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<body>
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<div id="content">
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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="#org0ce899e">1. Introduction</a></li>
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<li><a href="#org6b23ebf">2. Numerical evaluation of the energy</a>
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<ul>
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<li><a href="#org8338702">2.1. Local energy</a>
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<ul>
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<li><a href="#org8336725">2.1.1. Exercise 1</a></li>
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<li><a href="#org41ed066">2.1.2. Exercise 2</a></li>
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<li><a href="#org97eddc2">2.1.3. Exercise 3</a></li>
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<li><a href="#orgfabc08a">2.1.4. Exercise 4</a></li>
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</ul>
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</li>
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<li><a href="#org4f19dc5">2.2. Plot of the local energy along the \(x\) axis</a>
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<ul>
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<li><a href="#orgcf92129">2.2.1. Exercise</a></li>
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</ul>
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</li>
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<li><a href="#orgf95b6ea">2.3. Numerical estimation of the energy</a>
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<ul>
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<li><a href="#org051e985">2.3.1. Exercise</a></li>
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</ul>
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</li>
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<li><a href="#org81be2eb">2.4. Variance of the local energy</a>
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<ul>
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<li><a href="#org8fdf2fb">2.4.1. Exercise</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="#org367e803">3. Variational Monte Carlo</a>
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<ul>
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<li><a href="#orgc5ae535">3.1. Computation of the statistical error</a>
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<ul>
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<li><a href="#org49ca469">3.1.1. Exercise</a></li>
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</ul>
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</li>
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<li><a href="#org62371bd">3.2. Uniform sampling in the box</a>
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<ul>
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<li><a href="#org952f8d1">3.2.1. Exercise</a></li>
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</ul>
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</li>
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<li><a href="#org3423e74">3.3. Gaussian sampling</a>
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<ul>
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<li><a href="#org98a43ad">3.3.1. Exercise</a></li>
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</ul>
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</li>
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<li><a href="#orgb44eaed">3.4. Sampling with \(\Psi^2\)</a>
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<ul>
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<li><a href="#orgb9e3617">3.4.1. Importance sampling</a></li>
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<li><a href="#org9669b71">3.4.2. Metropolis algorithm</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="#orgac06adb">4. <span class="todo TODO">TODO</span> Diffusion Monte Carlo</a>
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<ul>
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<li><a href="#orgfc3b72d">4.1. Hydrogen atom</a></li>
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<li><a href="#org1f0b148">4.2. Dihydrogen</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org0ce899e" class="outline-2">
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<h2 id="org0ce899e"><span class="section-number-2">1</span> Introduction</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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We propose different exercises to understand quantum Monte Carlo (QMC)
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methods. In the first section, we propose to compute the energy of a
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hydrogen atom using numerical integration. The goal of this section is
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to introduce the <i>local energy</i>.
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Then we introduce the variational Monte Carlo (VMC) method which
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computes a statistical estimate of the expectation value of the energy
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associated with a given wave function.
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Finally, we introduce the diffusion Monte Carlo (DMC) method which
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gives the exact energy of the \(H_2\) molecule.
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</p>
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<p>
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Code examples will be given in Python and Fortran. Whatever language
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can be chosen.
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</p>
|
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<p>
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|
We consider the stationary solution of the Schrödinger equation, so
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|
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>
|
|
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|
<p>
|
|
<b>Note</b>
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|
</p>
|
|
<div class="important">
|
|
<p>
|
|
In Fortran, when you use a double precision constant, don't forget
|
|
to put d0 as a suffix (for example 2.0d0), or it will be
|
|
interpreted as a single precision value
|
|
</p>
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|
|
|
</div>
|
|
</div>
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|
</div>
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|
|
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|
<div id="outline-container-org6b23ebf" class="outline-2">
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|
<h2 id="org6b23ebf"><span class="section-number-2">2</span> Numerical evaluation of the energy</h2>
|
|
<div class="outline-text-2" id="text-2">
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|
<p>
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|
In this section we consider the Hydrogen atom with the following
|
|
wave function:
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</p>
|
|
|
|
<p>
|
|
\[
|
|
\Psi(\mathbf{r}) = \exp(-a |\mathbf{r}|)
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|
\]
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|
</p>
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|
|
|
<p>
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|
We will first verify that \(\Psi\) is an eigenfunction of the Hamiltonian
|
|
</p>
|
|
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|
<p>
|
|
\[
|
|
\hat{H} = \hat{T} + \hat{V} = - \frac{1}{2} \Delta - \frac{1}{|\mathbf{r}|}
|
|
\]
|
|
</p>
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|
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|
<p>
|
|
when \(a=1\), by checking that \(\hat{H}\Psi(\mathbf{r}) = E\Psi(\mathbf{r})\) for
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|
all \(\mathbf{r}\). We will check that the local energy, defined as
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})},
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|
\]
|
|
</p>
|
|
|
|
<p>
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|
is constant. We will also see that when \(a \ne 1\) the local energy
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|
is not constant, so \(\hat{H} \Psi \ne E \Psi\).
|
|
</p>
|
|
|
|
|
|
<p>
|
|
The probabilistic <i>expected value</i> of an arbitrary function \(f(x)\)
|
|
with respect to a probability density function \(p(x)\) is given by
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|
</p>
|
|
|
|
<p>
|
|
\[ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx \].
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</p>
|
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|
<p>
|
|
Recall that a probability density function \(p(x)\) is non-negative
|
|
and integrates to one:
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</p>
|
|
|
|
<p>
|
|
\[ \int_{-\infty}^\infty p(x)\,dx = 1 \].
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|
</p>
|
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|
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<p>
|
|
The electronic energy of a system is the expectation value of the
|
|
local energy \(E(\mathbf{r})\) with respect to the 3N-dimensional
|
|
electron density given by the square of the wave function:
|
|
</p>
|
|
|
|
\begin{eqnarray*}
|
|
E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle}
|
|
= \frac{\int \Psi(\mathbf{r})\, \hat{H} \Psi(\mathbf{r})\, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \\
|
|
& = & \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
|
|
= \frac{\int \left[\Psi(\mathbf{r})\right]^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
|
|
= \langle E_L \rangle_{\Psi^2}
|
|
\end{eqnarray*}
|
|
</div>
|
|
|
|
<div id="outline-container-org8338702" class="outline-3">
|
|
<h3 id="org8338702"><span class="section-number-3">2.1</span> Local energy</h3>
|
|
<div class="outline-text-3" id="text-2-1">
|
|
</div>
|
|
|
|
<div id="outline-container-org8336725" class="outline-4">
|
|
<h4 id="org8336725"><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 arguments
|
|
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: #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: #a020f0;">return</span> -1. / 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;">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>
|
|
potential = -1.d0 / dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
|
|
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">potential</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org41ed066" class="outline-4">
|
|
<h4 id="org41ed066"><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: #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 id="outline-container-org97eddc2" class="outline-4">
|
|
<h4 id="org97eddc2"><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 \[-\frac{1}{2}\frac{\Delta \Psi}{\Psi}.\]
|
|
</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>
|
|
So the local kinetic energy is
|
|
\[
|
|
-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
|
|
\]
|
|
</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: #a020f0;">return</span> -0.5 * (a**2 - (2.*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;">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>
|
|
kinetic = -0.5d0 * (a*a - (2.d0*a) / <span style="color: #a020f0;">&</span>
|
|
dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ) )
|
|
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">kinetic</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfabc08a" class="outline-4">
|
|
<h4 id="orgfabc08a"><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 \(\mathbf{r}\).
|
|
The function accepts <code>x,y,z</code> as input arguments and returns the
|
|
local 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: #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, 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-org4f19dc5" class="outline-3">
|
|
<h3 id="org4f19dc5"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-orgcf92129" class="outline-4">
|
|
<h4 id="orgcf92129"><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.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
<p>
|
|
<b>Python</b>
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-python"><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)
|
|
|
|
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">make_array</span>(a):
|
|
<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])
|
|
<span style="color: #a020f0;">return</span> y
|
|
|
|
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> = make_array(a)
|
|
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>
|
|
|
|
<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 class="figure">
|
|
<p><img src="plot.png" alt="plot.png" />
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf95b6ea" class="outline-3">
|
|
<h3 id="orgf95b6ea"><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 probability density 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-org051e985" class="outline-4">
|
|
<h4 id="org051e985"><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 in 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: #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>
|
|
|
|
<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>
|
|
|
|
<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>
|
|
|
|
<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 id="outline-container-org81be2eb" class="outline-3">
|
|
<h3 id="org81be2eb"><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 the average:
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\sigma^2(E_L) = \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \left[
|
|
E_L(\mathbf{r}) - E \right]^2 \, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}}
|
|
\]
|
|
</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-org8fdf2fb" class="outline-4">
|
|
<h4 id="org8fdf2fb"><span class="section-number-4">2.4.1</span> Exercise</h4>
|
|
<div class="outline-text-4" id="text-2-4-1">
|
|
<div class="exercise">
|
|
<p>
|
|
Compute a numerical estimate of the variance of the local energy
|
|
in 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: #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;">El</span> = e_loc(a, r)
|
|
<span style="color: #a0522d;">E</span> += w * El
|
|
<span style="color: #a0522d;">norm</span> += w
|
|
<span style="color: #a0522d;">E</span> = E / norm
|
|
<span style="color: #a0522d;">s2</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;">El</span> = e_loc(a, r)
|
|
<span style="color: #a0522d;">s2</span> += w * (El - E)**2
|
|
<span style="color: #a0522d;">s2</span> = s2 / norm
|
|
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"a = {a} \t E = {E:10.8f} \t \sigma^2 = {s2:10.8f}"</span>)
|
|
</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: #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, s2</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
|
|
|
|
s2 = 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
|
|
s2 = s2 + w * ( e_loc(a(j), r) - energy )**2
|
|
norm = norm + w
|
|
<span style="color: #a020f0;">end do</span>
|
|
<span style="color: #a020f0;">end do</span>
|
|
<span style="color: #a020f0;">end do</span>
|
|
s2 = s2 / 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: #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>
|
|
|
|
<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>
|
|
|
|
<pre class="example">
|
|
a = 0.10000000000000001 E = -0.24518438948809140 s2 = 2.6965218719733813E-002
|
|
a = 0.20000000000000001 E = -0.26966057967803236 s2 = 3.7197072370217653E-002
|
|
a = 0.50000000000000000 E = -0.38563576125173815 s2 = 5.3185967578488862E-002
|
|
a = 1.0000000000000000 E = -0.50000000000000000 s2 = 0.0000000000000000
|
|
a = 1.5000000000000000 E = -0.39242967082602065 s2 = 0.31449670909180444
|
|
a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814270851303
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org367e803" class="outline-2">
|
|
<h2 id="org367e803"><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 do a Monte Carlo sampling.
|
|
</p>
|
|
|
|
<p>
|
|
Moreover, a Monte Carlo sampling will alow us to remove the bias due
|
|
to the discretization of space, and compute a statistical confidence
|
|
interval.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc5ae535" class="outline-3">
|
|
<h3 id="orgc5ae535"><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 by the central limit theorem.
|
|
</p>
|
|
|
|
<p>
|
|
The estimate of the energy is
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
E = \frac{1}{M} \sum_{i=1}^M E_M
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
The variance of the average energies can be computed as
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\sigma^2 = \frac{1}{M-1} \sum_{i=1}^{M} (E_M - E)^2
|
|
\]
|
|
</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-org49ca469" class="outline-4">
|
|
<h4 id="org49ca469"><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: #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>1)
|
|
<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: #a020f0;">return</span> (average, sqrt(variance/M))
|
|
</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>
|
|
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 id="outline-container-org62371bd" class="outline-3">
|
|
<h3 id="org62371bd"><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 do our first Monte Carlo calculation to compute the
|
|
energy of the hydrogen atom.
|
|
</p>
|
|
|
|
<p>
|
|
At every Monte Carlo step:
|
|
</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>
|
|
One Monte Carlo run will consist of \(N\) Monte Carlo steps. Once all the
|
|
steps have been computed, the run returns the average energy
|
|
\(\bar{E}_k\) over the \(N\) steps of the run.
|
|
</p>
|
|
|
|
<p>
|
|
To compute the statistical error, perform \(M\) 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-org952f8d1" class="outline-4">
|
|
<h4 id="org952f8d1"><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 \(a=0.9\). 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="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):
|
|
<span style="color: #a0522d;">E</span> = 0.
|
|
<span style="color: #a0522d;">N</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;">N</span> += w
|
|
<span style="color: #a0522d;">E</span> += w * e_loc(a,r)
|
|
<span style="color: #a020f0;">return</span> E/N
|
|
|
|
<span style="color: #a0522d;">a</span> = 0.9
|
|
<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>
|
|
|
|
<p>
|
|
<b>Fortran</b>
|
|
</p>
|
|
<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>
|
|
|
|
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
|
|
norm = norm + w
|
|
energy = energy + w * e_loc(a,r)
|
|
<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 = 0.9</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>
|
|
|
|
<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>
|
|
|
|
<pre class="example">
|
|
E = -0.49588321986667677 +/- 7.1758863546737969E-004
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3423e74" class="outline-3">
|
|
<h3 id="org3423e74"><span class="section-number-3">3.3</span> Gaussian sampling</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
We will now improve the sampling and allow to sample in the whole
|
|
3D space, correcting the bias related to the sampling in the box.
|
|
</p>
|
|
|
|
<p>
|
|
Instead of drawing uniform random numbers, we will draw Gaussian
|
|
random numbers centered on 0 and with a variance of 1.
|
|
</p>
|
|
|
|
<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>
|
|
Here is a Fortran implementation returning a Gaussian-distributed
|
|
n-dimensional vector \(\mathbf{z}\);
|
|
</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>
|
|
Now the sampling probability can be inserted into the equation of the energy:
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
E = \frac{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r})\right]^2}{P(\mathbf{r})}\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r}) \right]^2}{P(\mathbf{r})} d\mathbf{r}}
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
with the Gaussian probability
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
P(\mathbf{r}) = \frac{1}{(2 \pi)^{3/2}}\exp\left( -\frac{\mathbf{r}^2}{2} \right).
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
As the coordinates are drawn with probability \(P(\mathbf{r})\), the
|
|
average energy can be computed as
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
E \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\;
|
|
w_i = \frac{\left[\Psi(\mathbf{r}_i)\right]^2}{P(\mathbf{r}_i)} \delta \mathbf{r}
|
|
\]
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org98a43ad" class="outline-4">
|
|
<h4 id="org98a43ad"><span class="section-number-4">3.3.1</span> Exercise</h4>
|
|
<div class="outline-text-4" id="text-3-3-1">
|
|
<div class="exercise">
|
|
<p>
|
|
Modify the exercise of the previous section to sample with
|
|
Gaussian-distributed random numbers. Can you see an 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: #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: #a0522d;">norm_gauss</span> = 1./(2.*np.pi)**(1.5)
|
|
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">gaussian</span>(r):
|
|
<span style="color: #a020f0;">return</span> norm_gauss * np.exp(-np.dot(r,r)*0.5)
|
|
|
|
<span style="color: #a020f0;">def</span> <span style="color: #0000ff;">MonteCarlo</span>(a,nmax):
|
|
<span style="color: #a0522d;">E</span> = 0.
|
|
<span style="color: #a0522d;">N</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.normal(loc=0., scale=1.0, size=(3))
|
|
<span style="color: #a0522d;">w</span> = psi(a,r)
|
|
<span style="color: #a0522d;">w</span> = w*w / gaussian(r)
|
|
<span style="color: #a0522d;">N</span> += w
|
|
<span style="color: #a0522d;">E</span> += w * e_loc(a,r)
|
|
<span style="color: #a020f0;">return</span> E/N
|
|
|
|
<span style="color: #a0522d;">a</span> = 0.9
|
|
<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>
|
|
|
|
|
|
<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;">gaussian</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: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> norm_gauss = 1.d0/(2.d0*dacos(-1.d0))**(1.5d0)</span>
|
|
gaussian = norm_gauss * dexp( -0.5d0 * (r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ))
|
|
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">gaussian</span>
|
|
|
|
|
|
<span style="color: #a020f0;">subroutine</span> <span style="color: #0000ff;">gaussian_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, gaussian</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_gauss</span>(r,3)
|
|
w = psi(a,r)
|
|
w = w*w / gaussian(r)
|
|
norm = norm + w
|
|
energy = energy + w * e_loc(a,r)
|
|
<span style="color: #a020f0;">end do</span>
|
|
energy = energy / norm
|
|
<span style="color: #a020f0;">end subroutine</span> <span style="color: #0000ff;">gaussian_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 = 0.9</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;">gaussian_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>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian
|
|
./qmc_gaussian
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
E = -0.49517104619091717 +/- 1.0685523607878961E-004
|
|
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb44eaed" class="outline-3">
|
|
<h3 id="orgb44eaed"><span class="section-number-3">3.4</span> Sampling with \(\Psi^2\)</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<p>
|
|
We will now use the square of the wave function to make the sampling:
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
P(\mathbf{r}) = \left[\Psi(\mathbf{r})\right]^2
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
The expression for the energy will be simplified to the average of
|
|
the local energies, each with a weight of 1.
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
E \approx \frac{1}{M}\sum_{i=1}^M E_L(\mathbf{r}_i)
|
|
\]
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-orgb9e3617" class="outline-4">
|
|
<h4 id="orgb9e3617"><span class="section-number-4">3.4.1</span> Importance sampling</h4>
|
|
<div class="outline-text-4" id="text-3-4-1">
|
|
<p>
|
|
To generate the probability density \(\Psi^2\), we consider a
|
|
diffusion process characterized by a time-dependent density
|
|
\([\Psi(\mathbf{r},t)]^2\), which obeys the Fokker-Planck equation:
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\frac{\partial \Psi^2}{\partial t} = \sum_i D
|
|
\frac{\partial}{\partial \mathbf{r}_i} \left(
|
|
\frac{\partial}{\partial \mathbf{r}_i} - F_i(\mathbf{r}) \right)
|
|
[\Psi(\mathbf{r},t)]^2.
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
\(D\) is the diffusion constant and \(F_i\) is the i-th component of a
|
|
drift velocity caused by an external potential. For a stationary
|
|
density, \( \frac{\partial \Psi^2}{\partial t} = 0 \), so
|
|
</p>
|
|
|
|
\begin{eqnarray*}
|
|
0 & = & \sum_i D
|
|
\frac{\partial}{\partial \mathbf{r}_i} \left(
|
|
\frac{\partial}{\partial \mathbf{r}_i} - F_i(\mathbf{r}) \right)
|
|
[\Psi(\mathbf{r})]^2 \\
|
|
0 & = & \sum_i D
|
|
\frac{\partial}{\partial \mathbf{r}_i} \left(
|
|
\frac{\partial [\Psi(\mathbf{r})]^2}{\partial \mathbf{r}_i} -
|
|
F_i(\mathbf{r})\,[\Psi(\mathbf{r})]^2 \right) \\
|
|
0 & = &
|
|
\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} -
|
|
\frac{\partial F_i }{\partial \mathbf{r}_i}[\Psi(\mathbf{r})]^2 -
|
|
\frac{\partial \Psi^2}{\partial \mathbf{r}_i} F_i(\mathbf{r})
|
|
\end{eqnarray*}
|
|
|
|
<p>
|
|
we search for a drift function which satisfies
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} =
|
|
[\Psi(\mathbf{r})]^2 \frac{\partial F_i }{\partial \mathbf{r}_i} +
|
|
\frac{\partial \Psi^2}{\partial \mathbf{r}_i} F_i(\mathbf{r})
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
to obtain a second derivative on the left, we need the drift to be
|
|
of the form
|
|
\[
|
|
F_i(\mathbf{r}) = g(\mathbf{r}) \frac{\partial \Psi^2}{\partial \mathbf{r}_i}
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\frac{\partial^2 \Psi^2}{\partial \mathbf{r}_i^2} =
|
|
[\Psi(\mathbf{r})]^2 \frac{\partial
|
|
g(\mathbf{r})}{\partial \mathbf{r}_i}\frac{\partial \Psi^2}{\partial \mathbf{r}_i} +
|
|
[\Psi(\mathbf{r})]^2 g(\mathbf{r}) \frac{\partial^2
|
|
\Psi^2}{\partial \mathbf{r}_i^2} +
|
|
\frac{\partial \Psi^2}{\partial \mathbf{r}_i}
|
|
g(\mathbf{r}) \frac{\partial \Psi^2}{\partial \mathbf{r}_i}
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
\(g = 1 / \Psi^2\) satisfies this equation, so
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
F(\mathbf{r}) = \frac{\nabla [\Psi(\mathbf{r})]^2}{[\Psi(\mathbf{r})]^2} = 2 \frac{\nabla
|
|
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} = 2 \nabla \left( \log \Psi(\mathbf{r}) \right)
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
In statistical mechanics, Fokker-Planck trajectories are generated
|
|
by a Langevin equation:
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\frac{\partial \mathbf{r}(t)}{\partial t} = 2D \frac{\nabla
|
|
\Psi(\mathbf{r}(t))}{\Psi} + \eta
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
where \(\eta\) is a normally-distributed fluctuating random force.
|
|
</p>
|
|
|
|
<p>
|
|
Discretizing this differential equation gives the following drifted
|
|
diffusion scheme:
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \tau\, 2D \frac{\nabla
|
|
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi
|
|
\]
|
|
where \(\chi\) is a Gaussian random variable with zero mean and
|
|
variance \(\tau\,2D\).
|
|
</p>
|
|
</div>
|
|
|
|
<ol class="org-ol">
|
|
<li><a id="org43b5cbc"></a>Exercise 1<br />
|
|
<div class="outline-text-5" id="text-3-4-1-1">
|
|
<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: #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>
|
|
</li>
|
|
|
|
<li><a id="orgeaad292"></a><span class="todo TODO">TODO</span> Exercise 2<br />
|
|
<div class="outline-text-5" id="text-3-4-1-2">
|
|
<div class="exercise">
|
|
<p>
|
|
Sample \(\Psi^2\) approximately using the drifted diffusion scheme,
|
|
with a diffusion constant \(D=1/2\). You can use a time step of
|
|
0.001 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,tau,nmax):
|
|
<span style="color: #a0522d;">sq_tau</span> = np.sqrt(tau)
|
|
|
|
# <span style="color: #b22222;">Initialization</span>
|
|
<span style="color: #a0522d;">E</span> = 0.
|
|
<span style="color: #a0522d;">N</span> = 0.
|
|
<span style="color: #a0522d;">r_old</span> = np.random.normal(loc=0., scale=1.0, size=(3))
|
|
|
|
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
|
|
<span style="color: #a0522d;">d_old</span> = drift(a,r_old)
|
|
<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 + tau * d_old + chi*sq_tau
|
|
<span style="color: #a0522d;">N</span> += 1.
|
|
<span style="color: #a0522d;">E</span> += e_loc(a,r_new)
|
|
<span style="color: #a0522d;">r_old</span> = r_new
|
|
<span style="color: #a020f0;">return</span> E/N
|
|
|
|
|
|
<span style="color: #a0522d;">a</span> = 0.9
|
|
<span style="color: #a0522d;">nmax</span> = 100000
|
|
<span style="color: #a0522d;">tau</span> = 0.2
|
|
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,tau,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>
|
|
|
|
<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,tau,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, 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</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_old(3), r_new(3), d_old(3), sq_tau, chi(3)</span>
|
|
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">external</span> ::<span style="color: #a0522d;"> e_loc</span>
|
|
|
|
sq_tau = dsqrt(tau)
|
|
|
|
! <span style="color: #b22222;">Initialization</span>
|
|
energy = 0.d0
|
|
norm = 0.d0
|
|
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(r_old,3)
|
|
|
|
<span style="color: #a020f0;">do</span> istep = 1,nmax
|
|
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">drift</span>(a,r_old,d_old)
|
|
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(chi,3)
|
|
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
|
|
norm = norm + 1.d0
|
|
energy = energy + e_loc(a,r_new)
|
|
r_old(:) = r_new(:)
|
|
<span style="color: #a020f0;">end do</span>
|
|
energy = energy / norm
|
|
<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 = 0.9</span>
|
|
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 0.2</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;">variational_montecarlo</span>(a,tau,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>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-sh">gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc
|
|
./vmc
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
E = -0.48584030499187431 +/- 1.0411743995438257E-004
|
|
|
|
</pre>
|
|
</div>
|
|
</li>
|
|
</ol>
|
|
</div>
|
|
|
|
<div id="outline-container-org9669b71" class="outline-4">
|
|
<h4 id="org9669b71"><span class="section-number-4">3.4.2</span> Metropolis algorithm</h4>
|
|
<div class="outline-text-4" id="text-3-4-2">
|
|
<p>
|
|
Discretizing the differential equation to generate the desired
|
|
probability density will suffer from a discretization error
|
|
leading to biases in the averages. The <a href="https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm">Metropolis-Hastings
|
|
sampling algorithm</a> removes exactly the discretization errors, so
|
|
large time steps can be employed.
|
|
</p>
|
|
|
|
<p>
|
|
After the new position \(\mathbf{r}_{n+1}\) has been computed, an
|
|
additional accept/reject step is performed. 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>
|
|
\[ A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \min \left( 1,
|
|
\frac{g(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) P(\mathbf{r}_{n+1})}
|
|
{g(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) P(\mathbf{r}_{n})}
|
|
\right)
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
In our Hydrogen atom example, \(P\) is \(\Psi^2\) and \(g\) is a
|
|
solution of the discretized Fokker-Planck equation:
|
|
</p>
|
|
|
|
\begin{eqnarray*}
|
|
P(r_{n}) &=& \Psi^2(\mathbf{r}_n) \\
|
|
g(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = &
|
|
\frac{1}{(4\pi\,D\,\tau)^{3/2}} \exp \left[ - \frac{\left(
|
|
\mathbf{r}_{n+1} - \mathbf{r}_{n} - 2D \frac{\nabla
|
|
\Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{4D\,\tau} \right]
|
|
\end{eqnarray*}
|
|
|
|
<p>
|
|
The accept/reject step is the following:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Compute \(A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1})\).</li>
|
|
<li>Draw a uniform random number \(u\)</li>
|
|
<li>if \(u \le A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1})\), accept
|
|
the move</li>
|
|
<li>if \(u>A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1})\), reject
|
|
the move: set \(\mathbf{r}_{n+1} = \mathbf{r}_{n}\), but <b>don't
|
|
remove the sample from the average!</b></li>
|
|
</ul>
|
|
|
|
<p>
|
|
The <i>acceptance rate</i> is the ratio of the number of accepted step
|
|
over the total number of steps. The time step should be adapted so
|
|
that the acceptance rate is around 0.5 for a good efficiency of
|
|
the simulation.
|
|
</p>
|
|
</div>
|
|
|
|
<ol class="org-ol">
|
|
<li><a id="org8c3385f"></a>Exercise<br />
|
|
<div class="outline-text-5" id="text-3-4-2-1">
|
|
<div class="exercise">
|
|
<p>
|
|
Modify the previous program to introduce the accept/reject step.
|
|
You should recover the unbiased result.
|
|
Adjust the time-step so that the acceptance rate is 0.5.
|
|
</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,tau,nmax):
|
|
<span style="color: #a0522d;">E</span> = 0.
|
|
<span style="color: #a0522d;">N</span> = 0.
|
|
<span style="color: #a0522d;">accep_rate</span> = 0.
|
|
<span style="color: #a0522d;">sq_tau</span> = np.sqrt(tau)
|
|
<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;">r_new</span> = r_old + tau * d_old + sq_tau * 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)*tau + 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;">accep_rate</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: #a0522d;">N</span> += 1.
|
|
<span style="color: #a0522d;">E</span> += e_loc(a,r_old)
|
|
<span style="color: #a020f0;">return</span> E/N, accep_rate/N
|
|
|
|
|
|
<span style="color: #a0522d;">a</span> = 0.9
|
|
<span style="color: #a0522d;">nmax</span> = 100000
|
|
<span style="color: #a0522d;">tau</span> = 1.0
|
|
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,tau,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[0] <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> X])
|
|
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error([x[1] <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> X])
|
|
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}\nA = {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,tau,nmax,energy,accep_rate)
|
|
<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, 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_rate</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, sq_tau, 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_tau = dsqrt(tau)
|
|
|
|
! <span style="color: #b22222;">Initialization</span>
|
|
energy = 0.d0
|
|
norm = 0.d0
|
|
accep_rate = 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) + d_old(2)*d_old(2) + d_old(3)*d_old(3)
|
|
psi_old = psi(a,r_old)
|
|
|
|
<span style="color: #a020f0;">do</span> istep = 1,nmax
|
|
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(chi,3)
|
|
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
|
|
<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) + d_new(2)*d_new(2) + 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)*tau + 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>
|
|
accep_rate = accep_rate + 1.d0
|
|
r_old(:) = r_new(:)
|
|
d_old(:) = d_new(:)
|
|
d2_old = d2_new
|
|
psi_old = psi_new
|
|
<span style="color: #a020f0;">end if</span>
|
|
norm = norm + 1.d0
|
|
energy = energy + e_loc(a,r_old)
|
|
<span style="color: #a020f0;">end do</span>
|
|
energy = energy / norm
|
|
accep_rate = accep_rate / norm
|
|
<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 = 0.9</span>
|
|
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 1.0</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,tau,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>
|
|
|
|
<pre class="example">
|
|
E = -0.49499990423528023 +/- 1.5958250761863871E-004
|
|
A = 0.78861366666666655 +/- 3.5096729498002445E-004
|
|
|
|
</pre>
|
|
</div>
|
|
</li>
|
|
</ol>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-orgac06adb" class="outline-2">
|
|
<h2 id="orgac06adb"><span class="section-number-2">4</span> <span class="todo TODO">TODO</span> Diffusion Monte Carlo</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
</div>
|
|
|
|
<div id="outline-container-orgfc3b72d" class="outline-3">
|
|
<h3 id="orgfc3b72d"><span class="section-number-3">4.1</span> Hydrogen atom</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
</div>
|
|
<ol class="org-ol">
|
|
<li><a id="orgb7d1c9f"></a>Exercise<br />
|
|
<div class="outline-text-5" id="text-4-1-0-1">
|
|
<div class="exercise">
|
|
<p>
|
|
Modify the Metropolis VMC program to introduce the PDMC weight.
|
|
In the limit \(\tau \rightarrow 0\), you should recover the exact
|
|
energy of H for any value of \(a\).
|
|
</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,tau,nmax,Eref):
|
|
<span style="color: #a0522d;">E</span> = 0.
|
|
<span style="color: #a0522d;">N</span> = 0.
|
|
<span style="color: #a0522d;">accep_rate</span> = 0.
|
|
<span style="color: #a0522d;">sq_tau</span> = np.sqrt(tau)
|
|
<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: #a0522d;">w</span> = 1.0
|
|
<span style="color: #a020f0;">for</span> istep <span style="color: #a020f0;">in</span> <span style="color: #483d8b;">range</span>(nmax):
|
|
<span style="color: #a0522d;">chi</span> = np.random.normal(loc=0., scale=1.0, size=(3))
|
|
<span style="color: #a0522d;">el</span> = e_loc(a,r_old)
|
|
<span style="color: #a0522d;">w</span> *= np.exp(-tau*(el - Eref))
|
|
<span style="color: #a0522d;">N</span> += w
|
|
<span style="color: #a0522d;">E</span> += w * el
|
|
|
|
<span style="color: #a0522d;">r_new</span> = r_old + tau * d_old + sq_tau * 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)*tau + 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: #b22222;">PDMC weight</span>
|
|
<span style="color: #a020f0;">if</span> np.random.uniform() < q:
|
|
<span style="color: #a0522d;">accep_rate</span> += w
|
|
<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> E/N, accep_rate/N
|
|
|
|
|
|
<span style="color: #a0522d;">a</span> = 0.9
|
|
<span style="color: #a0522d;">nmax</span> = 10000
|
|
<span style="color: #a0522d;">tau</span> = .1
|
|
<span style="color: #a0522d;">X</span> = [MonteCarlo(a,tau,nmax,-0.5) <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[0] <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> X])
|
|
<span style="color: #a0522d;">A</span>, <span style="color: #a0522d;">deltaA</span> = ave_error([x[1] <span style="color: #a020f0;">for</span> x <span style="color: #a020f0;">in</span> X])
|
|
<span style="color: #a020f0;">print</span>(f<span style="color: #8b2252;">"E = {E} +/- {deltaE}\nA = {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,tau,nmax,energy,accep_rate)
|
|
<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, 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_rate</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, sq_tau, 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_tau = dsqrt(tau)
|
|
|
|
! <span style="color: #b22222;">Initialization</span>
|
|
energy = 0.d0
|
|
norm = 0.d0
|
|
accep_rate = 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) + d_old(2)*d_old(2) + d_old(3)*d_old(3)
|
|
psi_old = psi(a,r_old)
|
|
|
|
<span style="color: #a020f0;">do</span> istep = 1,nmax
|
|
<span style="color: #a020f0;">call</span> <span style="color: #0000ff;">random_gauss</span>(chi,3)
|
|
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
|
|
<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) + d_new(2)*d_new(2) + 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)*tau + 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>
|
|
accep_rate = accep_rate + 1.d0
|
|
r_old(:) = r_new(:)
|
|
d_old(:) = d_new(:)
|
|
d2_old = d2_new
|
|
psi_old = psi_new
|
|
<span style="color: #a020f0;">end if</span>
|
|
norm = norm + 1.d0
|
|
energy = energy + e_loc(a,r_old)
|
|
<span style="color: #a020f0;">end do</span>
|
|
energy = energy / norm
|
|
accep_rate = accep_rate / norm
|
|
<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 = 0.9</span>
|
|
<span style="color: #228b22;">double precision</span>, <span style="color: #a020f0;">parameter</span> ::<span style="color: #a0522d;"> tau = 1.0</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,tau,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>
|
|
|
|
<pre class="example">
|
|
E = -0.49499990423528023 +/- 1.5958250761863871E-004
|
|
A = 0.78861366666666655 +/- 3.5096729498002445E-004
|
|
|
|
</pre>
|
|
</div>
|
|
</li>
|
|
</ol>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org1f0b148" class="outline-3">
|
|
<h3 id="org1f0b148"><span class="section-number-3">4.2</span> Dihydrogen</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
We will now consider the H<sub>2</sub> molecule in a minimal basis composed of the
|
|
\(1s\) orbitals of the hydrogen atoms:
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\Psi(\mathbf{r}_1, \mathbf{r}_2) =
|
|
\exp(-(\mathbf{r}_1 - \mathbf{R}_A)) +
|
|
\]
|
|
where \(\mathbf{r}_1\) and \(\mathbf{r}_2\) denote the electron
|
|
coordinates and \(\mathbf{R}_A\) and \(\mathbf{R}_B\) the coordinates of
|
|
the nuclei.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
|
|
<p class="date">Created: 2021-01-13 Wed 17:12</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|