\) which is close to the
|
||
fixed-node energy, and a good trial wave function. Moreover, we
|
||
can't let \(\tau\) become too large because the weight whether
|
||
explode or vanish: we need to have a fixed value of \(\tau\)
|
||
(projection time).
|
||
The big advantage of PDMC is that it is rather simple to implement
|
||
starting from a VMC code:
|
||
</p>
|
||
|
||
<ol class="org-ol">
|
||
<li>Start with \(W(\mathbf{r}_0)=1, \tau_0 = 0\)</li>
|
||
<li>Evaluate the local energy at \(\mathbf{r}_{n}\)</li>
|
||
<li>Compute the contribution to the weight \(w(\mathbf{r}_n) =
|
||
\exp(-\delta t(E_L(\mathbf{r}_n)-E_\text{ref}))\)</li>
|
||
<li>Update \(W(\mathbf{r}_{n}) = W(\mathbf{r}_{n-1}) \times w(\mathbf{r}_n)\)</li>
|
||
<li>Accumulate the weighted energy \(W(\mathbf{r}_n) \times
|
||
E_L(\mathbf{r}_n)\),
|
||
and the weight \(W(\mathbf{r}_n)\) for the normalization</li>
|
||
<li>Update \(\tau_n = \tau_{n-1} + \delta t\)</li>
|
||
<li>If \(\tau_{n} > \tau_\text{max}\) (\(\tau_\text{max}\) is an input parameter), the long projection time has
|
||
been reached and we can start an new trajectory from the current
|
||
position: reset \(W(r_n) = 1\) and \(\tau_n
|
||
= 0\)</li>
|
||
<li>Compute a new position \(\mathbf{r'} = \mathbf{r}_n +
|
||
\delta t\, \frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi\)</li>
|
||
<li>Evaluate \(\Psi(\mathbf{r}')\) and \(\frac{\nabla \Psi(\mathbf{r'})}{\Psi(\mathbf{r'})}\) at the new position</li>
|
||
<li>Compute the ratio \(A = \frac{T(\mathbf{r}' \rightarrow \mathbf{r}_{n}) P(\mathbf{r}')}{T(\mathbf{r}_{n} \rightarrow \mathbf{r}') P(\mathbf{r}_{n})}\)</li>
|
||
</ol>
|
||
<ol class="org-ol">
|
||
<li>Draw a uniform random number \(v \in [0,1]\)</li>
|
||
<li>if \(v \le A\), accept the move : set \(\mathbf{r}_{n+1} = \mathbf{r'}\)</li>
|
||
<li>else, reject the move : set \(\mathbf{r}_{n+1} = \mathbf{r}_n\)</li>
|
||
</ol>
|
||
|
||
|
||
<p>
|
||
Some comments are needed:
|
||
</p>
|
||
|
||
<ul class="org-ul">
|
||
<li><p>
|
||
You estimate the energy as
|
||
</p>
|
||
|
||
\begin{eqnarray*}
|
||
E = \frac{\sum_{k=1}^{N_{\rm MC}} E_L(\mathbf{r}_k) W(\mathbf{r}_k, k\delta t)}{\sum_{k=1}^{N_{\rm MC}} W(\mathbf{r}_k, k\delta t)}
|
||
\end{eqnarray*}</li>
|
||
|
||
<li><p>
|
||
The result will be affected by a time-step error
|
||
(the finite size of \(\delta t\)) due to the discretization of the
|
||
integral, and one has in principle to extrapolate to the limit
|
||
\(\delta t \rightarrow 0\). This amounts to fitting the energy
|
||
computed for multiple values of \(\delta t\).
|
||
</p>
|
||
|
||
<p>
|
||
Here, you will be using a small enough time-step and you should not worry about the extrapolation.
|
||
</p></li>
|
||
<li>The accept/reject step (steps 9-12 in the algorithm) is in principle not needed for the correctness of
|
||
the DMC algorithm. However, its use reduces significantly the time-step error.</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<div id="outline-container-orgb1e1f31" class="outline-3">
|
||
<h3 id="orgb1e1f31"><span class="section-number-3">4.5</span> Hydrogen atom</h3>
|
||
<div class="outline-text-3" id="text-4-5">
|
||
</div>
|
||
|
||
<div id="outline-container-org10928e0" class="outline-4">
|
||
<h4 id="org10928e0"><span class="section-number-4">4.5.1</span> Exercise</h4>
|
||
<div class="outline-text-4" id="text-4-5-1">
|
||
<div class="exercise">
|
||
<p>
|
||
Modify the Metropolis VMC program into a PDMC program.
|
||
In the limit \(\delta t \rightarrow 0\), you should recover the exact
|
||
energy of H for any value of \(a\), as long as the simulation is stable.
|
||
We choose here a time step of 0.05 a.u. and a fixed projection
|
||
time \(\tau_{\text{max}}\) =100 a.u.
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python"><span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a, nmax, dt, Eref):
|
||
# <span style="color: #b22222;">TODO</span>
|
||
|
||
# <span style="color: #b22222;">Run simulation</span>
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
<span style="color: #a0522d;">dt</span> = 0.05
|
||
<span style="color: #a0522d;">tau</span> = 100.
|
||
<span style="color: #a0522d;">E_ref</span> = -0.5
|
||
|
||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a, nmax, dt, E_ref) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
|
||
# <span style="color: #b22222;">Energy</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
|
||
|
||
# <span style="color: #b22222;">Acceptance rate</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (_, x) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"A = {A} +/- {deltaA}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">pdmc</span>(a, dt, nmax, energy, accep, tau, E_ref)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, dt, tau</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy, accep</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> E_ref</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_accep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> sq_dt, chi(3)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> psi_old, psi_new</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> d_old(3), d_new(3)</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
|
||
! <span style="color: #b22222;">TODO</span>
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">pdmc</span>
|
||
|
||
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 1.2d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> dt = 0.05d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> E_ref = -0.5d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 100.d0</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
|
||
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
|
||
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns), accep(nruns)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
<span style="color: #a020f0;">do</span> irun=1,nruns
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">pdmc</span>(a, dt, nmax, X(irun), accep(irun), tau, E_ref)
|
||
<span style="color: #a020f0;">enddo</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(accep,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'A = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 pdmc.f90 -o pdmc
|
||
./pdmc
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org834cb8c" class="outline-5">
|
||
<h5 id="org834cb8c"><span class="section-number-5">4.5.1.1</span> Solution   <span class="tag"><span class="solution">solution</span></span></h5>
|
||
<div class="outline-text-5" id="text-4-5-1-1">
|
||
<p>
|
||
<b>Python</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-python">#<span style="color: #b22222;">!/usr/bin/env python3</span>
|
||
|
||
<span style="color: #a020f0;">from</span> hydrogen <span style="color: #a020f0;">import</span> *
|
||
<span style="color: #a020f0;">from</span> qmc_stats <span style="color: #a020f0;">import</span> *
|
||
|
||
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a, nmax, dt, tau, Eref):
|
||
<span style="color: #a0522d;">sq_dt</span> = np.sqrt(dt)
|
||
|
||
<span style="color: #a0522d;">energy</span> = 0.
|
||
<span style="color: #a0522d;">N_accep</span> = 0
|
||
<span style="color: #a0522d;">normalization</span> = 0.
|
||
|
||
<span style="color: #a0522d;">w</span> = 1.
|
||
<span style="color: #a0522d;">tau_current</span> = 0.
|
||
|
||
<span style="color: #a0522d;">r_old</span> = np.random.normal(loc=0., scale=1.0, size=(3))
|
||
<span style="color: #a0522d;">d_old</span> = drift(a,r_old)
|
||
<span style="color: #a0522d;">d2_old</span> = np.dot(d_old,d_old)
|
||
<span style="color: #a0522d;">psi_old</span> = psi(a,r_old)
|
||
|
||
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
|
||
<span style="color: #a0522d;">el</span> = e_loc(a,r_old)
|
||
<span style="color: #a0522d;">w</span> *= np.exp(-dt*(el - Eref))
|
||
|
||
<span style="color: #a0522d;">normalization</span> += w
|
||
<span style="color: #a0522d;">energy</span> += w * el
|
||
|
||
<span style="color: #a0522d;">tau_current</span> += dt
|
||
|
||
# <span style="color: #b22222;">Reset when tau is reached</span>
|
||
<span style="color: #a020f0;">if</span> tau_current > tau:
|
||
<span style="color: #a0522d;">w</span> = 1.
|
||
<span style="color: #a0522d;">tau_current</span> = 0.
|
||
|
||
<span style="color: #a0522d;">chi</span> = np.random.normal(loc=0., scale=1.0, size=(3))
|
||
|
||
<span style="color: #a0522d;">r_new</span> = r_old + dt * d_old + sq_dt * chi
|
||
<span style="color: #a0522d;">d_new</span> = drift(a,r_new)
|
||
<span style="color: #a0522d;">d2_new</span> = np.dot(d_new,d_new)
|
||
<span style="color: #a0522d;">psi_new</span> = psi(a,r_new)
|
||
|
||
# <span style="color: #b22222;">Metropolis</span>
|
||
<span style="color: #a0522d;">prod</span> = np.dot((d_new + d_old), (r_new - r_old))
|
||
<span style="color: #a0522d;">argexpo</span> = 0.5 * (d2_new - d2_old)*dt + prod
|
||
|
||
<span style="color: #a0522d;">q</span> = psi_new / psi_old
|
||
<span style="color: #a0522d;">q</span> = np.exp(-argexpo) * q*q
|
||
|
||
<span style="color: #a020f0;">if</span> np.random.uniform() <= q:
|
||
<span style="color: #a0522d;">N_accep</span> += 1
|
||
<span style="color: #a0522d;">r_old</span> = r_new
|
||
<span style="color: #a0522d;">d_old</span> = d_new
|
||
<span style="color: #a0522d;">d2_old</span> = d2_new
|
||
<span style="color: #a0522d;">psi_old</span> = psi_new
|
||
|
||
<span style="color: #a020f0;">return</span> energy/normalization, N_accep/nmax
|
||
|
||
|
||
# <span style="color: #b22222;">Run simulation</span>
|
||
<span style="color: #a0522d;">a</span> = 1.2
|
||
<span style="color: #a0522d;">nmax</span> = 100000
|
||
<span style="color: #a0522d;">dt</span> = 0.05
|
||
<span style="color: #a0522d;">tau</span> = 100.
|
||
<span style="color: #a0522d;">E_ref</span> = -0.5
|
||
|
||
<span style="color: #a0522d;">X0</span> = [ MonteCarlo(a, nmax, dt, tau, E_ref) <span style="color: #a020f0;">for</span> i <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(30)]
|
||
|
||
# <span style="color: #b22222;">Energy</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (x, _) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">E</span>, <span style="color: #a0522d;">deltaE</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}"</span>)
|
||
|
||
# <span style="color: #b22222;">Acceptance rate</span>
|
||
<span style="color: #a0522d;">X</span> = [ x <span style="color: #a020f0;">for</span> (_, x) <span style="color: #a020f0;">in</span> X0 ]
|
||
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error(X)
|
||
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"A = {A} +/- {deltaA}"</span>)
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<b>Fortran</b>
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">pdmc</span>(a, dt, nmax, energy, accep, tau, E_ref)
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> a, dt, tau</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> nmax </span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(out) ::<span style="color: #a0522d;"> energy, accep</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">intent</span>(in) ::<span style="color: #a0522d;"> E_ref</span>
|
||
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> istep</span>
|
||
<span style="color: #228b22;">integer</span>*8 ::<span style="color: #a0522d;"> n_accep</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> sq_dt, chi(3), d2_old, prod, u</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> psi_old, psi_new, d2_new, argexpo, q</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> r_old(3), r_new(3)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> d_old(3), d_new(3)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> e, w, norm, tau_current</span>
|
||
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc, psi</span>
|
||
|
||
sq_dt = dsqrt(dt)
|
||
|
||
! <span style="color: #b22222;">Initialization</span>
|
||
energy = 0.d0
|
||
n_accep = 0_8
|
||
norm = 0.d0
|
||
|
||
w = 1.d0
|
||
tau_current = 0.d0
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(r_old,3)
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_old,d_old)
|
||
d2_old = d_old(1)*d_old(1) + <span style="color: #a020f0;">&</span>
|
||
d_old(2)*d_old(2) + <span style="color: #a020f0;">&</span>
|
||
d_old(3)*d_old(3)
|
||
|
||
psi_old = psi(a,r_old)
|
||
|
||
<span style="color: #a020f0;">do</span> istep = 1,nmax
|
||
e = e_loc(a,r_old)
|
||
w = w * dexp(-dt*(e - E_ref))
|
||
|
||
norm = norm + w
|
||
energy = energy + w*e
|
||
|
||
tau_current = tau_current + dt
|
||
|
||
! <span style="color: #b22222;">Reset when tau is reached</span>
|
||
<span style="color: #a020f0;">if</span> (tau_current > tau) <span style="color: #a020f0;">then</span>
|
||
w = 1.d0
|
||
tau_current = 0.d0
|
||
<span style="color: #a020f0;">endif</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(chi,3)
|
||
r_new(:) = r_old(:) + dt*d_old(:) + chi(:)*sq_dt
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_new,d_new)
|
||
d2_new = d_new(1)*d_new(1) + <span style="color: #a020f0;">&</span>
|
||
d_new(2)*d_new(2) + <span style="color: #a020f0;">&</span>
|
||
d_new(3)*d_new(3)
|
||
|
||
psi_new = psi(a,r_new)
|
||
|
||
! <span style="color: #b22222;">Metropolis</span>
|
||
prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + <span style="color: #a020f0;">&</span>
|
||
(d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + <span style="color: #a020f0;">&</span>
|
||
(d_new(3) + d_old(3))*(r_new(3) - r_old(3))
|
||
|
||
argexpo = 0.5d0 * (d2_new - d2_old)*dt + prod
|
||
|
||
q = psi_new / psi_old
|
||
q = dexp(-argexpo) * q*q
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_number</span>(u)
|
||
|
||
<span style="color: #a020f0;">if</span> (u <= q) <span style="color: #a020f0;">then</span>
|
||
|
||
n_accep = n_accep + 1_8
|
||
|
||
r_old(:) = r_new(:)
|
||
d_old(:) = d_new(:)
|
||
d2_old = d2_new
|
||
psi_old = psi_new
|
||
|
||
<span style="color: #a020f0;">end if</span>
|
||
|
||
<span style="color: #a020f0;">end do</span>
|
||
|
||
energy = energy / norm
|
||
accep = <span style="color: #a020f0;">dble</span>(n_accep) / <span style="color: #a020f0;">dble</span>(nmax)
|
||
|
||
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">pdmc</span>
|
||
|
||
<span style="color: #a020f0;">program</span> <span style="color: #0000ff;">qmc</span>
|
||
<span style="color: #a020f0;">implicit</span> <span style="color: #228b22;">none</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> a = 1.2d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> dt = 0.05d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> E_ref = -0.5d0</span>
|
||
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 100.d0</span>
|
||
<span style="color: #228b22;">integer</span>*8 , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nmax = 100000</span>
|
||
<span style="color: #228b22;">integer</span> , <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> nruns = 30</span>
|
||
|
||
<span style="color: #228b22;">integer</span> ::<span style="color: #a0522d;"> irun</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> X(nruns), accep(nruns)</span>
|
||
<span style="color: #228b22;">double precision</span> ::<span style="color: #a0522d;"> ave, err</span>
|
||
|
||
<span style="color: #a020f0;">do</span> irun=1,nruns
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">pdmc</span>(a, dt, nmax, X(irun), accep(irun), tau, E_ref)
|
||
<span style="color: #a020f0;">enddo</span>
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(X,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'E = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">ave_error</span>(accep,nruns,ave,err)
|
||
<span style="color: #a020f0;">print</span> *, <span style="color: #8b2252;">'A = '</span>, ave, <span style="color: #8b2252;">'+/-'</span>, err
|
||
|
||
<span style="color: #a020f0;">end program</span> <span style="color: #0000ff;">qmc</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<pre class="example">
|
||
E = -0.49963953547336709 +/- 6.8755513992017491E-004
|
||
A = 0.98963533333333342 +/- 6.3052128284666221E-005
|
||
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="outline-container-org3be55e0" class="outline-2">
|
||
<h2 id="org3be55e0"><span class="section-number-2">5</span> Project</h2>
|
||
<div class="outline-text-2" id="text-5">
|
||
<p>
|
||
Change your PDMC code for one of the following:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>the Helium atom, or</li>
|
||
<li>the H<sub>2</sub> molecule at \(R\) =1.401 bohr.</li>
|
||
</ul>
|
||
|
||
<p>
|
||
And compute the ground state energy.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="outline-container-orgfa0dc42" class="outline-2">
|
||
<h2 id="orgfa0dc42"><span class="section-number-2">6</span> Acknowledgments</h2>
|
||
<div class="outline-text-2" id="text-6">
|
||
|
||
<div class="figure">
|
||
<p><img src="https://trex-coe.eu/sites/default/files/inline-images/euflag.jpg" alt="euflag.jpg" />
|
||
</p>
|
||
</div>
|
||
|
||
<p>
|
||
<a href="https://trex-coe.eu">TREX</a> : Targeting Real Chemical Accuracy at the Exascale project
|
||
has received funding from the European Union’s Horizon 2020 - Research and
|
||
Innovation program - under grant agreement no. 952165. The content of this
|
||
document does not represent the opinion of the European Union, and the European
|
||
Union is not responsible for any use that might be made of such content.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="postamble" class="status">
|
||
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
||
<p class="date">Created: 2021-02-04 Thu 18:12</p>
|
||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||
</div>
|
||
</body>
|
||
</html>
|